ROV BIZSTATS User Manual. Johnathan Mun, Ph.D., MBA, MS, CFC, CRM, FRM, MIFC

Transcription

1 ROV BIZSTATS User Manual Johnahan Mun, Ph.D., MBA, MS, CFC, CRM, FRM, MIFC

2 ROV BIZSTATS 011 This manual and he sofware described in i are furnished under license and may only be used or copied in accordance wih he erms of he end user license agreemen. Informaion in his documen is provided for informaional purposes only, is subjec o change wihou noice, and does no represen a commimen as o merchanabiliy or finess for a paricular purpose by Real Opions Valuaion, Inc. No par of his manual may be reproduced or ransmied in any form or by any means, elecronic or mechanical, including phoocopying and recording, for any purpose wihou he express wrien permission of Real Opions Valuaion, Inc. Maerials based on copyrighed publicaions by Dr. Johnahan Mun, Founder and CEO, Real Opions Valuaion, Inc. Wrien by Dr. Johnahan Mun. Wrien, designed, and published in he Unied Saes of America. To purchase addiional copies of his documen, conac Real Opions Valuaion, Inc., a he following e- mail address: Admin@RealOpionsValuaion.com or visi by Dr. Johnahan Mun. All righs reserved. Microsof is a regisered rademark of Microsof Corporaion in he U.S. and oher counries. Oher produc names menioned herein may be rademarks and/or regisered rademarks of he respecive holders.

3 Table of Conens INSTALLATION AND QUICK GETTING STARTED GUIDE... 9 Insallaion Requiremens... 9 Summary of Saisical Mehods in ROV BizSas Quick Geing Sared Guide BASIC STATISTICS... 1 Descripive Saisics... 1 Correlaion Marix... 1 Variance-Covariance Marix... Basic Saisics... Absolue Values... Average... Coun... Difference... Lag... Lead... 3 LN... 3 Log... 3 Max... 3 Median... 3 Min... 3 Mode... 3 Power... 3 Rank Ascending... 3 Rank Descending... 3 Relaive LN Reurns... 3 Relaive Reurns... 3 Semi-Sandard Deviaion (Lower)... 3 Semi-Sandard Deviaion (Upper)... 4 Sandard Deviaion (Populaion)... 4 Sandard Deviaion (Sample)... 4 Variance (Populaion)... 4 Variance (Sample)... 4 Sum... 4 ROV BizSas User Manual 3 Copyrigh Dr. Johnahan Mun

4 . HYPOTHESIS TESTING: BASIC MODEL CHOOSER... 5 Two-Tailed Hypohesis Tes... 5 Righ-Tailed Hypohesis Tes... 6 Lef-Tailed Hypohesis Tes HYPOTHESIS TESTING: PARAMETRIC MODELS One-Variable Tesing for Means (T-Tes) One-Variable Tesing for Means (Z-Tes) One-Variable Tesing for Proporions (Z-Tes) Two Variables wih Dependen Means (T-Tes) Two (Independen) Variables wih Equal Variances (T-Tes)... 3 Two (Independen) Variables wih Unequal Variances (T-Tes) Two (Independen) Variables Tesing for Means (Z-Tes) Two (Independen) Variables Tesing for Proporions (Z-Tes) Two (Independen) Variables Tesing for Variances (F-Tes) NONPARAMETRIC ANALYSIS The Basics of Nonparameric Mehodologies Chi-Square Goodness-of-Fi Tes Chi-Square Tes of Independence Chi-Square Populaion Variance Tes Friedman Tes Kruskal-Wallis Tes Lilliefors Tes Runs Tes Wilcoxon Signed-Rank Tes (One Variable) Wilcoxon Signed-Rank Tes (Two Variables) ANOVA (MULTIVARIATE HYPOTHESIS TESTS)... 4 Single Facor Muliple Treamens ANOVA... 4 Randomized Block Muliple Treamens ANOVA... 4 Two-Way ANOVA... 4 ROV BizSas User Manual 4 Copyrigh Dr. Johnahan Mun

5 6. FORECASTING, REGRESSION, AND ECONOMETRICS ARIMA (Auoregressive Inegraed Moving Average) Auo ARIMA (Auomaic Auoregressive Inegraed Moving Average) Basic Muliple Regression Basic Economerics and Auoeconomerics Combinaorial Fuzzy Logic GARCH Volailiy Forecass J-Curve and S-Curve Forecass Markov Chains Neural Nework Forecasing Nonlinear Exrapolaion Principal Componens Analysis Spline (Cubic Spline Inerpolaion and Exrapolaion) Sepwise Regression Time-Series Decomposiion Mehodologies Single Moving Average... 5 Single Exponenial Smoohing... 5 Double Moving Average... 5 Double Exponenial Smoohing... 5 Seasonal Addiive... 5 Seasonal Muliplicaive Hol-Winer s Seasonal Addiive Hol-Winer s Seasonal Muliplicaive Trendlines Volailiy: Log Reurns Approach Yield Curves: Bliss and Nelson-Siegel Mehods STOCHASTIC PROCESSES The Basics of Forecasing wih Sochasic Processes Random Walk: Brownian Moion Jump-Diffusion Mean-Reversion ANALYTICAL MODELS Auocorrelaion Conrol Chars ROV BizSas User Manual 5 Copyrigh Dr. Johnahan Mun

6 X-Bar Char R-Bar Char XMR Char P Char NP Char C Char U Char Deseasonalizaion Disribuional Fiing Heeroskedasiciy Maximum Likelihood Models on Logi, Probi, and Tobi... 6 Mulicollineariy Parial Auocorrelaion Segmenaion Clusering Seasonaliy Tes Srucural Break PROBABILITY DISTRIBUTIONS AND SIMULATION The Basics of Disribuions Probabiliy Tables Sandard Normal (Parial Area) Probabiliy Tables Sandard Normal (Full Area) Probabiliy Tables Suden s T-Table (One and Two Tails) APPENDIX 1: DETAILS ON PROBABILITY DISTRIBUTIONS... 7 Discree Disribuions... 7 Bernoulli or Yes/No Disribuion... 7 Binomial Disribuion Discree Uniform Geomeric Disribuion Hypergeomeric Disribuion Negaive Binomial Disribuion Pascal Disribuion Poisson Disribuion Coninuous Disribuions Arcsine Disribuion Bea Disribuion Bea 3 and Bea 4 Disribuions Cauchy Disribuion, or Lorenzian or Brei-Wigner Disribuion... 8 Chi-Square Disribuion ROV BizSas User Manual 6 Copyrigh Dr. Johnahan Mun

7 Cosine Disribuion Double Log Disribuion Erlang Disribuion Exponenial Disribuion Exponenial Disribuion Exreme Value Disribuion, or Gumbel Disribuion F Disribuion, or Fisher-Snedecor Disribuion Gamma Disribuion (Erlang Disribuion) Laplace Disribuion Logisic Disribuion Lognormal Disribuion Lognormal 3 Disribuion... 9 Normal Disribuion... 9 Parabolic Disribuion Pareo Disribuion Pearson V Disribuion Pearson VI Disribuion PERT Disribuion Power Disribuion Power 3 Disribuion Suden s Disribuion Triangular Disribuion Uniform Disribuion Weibull Disribuion (Rayleigh Disribuion) Weibull 3 Disribuion APPENDIX : A PRIMER ON TIME-SERIES METHODOLOGIES No Trend and No Seasonaliy: Single Moving Average and Single Exponenial Smoohing Single Moving Average Error Esimaion (RMSE, MSE, MAD, MAPE, and Theil s U) Single Exponenial Smoohing Wih Trend and No Seasonaliy: Double Moving Average and Double Exponenial Smoohing Double Moving Average Double Exponenial Smoohing No Trend and Wih Seasonaliy: Addiive Seasonaliy and Muliplicaive Seasonaliy Addiive Seasonaliy Muliplicaive Seasonaliy Wih Seasonaliy and Trend: Hol-Winer s Addiive and Hol-Winer s Muliplicaive Hol-Winers Addiive Seasonaliy Hol-Winers Muliplicaive Seasonaliy ROV BizSas User Manual 7 Copyrigh Dr. Johnahan Mun

8 APPENDIX 3: A PRIMER ON REGRESSION ANALYSIS The Basics of Regression Analysis Regression Oupu... 1 Goodness of Fi Assumpions in Regression Analysis The Pifalls of Forecasing: Ouliers, Nonlineariy, Mulicollineariy, Heeroskedasiciy, Auocorrelaion, and Srucural Breaks Oher Technical Issues in Regression Analysis Lack of Independence in he Dependen Variable Random, No Fixed, Independen Variable Special Problems wih Few Daa Poins Special Problems wih Regression Through he Origin ROV BizSas User Manual 8 Copyrigh Dr. Johnahan Mun

9 INSTALLATION AND QUICK GETTING STARTED GUIDE Insallaion Requiremens ROV BizSas works on Windows XP, Visa, and Windows 7 and even runs on a MAC operaing sysem (wihin Virual Machine or Parallels). A minimum of 100MB hard disk space and 51MB RAM is required (GB RAM or more is recommended). Insall he seup file as you would any regular sofware. The insallaion will copy he relevan files o your compuer and regisry, and a he end of he insallaion, he seup will run a required prerequisie: he C++ redisribuable package from Microsof. During insallaion, you can op o include an icon on your deskop for quick access o he sofware; oherwise, you can access he sofware hrough he following shorcu locaion: Sar Programs Real Opions Valuaion ROV BizSas ROV BizSas 011. The defaul sofware license comes wih a 3-day fully funcional rial. A he end of he rial period, you will need a Name-Key combinaion o coninue using he sofware. You will be promped for a name and key combinaion when you sar he sofware afer he rial period, or simply sar ROV BizSas and click on Help Insall License o ener your license key a any ime. When his name-key promp comes up, you will also be provided an 8-digi Hardware Fingerprin ha is unique o your compuer. You can purchase he license from our websie a you will need o provide your 8-digi Hardware Fingerprin o obain a license for your compuer. ROV BizSas comes in 10 languages: English, Chinese Simplified, Chinese Tradiional, French, German, Ialian, Japanese, Korean, Poruguese, and Spanish. The sofware comes in 3 separae modules, all available wihin he Sar Programs Real Opions Valuaion ROV BizSas locaion. The following secion provides a summary of he saisical mehods available in ROV BizSas, and he secion afer ha is a quick geing sared guide explaining how he modules work. The remaining secions of his user manual briefly describe he analyical ools and saisical models ha are available in ROV BizSas. Because he user inerface is simple o use and fairly user friendly, hese laer secions do no repea he procedures and seps required; insead, i is given ha he procedures will be similar o hose explained in he quick geing sared secion. ROV BizSas can handle differen regional decimal and numerical seings (e.g., one housand dollars and fify cens can be wrien as 1, or 1.000,50 or 1 000,50, ec.). The decimal seings can be se in ROV BizSas menu Daa Decimal Seings. However, when in doub, change he compuer s regional seings o English USA and keep he defaul Norh America 1, in ROV BizSas (his seing is guaraneed o work wih ROV BizSas and he defaul examples). ROV BizSas User Manual 9 Copyrigh Dr. Johnahan Mun

10 Summary of Saisical Mehods in ROV BizSas Here are he saisical mehods available in ROV BizSas, arranged alphabeically: 1. Absolue Values. ANOVA: Randomized Blocks Muliple Treamens 3. ANOVA: Single Facor Muliple Treamens 4. ANOVA: Two Way Analysis 5. ARIMA 6. Auo ARIMA 7. Auocorrelaion and Parial Auocorrelaion 8. Auoeconomerics (Deailed) 9. Auoeconomerics (Quick) 10. Average 11. Combinaorial Fuzzy Logic Forecasing 1. Conrol Char: C 13. Conrol Char: NP 14. Conrol Char: P 15. Conrol Char: R 16. Conrol Char: U 17. Conrol Char: X 18. Conrol Char: XMR 19. Correlaion 0. Correlaion (Linear 1. Coun. Covariance 3. Cubic Spline 4. Cusom Economeric Model 5. Daa Descripive Saisics 6. Deseasonalize 7. Difference 8. Disribuional Fiing 9. Exponenial J Curve 30. GARCH 31. Heeroskedasiciy 3. Lag 33. Lead 34. Limied Dependen Variables (Logi) 35. Limied Dependen Variables (Probi) 36. Limied Dependen Variables (Tobi) 37. Linear Inerpolaion 38. Linear Regression 39. LN 40. Log 41. Logisic S Curve 4. Markov Chain 43. Max 44. Median 45. Min 46. Mode 47. Neural Nework 48. Nonlinear Regression 49. Nonlinear Models 50. Nonparameric: Chi-Square Goodness of Fi 51. Nonparameric: Chi-Square Independence 5. Nonparameric: Chi-Square Populaion Variance 53. Nonparameric: Friedman Tes 54. Nonparameric: Kruskal-Wallis Tes 55. Nonparameric: Lilliefors Tes 56. Nonparameric: Runs Tes 57. Nonparameric: Wilcoxon Signed-Rank (One Var) 58. Nonparameric: Wilcoxon Signed-Rank (Two Var) 59. Parameric: One Variable (T) Mean 60. Parameric: One Variable (Z) Mean 61. Parameric: One Variable (Z) Proporion 6. Parameric: Two Variable (F) Variances 63. Parameric: Two Variable (T) Dependen Means 64. Parameric: Two Variable (T) Independen Equal Variance 65. Parameric: Two Variable (T) Independen Unequal Variance 66. Parameric: Two Variable (Z) Independen Means 67. Parameric: Two Variable (Z) Independen Proporions ROV BizSas User Manual 10 Copyrigh Dr. Johnahan Mun

11 68. Power 69. Principal Componen Analysis 70. Rank Ascending 71. Rank Descending 7. Relaive LN Reurns 73. Relaive Reurns 74. Seasonaliy 75. Segmenaion Clusering 76. Semi-Sandard Deviaion (Lower) 77. Semi-Sandard Deviaion (Upper) 78. Sandard D Area 79. Sandard D Bar 80. Sandard D Line 81. Sandard D Poin 8. Sandard D Scaer 83. Sandard 3D Area 84. Sandard 3D Bar 85. Sandard 3D Line 86. Sandard 3D Poin 87. Sandard 3D Scaer 88. Sandard Deviaion (Populaion) 89. Sandard Deviaion (Sample) 90. Sepwise Regression (Backward) 91. Sepwise Regression (Correlaion) 9. Sepwise Regression (Forward) 93. Sepwise Regression (Forward- Backward) 94. Sochasic Processes (Exponenial Brownian Moion) 95. Sochasic Processes (Geomeric Brownian Moion) 96. Sochasic Processes (Jump Diffusion) 97. Sochasic Processes (Mean Reversion wih Jump Diffusion) 98. Sochasic Processes (Mean Reversion) 99. Srucural Break 100. Sum 101. Time-Series Analysis (Auo) 10. Time-Series Analysis (Double Exponenial Smoohing) 103. Time-Series Analysis (Double Moving Average) 104. Time-Series Analysis (Hol-Winer s Addiive) 105. Time-Series Analysis (Hol-Winer s Muliplicaive) 106. Time-Series Analysis (Seasonal Addiive) 107. Time-Series Analysis (Seasonal Muliplicaive) 108. Time-Series Analysis (Single Exponenial Smoohing) 109. Time-Series Analysis (Single Moving Average) 110. Trend Line (Difference Derended) 111. Trend Line (Exponenial Derended) 11. Trend Line (Exponenial) 113. Trend Line (Linear Derended) 114. Trend Line (Linear) 115. Trend Line (Logarihmic Derended) 116. Trend Line (Logarihmic) 117. Trend Line (Moving Average Derended) 118. Trend Line (Moving Average) 119. Trend Line (Polynomial Derended) 10. Trend Line (Polynomial) 11. Trend Line (Power Derended) 1. Trend Line (Power) 13. Trend Line (Rae Derended) 14. Trend Line (Saic Mean Derended) 15. Trend Line (Saic Median Derended) 16. Variance (Populaion) 17. Variance (Sample) 18. Volailiy: EGARCH 19. Volailiy: EGARCH-T 130. Volailiy: GARCH 131. Volailiy: GARCH-M 13. Volailiy: GJR GARCH 133. Volailiy: GJR TGARCH 134. Volailiy: Log Reurns Approach 135. Volailiy: TGARCH 136. Volailiy: TGARCH-M 137. Yield Curve (Bliss) 138. Yield Curve (Nelson-Siegel) ROV BizSas User Manual 11 Copyrigh Dr. Johnahan Mun

12 Quick Geing Sared Guide The ROV BizSas sofware is an applied saisics ool ha is focused on user-friendliness bu ye is powerful enough o solve mos day-o-day saisical problems. This new ROV BizSas ool is a very powerful and fas module ha is used for running business saisics and analyical models on your daa. I covers more han 130 business saisics and analyical models (Figures 1 hrough 4). The following provides a few quick geing sared seps on running he ROV BizSas module and deails on each of he elemens in he sofware, while he remaining secions of his user manual are dedicaed o explaining and exploring some of he mos criical saisical mehodologies available in ROV BizSas. Procedure Noes Run ROV BizSas a Sar Programs Real Opions Valuaion ROV BizSas ROV BizSas 011 and click on Example o load a sample daa and model profile [A] or ype in your daa or copy/pase from anoher sofware such as Excel or Word/ex file ino he daa grid in Sep 1 [D] (Figure 1). You can add your own noes or variable names in he firs Noes row [C]. Selec he relevan model [F] o run in Sep and using he example daa inpu seings [G], ener in he relevan variables [H]. Separae variables for he same parameer using semicolons and use a new line (hi Ener o creae a new line) for differen parameers. Click Run [I] o compue he resuls [J]. You can view any relevan analyical resuls, chars, or saisics from he various abs in Sep 3. If required, you can provide a model name o save ino he profile in Sep 4 [L]. Muliple models can be saved in he same profile. Exising models can be edied or deleed [M] and rearranged in order of appearance [N], and all he changes can be saved [O] ino a single profile wih he file name exension *.bizsas. The daa grid size can be se in he menu, where he grid can accommodae up o 1,000 variable columns wih 1 million rows of daa per variable. The menu also allows you o change he language seings and decimal seings for your daa. To ge sared, i is always a good idea o load he example file [A] ha comes complee wih some daa and pre-creaed models [S] (Figure ). You can double-click on any of hese models o run hem and he resuls are shown in he repor area [J], which someimes can be a char or model saisics [T/U]. Using his example file, you can now see how he inpu parameers [H] are enered based on he model descripion [G], and you can proceed o creae your own cusom models. Click on he variable headers [D] o selec one or muliple variables a once, and hen righ-click o add, delee, copy, pase, or visualize [P] he variables seleced. Models can also be enered using a Command console [V/W/X] (Figure 3). To see how his works, double-click o run a model [S] and go o he Command console [V]. You can replicae he model or creae your own and click Run Command [X] when ready. Each line in he console represens a model and is relevan parameers. The enire *.bizsas profile (where daa and muliple models are creaed and saved) can be edied direcly in XML [Z] by opening he XML Edior from he File menu (Figure 4). ROV BizSas User Manual 1 Copyrigh Dr. Johnahan Mun

13 Changes o he profile can be programmaically made here and ake effec once he file is saved. Tips Click on he daa grid s column header(s) o selec he enire column(s) or variable(s), and once seleced, you can righ-click on he header o Auo Fi he column, or o Cu, Copy, Delee, or Pase daa. You can also click on and selec muliple column headers o selec muliple variables and righ-click and selec Visualize o char he daa. If a cell has a large value ha is no compleely displayed, click on and hover your mouse over ha cell and you will see a pop-up commen showing he enire value, or simply resize he variable column (drag he column o make i wider, double-click on he column s edge o auo fi he column, or righ-click on he column header and selec Auo Fi). Use he up, down, lef, and righ keys o move around he grid, or use he Home and End keys on he keyboard o move o he far lef and far righ of a row. You can also use combinaion keys such as Crl+Home o jump o he op lef cell, Crl+End o he boom righ cell, Shif+Up/Down o selec a specific area, and so forh. You can ener shor noes for each variable on he Noes row. Remember o make your noes shor and simple. Try ou he various char icons on he Visualize ab o change he look and feel of he chars (e.g., roae, shif, zoom, change colors, add legend, ec.). The Copy buon is used o copy he Resuls, Chars, and Saisics abs in Sep 3 afer a model is run. If no models are run, hen he Copy funcion will only copy a blank page. The Repor buon will only run if here are saved models in Sep 4 or if here are daa in he grid, oherwise he repor generaed will be empy. You will also need Microsof Excel o be insalled o run he daa exracion and resuls repors, and Microsof PowerPoin available o run he char repors. When in doub abou how o run a specific model or saisical mehod, sar he Example profile and review how he daa is seup in Sep 1 or how he inpu parameers are enered in Sep. You can use hese as geing sared guides and emplaes for your own daa and models. The language can be changed in he Language menu. Noe ha currenly here are 10 languages available in he sofware wih more o be added laer. However, someimes cerain limied resuls will sill be shown in English. You can change how he lis of models in Sep is shown by changing he View dropdown lis. You can lis he models alphabeically, caegorically, and by daa inpu requiremens noe ha in cerain Unicode languages (e.g., Chinese, Japanese, and Korean), here is no alphabeical arrangemen and, herefore, he firs opion will be unavailable. The sofware can handle differen regional decimal and numerical seings (e.g., one housand dollars and fify cens can be wrien as 1, or 1.000,50 or 1 000,50 ec.). The decimal seings can be se in ROV BizSas menu Daa Decimal Seings. However, when in doub, please change he compuer s regional seings o English USA and keep he defaul Norh America 1, in ROV BizSas (his seing is guaraneed o work wih ROV BizSas and he defaul examples). ROV BizSas User Manual 13 Copyrigh Dr. Johnahan Mun

14 Figure 1 ROV BizSas (Saisical Analysis) Figure ROV BizSas (Daa Visualizaion and Resuls Chars) ROV BizSas User Manual 14 Copyrigh Dr. Johnahan Mun

15 Figure 3 ROV BizSas (Command Console) Figure 4 ROV BizSas (XML Edior) ROV BizSas User Manual 15 Copyrigh Dr. Johnahan Mun

16 Disribuional Chars and Tables is a Probabiliy Disribuion ool ha is a very powerful and fas module used for generaing disribuion chars and ables (Figures 5 hrough 8). Procedure Run ROV BizSas a Sar Programs Real Opions Valuaion ROV BizSas ROV Probabiliy Disribuions, click on he Apply Global Inpus buon o load a sample se of inpu parameers or ener your own inpus, and click Run o compue he resuls. The resuling four momens and CDF, ICDF, and PDF are compued for each of he 45 probabiliy disribuions (Figure 5). Figure 5 Probabiliy Disribuion Tool (45 Probabiliy Disribuions) Click on he Chars and Tables ab (Figure 6), selec a disribuion [A] (e.g., Arcsine), choose if you wish o run he CDF, ICDF, or PDF [B], ener he relevan inpus, and click Run Char or Run Table [C]. You can swich beween he Chars and Table ab o view he resuls as well as ry ou some of he char icons [E] o see he effecs on he char. You can also change wo parameers [H] o generae muliple chars and disribuion ables by enering he From/To/Sep inpu or using he Cusom inpus and hen hiing Run. For example, as illusraed in Figure 7, run he Bea disribuion and selec PDF [G], selec Alpha and Bea o change [H] using cusom [I] inpus and ener he relevan inpu parameers: ;5;5 for Alpha and 5;3;5 for Bea [J], and click Run Char. This will generae hree Bea disribuions [K]: Bea (,5), Bea (5,3), and Bea (5,5) [L]. Explore various char ypes, gridlines, language, and decimal seings [M], and ry rerunning he disribuion using heoreical versus empirically simulaed values [N]. Figure 8 illusraes he probabiliy ables generaed for a binomial disribuion where he probabiliy of success and number of successful rials (random variable X) are seleced o ROV BizSas User Manual 16 Copyrigh Dr. Johnahan Mun

17 vary [O] using he From/To/Sep opion. Try o replicae he calculaion as shown and click on he Table ab [P] o view he creaed probabiliy densiy funcion resuls. This example uses a binomial disribuion wih a saring inpu se of Trials = 0, Probabiliy (of success) = 0.5, and Random X, or Number of Successful Trials, = 10, where he Probabiliy of Success is allowed o change from 0., 0.5,, 0.50 and is shown as he row variable, and he Number of Successful Trials is also allowed o change from 0, 1,,, 8 and is shown as he column variable. PDF is chosen and, hence, he resuls in he able show he probabiliy ha he given evens occur. For insance, he probabiliy of geing exacly successes when 0 rials are run where each rial has a 5% chance of success is , or 6.69%. Figure 6 ROV Probabiliy Disribuion (PDF and CDF Chars) ROV BizSas User Manual 17 Copyrigh Dr. Johnahan Mun

18 Figure 7 ROV Probabiliy Disribuion (Muliple Overlay Chars) Figure 8 ROV Probabiliy Disribuion (Disribuion Tables) ROV BizSas User Manual 18 Copyrigh Dr. Johnahan Mun

19 The Percenile Disribuional Fiing ool (Figure 9) is an alernae way of fiing probabiliy disribuions. Sar he ool from Sar Programs ROV Probabiliy Fiing Perceniles, choose he probabiliy disribuion and ypes of inpus you wish o use, ener he parameers, and click Run o obain he resuls. Review he fied R-square resuls and compare he empirical versus heoreical fiing resuls o deermine if your disribuion is a good fi. There are several relaed ools in he ROV Risk Simulaor sofware and ROV Quaniaive Daa Miner sofware, and each has is own uses and advanages: Disribuional Fiing (Perceniles) using an alernae mehod of enry (perceniles and firs/second momen combinaions) o find he bes-fiing parameers of a specified disribuion wihou he need for having raw daa. This mehod is suiable for use when here are insufficien daa, only when perceniles and momens are available, or as a means o recover he enire disribuion wih only wo or hree daa poins bu he disribuion ype needs o be assumed or known. Disribuional Fiing (Single Variable) using saisical mehods o fi your raw daa o all 4 disribuions o find he bes-fiing disribuion and is inpu parameers. Muliple daa poins are required for a good fi, and he disribuion ype may or may no be known ahead of ime. Disribuional Fiing (Muliple Variables) using saisical mehods o fi your raw daa on muliple variables a he same ime. This mehod uses he same algorihms as he single variable fiing, bu incorporaes a pairwise correlaion marix beween he variables. Muliple daa poins are required for a good fi, and he disribuion ype may or may no be known ahead of ime. Cusom Disribuion (Se Assumpion) using nonparameric resampling echniques o generae a cusom disribuion wih he exising raw daa and o simulae he disribuion based on his empirical disribuion. Fewer daa poins are required, and he disribuion ype is no known ahead of ime. ROV BizSas User Manual 19 Copyrigh Dr. Johnahan Mun

20 Figure 9 Percenile Disribuional Fiing Tool ROV BizSas User Manual 0 Copyrigh Dr. Johnahan Mun

21 1. BASIC STATISTICS Descripive Saisics Almos all disribuions can be described wihin four momens (some disribuions require one momen, while ohers require wo momens, ec.). Descripive saisics quaniaively capures hese momens. The firs momen describes he locaion of a disribuion (i.e., mean, median, and mode) and is inerpreed as he expeced value, expeced reurns, or he average value of occurrences. The second momen measures a disribuion's spread, or widh, and is frequenly described using measures such as Sandard Deviaions, Variances, Quariles, and Iner-Quarile Ranges. Sandard deviaion is a popular measure indicaing he average deviaion of all daa poins from heir mean. I is a popular measure as i is frequenly associaed wih risk (higher sandard deviaions meaning a wider disribuion, higher risk, or wider dispersion of daa poins around he mean value) and is unis are idenical o he unis in he original daa se. Skewness is he hird momen in a disribuion. Skewness characerizes he degree of asymmery of a disribuion around is mean. Posiive skewness indicaes a disribuion wih an asymmeric ail exending oward more posiive values. Negaive skewness indicaes a disribuion wih an asymmeric ail exending oward more negaive values. Kurosis characerizes he relaive peakedness or flaness of a disribuion compared o he normal disribuion. I is he fourh momen in a disribuion. A posiive kurosis value indicaes a relaively peaked disribuion. A negaive kurosis indicaes a relaively fla disribuion. The kurosis measured here has been cenered o zero (cerain oher kurosis measures are cenered on 3.0). While boh are equally valid, cenering across zero makes he inerpreaion simpler. A high posiive kurosis indicaes a peaked disribuion around is cener and lepokuric or fa ails. This indicaes a higher probabiliy of exreme evens (e.g., caasrophic evens, erroris aacks, sock marke crashes) han is prediced in a normal disribuion. Correlaion Marix The Correlaion module liss he Pearson's produc momen correlaions (commonly referred o as he Pearson s R) beween variable pairs. The correlaion coefficien ranges beween 1.0 and +1.0 inclusive. The sign indicaes he direcion of associaion beween he variables, while he coefficien indicaes he magniude or srengh of associaion. The Pearson's R only measures a linear relaionship and is less effecive in measuring nonlinear relaionships. A hypohesis -es is performed on he Pearson s R and he p-values are repored. If he calculaed p-value is less han or equal o he significance level used in he es, hen rejec he null hypohesis and conclude ha here is a significan correlaion beween he wo variables in quesion. Oherwise, he correlaion is no saisically significan. Finally, a Spearman Rank-Based Correlaion is also included. The Spearman s R firs ranks he raw daa hen performs he correlaion calculaion, which allows i o beer capure nonlinear relaionships. The Pearson s R is a parameric es and he underlying daa is assumed o be normally disribued, hence, he -es can be applied. However, he Spearman s R is a ROV BizSas User Manual 1 Copyrigh Dr. Johnahan Mun

22 nonparameric es, where no underlying disribuions are assumed, and, hence, he -es canno be applied. Variance-Covariance Marix The Covariance measures he average of he producs of deviaions for each daa poin pair. Use covariance o deermine he relaionship beween wo variables. The covariance is relaed o he correlaion in ha he correlaion is he covariance divided by he produc of he wo variables sandard deviaion, sandardizing he correlaion measuremen o be uniless and beween 1 and +1. Covariance is used when he unis of he variables are similar, allowing for easy comparison of he magniude of variabiliy abou heir respecive means. The covariance of he same variable is also known as he variance. The variance of a variable is he square of is sandard deviaion. This is why sandardizing he variance hrough dividing i by he variable s sandard deviaion (wice) yields a correlaion of 1.0, indicaing ha a variable is perfecly correlaed o iself. I mus be sressed ha a high covariance does no imply causaion. Associaions beween variables in no way imply ha he change of one variable causes anoher variable o change. Two variables ha are moving independenly of each oher bu in a relaed pah may have a high covariance bu heir relaionship migh be spurious. In order o capure his relaionship, use regression analysis insead. Basic Saisics The following basic saisical funcions are also included in ROV BizSas and heir shor definiions are lised below: Absolue Values Compues he absolue value of a number where i is he number wihou is sign. Average Compues he average or arihmeic mean of he rows of daa for he seleced variable. Coun Compues how many numbers here are in he rows of daa for he seleced variable. Difference Compues he difference of he curren period from he previous period. Lag Reurns he value lagged some number of periods (he enire chronological daa se is shifed down he number of lagged periods specified). ROV BizSas User Manual Copyrigh Dr. Johnahan Mun

23 Lead Reurns he value leading by some number of periods (he enire chronological daa se is shifed up he number of lead periods specified). LN Compues he naural logarihm. Log Compues he logarihmic value of some specified base. Max Compues he maximum of he rows of daa for he seleced variable. Median Compues he median of he rows of daa for he seleced variable. Min Compues he minimum of he rows of daa for he seleced variable. Mode Compues he mode or mos frequenly occurring of daa poins for he seleced variable. Power Compues he resul of a number raised o a specified power. Rank Ascending Ranks he rows of daa for he seleced variable in ascending order. Rank Descending Ranks he rows of daa for he seleced variable in descending order. Relaive LN Reurns Compues he naural logarihm of he relaive reurns from one period o anoher, where he relaive reurn is compued as he curren value divided by is previous value. Relaive Reurns Compues he relaive reurn where he curren value is divided by is previous value. Semi-Sandard Deviaion (Lower) Compues he sample sandard deviaion of daa poins below a specified value. ROV BizSas User Manual 3 Copyrigh Dr. Johnahan Mun

24 Semi-Sandard Deviaion (Upper) Compues he sample sandard deviaion of daa poins above a specified value. Sandard Deviaion (Populaion) Compues he populaion sandard deviaion of he rows of daa for he seleced variable. Sandard Deviaion (Sample) Compues he sample sandard deviaion of he rows of daa for he seleced variable. Variance (Populaion) Compues he populaion variance of he rows of daa for he seleced variable. Variance (Sample) Compues he sample variance of he rows of daa for he seleced variable. Sum Compues he sum oal of all he rows of daa for he seleced variable. ROV BizSas User Manual 4 Copyrigh Dr. Johnahan Mun

25 . HYPOTHESIS TESTING: BASIC MODEL CHOOSER Using he correc model or saisical mehodology is clearly he firs sep in your analysis. Figures.1 hrough.4 illusrae some of he logic behind some basic saisical models and how o choose he correc mehodology. They also provide a grea se of quick guidelines for model selecion. A hypohesis es is a saisical es used o deermine he characerisics of a populaion by esing a small sample colleced from he populaion. In mos cases, he populaion o be sudied migh be oo large, difficul, or expensive o be compleely sampled (e.g., all 100 million regisered voers in he Unied Saes in a paricular elecion) and, hence, a smaller sample (e.g., a random sample of 1,100 voers from 0 ciies) is colleced and he sample saisics are abulaed. Then, using hypohesis ess, he characerisics of he enire populaion can be inferred from his small sample. ROV BizSas allows he user o es one-variable, wo-variable, and muliplevariable hypoheses ess. To perform a hypohesis es, firs se up he null hypohesis (H 0 ) and he alernae hypohesis (H a ). Here are some quick rules: a) Always se up he alernae hypohesis firs, hen he null hypohesis. b) The alernae hypohesis mus always have he following signs: > or < or. c) The null hypohesis mus always have he following signs: or or =. d) If he alernae hypohesis is, hen i s a wo-ailed es; if <, hen i s a lef (one) ailed; and if >, hen i s a righ (one) ailed es Then, collec he sample daa. Nex, using ROV BizSas, run he appropriae hypohesis ess. You can use he Model Chooser diagrams provided as Figures.1 hrough.4 o help you find he righ hypohesis es o run under differen condiions. Then, depending on he resuls obained from ROV BizSas, make he relevan conclusions abou he populaion based on he sample daa colleced. Tha is, if he p-value is less han he significance level (he significance level is seleced by he user and is usually 0.10, 0.05, or 0.01) esed, rejec he null hypohesis and accep he alernae hypohesis. Two-Tailed Hypohesis Tes A wo-ailed hypohesis ess he null hypohesis such ha he populaion median of he sample daa se is saisically idenical o he hypohesized median. The alernaive hypohesis is ha he real populaion median is saisically differen from he hypohesized median when esed using he sample daa se. If he calculaed p-value is less han or equal o he alpha significance value, hen rejec he null hypohesis and accep he alernae hypohesis. Oherwise, if he p-value is higher han he alpha significance value, do no rejec he null hypohesis. ROV BizSas User Manual 5 Copyrigh Dr. Johnahan Mun

26 Righ-Tailed Hypohesis Tes A righ-ailed hypohesis ess he null hypohesis such ha he populaion median of he sample daa se is saisically less han or equal o he hypohesized median. The alernaive hypohesis is ha he real populaion median is saisically greaer han he hypohesized median when esed using he sample daa se. If he calculaed p-value is less han or equal o he alpha significance value, hen rejec he null hypohesis and accep he alernae hypohesis. Oherwise, if he p- value is higher han he alpha significance value, do no rejec he null hypohesis. Lef-Tailed Hypohesis Tes A lef-ailed hypohesis ess he null hypohesis such ha he populaion median of he sample daa se is saisically greaer han or equal o he hypohesized median. The alernaive hypohesis is ha he real populaion median is saisically less han he hypohesized median when esed using he sample daa se. If he calculaed p-value is less han or equal o he alpha significance value, hen rejec he null hypohesis and accep he alernae hypohesis. Oherwise, if he p-value is higher han he alpha significance value, do no rejec he null hypohesis. ROV BizSas User Manual 6 Copyrigh Dr. Johnahan Mun

27 MODEL SELECTION Wha do you wan o do? Describe saisical properies of daa Make inference abou populaion hrough samples Find he probabiliies of evens occurring Forecas or predic based on hisorical or generaed daa Explain or model saisical relaionships among variables Show only he basic saisics of one variable a a ime? See Hypohesis Tess Simulae possible oucomes Creae probabiliy ables Hisorical daa exiss and evens are predicable? YES NO Sochasic Processes YES Descripive Saisics NO Show pairwise relaionships? Mone Carlo Probabiliy Tables Are daa all ime series? YES NO Muliple or Sepwise Regression Do you wish o sandardize he measuremen? YES Correlaion YES NO Covariance Find exac probabiliies of oucomes Discree Evens Discree Disribuion Coninuous Evens Coninuous Disribuion Basic Time Series, Auocorrelaion, Muliple Figure.1 Model Selecion Flowchar ROV BizSas User Manual 7 Copyrigh Dr. Johnahan Mun

28 Hypohesis Tess: One-Variable How many groups of variables are you comparing? ONE Are you comparing Means or Proporions? Proporions Is n 5 and n(1-) 5? NO Binomial Disribuion Means Is populaion sandard deviaion known? YES 1 Sample Z- Tes for Proporions YES NO Is populaion approximaely normal or sample size n 30? Is populaion approximaely normal or sample size n 30? YES NO YES NO 1 Sample Z- Tes for Means Wilcoxon Signed- Rank Tess 1 Sample T- Tes for Means Wilcoxon Signed- Rank Tess Figure. One-Variable Hypohesis Tes ROV BizSas User Manual 8 Copyrigh Dr. Johnahan Mun

29 Hypohesis Tess: Two-Variable -Variable Dependen T-Tes How many groups of variables are you comparing? TWO Are he wo samples Dependen? Dependen Is he populaion normally disribued or n 30? Independen Wilcoxon Signed- Rank Tes -Variable F-Tes Variances Are you comparing Means, Proporions, or Variances? Proporions -Variable Proporion Z- Tes Means Are populaion sandard deviaions equal? YES -Variable T- Tes Equal Variances NO Do boh populaions have sample size n 30 each? YES NO -Variable Z-Tes for Means -Variable T-Tes Unequal Figure.3 Two-Variable Hypohesis Tes ROV BizSas User Manual 9 Copyrigh Dr. Johnahan Mun

30 Hypohesis Tess: Muli-Variable YES Is he populaion normally disribued or n 30? YES NO Randomized Block ANOVA Friedman s Tes How many groups of variables are you comparing? THREE How many Facors are you esing? ONE Are you Conrolling or Blocking for an exogenous effec? TWO Two-Facor ANOVA NO Is he populaion normally disribued or n 30? YES NO Single-Facor ANOVA Kruskal-Wallis Tes Figure.4 Muliple-Variable Hypohesis Tes ROV BizSas User Manual 30 Copyrigh Dr. Johnahan Mun

31 3. HYPOTHESIS TESTING: PARAMETRIC MODELS One-Variable Tesing for Means (T-Tes) This one-variable -es of means is appropriae when he populaion sandard deviaion is no known bu he sampling disribuion is assumed o be approximaely normal (he -es is used when he sample size is less han 30). This -es can be applied o hree ypes of hypohesis ess o be examined a wo-ailed es, a righ-ailed es, and a lef-ailed es based on he sample daa se if he populaion mean is equal o, less han, or greaer han he hypohesized mean. If he calculaed p-value is less han or equal o he significance level in he es, hen rejec he null hypohesis and conclude ha he rue populaion mean is no equal o (wo-ailed es), less han (lef-ailed es), or greaer han (righ-ailed es) he hypohesized mean based on he sample esed. Oherwise, he rue populaion mean is saisically similar o he hypohesized mean. One-Variable Tesing for Means (Z-Tes) The one-variable Z-es is appropriae when he populaion sandard deviaion is known, and he sampling disribuion is assumed o be approximaely normal (his applies when he number of daa poins exceeds 30). This Z-es can be applied o hree ypes of hypohesis ess o be examined a wo-ailed es, a righ-ailed es, and a lef-ailed es based on he sample daa se if he populaion mean is equal o, less han, or greaer han he hypohesized mean. If he calculaed p-value is less han or equal o he significance level in he es, hen rejec he null hypohesis and conclude ha he rue populaion mean is no equal o (wo-ailed es), less han (lef-ailed es), or greaer han (righ-ailed es) he hypohesized mean based on he sample esed. Oherwise, he rue populaion mean is saisically similar o he hypohesized mean. One-Variable Tesing for Proporions (Z-Tes) The one-variable Z-es for proporions is appropriae when he sampling disribuion is assumed o be approximaely normal (his applies when he number of daa poins exceeds 30, and when he number of daa poins, N, muliplied by he hypohesized populaion proporion mean, P, is greaer han or equal o five, or NP 5 ). The daa used in he analysis have o be proporions and be beween 0 and 1. This Z-es can be applied o hree ypes of hypohesis ess o be examined a wo-ailed es, a righ-ailed es, and a lef-ailed es based on he sample daa se if he populaion mean is equal o, less han, or greaer han he hypohesized mean. If he calculaed p-value is less han or equal o he significance level in he es, hen rejec he null hypohesis and conclude ha he rue populaion mean is no equal o (wo-ailed es), less han (lef-ailed es), or greaer han (righ-ailed es) he hypohesized mean based on he sample esed. Oherwise, he rue populaion mean is saisically similar o he hypohesized mean. Two Variables wih Dependen Means (T-Tes) The wo-variable dependen -es is appropriae when he populaion sandard deviaion is no known bu he sampling disribuion is assumed o be approximaely normal (he -es is used when he sample size is less han 30). In addiion, his es is specifically formulaed for esing ROV BizSas User Manual 31 Copyrigh Dr. Johnahan Mun

32 he same or similar samples before and afer an even (e.g., measuremens aken before a medical reamen are compared agains hose measuremens aken afer he reamen o see if here is a difference). This -es can be applied o hree ypes of hypohesis ess: a wo-ailed es, a righailed es, and a lef-ailed es. Suppose ha a new hear medicaion was adminisered o 100 paiens (N = 100) and he hear raes before and afer he medicaion was adminisered were measured. The wo dependen variables -es can be applied o deermine if he new medicaion is effecive by esing o see if here are saisically differen "before and afer" averages. The dependen variables es is used here because here is only a single sample colleced (he same paiens' hearbeas were measured before and afer he new drug adminisraion). The wo-ailed null hypohesis ess ha he rue populaion s mean of he difference beween he wo variables is zero, versus he alernae hypohesis ha he difference is saisically differen from zero. The righ-ailed null hypohesis es is such ha he differences in he populaion means (firs mean less second mean) is saisically less han or equal o zero (which is idenical o saying ha mean of he firs sample is less han or equal o he mean of he second sample). The alernaive hypohesis is ha he real populaions mean difference is saisically greaer han zero when esed using he sample daa se (which is idenical o saying ha he mean of he firs sample is greaer han he mean of he second sample). The lef-ailed null hypohesis es is such ha he differences in he populaion means (firs mean less second mean) is saisically greaer han or equal o zero (which is idenical o saying ha he mean of he firs sample is greaer han or equal o he mean of he second sample). The alernaive hypohesis is ha he real populaions mean difference is saisically less han zero when esed using he sample daa se (which is idenical o saying ha he mean of he firs sample is less han he mean of he second sample). If he calculaed p-value is less han or equal o he significance level in he es, hen rejec he null hypohesis and conclude ha he rue populaion difference of he populaion means is no equal o (wo-ailed es), less han (lef-ailed es), or greaer han (righ-ailed es) zero based on he sample esed. Oherwise, he rue populaion mean is saisically similar o he hypohesized mean. Two (Independen) Variables wih Equal Variances (T-Tes) The wo-variable -es wih equal variances is appropriae when he populaion sandard deviaion is no known bu he sampling disribuion is assumed o be approximaely normal (he -es is used when he sample size is less han 30). In addiion, he wo independen samples are assumed o have similar variances. For illusraion, suppose ha a new engine design is esed agains an exising engine design o see if here is a saisically significan differen beween he wo. The -es on wo (independen) variables wih equal variances can be applied. This es is used because here are wo disincly differen samples colleced here (new engine and exising engine) bu he variances of boh samples are assumed o be similar (he means may or may no be similar, bu he flucuaions around he mean are assumed o be similar). This -es can be applied o hree ypes of hypohesis ess: a wo-ailed es, a righ-ailed es, and a lef-ailed es. A wo-ailed hypohesis ess he null hypohesis, H 0, such ha he populaions mean difference (HMD) beween he wo variables is saisically idenical o he ROV BizSas User Manual 3 Copyrigh Dr. Johnahan Mun

33 hypohesized mean differences. If HMD is se o zero, his is he same as saying ha he firs mean equals he second mean. The alernaive hypohesis, H a, is ha he difference beween he real populaion means is saisically differen from he hypohesized mean differences when esed using he sample daa se. If HMD is se o zero, his is he same as saying ha he firs mean does no equal he second mean. A righ-ailed hypohesis ess he null hypohesis, H 0, such ha he populaion mean differences beween he wo variables is saisically less han or equal o he hypohesized mean differences. If HMD is se o zero, his is he same as saying ha he firs mean is less han or equals he second mean. The alernaive hypohesis, H a, is ha he real difference beween populaion means is saisically greaer han he hypohesized mean differences when esed using he sample daa se. If HMD is se o zero, his is he same as saying ha he firs mean is greaer han he second mean. A lef-ailed hypohesis ess he null hypohesis, H 0, such ha he differences beween he populaion means of he wo variables is saisically greaer han or equal o he hypohesized mean differences. If HMD is se o zero, his is he same as saying ha he firs mean is greaer han or equals he second mean. The alernaive hypohesis, H a, is ha he real difference beween populaion means is saisically less han he hypohesized mean difference when esed using he sample daa se. If HMD is se o zero, his is he same as saying ha he firs mean is less han he second mean. If he calculaed p-value is less han or equal o he significance level in he es, hen rejec he null hypohesis and conclude ha he rue populaion difference of he populaion means is no equal o (wo-ailed es), less han (lef-ailed es), or greaer han (righ-ailed es) HMD based on he sample esed. Oherwise, he rue difference of he populaion means is saisically similar o he HMD. For daa requiremens, see he preceding secion, Two Variables wih Dependen Means (T-Tes). Two (Independen) Variables wih Unequal Variances (T-Tes) The wo-variable -es wih unequal variances (he populaion variance of sample 1 is expeced o be differen from he populaion variance of sample ) is appropriae when he populaion sandard deviaion is no known bu he sampling disribuion is assumed o be approximaely normal (he -es is used when he sample size is less han 30). In addiion, he wo independen samples are assumed o have similar variances. To illusraed, suppose ha a new cusomer relaionship managemen (CRM) process is being evaluaed for is effeciveness, and he cusomer saisfacion rankings beween wo hoels (one wih and he oher wihou CRM implemened) are colleced. The -es on wo (independen) variables wih unequal variances can be applied. This es is used here because here are wo disincly differen samples colleced (cusomer survey resuls of wo differen hoels) and he variances of boh samples are assumed o be dissimilar (due o he difference in geographical locaion, plus he demographics and psychographics of he cusomers are differen on boh properies). This -es can be applied o hree ypes of hypohesis ess: a wo-ailed es, a righ-ailed es, and a lef-ailed es. A wo-ailed hypohesis ess he null hypohesis, H 0, such ha he populaion mean differences beween he wo variables are saisically idenical o he ROV BizSas User Manual 33 Copyrigh Dr. Johnahan Mun

34 hypohesized mean differences. If HMD is se o zero, his is he same as saying ha he firs mean equals he second mean. The alernaive hypohesis, H a, is ha he real difference beween he populaion means is saisically differen from he hypohesized mean differences when esed using he sample daa se. If HMD is se o zero, his is he same as saying ha he firs mean does no equal he second mean. A righ-ailed hypohesis ess he null hypohesis, H 0, such ha he difference beween he wo variables populaion means is saisically less han or equal o he hypohesized mean differences. If HMD is se o zero, his is he same as saying ha he firs mean is less han or equals he second mean. The alernaive hypohesis, H a, is ha he real populaions mean difference is saisically greaer han he hypohesized mean differences when esed using he sample daa se. If HMD is se o zero, his is he same as saying ha he firs mean is greaer han he second mean. A lef-ailed hypohesis ess he null hypohesis, H 0, such ha he difference beween he wo variables populaion means is saisically greaer han or equal o he hypohesized mean differences. If HMD is se o zero, his is he same as saying ha he firs mean is greaer han or equals he second mean. The alernaive hypohesis, H a, is ha he real difference beween populaion means is saisically less han he hypohesized mean difference when esed using he sample daa se. If HMD is se o zero, his is he same as saying ha he firs mean is less han he second mean. If he calculaed p-value is less han or equal o he significance level in he es, hen rejec he null hypohesis and conclude ha he rue populaion difference of he populaion means is no equal o (wo-ailed es), less han (lef-ailed es), or greaer han (righ-ailed es) he hypohesized mean based on he sample esed. Oherwise, he rue difference of he populaion means is saisically similar o he hypohesized mean. Two (Independen) Variables Tesing for Means (Z-Tes) The wo-variable Z-es is appropriae when he populaion sandard deviaions are known for he wo samples, and he sampling disribuion of each variable is assumed o be approximaely normal (his applies when he number of daa poins of each variable exceeds 30). To illusrae, suppose ha a marke survey was conduced on wo differen markes, he sample colleced is large (N mus exceed 30 for boh variables), and he researcher is ineresed in esing wheher here is a saisically significan difference beween he wo markes. Furher suppose ha such a marke survey has been performed many imes in he pas and he populaion sandard deviaions are known. A wo independen variables Z-es can be applied because he sample size exceeds 30 on each marke and he populaion sandard deviaions are known. This Z-es can be applied o hree ypes of hypohesis ess: a wo-ailed es, a righ-ailed es, and a lef-ailed es. A wo-ailed hypohesis ess he null hypohesis, H 0, such ha he difference beween he wo populaion means is saisically idenical o he hypohesized mean. The alernaive hypohesis, H a, is ha he real difference beween he wo populaion means is saisically differen from he hypohesized mean when esed using he sample daa se. A righ-ailed hypohesis ess he null hypohesis, H 0, such ha he difference beween he wo populaion means is saisically less han or equal o he hypohesized mean. The alernaive ROV BizSas User Manual 34 Copyrigh Dr. Johnahan Mun

35 hypohesis, H a, is ha he real difference beween he wo populaion means is saisically greaer han he hypohesized mean when esed using he sample daa se. A lef-ailed hypohesis ess he null hypohesis, H 0, such ha he difference beween he wo populaion means is saisically greaer han or equal o he hypohesized mean. The alernaive hypohesis, H a, is ha he real difference beween he wo populaion means is saisically less han he hypohesized mean when esed using he sample daa se. Two (Independen) Variables Tesing for Proporions (Z-Tes) The wo-variable Z-es on proporions is appropriae when he sampling disribuion is assumed o be approximaely normal (his applies when he number of daa poins of boh samples exceeds 30). Furher, he daa should all be proporions and be beween 0 and 1. To illusrae, suppose ha a brand research was conduced on wo differen headache pills, he sample colleced is large (N mus exceed 30 for boh variables), and he researcher is ineresed in esing wheher here is a saisically significan difference beween he proporion of headache sufferers of boh samples using he differen headache medicaion. A wo independen variables Z-es for proporions can be applied because he sample size exceeds 30 on each marke, and he daa colleced are proporions. This Z-es can be applied o hree ypes of hypohesis ess: a wo-ailed es, a righ-ailed es, and a lef-ailed es. A wo-ailed hypohesis ess he null hypohesis, H 0, ha he difference in he populaion proporion is saisically idenical o he hypohesized difference (if he hypohesized difference is se o zero, he null hypohesis ess if he populaion proporions of he wo samples are idenical). The alernaive hypohesis, H a, is ha he real difference in populaion proporions is saisically differen from he hypohesized difference when esed using he sample daa se. A righ-ailed hypohesis ess he null hypohesis, H 0, ha he difference in he populaion proporion is saisically less han or equal o he hypohesized difference (if he hypohesized difference is se o zero, he null hypohesis ess if populaion proporion of sample 1 is equal o or less han he populaion proporion of sample ). The alernaive hypohesis, H a, is ha he real difference in populaion proporions is saisically greaer han he hypohesized difference when esed using he sample daa se. A lef-ailed hypohesis ess he null hypohesis, H 0, ha he difference in he populaion proporion is saisically greaer han or equal o he hypohesized difference (if he hypohesized difference is se o zero, he null hypohesis ess if populaion proporion of sample 1 is equal o or greaer han he populaion proporion of sample ). The alernaive hypohesis, H a, is ha he real difference in populaion proporions is saisically less han he hypohesized difference when esed using he sample daa se. Two (Independen) Variables Tesing for Variances (F-Tes) The wo-variable F-es analyzes he variances from wo samples (he populaion variance of sample 1 is esed wih he populaion variance of sample o see if hey are equal) and is appropriae when he populaion sandard deviaion is no known bu he sampling disribuion is assumed o be approximaely normal. The measuremen of variaion is a key issue in Six Sigma and qualiy conrol applicaions. In his illusraion, suppose ha he variaion or variance around ROV BizSas User Manual 35 Copyrigh Dr. Johnahan Mun

36 he unis produced in a manufacuring process is compared o anoher process o deermine which process is more variable and, hence, less predicable in qualiy. This F-es can ypically be applied o a single hypohesis es: a wo-ailed es. A wo-ailed hypohesis ess he null hypohesis, H 0, such ha he populaion variance of he wo variables is saisically idenical. The alernaive hypohesis, H a, is ha he populaion variances are saisically differen from one anoher when esed using he sample daa se. If he calculaed p-value is less han or equal o he significance level in he es, hen rejec he null hypohesis and conclude ha he rue populaion variances of he wo variables are no saisically equal o one anoher. Oherwise, he rue populaion variances are saisically similar o each oher. ROV BizSas User Manual 36 Copyrigh Dr. Johnahan Mun

37 4. NONPARAMETRIC ANALYSIS The Basics of Nonparameric Mehodologies Nonparameric echniques make no assumpions abou he specific shape or disribuion from which he sample is drawn. This lack of assumpions makes i differen from he oher hypoheses ess such as ANOVA or -ess (parameric ess) where he sample is assumed o be drawn from a populaion ha is normally or approximaely normally disribued. If normaliy is assumed, he power of he es is higher due o his normaliy resricion. However, if flexibiliy on disribuional requiremens is needed, hen nonparameric echniques are superior. In general, nonparameric mehodologies provide he following advanages over oher parameric ess: Normaliy or approximae normaliy does no have o be assumed. Fewer assumpions abou he populaion are required; ha is, nonparameric ess do no require ha he populaion assume any specific disribuion. Smaller sample sizes can be analyzed. Samples wih nominal and ordinal scales of measuremen can be esed. Sample variances do no have o be equal, whereas equaliy is required in parameric ess. However, several caveas are worhy of menion: Compared o parameric ess, nonparameric ess use daa less efficienly. The power of he es is lower han ha of he parameric ess. Therefore, if all he required assumpions are saisfied, i is beer o use parameric ess. However, in realiy, i may be difficul o jusify hese disribuional assumpions, or small sample sizes may exis, requiring he need for nonparameric ess. Thus, nonparameric ess should be used when he daa are nominal or ordinal, or when he daa are inerval or raio bu he normaliy assumpion is no me. The following covers each of he nonparameric ess available for use in he sofware. Chi-Square Goodness-of-Fi Tes The Chi-Square es for goodness of fi is used o deermine wheher a sample daa se could have been drawn from a populaion having a specified probabiliy disribuion. The probabiliy disribuion esed here is he normal disribuion. The null hypohesis (H 0 ) esed is such ha he sample is randomly drawn from he normal disribuion, versus he alernae hypohesis (H a ) ha he sample is no from a normal disribuion. If he calculaed p-value is less han or equal o he alpha significance value, hen rejec he null hypohesis and accep he alernae hypohesis. Oherwise, if he p-value is higher han he alpha significance value, do no rejec he null hypohesis. For he Chi-Square goodness-of-fi es, creae daa ables such as he one below, and selec he daa in he blue area (e.g., selec he daa from D6 o E13, or daa poins 800 o 4). To exend he daa se, jus add more observaions (rows). ROV BizSas User Manual 37 Copyrigh Dr. Johnahan Mun

38 Chi-Square Tes of Independence The Chi-Square es for independence examines wo variables o see if here is some saisical relaionship beween hem. This es is no used o find he exac naure of he relaionship beween he wo variables, bu o simply es if he variables could be independen of each oher. The null hypohesis (H 0 ) esed is such ha he variables are independen of each oher, versus he alernae hypohesis (H a ) ha he variables are no independen of each oher. The Chi-Square es looks a a able of observed frequencies and a able of expeced frequencies. The amoun of dispariy beween hese wo ables is calculaed and compared wih he Chi-Square es saisic. The observed frequencies reflec he cross-classificaion for members of a single sample, and he able of expeced frequencies is consruced under he assumpion ha he null hypohesis is rue. Chi-Square Populaion Variance Tes The Chi-Square es for populaion variance is used for hypohesis esing and confidence inerval esimaion for a populaion variance. The populaion variance of a sample is ypically unknown, and, hence, he need for quanifying his confidence inerval. The populaion is assumed o be normally disribued. Friedman Tes The Friedman es is a form of nonparameric es, which makes no assumpions abou he specific shape of he populaion from which he sample is drawn, allowing for smaller sample daa ses o be analyzed. This mehod is he exension of he Wilcoxon Signed Rank es for paired samples. The corresponding parameric es is he Randomized Block Muliple Treamen ANOVA, bu unlike he ANOVA, he Friedman es does no require ha he daa se be randomly sampled from normally disribued populaions wih equal variances. The Friedman es uses a wo-ailed hypohesis es where he null hypohesis (H 0 ) is such ha he populaion medians of each reamen are saisically idenical o he res of he group. Tha is, here is no effec among he differen reamen groups. The alernaive hypohesis (H a ) is such ROV BizSas User Manual 38 Copyrigh Dr. Johnahan Mun

39 ha he real populaion medians are saisically differen from one anoher when esed using he sample daa se. Tha is, he medians are saisically differen, which means ha here is a saisically significan effec among he differen reamen groups. If he calculaed p-value is less han or equal o he alpha significance value, hen rejec he null hypohesis and accep he alernae hypohesis. Oherwise, if he p-value is higher han he alpha significance value, do no rejec he null hypohesis. For he Friedman es, creae daa ables such as he one below, and selec he daa in he blue area (e.g., selec he daa from C o F3, or daa poins Treamen 1 o 80). Kruskal-Wallis Tes The Kruskal-Wallis es is a form of nonparameric es, which makes no assumpions abou he specific shape of he populaion from which he sample is drawn, allowing for smaller sample daa ses o be analyzed. This mehod is he exension of he Wilcoxon Signed Rank es by comparing more han wo independen samples. The corresponding parameric es is he One- Way ANOVA, bu unlike he ANOVA, he Kruskal-Wallis does no require ha he daa se be randomly sampled from normally disribued populaions wih equal variances. The Kruskal- Wallis es is a wo-ailed hypohesis es where he null hypohesis (H 0 ) is such ha he populaion medians of each reamen are saisically idenical o he res of he group. Tha is, here is no effec among he differen reamen groups. The alernaive hypohesis (H a ) is such ha he real populaion medians are saisically differen from one anoher when esed using he sample daa se. Tha is, he medians are saisically differen, which means ha here is a saisically significan effec among he differen reamen groups. If he calculaed p-value is less han or equal o he alpha significance value, hen rejec he null hypohesis and accep he alernae hypohesis. Oherwise, if he p-value is higher han he alpha significance value, do no rejec he null hypohesis. The benefi of he Kruskal-Wallis es is ha i can be applied o ordinal, inerval, and raio daa while ANOVA is only applicable for inerval and raio daa. Also, he Friedman es can be run wih fewer daa poins. ROV BizSas User Manual 39 Copyrigh Dr. Johnahan Mun

40 To illusrae, suppose ha hree differen drug indicaions (T = 3) were developed and esed on 100 paiens each (N = 100). The Kruskal-Wallis es can be applied o es if hese hree drugs are all equally effecive saisically. If he calculaed p-value is less han or equal o he significance level used in he es, hen rejec he null hypohesis and conclude ha here is a significan difference among he differen reamens. Oherwise, he reamens are all equally effecive. For he Kruskal-Wallis es, creae daa ables such as he one below, and selec he daa in he blue area (e.g., selec he daa from C40 o F50, or daa poins Treamen 1 o 80). To exend he daa se, jus add more observaions (rows) or more reamen variables o compare (columns). Lilliefors Tes The Lilliefors es is a form of nonparameric es, which makes no assumpions abou he specific shape of he populaion from which he sample is drawn, allowing for smaller sample daa ses o be analyzed. This es evaluaes he null hypohesis (H 0 ) of wheher he daa sample was drawn from a normally disribued populaion, versus an alernae hypohesis (H a ) ha he daa sample is no normally disribued. If he calculaed p-value is less han or equal o he alpha significance value, hen rejec he null hypohesis and accep he alernae hypohesis. Oherwise, if he p- value is higher han he alpha significance value, do no rejec he null hypohesis. This es relies on wo cumulaive frequencies: one derived from he sample daa se and one from a heoreical disribuion based on he mean and sandard deviaion of he sample daa. An alernaive o his es is he Chi-Square es for normaliy. The Chi-Square es requires more daa poins o run compared o he Lilliefors es. Runs Tes The runs es is a form of nonparameric es, which makes no assumpions abou he specific shape of he populaion from which he sample is drawn, allowing for smaller sample daa ses o be analyzed. This es evaluaes he randomness of a series of observaions by analyzing he number of runs i conains. A run is a consecuive appearance of one or more observaions ha are similar. The null hypohesis (H 0 ) esed is wheher he daa sequence is random, versus he alernae hypohesis (H a ) ha he daa sequence is no random. If he calculaed p-value is less han or equal o he alpha significance value, hen rejec he null hypohesis and accep he alernae hypohesis. Oherwise, if he p-value is higher han he alpha significance value, do no rejec he null hypohesis. ROV BizSas User Manual 40 Copyrigh Dr. Johnahan Mun

41 Wilcoxon Signed-Rank Tes (One Variable) The single variable Wilcoxon Signed Rank es is a form of nonparameric es, which makes no assumpions abou he specific shape of he populaion from which he sample is drawn, allowing for smaller sample daa ses o be analyzed. This mehod looks a wheher a sample daa se could have been randomly drawn from a paricular populaion whose median is being hypohesized. The corresponding parameric es is he one-sample -es, which should be used if he underlying populaion is assumed o be normal, providing a higher power on he es. The Wilcoxon Signed Rank es can be applied o hree ypes of hypohesis ess: a wo-ailed es, a righ-ailed es, and a lef-ailed es. If he calculaed Wilcoxon saisic is ouside he criical limis for he specific significance level in he es, rejec he null hypohesis and conclude ha he rue populaion median is no equal o (wo-ailed es), less han (lef-ailed es), or greaer han (righ-ailed es) he hypohesized median based on he sample esed. Oherwise, he rue populaion median is saisically similar o he hypohesized median. Wilcoxon Signed-Rank Tes (Two Variables) The Wilcoxon Signed Rank es for paired variables is a form of nonparameric es, which makes no assumpions abou he specific shape of he populaion from which he sample is drawn, allowing for smaller sample daa ses o be analyzed. This mehod looks a wheher he median of he differences beween he wo paired variables are equal. This es is specifically formulaed for esing he same or similar samples before and afer an even (e.g., measuremens aken before a medical reamen are compared agains hose measuremens aken afer he reamen o see if here is a difference). The corresponding parameric es is he wo-sample -es wih dependen means, which should be used if he underlying populaion is assumed o be normal, providing a higher power on he es. The Wilcoxon Signed Rank es can be applied o hree ypes of hypohesis ess: a wo-ailed es, a righ-ailed es, and a lef-ailed es. To illusrae, suppose ha a new engine design is esed agains an exising engine design o see if here is a saisically significan differen beween he wo. The paired variable Wilcoxon Signed- Rank es can be applied. If he calculaed Wilcoxon saisic is ouside he criical limis for he specific significance level in he es, rejec he null hypohesis and conclude ha he difference beween he rue populaion medians is no equal o (wo-ailed es), less han (lef-ailed es), or greaer han (righ-ailed es) he hypohesized median difference based on he sample esed. Oherwise, he rue populaion median is saisically similar o he hypohesized median. ROV BizSas User Manual 41 Copyrigh Dr. Johnahan Mun

42 5. ANOVA (MULTIVARIATE HYPOTHESIS TESTS) Single Facor Muliple Treamens ANOVA The one-way ANOVA for single facor wih muliple reamens es is an exension of he wovariable -es, looking a muliple variables simulaneously. The ANOVA is appropriae when he sampling disribuion is assumed o be approximaely normal. ANOVA can be applied o only he wo-ailed hypohesis es. A wo-ailed hypohesis ess he null hypohesis (H 0 ) such ha he populaion means of each reamen is saisically idenical o he res of he group, which means ha here is no effec among he differen reamen groups. The alernaive hypohesis (H a ) is such ha he real populaion means are saisically differen from one anoher when esed using he sample daa se. To illusrae, suppose ha hree differen drug indicaions (T = 3) were developed and esed on 100 paiens each (N = 100). The one-way ANOVA can be applied o es if hese hree drugs are all equally effecive saisically. If he calculaed p-value is less han or equal o he significance level used in he es, hen rejec he null hypohesis and conclude ha here is a significan difference among he differen reamens. Oherwise, he reamens are all equally effecive. Randomized Block Muliple Treamens ANOVA The one-way randomized block ANOVA is appropriae when he sampling disribuion is assumed o be approximaely normal and when here exiss a block variable for which ANOVA will conrol (block he effecs of his variable by conrolling i in he experimen). ANOVA can be applied o only he wo-ailed hypohesis es. This analysis can es for he effecs of boh he reamens as well as he effeciveness of he conrol, or block, variable. If he calculaed p-value for he reamen is less han or equal o he significance level used in he es, hen rejec he null hypohesis and conclude ha here is a significan difference among he differen reamens. If he calculaed p-value for he block variable is less han or equal o he significance level used in he es, hen rejec he null hypohesis and conclude ha here is a significan difference among he differen block variables. To illusrae, suppose ha hree differen headlamp designs (T = 3) were developed and esed on four groups of voluneer drivers grouped by heir age (B = 4). The one-way randomized block ANOVA can be applied o es if hese hree headlamps are all equally effecive saisically when esed using he voluneers' driving es grades. Oherwise, he reamens are all equally effecive. This es can deermine if he differences occur because of he reamen (ha he ype of headlamp will deermine differences in driving es scores) or from he block, or conrolled, variable (ha age may yield differen driving abiliies). Two-Way ANOVA The wo-way ANOVA is an exension of he single facor and randomized block ANOVA by simulaneously examining he effecs of wo facors on he dependen variable, along wih he effecs of ineracions beween he differen levels of hese wo facors. Unlike he randomized ROV BizSas User Manual 4 Copyrigh Dr. Johnahan Mun

43 block design, his model examines he ineracions beween differen levels of he facors, or independen variables. In a wo-facor experimen, ineracion exiss when he effec of a level for one facor depends on which level of he oher facor is presen. There are hree ses of null (H 0 ) and alernae (H a ) hypoheses o be esed in he wo-way analysis of variance. The firs es is on he firs independen variable, where he null hypohesis is ha no level of he firs facor has an effec on he dependen variable. The alernae hypohesis is ha here is a leas one level of he firs facor having an effec on he dependen variable. If he calculaed p-value is less han or equal o he alpha significance value, hen rejec he null hypohesis and accep he alernae hypohesis. Oherwise, if he p-value is higher han he alpha significance value, do no rejec he null hypohesis. The second es is on he second independen variable, where he null hypohesis is ha no level of he second facor has an effec on he dependen variable. The alernae hypohesis is ha here is a leas one level of he second facor having an effec on he dependen variable. If he calculaed p-value is less han or equal o he alpha significance value, hen rejec he null hypohesis and accep he alernae hypohesis. Oherwise, if he p-value is higher han he alpha significance value, do no rejec he null hypohesis. The hird es is on he ineracion of boh he firs and second independen variables, where he null hypohesis is ha here are no ineracing effecs beween levels of he firs and second facors. The alernae hypohesis is ha here is a leas one combinaion of levels of he firs and second facors having an effec on he dependen variable. If he calculaed p-value is less han or equal o he alpha significance value, hen rejec he null hypohesis and accep he alernae hypohesis. Oherwise, if he p-value is higher han he alpha significance value, do no rejec he null hypohesis. For he Two-Way ANOVA module, creae ables such as he one below, and selec he daa in he blue area (804 o 835). You can exend he daa by adding rows of facors and columns of reamens. Noe ha he number of replicaions in he able above is (i.e., wo rows of observaions per Facor A ype). Of course, you can increase he number of replicaions as required. The number of replicaions has o be consisen if you wish o exend he daa se. ROV BizSas User Manual 43 Copyrigh Dr. Johnahan Mun

44 6. FORECASTING, REGRESSION, AND ECONOMETRICS ARIMA (Auoregressive Inegraed Moving Average) One very powerful advanced imes-series forecasing ool is he ARIMA or Auo Regressive Inegraed Moving Average approach. ARIMA forecasing assembles hree separae ools ino a comprehensive model. The firs ool segmen is he auoregressive or AR erm, which corresponds o he number of lagged value of he residual in he uncondiional forecas model. In essence, he model capures he hisorical variaion of acual daa o a forecasing model and uses his variaion, or residual, o creae a beer predicing model. The second ool segmen is he inegraion order or he I erm. This inegraion erm corresponds o he number of differencing he ime series o be forecased goes hrough. This elemen accouns for any nonlinear growh raes exising in he daa. The hird ool segmen is he moving average or MA erm, which is essenially he moving average of lagged forecas errors. By incorporaing his average of lagged forecas errors, he model, in essence, learns from is forecas errors or misakes and correcs for hem hrough a moving average calculaion. Auo ARIMA (Auomaic Auoregressive Inegraed Moving Average) ARIMA is an advanced modeling echnique used o model and forecas ime-series daa (daa ha have a ime componen o hem, e.g., ineres raes, inflaion, sales revenues, gross domesic produc). The ARIMA Auo Model selecion will analyze all combinaions of ARIMA (p,d,q) for he mos common values of 0, 1, and, and repors he relevan Akaike Informaion Crierion (AIC) and Schwarz Crierion (SC). The lowes AIC and SC model is hen chosen and run. You can also add in exogenous variables ino he model selecion. In addiion, in order o forecas ARIMA models wih exogenous variables, make sure ha he exogenous variables have enough daa poins o cover he addiional number of periods o forecas. Finally, be aware ha due o he complexiy of he models, his module may ake several minues o run. Please be paien. Auoregressive Inegraed Moving Average, or ARIMA(p,d,q), models are he exension of he AR model ha uses hree componens for modeling he serial correlaion in he ime-series daa. The firs componen is he auoregressive (AR) erm. The AR(p) model uses he p lags of he ime series in he equaion. An AR(p) model has he form: y = a 1 y a p y -p + e. The second componen is he inegraion (d) order erm. Each inegraion order corresponds o differencing he ime series. I(1) means differencing he daa once; I(d) means differencing he daa d imes. The hird componen is he moving average (MA) erm. The MA(q) model uses he q lags of he forecas errors o improve he forecas. An MA(q) model has he form: y = e + b 1 e b q e -q. Finally, an ARMA(p,q) model has he combined form: y = a 1 y a p y -p + e + b 1 e b q e -q. Basic Muliple Regression I is assumed ha he user is familiar wih regression analysis. If no, refer o Dr. Johnahan Mun s Modeling Risk (Second Ediion, Wiley 010), or his manual s Appendix 3: A Primer on ROV BizSas User Manual 44 Copyrigh Dr. Johnahan Mun

45 Regression Analysis before coninuing. Muliple Regression analysis is used o find a saisical and mahemaical relaionship beween a single dependen variable and muliple independen variables. Regression is useful for deermining he relaionship as well as for forecasing. To illusrae, suppose you wan o deermine if sales of a produc can be aribued o an adverisemen in a local paper. In his case, sales revenue is he dependen variable, Y (i is dependen on size of he adverisemen and how frequenly is appears a week), while adverisemen size and frequency are he independen variables X1 and X (hey are independen of sales). Inerpreing he regression analysis is more complex (his may include hypohesis - ess, F-ess, ANOVA, correlaions, auocorrelaions, ec.). Basic Economerics and Auoeconomerics Economerics refers o a branch of business analyics, modeling, and forecasing echniques for modeling he behavior or forecasing cerain business, financial, economic, physical science, and oher variables. Running he Basic Economerics models is similar o regular regression analysis excep ha he dependen and independen variables are allowed o be modified before a regression is run. The repor generaed is he same as shown in he Muliple Regression secion previously and he inerpreaions are idenical o hose described previously Combinaorial Fuzzy Logic In conras, he erm fuzzy logic is derived from fuzzy se heory o deal wih reasoning ha is approximae raher han accurae. As opposed o crisp logic, where binary ses have binary logic, fuzzy logic variables may have a ruh value ha ranges beween 0 and 1 and is no consrained o he wo ruh values of classic proposiional logic. This fuzzy weighing schema is used ogeher wih a combinaorial mehod o yield ime-series forecas resuls. Noe ha neiher neural neworks nor fuzzy logic echniques have ye been esablished as valid and reliable mehods in he business forecasing domain, on eiher a sraegic, acical, or operaional level. Much research is sill required in hese advanced forecasing fields. Noneheless, ROV BizSas provides he fundamenals of hese wo echniques for he purposes of running ime-series forecass. We recommend ha you do no use any of hese echniques in isolaion, bu, raher, in combinaion wih he oher ROV BizSas forecasing mehodologies o build more robus models. GARCH Volailiy Forecass The Generalized Auoregressive Condiional Heeroskedasiciy (GARCH) Model is used o model hisorical and forecas fuure volailiy levels of a markeable securiy (e.g., sock prices, commodiy prices, oil prices, ec.). The daa se has o be a ime series of raw price levels. GARCH will firs conver he prices ino relaive reurns and hen run an inernal opimizaion o fi he hisorical daa o a mean-revering volailiy erm srucure, while assuming ha he volailiy is heeroskedasic in naure (changes over ime according o some economeric characerisics). The heoreical specifics of a GARCH model are ouside he purview of his user manual. Noes The ypical volailiy forecas siuaion requires P = 1, Q = 1; Periodiciy = number of periods per year (1 for monhly daa, 5 for weekly daa, 5 or 365 for daily daa); Base = minimum of 1 and up o he periodiciy value; and Forecas Periods = number of annualized volailiy forecass ROV BizSas User Manual 45 Copyrigh Dr. Johnahan Mun

46 you wish o obain. There are several GARCH models available in ROV BizSas, including EGARCH, EGARCH-T, GARCH-M, GJR-GARCH, GJR-GARCH-T, IGARCH, and T- GARCH. GARCH models are used mainly in analyzing financial ime-series daa o ascerain heir condiional variances and volailiies. These volailiies are hen used o value he opions as usual, bu he amoun of hisorical daa necessary for a good volailiy esimae remains significan. Usually, several dozen and even up o hundreds of daa poins are required o obain good GARCH esimaes. GARCH is a erm ha incorporaes a family of models ha can ake on a variey of forms, known as GARCH(p,q), where p and q are posiive inegers ha define he resuling GARCH model and is forecass. In mos cases for financial insrumens, a GARCH(1,1) is sufficien and is mos generally used. For insance, a GARCH (1,1) model akes he form of: y x 1 1 where he firs equaion s dependen variable (y ) is a funcion of exogenous variables (x ) wih an error erm ( ). The second equaion esimaes he variance (squared volailiy ) a ime, which depends on a hisorical mean (); news abou volailiy from he previous period, measured as a lag of he squared residual from he mean equaion ( -1 ); and volailiy from he previous period ( -1 ). The exac modeling specificaion of a GARCH model is beyond he scope of his manual. Suffice i o say ha deailed knowledge of economeric modeling (model specificaion ess, srucural breaks, and error esimaion) is required o run a GARCH model, making i less accessible o he general analys. Anoher problem wih GARCH models is ha he model usually does no provide a good saisical fi. Tha is, i is impossible o predic he sock marke and, of course, equally if no harder o predic a sock s volailiy over ime. Noe ha he GARCH funcion has several inpus as follow: Time-Series Daa. The ime series of daa in chronological order (e.g., sock prices). Typically, dozens of daa poins are required for a decen volailiy forecas. Periodiciy. A posiive ineger indicaing he number of periods per year (e.g., 1 for monhly daa, 5 for daily rading daa, ec.), assuming you wish o annualize he volailiy. For geing periodic volailiy, ener 1. Predicive Base. The number of periods back (of he ime-series daa) o use as a base o forecas volailiy. The higher his number, he longer he hisorical base is used o forecas fuure volailiy. Forecas Period. A posiive ineger indicaing how many fuure periods beyond he hisorical sock prices you wish o forecas. Variance Targeing. This variable is se as False by defaul (even if you do no ener anyhing here) bu can be se as True. False means he omega variable is auomaically opimized and compued. The suggesion is o leave his variable empy. If you wish o creae mean-revering volailiy wih variance argeing, se his variable as True. P. The number of previous lags on he mean equaion. Q. The number of previous lags on he variance equaion. ROV BizSas User Manual 46 Copyrigh Dr. Johnahan Mun

47 The accompanying able liss some of he GARCH specificaions used in Risk Simulaor wih wo underlying disribuional assumpions: one for normal disribuion and he oher for he disribuion. GARCH-M Variance in Mean Equaion GARCH-M Sandard Deviaion in Mean Equaion GARCH-M Log Variance in Mean Equaion GARCH EGARCH z ~ Normal Disribuion y c z y 1 1 c z y 1 1 c ln( ) z 1 1 y x y ln z 1 1 ln 1 E( ) E( ) r z ~ T Disribuion y c z y 1 1 c z y 1 1 c ln( ) z y 1 1 z y ln 1 1 z ln 1 E( ) r (( 1) / ) E( ) ( 1) ( / ) GJR-GARCH y z 1 r d d 1 if 1 0 oherwise y z 1 r d d 1 if 1 0 oherwise For he GARCH-M models, he condiional variance equaions are he same in he six variaions, bu he mean quesions are differen and assumpion on z can be eiher normal disribuion or ROV BizSas User Manual 47 Copyrigh Dr. Johnahan Mun

48 disribuion. The esimaed parameers for GARCH-M wih normal disribuion are hose five parameers in he mean and condiional variance equaions. The esimaed parameers for GARCH-M wih he disribuion are hose five parameers in he mean and condiional variance equaions plus anoher parameer, he degrees of freedom for he disribuion. In conras, for he GJR models, he mean equaions are he same in he six variaions and he differences are ha he condiional variance equaions and he assumpion on z can be eiher a normal disribuion or disribuion. The esimaed parameers for EGARCH and GJR-GARCH wih normal disribuion are hose four parameers in he condiional variance equaion. The esimaed parameers for GARCH, EARCH, and GJR-GARCH wih disribuion are hose parameers in he condiional variance equaion plus he degrees of freedom for he disribuion. More echnical deails of GARCH mehodologies fall ouside of he scope of his manual. J-Curve and S-Curve Forecass The J curve, or exponenial growh curve, is one where he growh of he nex period depends on he curren period s level and he increase is exponenial. This phenomenon means ha over ime, he values will increase significanly, from one period o anoher. This model is ypically used in forecasing biological growh and chemical reacions over ime. The S curve, or logisic growh curve, sars off like a J curve, wih exponenial growh raes. Over ime, he environmen becomes sauraed (e.g., marke sauraion, compeiion, overcrowding), he growh slows, and he forecas value evenually ends up a a sauraion or maximum level. The S-curve model is ypically used in forecasing marke share or sales growh of a new produc from marke inroducion unil mauriy and decline, populaion dynamics, growh of bacerial culures, and oher naurally occurring variables. Markov Chains A Markov chain exiss when he probabiliy of a fuure sae depends on a previous sae and when linked ogeher forms a chain ha revers o a long-run seady sae level. This Markov approach is ypically used o forecas he marke share of wo compeiors. The required inpus are he saring probabiliy of a cusomer in he firs sore (he firs sae) reurning o he same sore in he nex period versus he probabiliy of swiching o a compeior s sore in he nex sae. Neural Nework Forecasing The erm Neural Nework is ofen used o refer o a nework or circui of biological neurons, while modern usage of he erm ofen refers o arificial neural neworks comprising arificial neurons, or nodes, recreaed in a sofware environmen. Such neworks aemp o mimic he neurons in he human brain in ways of hinking and idenifying paerns and, in our siuaion, idenifying paerns for he purposes of forecasing ime-series daa. Noe ha he number of hidden layers in he nework is an inpu parameer and will need o be calibraed wih your daa. Typically, he more complicaed he daa paern, he higher he number of hidden layers you would need and he longer i would ake o compue. I is recommended ha you sar a 3 layers. The esing period is simply he number of daa poins used in he final calibraion of he Neural Nework model, and we recommend using a leas he same number of periods you wish o forecas as he esing period. ROV BizSas User Manual 48 Copyrigh Dr. Johnahan Mun

49 Nonlinear Exrapolaion Exrapolaion involves making saisical forecass by using hisorical rends ha are projeced for a specified period of ime ino he fuure. I is only used for ime-series forecass. For crosssecional or mixed panel daa (ime-series wih cross-secional daa), mulivariae regression is more appropriae. This mehodology is useful when major changes are no expeced; ha is, causal facors are expeced o remain consan or when he causal facors of a siuaion are no clearly undersood. I also helps discourage he inroducion of personal biases ino he process. Exrapolaion is fairly reliable, relaively simple, and inexpensive. However, exrapolaion, which assumes ha recen and hisorical rends will coninue, produces large forecas errors if disconinuiies occur wihin he projeced ime period; ha is, pure exrapolaion of ime series assumes ha all we need o know is conained in he hisorical values of he series being forecased. If we assume ha pas behavior is a good predicor of fuure behavior, exrapolaion is appealing. This makes i a useful approach when all ha is needed are many shor-erm forecass. This mehodology esimaes he f(x) funcion for any arbirary x value, by inerpolaing a smooh nonlinear curve hrough all he x values and, using his smooh curve, exrapolaes fuure x values beyond he hisorical daa se. The mehodology employs eiher he polynomial funcional form or he raional funcional form (a raio of wo polynomials). Typically, a polynomial funcional form is sufficien for well-behaved daa, however, raional funcional forms are someimes more accurae (especially wih polar funcions, i.e., funcions wih denominaors approaching zero). Principal Componens Analysis Principal Componens Analysis is a way of idenifying paerns in daa and recasing he daa in such as way as o highligh heir similariies and differences. Paerns of daa are very difficul o find in high dimensions when muliple variables exis, and higher dimensional graphs are very difficul o represen and inerpre. Once he paerns in he daa are found, hey can be compressed, resuling in a reducion of he number of dimensions. This reducion of daa dimensions does no mean much loss of informaion. Insead, similar levels of informaion can now be obained by fewer variables. The analysis provides he Eigenvalues and Eigenvecors of he daa se. The Eigenvecor wih he highes Eigenvalue is he principle componen of he daa se. Ranking he Eigenvalues from highes o lowes provides he componens in order of saisical significance. If he Eigenvalues are small, you do no lose much informaion. I is up o you o decide how many componens o ignore based on heir Eigenvalues. The proporions and cumulaive proporions ell you how much of he variaion in he daa se can be explained by incorporaing ha componen. Finally, he daa is hen ransformed o accoun for only he number of componens you decide o keep. Spline (Cubic Spline Inerpolaion and Exrapolaion) Someimes here are missing values in a ime-series daa se. For insance, ineres raes for years 1 o 3 may exis, followed by years 5 o 8, and hen year 10. Spline curves can be used o inerpolae he missing years ineres rae values based on he daa ha exis. Spline curves can also be used o forecas or exrapolae values of fuure ime periods beyond he ime period of available daa. The daa can be linear or nonlinear. The Known X values represen he values on he x-axis of a char (in our example, his is Years of he known ineres raes, and, usually, he x- axis are he values ha are known in advance such as ime or years) and he Known Y values ROV BizSas User Manual 49 Copyrigh Dr. Johnahan Mun

50 represen he values on he y-axis (in our case, he known Ineres Raes). The y-axis variable is ypically he variable you wish o inerpolae missing values from or exrapolae he values ino he fuure. Sepwise Regression One powerful auomaed approach o regression analysis is Sepwise Regression. Based on is namesake, he regression process proceeds in muliple seps. There are several ways o se up hese sepwise algorihms, including he correlaion approach, forward mehod, backward mehod, and he forward and backward mehod (hese mehods are all available in ROV BizSas). In he correlaion mehod, he dependen variable (Y) is correlaed o all he independen variables (X), and a regression is run, saring wih he X variable wih he highes absolue correlaion value. Then subsequen X variables are added unil he p-values indicae ha he new X variable is no longer saisically significan. This approach is quick and simple bu does no accoun for ineracions among variables, and an X variable, when added, will saisically overshadow oher variables. In he forward mehod, we firs correlae Y wih all X variables, run a regression for Y on he highes absolue value correlaion of X, and obain he fiing errors. Then, correlae hese errors wih he remaining X variables and choose he highes absolue value correlaion among his remaining se and run anoher regression. Repea he process unil he p-value for he laes X variable coefficien is no longer saisically significan hen sop he process. In he backward mehod, run a regression wih Y on all X variables and, reviewing each variable s p-value, sysemaically eliminae he variable wih he larges p-value. Then run a regression again, repeaing each ime unil all p-values are saisically significan. In he forward and backward mehod, apply he forward mehod o obain hree X variables, and hen apply he backward approach o see if one of hem needs o be eliminaed because i is saisically insignifican. Repea he forward mehod, and hen he backward mehod unil all remaining X variables are considered. The Sepwise Regression is an auomaic search process ieraing hrough all he independen variables, and i models he variables ha are saisically significan in explaining he variaions in he dependen variable. Sepwise Regression is very powerful when here are many independen variables and a large combinaion of models can be buil. To illusrae, suppose you wan o deermine if sales of a produc can be aribued o an adverisemen in a local paper. In his case, sales revenue is he dependen variable Y, while he independen variables X1 o X5 are he size of he adverisemen, cos of he ad, number of readers, day of he week, and how frequenly i appears a week. Sepwise Regression will auomaically ierae hrough hese X variables o find hose ha are saisically significan in he regression model. Inerpreing he regression analysis is more complex (his may include hypohesis -ess, F-ess, ANOVA, correlaions, auocorrelaions, ec.). Time-Series Decomposiion Mehodologies I is assumed ha he user is familiar wih basic ime-series analysis mehodologies. If no, refer o Dr. Johnahan Mun s Modeling Risk (Second Ediion, Wiley 010), or his manual s Appendix ROV BizSas User Manual 50 Copyrigh Dr. Johnahan Mun

51 : A Primer on Time-Series Mehodologies before coninuing. Forecasing is he ac of predicing he fuure wheher i is based on hisorical daa or speculaion abou he fuure when no hisory exiss. When hisorical daa exis, a quaniaive or saisical approach is bes, bu if no hisorical daa exis, hen a qualiaive or judgmenal approach is usually he only recourse. Figure 6.1 liss he eigh mos common ime-series models, segregaed by seasonaliy and rend. For insance, if he daa variable has no rend or seasonaliy, hen a single moving-average model or a single exponenial-smoohing model would suffice. However, if seasonaliy exiss bu no discernable rend is presen, eiher a seasonal addiive or seasonal muliplicaive model would be beer, and so forh. The following subsecions explore hese models in more deail hrough compuaional examples. No Seasonaliy Wih Seasonaliy No Trend Wih Trend Single Moving Average Single Exponenial Smoohing Double Moving Average Double Exponenial Smoohing Seasonal Addiive Seasonal Muliplicaive Hol-Winer's Addiive Hol-Winer's Muliplicaive Figure 6.1 The Eigh Mos Common Time-Series Mehods The bes-fiing es for he moving average forecas uses he Roo Mean Squared Errors (RMSE). The RMSE calculaes he square roo of he average squared deviaions of he fied values versus he acual daa poins. Mean Squared Error (MSE) is an absolue error measure ha squares he errors (he difference beween he acual hisorical daa and he forecas-fied daa prediced by he model) o keep he posiive and negaive errors from canceling each oher ou. This measure also ends o exaggerae large errors by weighing he large errors more heavily han smaller errors by squaring hem, which can help when comparing differen ime-series models. Roo Mean Square Error (RMSE) is he square roo of MSE and is he mos popular error measure, also known as he quadraic loss funcion. RMSE can be defined as he average of he absolue values of he forecas errors and is highly appropriae when he cos of he forecas errors is proporional o he absolue size of he forecas error. The RMSE is used as he selecion crieria for he bes-fiing ime-series model. Mean Absolue Deviaion (MAD) is an error saisic ha averages he disance (absolue value of he difference beween he acual hisorical daa and he forecas-fied daa prediced by he model) beween each pair of acual and fied forecas daa poins and is mos appropriae when he cos of forecas errors is proporional o he absolue size of he forecas errors. Mean Absolue Percenage Error (MAPE) is a relaive error saisic measured as an average percen error of he hisorical daa poins and is mos appropriae when he cos of he forecas error is more closely relaed o he percenage error han he numerical size of he error. Finally, an associaed measure is he Theil's U saisic, which measures he naivey of he model's ROV BizSas User Manual 51 Copyrigh Dr. Johnahan Mun

52 forecas. Tha is, if he Theil's U saisic is less han 1.0, hen he forecas mehod used provides an esimae ha is saisically beer han guessing. Single Moving Average The single moving average is applicable when ime-series daa wih no rend and seasonaliy exis. This model is no appropriae when used o predic cross-secional daa. The single moving average simply uses an average of he acual hisorical daa o projec fuure oucomes. This average is applied consisenly moving forward, hence he erm moving average. The value of he moving average for a specific lengh is simply he summaion of acual hisorical daa arranged and indexed in a ime sequence. The sofware finds he opimal moving average lag auomaically hrough an opimizaion process ha minimizes he forecas errors. Single Exponenial Smoohing The single exponenial smoohing approach is used when no discernable rend or seasonaliy exiss in he ime-series daa. This model is no appropriae when used o predic cross-secional daa. This mehod weighs pas daa wih exponenially decreasing weighs going ino he pas; ha is, he more recen he daa value, he greaer is weigh. This weighing largely overcomes he limiaions of moving averages or percenage-change models. The weigh used is ermed he alpha measure. The sofware finds he opimal alpha parameer auomaically hrough an opimizaion process ha minimizes he forecas errors. Double Moving Average The double moving average mehod will smooh ou pas daa by performing a moving average on a subse of daa ha represens a moving average of an original se of daa. Tha is, a second moving average is performed on he firs moving average. The second moving average applicaion capures he rending effec of he daa. The resuls are hen weighed and forecass are creaed. The sofware finds he opimal moving average lag auomaically hrough an opimizaion process ha minimizes he forecas errors. Double Exponenial Smoohing The double exponenial smoohing mehod is used when he daa exhibi a rend bu no seasonaliy. This model is no appropriae when used o predic cross-secional daa. Double exponenial smoohing applies single exponenial smoohing wice, once o he original daa and hen o he resuling single exponenial smoohing daa. An alpha weighing parameer is used on he firs or single exponenial smoohing (SES), while a bea weighing parameer is used on he second or double exponenial smoohing (DES). This approach is useful when he hisorical daa series is no saionary. The sofware finds he opimal alpha and bea parameers auomaically hrough an opimizaion process ha minimizes he forecas errors. Seasonal Addiive If he ime-series daa has no appreciable rend bu exhibis seasonaliy, hen he addiive seasonaliy and muliplicaive seasonaliy mehods apply. The addiive seasonaliy model breaks he hisorical daa ino a level (L), or base-case, componen as measured by he alpha parameer, and a seasonaliy (S) componen measured by he gamma parameer. The resuling forecas value is simply he addiion of his base-case level o he seasonaliy value. The sofware finds he opimal alpha and gamma parameers auomaically hrough an opimizaion process ha minimizes he forecas errors. ROV BizSas User Manual 5 Copyrigh Dr. Johnahan Mun

53 Seasonal Muliplicaive If he ime-series daa has no appreciable rend bu exhibis seasonaliy, hen he addiive seasonaliy and muliplicaive seasonaliy mehods apply. The muliplicaive seasonaliy model breaks he hisorical daa ino a level (L), or base-case, componen as measured by he alpha parameer, and a seasonaliy (S) componen measured by he gamma parameer. The resuling forecas value is simply he muliplicaion of his base-case level by he seasonaliy value. The sofware finds he opimal alpha and gamma parameers auomaically hrough an opimizaion process ha minimizes he forecas errors. Hol-Winer s Seasonal Addiive When boh seasonaliy and rend exis, more advanced models are required o decompose he daa ino heir base elemens: a base-case level (L) weighed by he alpha parameer; a rend componen (b) weighed by he bea parameer; and a seasonaliy componen (S) weighed by he gamma parameer. Several mehods exis, bu he wo mos common are he Hol-Winer s addiive seasonaliy and Hol-Winer s muliplicaive seasonaliy mehods. In he Hol-Winer s addiive model, he base-case level, seasonaliy, and rend are added ogeher o obain he forecas fi. Hol-Winer s Seasonal Muliplicaive When boh seasonaliy and rend exis, more advanced models are required o decompose he daa ino heir base elemens: a base-case level (L) weighed by he alpha parameer; a rend componen (b) weighed by he bea parameer; and a seasonaliy componen (S) weighed by he gamma parameer. Several mehods exis, bu he wo mos common are he Hol-Winer s addiive seasonaliy and Hol-Winer s muliplicaive seasonaliy mehods. In he Hol-Winer s muliplicaive model, he base-case level and rend are added ogeher and muliplied by he seasonaliy facor o obain he forecas fi. Trendlines Trendlines can be used o deermine if a se of ime-series daa follows any appreciable rend. Trends can be linear or nonlinear (such as exponenial, logarihmic, moving average, polynomial, or power). In forecasing models, he process usually includes removing he effecs of accumulaing daa ses from seasonaliy and rend o show only he absolue changes in values and o allow poenial cyclical paerns o be idenified afer removing he general drif, endency, wiss, bends, and effecs of seasonal cycles of a se of ime-series daa. For example, a derended daa se may be necessary o see a more accurae accoun of a company's sales in a given year by shifing he enire daa se from a slope o a fla surface o beer expose he underlying cycles and flucuaions. Volailiy: Log Reurns Approach There are several ways o esimae he volailiy used in forecasing and opion valuaion models. The mos common approach is he Logarihmic Reurns Approach. This mehod is used mainly for compuing he volailiy on liquid and radable asses, such as socks in financial opions. However, someimes i is used for oher raded asses, such as he price of oil or elecriciy. This mehod canno be used when negaive cash flows or prices occur, which means i is used only on posiive daa, making i mos appropriae for compuing he volailiy of raded asses. The approach is simply o ake he annualized sandard deviaion of he logarihmic relaive reurns of ROV BizSas User Manual 53 Copyrigh Dr. Johnahan Mun

54 he ime-series daa as he proxy for volailiy. See he secion on GARCH models for more advanced volailiy compuaions. Yield Curves: Bliss and Nelson-Siegel Mehods The Bliss inerpolaion model is used for generaing he erm srucure of ineres raes and yield curve esimaion. Economeric modeling echniques are required o calibrae he values of several inpu parameers in his model. The Bliss approach modifies he Nelson-Siegel mehod by adding an addiional generalized parameer. Virually any yield curve shape can be inerpolaed using hese wo models, which are widely used a banks around he world. In conras, he Nelson- Siegel model is run wih four curve esimaion parameers. If properly modeled, i can be made o fi almos any yield curve shape. Calibraing he inpus in hese models requires faciliy wih economeric modeling and error opimizaion echniques. Typically, if some ineres raes exis, a beer approach is o use a spline inerpolaion mehod such as cubic spline and so forh. ROV BizSas User Manual 54 Copyrigh Dr. Johnahan Mun

55 7. STOCHASTIC PROCESSES The Basics of Forecasing wih Sochasic Processes A sochasic process is nohing bu a mahemaically defined equaion ha can creae a series of oucomes over ime, oucomes ha are no deerminisic in naure. Tha is, i does no follow any simple discernible rule such as price will increase X percen every year or revenues will increase by his facor of X plus Y percen. A sochasic process is, by definiion, nondeerminisic, and one can plug numbers ino a sochasic process equaion and obain differen resuls every ime. For insance, he pah of a sock price is sochasic in naure, and one canno reliably predic he sock price pah wih any cerainy. However, he price evoluion over ime is enveloped in a process ha generaes hese prices. The process is fixed and predeermined, bu he oucomes are no. Hence, by sochasic simulaion, we creae muliple pahways of prices, obain a saisical sampling of hese simulaions, and make inferences on he poenial pahways ha he acual price may underake given he naure and parameers of he sochasic process used o generae he ime series. Random Walk: Brownian Moion Assume a process X, where X [ X : 0] if and only if X is coninuous, where he saring poin is X 0 0, where X is normally disribued wih mean zero and variance one or X N(0, 1), and where each incremen in ime is independen of each oher previous incremen and is iself normally disribued wih mean zero and variance, such ha X a X N( 0, ). Then, he process dx X d X dz follows a Geomeric Brownian Moion, where is a dx drif parameer, he volailiy measure, and dz d such ha ln N(, ) X or X and dx are lognormally disribued. If a ime zero, X(0) = 0, hen he expeced value of he ROV BizSas User Manual 55 Copyrigh Dr. Johnahan Mun

56 process X a any ime is such ha E [ X ( )] X 0e and he variance of he process X a ime is V[ X ( )] X e ( e 1). In he coninuous case where here is a drif parameer, he 0 expeced value hen becomes E Jump-Diffusion 0 r ( r ) X ( ) e d 0 X 0e d X 0. ( r ) Sar-up venures and research and developmen iniiaives usually follow a jump-diffusion process. Business operaions may be saus quo for a few monhs or years, and hen a produc or iniiaive becomes highly successful and akes off. An iniial public offering of equiies is a exbook example of his. Assuming ha he probabiliy of he jumps follows a Poisson disribuion, we have a process dx f ( X, ) d g( X, ) dq, where he funcions f and g are 0 wih P( X ) 1 d known and where he probabiliy process is dq. wih P( X ) Xd ROV BizSas User Manual 56 Copyrigh Dr. Johnahan Mun

57 Mean-Reversion If a sochasic process has a long-run aracor such as a long-run producion cos or long-run seady sae inflaionary price level, hen a mean-reversion process is more likely. The process revers o a long-run average such ha he expeced value is E[ X ] X ( X 0 X ) e and he variance isv[ X X ]. The special circumsance ha becomes useful is ha in (1 e ) he limiing case when he ime change becomes insananeous or when d 0, we have he condiion where X X 1 X ( 1 e ) X 1( e 1), which is he firs order auoregressive process, and can be esed economerically in a uni roo conex. ROV BizSas User Manual 57 Copyrigh Dr. Johnahan Mun

58 8. ANALYTICAL MODELS Auocorrelaion Auocorrelaion can be defined as he correlaion of a daa se o iself in he pas. I is he correlaion beween observaions of a ime series separaed by specified ime unis. Cerain imeseries daa follow an auocorrelaed series as fuure oucomes rely heavily on pas oucomes (e.g., revenues or sales ha follow a weekly, monhly, quarerly, or annual seasonal cycle; inflaion and ineres raes ha follow some economic or business cycle, ec.). The erm auocorrelaion describes a relaionship or correlaion beween values of he same daa series a differen ime periods. The erm lag defines he offse when comparing a daa series wih iself. For auocorrelaion, lag refers o he offse of daa ha you choose when correlaing a daa series wih iself. In ROV BizSas, he auocorrelaion funcion is calculaed, ogeher wih he Q-saisic and relevan p-values. If he p-values are below he esed significance level, hen he null hypohesis (H 0 ) of no auocorrelaion is rejeced, and i is concluded ha here is auocorrelaion ha ha paricular lag. Conrol Chars Someimes he specificaion limis are no se; insead, saisical conrol limis are compued based on he acual daa colleced (e.g., he number of defecs in a manufacuring line). For insance, in Figure 8.1, we see 0 sample experimens or samples aken a various imes of a manufacuring process. The number of samples aken varied over ime, and he number of defecive pars were also gahered. The upper conrol limi (UCL) and lower conrol limi (LCL) are compued, as are he cenral line (CL) and oher sigma levels. The resuling char is called a conrol char, and if he process is ou of conrol, he acual defec line will be ouside of he UCL and LCL lines. Typically, when he LCL is a negaive value, we se he floor as zero, as illusraed in Figure 8.1. In he inerpreaion of a conrol char, by adding in he ±1 and sigma lines, we can divide he conrol chars ino several areas or zones, as illusraed in Figure 8.. The following are rules of humb ha ypically apply o conrol chars o deermine if he process is ou of conrol: If one poin is beyond Area A If wo ou of hree consecuive poins are in Area A or beyond If four ou of five consecuive poins are in Area B or beyond If eigh consecuive poins are in Area C or beyond Addiionally, a poenial srucural shif can be deeced if any one of he following occurs: A leas 10 ou of 11 sequenial poins are on one side of he CL A leas 1 ou of 14 sequenial poins are on one side of he CL A leas 14 ou of 17 sequenial poins are on one side of he CL A leas 16 ou of 0 sequenial poins are on one side of he CL ROV BizSas User Manual 58 Copyrigh Dr. Johnahan Mun

59 Figure 8.1 Example qualiy conrol p-char X-Bar Char Used when he variable has raw daa values and here are muliple measuremens in a sample experimen, muliple experimens are run, and he average of he colleced daa is of ineres. R-Bar Char Used when he variable has raw daa values and here are muliple measuremens in a sample experimen, muliple experimens are run, and he range of he colleced daa is of ineres. XMR Char Used when he variable has raw daa values and is a single measuremen aken in each sample experimen, muliple experimens are run, and he acual value of he colleced daa is of ineres. P Char Used when he variable of ineres is an aribue (e.g., defecive or nondefecive) and he daa colleced are in proporions of defecs (or number of defecs in a specific sample), here are muliple measuremens in a sample experimen, muliple experimens are run wih differing numbers of samples colleced in each, and he average proporion of defecs of he colleced daa is of ineres. NP Char Used when he variable of ineres is an aribue (e.g., defecive or nondefecive) and he daa colleced are in proporions of defecs (or number of defecs in a specific sample), here are muliple measuremens in a sample experimen, muliple experimens are run wih a consan number of samples in each, and he average proporion of defecs of he colleced daa is of ineres. ROV BizSas User Manual 59 Copyrigh Dr. Johnahan Mun

60 C Char Used when he variable of ineres is an aribue (e.g., defecive or nondefecive) and he daa colleced are in oal number of defecs (acual coun in unis), here are muliple measuremens in a sample experimen, muliple experimens are run wih he same number of samples colleced in each, and he average number of defecs of he colleced daa is of ineres. U Char Used when he variable of ineres is an aribue (e.g., defecive or nondefecive) and he daa colleced are in oal number of defecs (acual coun in unis), here are muliple measuremens in a sample experimen, muliple experimens are run wih differing numbers of samples colleced in each, and he average number of defecs of he colleced daa is of ineres. Figure 8. Inerpreing conrol chars Deseasonalizaion The daa deseasonalizaion mehod removes any seasonal componens in your original daa. In forecasing models, he process usually includes removing he effecs of accumulaing daa ses from seasonaliy and rend o show only he absolue changes in values and o allow poenial cyclical paerns o be idenified afer removing he general drif, endency, wiss, bends, and effecs of seasonal cycles of a se of ime-series daa. Many ime-series daa exhibi seasonaliy where cerain evens repea hemselves afer some ime period or seasonaliy period (e.g., ski resors revenues are higher in winer han in summer, and his predicable cycle will repea iself every winer). Seasonaliy periods represen how many periods would have o pass before he cycle repeas iself (e.g., 4 hours in a day, 1 monhs in a year, 4 quarers in a year, 60 minues in an hour, ec.). For deseasonalized and derended daa, a seasonal index greaer han 1 indicaes a high period or peak wihin he seasonal cycle, and a value below 1 indicaes a dip in he cycle. ROV BizSas User Manual 60 Copyrigh Dr. Johnahan Mun

61 Disribuional Fiing Anoher powerful simulaion ool is disribuional fiing or deermining which disribuion o use for a paricular inpu variable in a model and wha he relevan disribuional parameers are. If no hisorical daa exis, hen he analys mus make assumpions abou he variables in quesion. One approach is o use he Delphi mehod where a group of expers is asked wih esimaing he behavior of each variable. For insance, a group of mechanical engineers can be asked wih evaluaing he exreme possibiliies of a spring coil s diameer hrough rigorous experimenaion or guessimaes. These values can be used as he variable s inpu parameers (e.g., uniform disribuion wih exreme values beween 0.5 and 1.). When esing is no possible (e.g., marke share and revenue growh rae), managemen can sill make esimaes of poenial oucomes and provide he bes-case, mos-likely case, and wors-case scenarios. However, if reliable hisorical daa are available, disribuional fiing can be accomplished. Assuming ha hisorical paerns hold and ha hisory ends o repea iself, hen hisorical daa can be used o find he bes-fiing disribuion wih heir relevan parameers o beer define he variables o be simulaed. Heeroskedasiciy A common violaion in regression, economeric modeling, and some ime-series forecas mehods is heeroskedasiciy. Heeroskedasiciy is defined as he variance of he forecas errors increasing over ime. If picured graphically, he widh of he verical daa flucuaions increases or fans ou over ime. In his example, he daa poins have been changed o exaggerae he effec. However, in mos ime-series analysis, checking for heeroskedasiciy is a much more difficul ask. The coefficien of deerminaion, or R-squared, in a muliple regression analysis drops significanly when heeroskedasiciy exiss. As is, he curren regression model is insufficien and incomplee. If he variance of he dependen variable is no consan, hen he error s variance will no be consan. The mos common form of such heeroskedasiciy in he dependen variable is ha he variance of he dependen variable may increase as he mean of he dependen variable increases for daa wih posiive independen and dependen variables. Unless he heeroskedasiciy of he dependen variable is pronounced, is effec will no be severe: he leas-squares esimaes will sill be unbiased, and he esimaes of he slope and inercep will eiher be normally disribued if he errors are normally disribued, or a leas normally disribued asympoically (as he number of daa poins becomes large) if he errors are no normally disribued. The esimae for he variance of he slope and overall variance will be inaccurae, bu he inaccuracy is no likely o be subsanial if he independen-variable values are symmeric abou heir mean. Heeroskedasiciy of he dependen variable is usually deeced informally by examining he X-Y scaer plo of he daa before performing he regression. If boh nonlineariy and unequal variances are presen, employing a ransformaion of he dependen variable may have he effec of simulaneously improving he lineariy and promoing equaliy of he variances. Oherwise, a weighed leas-squares linear regression may be he preferred mehod of dealing wih nonconsan variance of he dependen variable. ROV BizSas User Manual 61 Copyrigh Dr. Johnahan Mun

62 Maximum Likelihood Models on Logi, Probi, and Tobi Limied Dependen Variables describe he siuaion where he dependen variable conains daa ha are limied in scope and range, such as binary responses (0 or 1), runcaed, ordered, or censored daa. For insance, given a se of independen variables (e.g., age, income, educaion level of credi card or morgage loan holders), we can model he probabiliy of defaul using maximum likelihood esimaion (MLE). The response or dependen variable Y is binary, ha is, i can have only wo possible oucomes ha we denoe as 1 and 0 (e.g., Y may represen presence/absence of a cerain condiion, defauled/no defauled on previous loans, success/failure of some device, answer yes/no on a survey, ec.) and we also have a vecor of independen variable regressors X, which are assumed o influence he oucome Y. A ypical ordinary leas squares regression approach is invalid because he regression errors are heeroskedasic and nonnormal, and he resuling esimaed probabiliy esimaes will reurn nonsensical values of above 1 or below 0. MLE analysis handles hese problems using an ieraive opimizaion rouine o maximize a log likelihood funcion when he dependen variables are limied. A Logi or Logisic regression is used for predicing he probabiliy of occurrence of an even by fiing daa o a logisic curve. I is a generalized linear model used for binomial regression, and like many forms of regression analysis, i makes use of several predicor variables ha may be eiher numerical or caegorical. MLE applied in a binary mulivariae logisic analysis is used o model dependen variables o deermine he expeced probabiliy of success of belonging o a cerain group. The esimaed coefficiens for he Logi model are he logarihmic odds raios and canno be inerpreed direcly as probabiliies. A quick compuaion is firs required and he approach is simple. Specifically, he Logi model is specified as Esimaed Y = LN[P i /(1 P i )] or, conversely, P i = EXP(Esimaed Y)/(1+EXP(Esimaed Y)), and he coefficiens β i are he log odds raios. So, aking he anilog, or EXP(β i ), we obain he odds raio of P i /(1 P i ). This means ha wih an increase in a uni of β i he log odds raio increases by his amoun. Finally, he rae of change in he probabiliy is dp/dx = β i P i (1 P i ). The Sandard Error measures how accurae he prediced Coefficiens are, and he -Saisics are he raios of each prediced Coefficien o is Sandard Error and are used in he ypical regression hypohesis es of he significance of each esimaed parameer. To esimae he probabiliy of success of belonging o a cerain group (e.g., predicing if a smoker will develop ches complicaions given he amoun smoked per year), simply compue he Esimaed Y value using he MLE coefficiens. For example, if he model is Y = (Cigarees), hen someone smoking 100 packs per year has an Esimaed Y of (100) = 1.6. Nex, compue he inverse anilog of he odds raio: EXP(Esimaed Y)/[1 + EXP(Esimaed Y)] = EXP(1.6)/(1+ EXP(1.6)) = So, such a person has an 83.0% chance of developing some ches complicaions in his or her lifeime. A Probi model (someimes also known as a Normi model) is a popular alernaive specificaion for a binary response model, which employs a Probi funcion esimaed using maximum likelihood esimaion and is called Probi regression. The Probi and Logisic regression models end o produce very similar predicions where he parameer esimaes in a logisic regression end o be 1.6 o 1.8 imes higher han hey are in a corresponding Probi model. The choice of using a Probi or Logi is enirely up o convenience, and he main disincion is ha he logisic disribuion has a higher kurosis (faer ails) o accoun for exreme values. For example, suppose ha house ownership is he decision o be modeled, and his response variable is binary (home purchase or no home purchase) and depends on a series of independen variables X i such as income, age, and so forh, such ha I i = β 0 + β 1 X β n X n, where he larger he value of I i, ROV BizSas User Manual 6 Copyrigh Dr. Johnahan Mun

63 he higher he probabiliy of home ownership. For each family, a criical I* hreshold exiss, where if exceeded, he house is purchased, oherwise, no home is purchased, and he oucome probabiliy (P) is assumed o be normally disribued such ha P i = CDF(I) using a sandard normal cumulaive disribuion funcion (CDF). Therefore, use he esimaed coefficiens exacly like hose of a regression model and using he Esimaed Y value, apply a sandard normal disribuion (you can use Excel s NORMSDIST funcion or Risk Simulaor's Disribuional Analysis ool by selecing Normal disribuion and seing he mean o be 0 and sandard deviaion o be 1). Finally, o obain a Probi or probabiliy uni measure, se I i + 5 (his is because whenever he probabiliy P i < 0.5, he esimaed I i is negaive, due o he fac ha he normal disribuion is symmerical around a mean of zero). The Tobi model (Censored Tobi) is an economeric and biomeric modeling mehod used o describe he relaionship beween a non-negaive dependen variable Y i and one or more independen variables X i. A Tobi model is an economeric model in which he dependen variable is censored; ha is, he dependen variable is censored because values below zero are no observed. The Tobi model assumes ha here is a laen unobservable variable Y*. This variable is linearly dependen on he X i variables via a vecor of β i coefficiens ha deermine heir inerrelaionships. In addiion, here is a normally disribued error erm, U i, o capure random influences on his relaionship. The observable variable Y i is defined o be equal o he laen variables whenever he laen variables are above zero and Y i is assumed o be zero oherwise. Tha is, Y i = Y* if Y* > 0 and Y i = 0 if Y* = 0. If he relaionship parameer β i is esimaed by using ordinary leas squares regression of he observed Y i on X i, he resuling regression esimaors are inconsisen and yield downward-biased slope coefficiens and an upward-biased inercep. Only MLE would be consisen for a Tobi model. In he Tobi model, here is an ancillary saisic called sigma, which is equivalen o he sandard error of esimae in a sandard ordinary leas squares regression, and he esimaed coefficiens are used he same way as a regression analysis. Mulicollineariy Mulicollineariy exiss when here is a linear relaionship beween he independen variables in a regression analysis. When his occurs, he regression equaion canno be esimaed a all. In nearcollineariy siuaions, he esimaed regression equaion will be biased and provide inaccurae resuls. This siuaion is especially rue when a sepwise regression approach is used, where he saisically significan independen variables will be hrown ou of he regression mix earlier han expeced, resuling in a regression equaion ha is neiher efficien nor accurae. As an example, suppose he following muliple regression analysis exiss, where Y i 1 X, i 3 X 3, i i hen he esimaed slopes can be calculaed hrough ˆ ˆ 3 Y X i Y X i, i X 3, i X X, i X, i 3, i X, i X 3, i 3, i Yi X 3, i X, i X, i X 3, i Yi X, i X, i X, i X 3, i X X 3, i 3, i ROV BizSas User Manual 63 Copyrigh Dr. Johnahan Mun

64 Now suppose ha here is perfec mulicollineariy, ha is, here exiss a perfec linear relaionship beween X and X 3, such ha X for all posiive values of. Subsiuing X 3, i, i his linear relaionship ino he slope calculaions for, he resul is indeerminae. In oher words, we have ˆ Y X Yi X, i X, i 0 X 0, i i, i, i X, i X, i X The same calculaion and resuls apply o 3, which means ha he muliple regression analysis breaks down and canno be esimaed given a perfec collineariy condiion. One quick es of he presence of mulicollineariy in a muliple regression equaion is ha he R-squared value is relaively high while he -saisics are relaively low. Anoher quick es is o creae a correlaion marix beween he independen variables. A high cross-correlaion indicaes a poenial for auocorrelaion. The rule of humb is ha a correlaion wih an absolue value greaer han 0.75 is indicaive of severe mulicollineariy. Anoher es for mulicollineariy is he use of he variance inflaion facor (VIF), obained by regressing each independen variable o all he oher independen variables, obaining he R- squared value, and calculaing he VIF of ha variable by esimaing: VIF i 1 (1 R i ) A high VIF value indicaes a high R-squared near uniy. As a rule of humb, a VIF value greaer han 10 is usually indicaive of desrucive mulicollineariy. Parial Auocorrelaion Auocorrelaion can be defined as he correlaion of a daa se o iself in he pas. I is he correlaion beween observaions of a ime series separaed by specified ime unis. Cerain imeseries daa follow an auocorrelaed series as fuure oucomes rely heavily on pas oucomes (e.g., revenues or sales ha follow a weekly, monhly, quarerly, or annual seasonal cycle; inflaion and ineres raes ha follow some economic or business cycle, ec.). Parial Auocorrelaions (PAC), in conras, are used o measure he degree of associaion beween each daa poin a a paricular ime Y and a ime lag Y k when he cumulaive effecs of all oher ime lags (1,, 3,..., k 1) have been removed. The erm lag defines he offse when comparing a daa series wih iself. In his module, he Parial Auocorrelaion funcion is calculaed, ogeher wih he Q-saisic and relevan p-values. If he p-values are below he esed significance level, hen he null hypohesis (H 0 ) of no auocorrelaion is rejeced and i is concluded ha here is auocorrelaion ha ha paricular lag. Segmenaion Clusering Segmenaion clusering akes he original daa se and runs some inernal algorihms (a combinaion or k-means hierarchical clusering and oher mehod of momens in order o find he bes-fiing groups or naural saisical clusers) o saisically divide, or segmen, he original ROV BizSas User Manual 64 Copyrigh Dr. Johnahan Mun

65 daa se ino muliple groups. This echnique is valuable in a variey of seings including markeing (such as marke segmenaion of cusomers ino various cusomer relaionship managemen groups), physical sciences, engineering, and ohers. Seasonaliy Tes Many ime-series daa exhibi seasonaliy where cerain evens repea hemselves afer some ime period or seasonaliy period (e.g., ski resors revenues are higher in winer han in summer, and his predicable cycle will repea iself every winer). Seasonaliy periods represen how many periods would have o pass before he cycle repeas iself (e.g., 4 hours in a day, 1 monhs in a year, 4 quarers in a year, 60 minues in an hour, ec.). For deseasonalized and derended daa, a seasonal index greaer han 1 indicaes a high period or peak wihin he seasonal cycle, and a value below 1 indicaes a dip in he cycle. Ener in he maximum seasonaliy period o es. Tha is, if you ener 6, he ool will es he following seasonaliy periods: 1,, 3, 4, 5, and 6. Period 1, of course, implies no seasonaliy in he daa. Review he repor generaed for more deails on he mehodology, applicaion, and resuling chars and seasonaliy es resuls. The bes seasonaliy periodiciy is lised firs (ranked by he lowes RMSE error measure), and all he relevan error measuremens are included for comparison: roo mean squared error (RMSE), mean squared error (MSE), mean absolue deviaion (MAD), and mean absolue percenage error (MAPE). Srucural Break A srucural break ess wheher he coefficiens in differen daa ses are equal, and his es is mos commonly used in ime-series analysis o es for he presence of a srucural break. A imeseries daa se can be divided ino wo subses. Srucural break analysis is used o es each subse individually and on one anoher and on he enire daa se o saisically deermine if, indeed, here is a break saring a a paricular ime period. The srucural break es is ofen used o deermine wheher he independen variables have differen impacs on differen subgroups of he populaion, such as o es if a new markeing campaign, aciviy, major even, acquisiion, divesiure, and so forh have an impac on he ime-series daa. Suppose, for example, a daa se has 100 ime-series daa poins. You can se various breakpoins o es, for insance, daa poins 10, 30, and 51. (This means ha hree srucural break ess will be performed: daa poins 1 9 compared wih ; daa poins 1 9 compared wih ; and 1 50 compared wih o see if here is a break in he underlying srucure a he sar of daa poins 10, 30, and 51.). A one-ailed hypohesis es is performed on he null hypohesis (H 0 ) such ha he wo daa subses are saisically similar o one anoher, ha is, here is no saisically significan srucural break. The alernaive hypohesis (H a ) is ha he wo daa subses are saisically differen from one anoher, indicaing a possible srucural break. If he calculaed p-values are less han or equal o 0.01, 0.05, or 0.10, hen he hypohesis is rejeced, which implies ha he wo daa subses are saisically significanly differen a he 1%, 5%, and 10% significance levels. High p-values indicae ha here is no saisically significan srucural break. ROV BizSas User Manual 65 Copyrigh Dr. Johnahan Mun

66 9. PROBABILITY DISTRIBUTIONS AND SIMULATION The Basics of Disribuions The ROV BizSas sofware includes 45 probabiliy disribuions common and no so common ones and hey are divided beween discree and coninuous disribuions. This secion demonsraes he power of Mone Carlo simulaion, bu o ge sared wih simulaion, one firs needs o undersand he concep of probabiliy disribuions. To begin o undersand probabiliy, consider his example: You wan o look a he disribuion of nonexemp wages wihin one deparmen of a large company. Firs, you gaher raw daa in his case, he wages of each nonexemp employee in he deparmen. Second, you organize he daa ino a meaningful forma and plo he daa as a frequency disribuion on a char. To creae a frequency disribuion, you divide he wages ino group inervals and lis hese inervals on he char s horizonal axis. Then you lis he number, or frequency, of employees in each inerval on he char s verical axis. Now you can easily see he disribuion of nonexemp wages wihin he deparmen. A glance a he char illusraed in Figure 9.1 reveals ha mos of he employees (approximaely 60 ou of a oal of 180) earn from $7.00 o $9.00 per hour Number of Employees Hourly Wage Ranges in Dollars Figure 9.1 Frequency Hisogram I You can char his daa as a probabiliy disribuion. A probabiliy disribuion shows he number of employees in each inerval as a fracion of he oal number of employees. To creae a probabiliy disribuion, you divide he number of employees in each inerval by he oal number of employees and lis he resuls on he char s verical axis. The char in Figure 9. shows you he number of employees in each wage group as a fracion of all employees; you can esimae he likelihood or probabiliy ha an employee drawn a random from he whole group earns a wage wihin a given inerval. For example, assuming he same condiions exis a he ime he sample was aken, he probabiliy is 0.33 (a one in hree chance) ha an employee drawn a random from he whole group earns beween $8.00 and $8.50 an hour. ROV BizSas User Manual 66 Copyrigh Dr. Johnahan Mun

67 0.33 Probabiliy Hourly Wage Ranges in Dollars Figure 9. Frequency Hisogram II Probabiliy disribuions are eiher discree or coninuous. Discree probabiliy disribuions describe disinc values, usually inegers, wih no inermediae values and are shown as a series of verical bars. A discree disribuion, for example, migh describe he number of heads in four flips of a coin as 0, 1,, 3, or 4. Coninuous disribuions are acually mahemaical absracions because hey assume he exisence of every possible inermediae value beween wo numbers. Tha is, a coninuous disribuion assumes here is an infinie number of values beween any wo poins in he disribuion. However, in many siuaions, you can effecively use a coninuous disribuion o approximae a discree disribuion even hough he coninuous model does no necessarily describe he siuaion exacly. Selecing he Righ Probabiliy Disribuion Ploing daa is one guide o selecing a probabiliy disribuion. The following seps provide anoher process for selecing probabiliy disribuions ha bes describe he uncerain variables in your spreadshees: Look a he variable in quesion. Lis everyhing you know abou he condiions surrounding his variable. You migh be able o gaher valuable informaion abou he uncerain variable from hisorical daa. If hisorical daa are no available, use your own judgmen, based on experience, lising everyhing you know abou he uncerain variable. Review he descripions of he probabiliy disribuions. Selec he disribuion ha characerizes his variable. A disribuion characerizes a variable when he condiions of he disribuion mach hose of he variable. Mone Carlo Simulaion Mone Carlo simulaion in is simples form is a random number generaor ha is useful for forecasing, esimaion, and risk analysis. A simulaion calculaes numerous scenarios of a model by repeaedly picking values from a user-predefined probabiliy disribuion for he uncerain variables and using hose values for he model. As all hose scenarios produce associaed resuls in a model, each scenario can have a forecas. Forecass are evens (usually wih formulas or funcions) ha you define as imporan oupus of he model. These usually are evens such as oals, ne profi, or gross expenses. ROV BizSas User Manual 67 Copyrigh Dr. Johnahan Mun

68 Simplisically, hink of he Mone Carlo simulaion approach as repeaedly picking golf balls ou of a large baske wih replacemen. The size and shape of he baske depend on he disribuional inpu assumpion (e.g., a normal disribuion wih a mean of 100 and a sandard deviaion of 10, versus a uniform disribuion or a riangular disribuion) where some baskes are deeper or more symmerical han ohers, allowing cerain balls o be pulled ou more frequenly han ohers. The number of balls pulled repeaedly depends on he number of rials simulaed. For a large model wih muliple relaed assumpions, imagine a very large baske wherein many smaller baskes reside. Each small baske has is own se of golf balls ha are bouncing around. Someimes hese small baskes are linked wih each oher (if here is a correlaion beween he variables) and he golf balls are bouncing in andem, while oher imes he balls are bouncing independen of one anoher. The balls ha are picked each ime from hese ineracions wihin he model (he large cenral baske) are abulaed and recorded, providing a forecas oupu resul of he simulaion. Wih Mone Carlo simulaion, ROV BizSas generaes random values for each assumpion s probabiliy disribuion ha are oally independen. In oher words, he random value seleced for one rial has no effec on he nex random value generaed. Use Mone Carlo sampling when you wan o simulae real-world wha-if scenarios for your spreadshee model. The Appendix provides a deailed lising of he differen ypes of discree and coninuous probabiliy disribuions ha can be used in Mone Carlo simulaion. ROV BizSas User Manual 68 Copyrigh Dr. Johnahan Mun

69 Probabiliy Tables Sandard Normal (Parial Area) Figure 9.3 shows he Sandard Normal Z Disribuion able for parial areas. Tha is, he able liss he probabiliy under he curve of he shaded area seen in Figure 9.4. To illusrae, for a Z-value of 1.96, refer o he 1.9 row and 0.06 column o obain an area of This means here is 47.50% in he shaded region and.50% in he single-ail. Similarly, here is 95% in he body or 5% in boh ails. Thus, a Z-score of 1.96 represens a 95% confidence inerval on a sandard normal Z-disribuion, leaving.5% of he area in each ail. Sandard Normal Disribuion (Parial Area) Z Example: For a Z-value of 1.96, refer o he 1.9 row and 0.06 column for he area of This means here is 47.50% in he shaded region and.50% in he single-ail. Similarly, here is 95% in he body or 5% in boh ails. Figure 9.3 Sandard Normal Z Disribuion (Parial Area) Z = 0 Z = 1.96 Figure 9.4 Visual Represenaion of he Sandard Normal Parial Area ROV BizSas User Manual 69 Copyrigh Dr. Johnahan Mun

70 Probabiliy Tables Sandard Normal (Full Area) Figure 9.5 shows he Sandard Normal Z Disribuion able for he full area. Tha is, he able liss he probabiliy under he curve from he end of he lef ail unil he specified value, as shown in Figure 9.6. To illusrae, for a Z-value of 1.96, refer o he 1.9 row and 0.06 column o obain an area of This means here is 97.50% from he lef ail of he disribuion unil he Z-score of 1.96, leaving.50% in he single-ail. This is consisen wih Figure 9.4 s parial area, where for he same Z-score, here is.5% in each ail. Sandard Normal Disribuion (Full Area) Z Example: For a Z-value of.33, refer o he.3 row and 0.03 column for he area of This means here is 99% in he shaded region and 1% in he one-sided lef or righ ail. Figure 9.5 Sandard Normal Z Disribuion (Full Area) Z = 1.96 Figure 9.6 Visual Represenaion of he Sandard Normal Full Area ROV BizSas User Manual 70 Copyrigh Dr. Johnahan Mun

71 Probabiliy Tables Suden s T-Table (One and Two Tails) Figure 9.7 shows he Suden s T Disribuion able for one and wo ails. Tha is, he able liss he probabiliy under he curve as shown in Figure 9.8. To illusrae, for degrees of freedom (df), an alpha () significance of 10% yields -values of (for he one-ail disribuion) and.900 (for he wo-ail). This means here is a 10% probabiliy of as seen in Figure 19 (he 10% is on one end of he disribuion for he one-ail, while i is divided ino wo porions of 5% in he wo-ail disribuion). Suden's -Disribuion one-ail wo-ail alpha alpha df = df = Example: For an alpha in he single righ ail area of.5% wih 15 degrees of freedom, he criical- value is Figure 9.7 Suden s T-Table Figure 9.8 Visual Represenaion of he Suden s T Disribuion (One and Two Tails) ROV BizSas User Manual 71 Copyrigh Dr. Johnahan Mun

72 APPENDIX 1: DETAILS ON PROBABILITY DISTRIBUTIONS Probabiliy disribuions are eiher discree or coninuous. Discree probabiliy disribuions describe disinc values, usually inegers, wih no inermediae values and are shown as a series of verical bars. A discree disribuion, for example, migh describe he number of heads in four flips of a coin as 0, 1,, 3, or 4. Coninuous disribuions are acually mahemaical absracions because hey assume he exisence of every possible inermediae value beween wo numbers. Tha is, a coninuous disribuion assumes here is an infinie number of values beween any wo poins in he disribuion. However, in many siuaions, you can effecively use a coninuous disribuion o approximae a discree disribuion even hough he coninuous model does no necessarily describe he siuaion exacly. Discree Disribuions Bernoulli or Yes/No Disribuion The Bernoulli disribuion is a discree disribuion wih wo oucomes (e.g., head or ails, success or failure, 0 or 1). I is he binomial disribuion wih one rial and can be used o simulae Yes/No or Success/Failure condiions. This disribuion is he fundamenal building block of oher more complex disribuions. For insance: Binomial disribuion: a Bernoulli disribuion wih higher number of n oal rials ha compues he probabiliy of x successes wihin his oal number of rials. Geomeric disribuion: a Bernoulli disribuion wih higher number of rials ha compues he number of failures required before he firs success occurs. Negaive binomial disribuion: a Bernoulli disribuion wih higher number of rials ha compues he number of failures before he Xh success occurs. The mahemaical consrucs for he Bernoulli disribuion are as follows: 1 p P( n) p or P( n) p Mean p x (1 p) for x 0 for x 1 1x Sandard Deviaion p(1 p) Skewness = 1 p p(1 p) ROV BizSas User Manual 7 Copyrigh Dr. Johnahan Mun

73 Excess Kurosis = 6 p 6 p 1 p(1 p) Probabiliy of success (p) is he only disribuional parameer. Also, i is imporan o noe ha here is only one rial in he Bernoulli disribuion, and he resuling simulaed value is eiher 0 or 1. Inpu requiremens: Probabiliy of success > 0 and < 1 (i.e., p ). Binomial Disribuion The binomial disribuion describes he number of imes a paricular even occurs in a fixed number of rials, such as he number of heads in 10 flips of a coin or he number of defecive iems ou of 50 iems chosen. Condiions The hree condiions underlying he binomial disribuion are: For each rial, only wo oucomes are possible ha are muually exclusive. The rials are independen wha happens in he firs rial does no affec he nex rial. The probabiliy of an even occurring remains he same from rial o rial. The mahemaical consrucs for he binomial disribuion are as follows: P( x) Mean np n! p x!( n x)! x (1 p) ( nx) Sandard Deviaion np(1 p) for n 0; x 0,1,,... n; and 0 p 1 Skewness = 1 p np(1 p) Excess Kurosis = 6 p 6 p 1 np(1 p) Probabiliy of success (p) and he ineger number of oal rials (n) are he disribuional parameers. The number of successful rials is denoed x. I is imporan o noe ha probabiliy of success (p) of 0 or 1 are rivial condiions ha do no require any simulaions and, hence, are no allowed in he sofware. Inpu requiremens: ROV BizSas User Manual 73 Copyrigh Dr. Johnahan Mun

74 Probabiliy of success > 0 and < 1 (i.e., p ). Number of rials 1 or posiive inegers and 1000 (for larger rials, use he normal disribuion wih he relevan compued binomial mean and sandard deviaion as he normal disribuion s parameers). Discree Uniform The discree uniform disribuion is also known as he equally likely oucomes disribuion, where he disribuion has a se of N elemens and each elemen has he same probabiliy. This disribuion is relaed o he uniform disribuion bu is elemens are discree and no coninuous. The mahemaical consrucs for he discree uniform disribuion are as follows: P( x) Mean = 1 N N 1 Sandard Deviaion = ranked value ( N 1)( N 1) 1 ranked value Skewness = 0 (i.e., he disribuion is perfecly symmerical) Excess Kurosis = 6( N 1) 5( N 1)( N 1) ranked value Inpu requiremens: Minimum < maximum and boh mus be inegers (negaive inegers and zero are allowed). Geomeric Disribuion The geomeric disribuion describes he number of rials unil he firs successful occurrence, such as he number of imes you need o spin a roulee wheel before you win. Condiions The hree condiions underlying he geomeric disribuion are: The number of rials is no fixed. The rials coninue unil he firs success. The probabiliy of success is he same from rial o rial. The mahemaical consrucs for he geomeric disribuion are as follows: P( x) p(1 p) x1 for 0 p 1and x 1,,..., n ROV BizSas User Manual 74 Copyrigh Dr. Johnahan Mun

75 1 Mean 1 p Sandard Deviaion p Skewness = 1 p 1 p p Excess Kurosis = p 6 p 6 1 p Probabiliy of success (p) is he only disribuional parameer. The number of successful rials simulaed is denoed x, which can only ake on posiive inegers. Inpu requiremens: Probabiliy of success > 0 and < 1 (i.e., p ). I is imporan o noe ha probabiliy of success (p) of 0 or 1 are rivial condiions ha do no require any simulaions and, hence, are no allowed in he sofware. Hypergeomeric Disribuion The hypergeomeric disribuion is similar o he binomial disribuion in ha boh describe he number of imes a paricular even occurs in a fixed number of rials. The difference is ha binomial disribuion rials are independen, whereas hypergeomeric disribuion rials change he probabiliy for each subsequen rial and are called rials wihou replacemen. For example, suppose a box of manufacured pars is known o conain some defecive pars. You choose a par from he box, find i is defecive, and remove he par from he box. If you choose anoher par from he box, he probabiliy ha i is defecive is somewha lower han for he firs par because you have already removed a defecive par. If you had replaced he defecive par, he probabiliies would have remained he same, and he process would have saisfied he condiions for a binomial disribuion. Condiions The hree condiions underlying he hypergeomeric disribuion are: The oal number of iems or elemens (he populaion size) is a fixed number, a finie populaion. The populaion size mus be less han or equal o 1,750. The sample size (he number of rials) represens a porion of he populaion. The known iniial probabiliy of success in he populaion changes afer each rial. The mahemaical consrucs for he hypergeomeric disribuion are as follows: ROV BizSas User Manual 75 Copyrigh Dr. Johnahan Mun

76 ( N x )! ( N N x )! x!( N x x)! ( n x)!( N N x n x)! P( x) for x Max( n ( N N x ),0),..., Min( n, N x ) N! n!( N n)! N Mean = x n N ( N N Sandard Deviaion = N Skewness = ( N N x ) N N 1 ) N n( N n) Excess Kurosis = complex funcion x x x n( N n) ( N 1) The number of iems in he populaion or Populaion Size (N), rials sampled or Sample Size (n), and number of iems in he populaion ha have he successful rai or Populaion Successes (N x ) are he disribuional parameers. The number of successful rials is denoed x. Inpu requiremens: Populaion Size and ineger. Sample Size > 0 and ineger. Populaion Successes > 0 and ineger. Populaion Size > Populaion Successes. Sample Size < Populaion Successes. Populaion Size < ROV BizSas User Manual 76 Copyrigh Dr. Johnahan Mun

77 Negaive Binomial Disribuion The negaive binomial disribuion is useful for modeling he disribuion of he number of addiional rials required in addiion o he number of successful occurrences required (R). For insance, in order o close a oal of 10 sales opporuniies, how many exra sales calls would you need o make above 10 calls given some probabiliy of success in each call? The x-axis shows he number of addiional calls required or he number of failed calls. The number of rials is no fixed, he rials coninue unil he Rh success, and he probabiliy of success is he same from rial o rial. Probabiliy of success (p) and number of successes required (R) are he disribuional parameers. I is essenially a superdisribuion of he geomeric and binomial disribuions. This disribuion shows he probabiliies of each number of rials in excess of R o produce he required success R. Condiions The hree condiions underlying he negaive binomial disribuion are: The number of rials is no fixed. The rials coninue unil he rh success. The probabiliy of success is he same from rial o rial. The mahemaical consrucs for he negaive binomial disribuion are as follows: ( x r 1)! r x P( x) p (1 p) for x r, r 1,...;and 0 p 1 ( r 1)! x! r(1 p) Mean p r(1 p) Sandard Deviaion p p Skewness = r(1 p) Excess Kurosis = p 6 p 6 r(1 p) Probabiliy of success (p) and required successes (R) are he disribuional parameers. Inpu requiremens: Successes required mus be posiive inegers > 0 and < Probabiliy of success > 0 and < 1 (ha is, p ). I is imporan o noe ha probabiliy of success (p) of 0 or 1 are rivial condiions ha do no require any simulaions and, hence, are no allowed in he sofware. ROV BizSas User Manual 77 Copyrigh Dr. Johnahan Mun

78 Pascal Disribuion The Pascal disribuion is useful for modeling he disribuion of he number of oal rials required o obain he number of successful occurrences required. For insance, o close a oal of 10 sales opporuniies, how many oal sales calls would you need o make given some probabiliy of success in each call? The x-axis shows he oal number of calls required, which includes successful and failed calls. The number of rials is no fixed, he rials coninue unil he Rh success, and he probabiliy of success is he same from rial o rial. Pascal disribuion is relaed o he negaive binomial disribuion. Negaive binomial disribuion compues he number of evens required in addiion o he number of successes required given some probabiliy (in oher words, he oal failures), whereas he Pascal disribuion compues he oal number of evens required (in oher words, he sum of failures and successes) o achieve he successes required given some probabiliy. Successes required and probabiliy are he disribuional parameers. Condiions The hree condiions underlying he negaive binomial disribuion are: The number of rials is no fixed. The rials coninue unil he rh success. The probabiliy of success is he same from rial o rial. The mahemaical consrucs for he Pascal disribuion are shown below: ( x 1)! S X S p (1 p) for all x s f ( x) ( x s)!( s 1)! 0 oherwise k ( x 1)! S X S p (1 p) for all x s F( x) x1 ( x s)!( s 1)! 0 oherwise s Mean p Sandard Deviaion s(1 p) p p Skewness = r(1 p) p 6 p 6 Excess Kurosis = r(1 p) Successes Required and Probabiliy are he disribuional parameers. Inpu requiremens: ROV BizSas User Manual 78 Copyrigh Dr. Johnahan Mun

79 Successes required > 0 and is an ineger. 0 Probabiliy 1. Poisson Disribuion The Poisson disribuion describes he number of imes an even occurs in a given inerval, such as he number of elephone calls per minue or he number of errors per page in a documen. Condiions The hree condiions underlying he Poisson disribuion are: The number of possible occurrences in any inerval is unlimied. The occurrences are independen. The number of occurrences in one inerval does no affec he number of occurrences in oher inervals. The average number of occurrences mus remain he same from inerval o inerval. The mahemaical consrucs for he Poisson are as follows: x e P( x) for x and 0 x! Mean Sandard Deviaion = 1 Skewness = 1 Excess Kurosis = Rae, or Lambda (), is he only disribuional parameer. Inpu requiremens: Rae > 0 and 1000 (i.e., rae 1000). ROV BizSas User Manual 79 Copyrigh Dr. Johnahan Mun

80 Coninuous Disribuions Arcsine Disribuion The arcsine disribuion is U-shaped and is a special case of he bbea disribuion when boh shape and scale are equal o 0.5. Values close o he minimum and maximum have high probabiliies of occurrence whereas values beween hese wo exremes have very small probabiliies of occurrence. Minimum and maximum are he disribuional parameers. The mahemaical consrucs for he Arcsine disribuion are shown below. The probabiliy densiy funcion (PDF) is denoed f(x) and he cumulaive disribuion funcion (CDF) is denoed F(x). f ( x) 0 1 x(1 x) for 0 x 1 oherwise 0 x 0 F x x for x 1 x 1 1 ( ) sin ( ) 0 1 Min Max Mean Sandard Deviaion ( Max Min) 8 Skewness = 0 for all inpus Excess Kurosis = 1.5 for all inpus Minimum and maximum are he disribuional parameers. Inpu requiremens: Maximum > minimum (eiher inpu parameer can be posiive, negaive, or zero). Bea Disribuion The bea disribuion is very flexible and is commonly used o represen variabiliy over a fixed range. One of he more imporan applicaions of he bea disribuion is is use as a conjugae disribuion for he parameer of a Bernoulli disribuion. In his applicaion, he bea disribuion is used o represen he uncerainy in he probabiliy of occurrence of an even. I is also used o describe empirical daa and predic he random behavior of percenages and fracions, as he range of oucomes is ypically beween 0 and 1. ROV BizSas User Manual 80 Copyrigh Dr. Johnahan Mun

81 The value of he bea disribuion lies in he wide variey of shapes i can assume when you vary he wo parameers, alpha and bea. If he parameers are equal, he disribuion is symmerical. If eiher parameer is 1 and he oher parameer is greaer han 1, he disribuion is J-shaped. If alpha is less han bea, he disribuion is said o be posiively skewed (mos of he values are near he minimum value). If alpha is greaer han bea, he disribuion is negaively skewed (mos of he values are near he maximum value). The mahemaical consrucs for he bea disribuion are as follows: f ( x) Mean ( 1) x 1 x ( 1) ( ) ( ) ( ) Sandard Deviaion Skewness = ( ) ( ) for 0; 0; x 0 ( ) (1 ) 1 3( 1)[ ( 6) ( ) ] Excess Kurosis = 3 ( )( 3) Alpha () and bea () are he wo disribuional shape parameers, and is he Gamma funcion. Condiions The wo condiions underlying he bea disribuion are: The uncerain variable is a random value beween 0 and a posiive value. The shape of he disribuion can be specified using wo posiive values. Inpu requiremens: Alpha and bea boh > 0 and can be any posiive value. Bea 3 and Bea 4 Disribuions The original Bea disribuion only akes wo inpus, Alpha and Bea shape parameers. However, he oupu of he simulaed value is beween 0 and 1. In he Bea 3 disribuion, we add an exra parameer called Locaion or Shif, where we are no free o move away from his 0 o 1 oupu limiaion, herefore he Bea 3 disribuion is also known as a Shifed Bea disribuion. Similarly, he Bea 4 disribuion adds wo inpu parameers, Locaion or Shif, and Facor. The original Bbea disribuion is muliplied by he facor and shifed by he locaion, and, herefore he Bea 4 ROV BizSas User Manual 81 Copyrigh Dr. Johnahan Mun

82 is also known as he Muliplicaive Shifed Bea disribuion. The mahemaical consrucs for he Bea 3 and Bea 4 disribuions are based on hose in he Bea disribuion, wih he relevan shifs and facorial muliplicaion (e.g., he PDF and CDF will be adjused by he shif and facor, and some of he momens, such as he mean, will similarly be affeced; he sandard deviaion, in conras, is only affeced by he facorial muliplicaion, whereas he remaining momens are no affeced a all). Inpu requiremens: Locaion >=< 0 (locaion can ake on any posiive or negaive value including zero). Facor > 0. Cauchy Disribuion, or Lorenzian or Brei-Wigner Disribuion The Cauchy disribuion, also called he Lorenzian or Brei-Wigner disribuion, is a coninuous disribuion describing resonance behavior. I also describes he disribuion of horizonal disances a which a line segmen iled a a random angle cus he x-axis. The mahemaical consrucs for he cauchy or Lorenzian disribuion are as follows: f 1 / x) ( x m) ( / 4 The Cauchy disribuion is a special case because i does no have any heoreical momens (mean, sandard deviaion, skewness, and kurosis) as hey are all undefined. Mode locaion () and scale ( are he only wo parameers in his disribuion. The locaion parameer specifies he peak or mode of he disribuion, while he scale parameer specifies he half-widh a half-maximum of he disribuion. In addiion, he mean and variance of a Cauchy, or Lorenzian, disribuion are undefined. In addiion, he Cauchy disribuion is he Suden s T disribuion wih only 1 degree of freedom. This disribuion is also consruced by aking he raio of wo sandard normal disribuions (normal disribuions wih a mean of zero and a variance of one) ha are independen of one anoher. ROV BizSas User Manual 8 Copyrigh Dr. Johnahan Mun

83 Inpu requiremens: Locaion (Alpha) can be any value. Scale (Bea) > 0 and can be any posiive value. Chi-Square Disribuion The chi-square disribuion is a probabiliy disribuion used predominaly in hypohesis esing, and is relaed o he gamma and sandard normal disribuions. For insance, he sum of independen normal disribuions is disribued as a chi-square ( ) wih k degrees of freedom: Z d 1 Z Z k ~... k The mahemaical consrucs for he chi-square disribuion are as follows: f ( x) Mean = k 0.5 ( k / ) k / k / 1 x / x e for all x > 0 Sandard Deviaion = Skewness = k 1 Excess Kurosis = k k is he gamma funcion. Degrees of freedom, k, is he only disribuional parameer. The chi-square disribuion can also be modeled using a gamma disribuion by seing he k shape parameer equal o and he scaleequal o S where S is he scale. Inpu requiremens: Degrees of freedom > 1 and mus be an ineger < 300. Cosine Disribuion The cosine disribuion looks like a logisic disribuion where he median value beween he minimum and maximum have he highes peak or mode, carrying he maximum probabiliy of occurrence, while he exreme ails close o he minimum and maximum values have lower probabiliies. Minimum and maximum are he disribuional parameers. The mahemaical consrucs for he Cosine disribuion are shown below: ROV BizSas User Manual 83 Copyrigh Dr. Johnahan Mun

84 1 x a cos for min x max f ( x) b b 0 oherwise min max max min where a and b 1 x a 1 sin for min x max F ( x) b 1 for x > max Min Max Mean Sandard Deviaion = Skewness is always equal o 0 4 6(90 ) Excess Kurosis = 5( 6) ( Max Min) ( 8) 4 Minimum and maximum are he disribuional parameers. Inpu requiremens: Maximum > minimum (eiher inpu parameer can be posiive, negaive, or zero). Double Log Disribuion The double log disribuion looks like he Cauchy disribuion where he cenral endency is peaked and carries he maximum value probabiliy densiy bu declines faser he furher i ges away from he cener, creaing a symmerical disribuion wih an exreme peak in beween he minimum and maximum values. Minimum and maximum are he disribuional parameers. The mahemaical consrucs for he Double Log disribuion are shown below: 1 x a ln for min x max f ( x) b b 0 oherwise min max max min where a and b 1 x a x a 1 ln for min x a b b F( x) 1 x a x a 1 ln for a x max b b ROV BizSas User Manual 84 Copyrigh Dr. Johnahan Mun

85 Min Max Mean = Sandard Deviaion = ( Max Min) 36 Skewness is always equal o 0 Excess Kurosis is a complex funcion and no easily represened Minimum and maximum are he disribuional parameers. Inpu requiremens: Maximum > minimum (eiher inpu parameer can be posiive, negaive, or zero). Erlang Disribuion The Erlang disribuion is he same as he Gamma disribuion wih he requiremen ha he Alpha or shape parameer mus be a posiive ineger. An example applicaion of he Erlang disribuion is he calibraion of he rae of ransiion of elemens hrough a sysem of comparmens. Such sysems are widely used in biology and ecology (e.g., in epidemiology, an individual may progress a an exponenial rae from being healhy o becoming a disease carrier, and coninue exponenially from being a carrier o being infecious). Alpha (also known as shape) and Bea (also known as scale) are he disribuional parameers. The mahemaical consrucs for he Erlang disribuion are shown below: 1 x x/ e f ( x) for x 0 ( 1) 0 oherwise 1 i x/ ( x / ) 1 e for x 0 F( x) i0 i! 0 oherwise Mean Sandard Deviaion Skew 6 Excess Kurosis 3 Alpha and Bea are he disribuional parameers. ROV BizSas User Manual 85 Copyrigh Dr. Johnahan Mun

86 Inpu requiremens: Alpha (Shape) > 0 and is an Ineger Bea (Scale) > 0 Exponenial Disribuion The exponenial disribuion is widely used o describe evens recurring a random poins in ime, such as he ime beween failures of elecronic equipmen or he ime beween arrivals a a service booh. I is relaed o he Poisson disribuion, which describes he number of occurrences of an even in a given inerval of ime. An imporan characerisic of he exponenial disribuion is he memoryless propery, which means ha he fuure lifeime of a given objec has he same disribuion regardless of he ime i exised. In oher words, ime has no effec on fuure oucomes. Condiions The condiion underlying he exponenial disribuion is: The exponenial disribuion describes he amoun of ime beween occurrences. The mahemaical consrucs for he exponenial disribuion are as follows: ( ) x f x e for x 0; 0 1 Mean = 1 Sandard Deviaion = Skewness = (his value applies o all success rae inpus) Excess Kurosis = 6 (his value applies o all success rae inpus) Success rae () is he only disribuional parameer. The number of successful rials is denoed x. Inpu requiremens: Rae > 0. Exponenial Disribuion The Exponenial disribuion uses he same consrucs as he original Exponenial disribuion bu adds a Locaion or Shif parameer. The Exponenial disribuion sars from a minimum value of 0, whereas his Exponenial or Shifed Exponenial, disribuion shifs he saring locaion o any oher value. ROV BizSas User Manual 86 Copyrigh Dr. Johnahan Mun

87 Rae, or Lambda, and Locaion, or Shif, are he disribuional parameers. Inpu requiremens: Rae (Lambda) > 0. Locaion can be any posiive or negaive value including zero. Exreme Value Disribuion, or Gumbel Disribuion The exreme value disribuion (Type 1) is commonly used o describe he larges value of a response over a period of ime, for example, in flood flows, rainfall, and earhquakes. Oher applicaions include he breaking srenghs of maerials, consrucion design, and aircraf loads and olerances. The exreme value disribuion is also known as he Gumbel disribuion. The mahemaical consrucs for he exreme value disribuion are as follows: x 1 Z f ( x) ze where z e for Mean = Sandard Deviaion = 1 6 0; and any value of x and 1 6( ) Skewness = (his applies for all values of mode and scale) 3 Excess Kurosis = 5.4 (his applies for all values of mode and scale) Mode () and scale () are he disribuional parameers. Calculaing Parameers There are wo sandard parameers for he exreme value disribuion: mode and scale. The mode parameer is he mos likely value for he variable (he highes poin on he probabiliy disribuion). Afer you selec he mode parameer, you can esimae he scale parameer. The scale parameer is a number greaer han 0. The larger he scale parameer, he greaer he variance. Inpu requiremens: Mode Alpha can be any value. Scale Bea > 0. F Disribuion, or Fisher-Snedecor Disribuion The F disribuion, also known as he Fisher-Snedecor disribuion, is anoher coninuous disribuion used mos frequenly for hypohesis esing. Specifically, i is used o es he ROV BizSas User Manual 87 Copyrigh Dr. Johnahan Mun

88 saisical difference beween wo variances in analysis of variance ess and likelihood raio ess. The F disribuion wih he numeraor degree of freedom n and denominaor degree of freedom m is relaed o he chi-square disribuion in ha: / n d ~ F / m n m Mean = n, m m m Sandard Deviaion = m ( m n ) n( m ) ( m 4) for all m > 4 ( m n ) ( m 4) Skewness = m 6 n( m n ) 3 1( 16 0m 8m m 44n 3mn 5m Excess Kurosis = n( m 6)( m 8)( n m ) n n 5mn The numeraor degree of freedom n and denominaor degree of freedom m are he only disribuional parameers. Inpu requiremens: Degrees of freedom numeraor and degrees of freedom denominaor mus boh be inegers > 0 Gamma Disribuion (Erlang Disribuion) The gamma disribuion applies o a wide range of physical quaniies and is relaed o oher disribuions: lognormal, exponenial, Pascal, Erlang, Poisson, and chi-square. I is used in meeorological processes o represen polluan concenraions and precipiaion quaniies. The gamma disribuion is also used o measure he ime beween he occurrence of evens when he even process is no compleely random. Oher applicaions of he gamma disribuion include invenory conrol, economic heory, and insurance risk heory. Condiions The gamma disribuion is mos ofen used as he disribuion of he amoun of ime unil he rh occurrence of an even in a Poisson process. When used in his fashion, he hree condiions underlying he gamma disribuion are: The number of possible occurrences in any uni of measuremen is no limied o a fixed number. The occurrences are independen. The number of occurrences in one uni of measuremen does no affec he number of occurrences in oher unis. ROV BizSas User Manual 88 Copyrigh Dr. Johnahan Mun

89 The average number of occurrences mus remain he same from uni o uni. The mahemaical consrucs for he gamma disribuion are as follows: 1 x e ( ) f x ( ) Mean = Sandard Deviaion = Skewness = x Excess Kurosis = 6 wih any value of 0 and 0 Shape parameer alpha () and scale parameer bea () are he disribuional parameers, and is he Gamma funcion. When he alpha parameer is a posiive ineger, he gamma disribuion is called he Erlang disribuion, used o predic waiing imes in queuing sysems, where he Erlang disribuion is he sum of independen and idenically disribued random variables each having a memoryless exponenial disribuion. Seing n as he number of hese random variables, he mahemaical consruc of he Erlang disribuion is: f ( x) n1 x x e for all x > 0 and all posiive inegers of n ( n 1)! Inpu requiremens: Scale bea > 0 and can be any posiive value. Shape alpha 0.05 and any posiive value. Locaion can be any value. Laplace Disribuion The Laplace disribuion is also someimes called he double exponenial disribuion because i can be consruced wih wo exponenial disribuions (wih an addiional locaion parameer) spliced ogeher back-o-back, creaing an unusual peak in he middle. The probabiliy densiy funcion of he Laplace disribuion is reminiscen of he normal disribuion. However, whereas he normal disribuion is expressed in erms of he squared difference from he mean, he Laplace densiy is expressed in erms of he absolue difference from he mean, making he Laplace disribuion s ails faer han hose of he normal disribuion. When he locaion parameer is se o zero, he Laplace disribuion s random variable is exponenially disribued wih an inverse of ROV BizSas User Manual 89 Copyrigh Dr. Johnahan Mun

90 he scale parameer. Alpha (also known as locaion) and Bea (also known as scale) are he disribuional parameers. The mahemaical consrucs for he Laplace disribuion are shown below: 1 x f ( x) exp 1 x exp when x F( x) 1 x 1 exp when x Mean Sandard Deviaion Skewness is always equal o 0 as i is a symmerical disribuion Excess Kurosis is always equal o 3 Inpu requiremens: Alpha (Locaion) can ake on any posiive or negaive value including zero. Bea (Scale) > 0. Logisic Disribuion The logisic disribuion is commonly used o describe growh, ha is, he size of a populaion expressed as a funcion of a ime variable. I also can be used o describe chemical reacions and he course of growh for a populaion or individual. The mahemaical consrucs for he logisic disribuion are as follows: x e f ( x) for any value of and x 1 e Mean Sandard Deviaion 1 3 Skewness = 0 (his applies o all mean and scale inpus) Excess Kurosis = 1. (his applies o all mean and scale inpus) Mean () and scale () are he disribuional parameers. Calculaing Parameers ROV BizSas User Manual 90 Copyrigh Dr. Johnahan Mun

91 There are wo sandard parameers for he logisic disribuion: mean and scale. The mean parameer is he average value, which for his disribuion is he same as he mode because his is a symmerical disribuion. Afer you selec he mean parameer, you can esimae he scale parameer. The scale parameer is a number greaer han 0. The larger he scale parameer, he greaer he variance. Inpu requiremens: Scale Bea > 0 and can be any posiive value. Mean Alpha can be any value. Lognormal Disribuion The lognormal disribuion is widely used in siuaions where values are posiively skewed, for example, in financial analysis for securiy valuaion or in real esae for propery valuaion, and where values canno fall below zero. Sock prices are usually posiively skewed raher han normally (symmerically) disribued. Sock prices exhibi his rend because hey canno fall below he lower limi of zero bu migh increase o any price wihou limi. Similarly, real esae prices illusrae posiive skewness as propery values canno become negaive. Condiions The hree condiions underlying he lognormal disribuion are: The uncerain variable can increase wihou limis bu canno fall below zero. The uncerain variable is posiively skewed, wih mos of he values near he lower limi. The naural logarihm of he uncerain variable yields a normal disribuion. Generally, if he coefficien of variabiliy is greaer han 30%, use a lognormal disribuion. Oherwise, use he normal disribuion. The mahemaical consrucs for he lognormal disribuion are as follows: f ( x) x 1 e ln( ) Mean exp [ln( x) ln( )] [ln( )] Sandard Deviaion = exp exp Skewness = exp 1 ( exp( )) for x 0; 0 and 0 ROV BizSas User Manual 91 Copyrigh Dr. Johnahan Mun 1 Excess Kurosis = exp4 exp3 3exp 6

92 Mean () and sandard deviaion () are he disribuional parameers. Inpu requiremens: Mean and sandard deviaion boh > 0 and can be any posiive value. Lognormal Parameer Ses By defaul, he lognormal disribuion uses he arihmeic mean and sandard deviaion. For applicaions for which hisorical daa are available, i is more appropriae o use eiher he logarihmic mean and sandard deviaion, or he geomeric mean and sandard deviaion. Lognormal 3 Disribuion The Lognormal 3 disribuion uses he same consrucs as he original Lognormal disribuion bu adds a Locaion, or Shif, parameer. The Lognormal disribuion sars from a minimum value of 0, whereas his Lognormal 3, or Shifed Lognormal disribuion shifs he saring locaion o any oher value. Mean, Sandard Deviaion, and Locaion (Shif) are he disribuional parameers. Inpu requiremens: Mean > 0. Sandard Deviaion > 0. Locaion can be any posiive or negaive value including zero. Normal Disribuion The normal disribuion is he mos imporan disribuion in probabiliy heory because i describes many naural phenomena, such as people s IQs or heighs. Decision makers can use he normal disribuion o describe uncerain variables such as he inflaion rae or he fuure price of gasoline. Condiions The hree condiions underlying he normal disribuion are: Some value of he uncerain variable is he mos likely (he mean of he disribuion). The uncerain variable could as likely be above he mean as i could be below he mean (symmerical abou he mean). The uncerain variable is more likely o be in he viciniy of he mean han furher away. The mahemaical consrucs for he normal disribuion are as follows: ROV BizSas User Manual 9 Copyrigh Dr. Johnahan Mun

93 ( x) 1 f ( x) e for all values of x and ; while > 0 Mean Sandard Deviaion Skewness = 0 (his applies o all inpus of mean and sandard deviaion) Excess Kurosis = 0 (his applies o all inpus of mean and sandard deviaion) Mean () and sandard deviaion () are he disribuional parameers. Inpu requiremens: Sandard deviaion > 0 and can be any posiive value. Mean can ake on any value. Parabolic Disribuion The parabolic disribuion is a special case of he bea disribuion when Shape = Scale =. Values close o he minimum and maximum have low probabiliies of occurrence, whereas values beween hese wo exremes have higher probabiliies or occurrence. Minimum and maximum are he disribuional parameers. The mahemaical consrucs for he Parabolic disribuion are shown below: f ( x) ( 1) x 1 x ( 1) ( ) ( ) ( ) for 0; 0; x 0 Where he funcional form above is for a Bea disribuion, and for a Parabolic funcion, we se Alpha = Bea = and a shif of locaion in Minimum, wih a muliplicaive facor of (Maximum Minimum). Min Max Mean = Sandard Deviaion = Skewness = 0 Excess Kurosis = ( Max Min) 0 Minimum and Maximum are he disribuional parameers. Inpu requiremens: Maximum > minimum (eiher inpu parameer can be posiive, negaive, or zero). ROV BizSas User Manual 93 Copyrigh Dr. Johnahan Mun

94 Pareo Disribuion The Pareo disribuion is widely used for he invesigaion of disribuions associaed wih such empirical phenomena as ciy populaion sizes, he occurrence of naural resources, he size of companies, personal incomes, sock price flucuaions, and error clusering in communicaion circuis. The mahemaical consrucs for he Pareo are as follows: L f ( x) for x L ( 1 ) x mean L 1 sandard deviaion skewness = excess kurosis = ( 1) 3 L ( 1) ( ) 3 6( 6 ) ( 3)( 4) Shape () and Locaion () are he disribuional parameers. Calculaing Parameers There are wo sandard parameers for he Pareo disribuion: locaion and shape. The locaion parameer is he lower bound for he variable. Afer you selec he locaion parameer, you can esimae he shape parameer. The shape parameer is a number greaer han 0, usually greaer han 1. The larger he shape parameer, he smaller he variance and he hicker he righ ail of he disribuion. Inpu requiremens: Locaion > 0 and can be any posiive value Shape Pearson V Disribuion The Pearson V disribuion is relaed o he Inverse Gamma disribuion, where i is he reciprocal of he variable disribued according o he Gamma disribuion. Pearson V disribuion is also used o model ime delays where here is almos cerainy of some minimum delay and he maximum delay is unbounded, for example, delay in arrival of emergency services and ime o repair a machine. Alpha (also known as shape) and Bea (also known as scale) are he disribuional parameers. ROV BizSas User Manual 94 Copyrigh Dr. Johnahan Mun

95 The mahemaical consrucs for he Pearson V disribuion are shown below: x f ( x) e ( ) ( 1) / x (, / x) F( x) ( ) Mean 1 Sandard Deviaion 4 Skew 3 ( 1) ( ) Excess Kurosis 3 ( 3)( 4) Inpu requiremens: Alpha (Shape) > 0. Bea (Scale) > 0. Pearson VI Disribuion The Pearson VI disribuion is relaed o he Gamma disribuion, where i is he raional funcion of wo variables disribued according o wo Gamma disribuions. Alpha 1 (also known as shape 1), Alpha (also known as shape ), and Bea (also known as scale) are he disribuional parameers. The mahemaical consrucs for he Pearson VI disribuion are shown below: 1 1 ( x / ) f ( x) (, )[1 ( x / )] 1 x F( x) FB x Mean 1 1 Sandard Deviaion = Skew ( 1) ( 1) ( ) 1 1 1( 1 1) 3 ROV BizSas User Manual 95 Copyrigh Dr. Johnahan Mun

96 Excess Kurosis 3( ) ( 1) ( 5) 3 ( 3)( 4) 1( 1 1) Inpu requiremens: Alpha 1 (Shape 1) > 0. Alpha (Shape ) > 0. Bea (Scale) > 0. PERT Disribuion The PERT disribuion is widely used in projec and program managemen o define he worscase, nominal-case, and bes-case scenarios of projec compleion ime. I is relaed o he Bea and Triangular disribuions. PERT disribuion can be used o idenify risks in projec and cos models based on he likelihood of meeing arges and goals across any number of projec componens using minimum, mos likely, and maximum values, bu i is designed o generae a disribuion ha more closely resembles realisic probabiliy disribuions. The PERT disribuion can provide a close fi o he normal or lognormal disribuions. Like he riangular disribuion, he PERT disribuion emphasizes he "mos likely" value over he minimum and maximum esimaes. However, unlike he riangular disribuion, he PERT disribuion consrucs a smooh curve ha places progressively more emphasis on values around (near) he mos likely value, in favor of values around he edges. In pracice, his means ha we "rus" he esimae for he mos likely value, and we believe ha even if i is no exacly accurae (as esimaes seldom are), we have an expecaion ha he resuling value will be close o ha esimae. Assuming ha many real-world phenomena are normally disribued, he appeal of he PERT disribuion is ha i produces a curve similar o he normal curve in shape, wihou knowing he precise parameers of he relaed normal curve. Minimum, Mos Likely, and Maximum are he disribuional parameers. The mahemaical consrucs for he PERT disribuion are shown below: ( x min) (max x) f ( x) B( A1, A)(max min) A11 A1 A1 A1 min 4(likely) max min 4(likely) max min max where A1 6 6 and A 6 6 max min max min and B is he Bea funcion Min 4Mode Max Mean 6 ( Min)( Max ) Sandard Deviaion 7 ROV BizSas User Manual 96 Copyrigh Dr. Johnahan Mun

97 7 Min Max Skew ( Min)( Max ) 4 Inpu requiremens: Minimum Mos Likely Maximum and can be posiive, negaive, or zero. Power Disribuion The Power disribuion is relaed o he exponenial disribuion in ha he probabiliy of small oucomes is large bu exponenially decreases as he oucome value increases. Alpha (also known as shape) is he only disribuional parameer. The mahemaical consrucs for he Power disribuion are shown below: f ( x) x F( x) x 1 Mean 1 Sandard Deviaion (1 ) ( ) ( 1) Skew 3 Excess Kurosis is a complex funcion and canno be readily compued Inpu requiremens: Alpha > 0. Power 3 Disribuion The Power 3 disribuion uses he same consrucs as he original Power disribuion bu adds a Locaion, or Shif, parameer, and a muliplicaive Facor parameer. The Power disribuion sars from a minimum value of 0, whereas his Power 3, or Shifed Muliplicaive Power, disribuion shifs he saring locaion o any oher value. Alpha, Locaion or Shif, and Facor are he disribuional parameers. Inpu requiremens: Alpha > Locaion, or Shif, can be any posiive or negaive value including zero. Facor > 0. ROV BizSas User Manual 97 Copyrigh Dr. Johnahan Mun

98 Suden s Disribuion The Suden s disribuion is he mos widely used disribuion in hypohesis es. This disribuion is used o esimae he mean of a normally disribued populaion when he sample size is small o es he saisical significance of he difference beween wo sample means or confidence inervals for small sample sizes. The mahemaical consrucs for he disribuion are as follows: [( r 1) / ] ( r 1) / f ( ) (1 / r) r [ r / ] Mean = 0 (his applies o all degrees of freedom r excep if he disribuion is shifed o anoher nonzero cenral locaion) Sandard Deviaion = r r Skewness = 0 (his applies o all degrees of freedom r) 6 Excess Kurosis = for all r 4 r 4 x x where and is he gamma funcion. s Degrees of freedom r is he only disribuional parameer. The disribuion is relaed o he F disribuion as follows: he square of a value of wih r degrees of freedom is disribued as F wih 1 and r degrees of freedom. The overall shape of he probabiliy densiy funcion of he disribuion also resembles he bell shape of a normally disribued variable wih mean 0 and variance 1, excep ha i is a bi lower and wider or is lepokuric (fa ails a he ends and peaked cener). As he number of degrees of freedom grows (say, above 30), he disribuion approaches he normal disribuion wih mean 0 and variance 1. Inpu requiremens: Degrees of freedom 1 and mus be an ineger. Triangular Disribuion The riangular disribuion describes a siuaion where you know he minimum, maximum, and mos likely values o occur. For example, you could describe he number of cars sold per week when pas sales show he minimum, maximum, and usual number of cars sold. Condiions The hree condiions underlying he riangular disribuion are: The minimum number of iems is fixed. ROV BizSas User Manual 98 Copyrigh Dr. Johnahan Mun

99 The maximum number of iems is fixed. The mos likely number of iems falls beween he minimum and maximum values, forming a riangular-shaped disribuion, which shows ha values near he minimum and maximum are less likely o occur han hose near he mos-likely value. The mahemaical consrucs for he riangular disribuion are as follows: ( x Min) ( Max Min)( Likely min) f ( x) ( Max x) ( Max Min)( Max Likely) for Min x Likely for Likely x Max 1 Mean = ( Min Likely Max) 3 1 Sandard Deviaion = ( Min Likely Max Min Max Min Likely Max Likely) 18 ( Min Max Likely)(Min Max Likely)( Min Max Likely) Skewness = 3 / 5( Min Max Likely MinMax MinLikely MaxLikely ) Excess Kurosis = 0.6 (his applies o all inpus of Min, Max, and Likely) Minimum value (Min), mos-likely value (Likely), and maximum value (Max) are he disribuional parameers. Inpu requiremens: Min Mos Likely Max and can ake any value. However, Min < Max and can ake any value. Uniform Disribuion Wih he uniform disribuion, all values fall beween he minimum and maximum and occur wih equal likelihood. Condiions The hree condiions underlying he uniform disribuion are: The minimum value is fixed. The maximum value is fixed. All values beween he minimum and maximum occur wih equal likelihood. The mahemaical consrucs for he uniform disribuion are as follows: 1 f ( x) Max Min for all values such ha Min Max ROV BizSas User Manual 99 Copyrigh Dr. Johnahan Mun

100 Min Max Mean Sandard Deviaion ( Max Min) 1 Skewness = 0 (his applies o all inpus of Min and Max) Excess Kurosis = 1. (his applies o all inpus of Min and Max) Maximum value (Max) and minimum value (Min) are he disribuional parameers. Inpu requiremens: Min < Max and can ake any value. Weibull Disribuion (Rayleigh Disribuion) The Weibull disribuion describes daa resuling from life and faigue ess. I is commonly used o describe failure ime in reliabiliy sudies as well as he breaking srenghs of maerials in reliabiliy and qualiy conrol ess. Weibull disribuions are also used o represen various physical quaniies, such as wind speed. The Weibull disribuion is a family of disribuions ha can assume he properies of several oher disribuions. For example, depending on he shape parameer you define, he Weibull disribuion can be used o model he exponenial and Rayleigh disribuions, among ohers. The Weibull disribuion is very flexible. When he Weibull shape parameer is equal o 1.0, he Weibull disribuion is idenical o he exponenial disribuion. The Weibull locaion parameer les you se up an exponenial disribuion o sar a a locaion oher han 0.0. When he shape parameer is less han 1.0, he Weibull disribuion becomes a seeply declining curve. A manufacurer migh find his effec useful in describing par failures during a burn-in period. The mahemaical consrucs for he Weibull disribuion are as follows: x f ( x) Mean 1 e 1 (1 ) x 1 1 Sandard Deviaion (1 ) (1 ) Skewness = Excess Kurosis = 6 4 (1 1 3 (1 ) 1 1 ) 3(1 1 ) ( (1 ) (1 ) 3/ (1 1 ) (1 1 ) 3 ) (1 3 ( (1 ) (1 ) ) ) 4(1 1 ) (1 3 1 ) (1 4 1 ) ROV BizSas User Manual 100 Copyrigh Dr. Johnahan Mun

101 Shape () and cenral locaion scale () are he disribuional parameers, and is he Gamma funcion. Inpu requiremens: Shape Alpha Scale Bea > 0 and can be any posiive value. Weibull 3 Disribuion The Weibull 3 disribuion uses he same consrucs as he original Weibull disribuion bu adds a Locaion, or Shif, parameer. The Weibull disribuion sars from a minimum value of 0, whereas his Weibull 3, or Shifed Weibull, disribuion shifs he saring locaion o any oher value. Alpha, Bea, and Locaion or Shif are he disribuional parameers. Inpu requiremens: Alpha (Shape) Bea (Cenral Locaion Scale) > 0 and can be any posiive value. Locaion can be any posiive or negaive value including zero. ROV BizSas User Manual 101 Copyrigh Dr. Johnahan Mun

102 APPENDIX : A PRIMER ON TIME-SERIES METHODOLOGIES The BizSas sofware suppors eigh basic ime-series mehodologies as well as more advanced ime-series echniques such as ARIMA, Auo ARIMA, Muliple Regression, and Sepwise Regression. This appendix looks a he eigh basic ime series mehodologies in more deail han is given in he main ex of his manual, while Appendix 3 provides deails on regression analysis. No Trend and No Seasonaliy: Single Moving Average and Single Exponenial Smoohing Single Moving Average The single moving average is applicable when ime-series daa wih no rend and seasonaliy exis. The approach simply uses an average of he acual hisorical daa o projec fuure oucomes. This average is applied consisenly moving forward, hence he erm moving average. The value of he moving average (MA) for a specific lengh (n) is simply he summaion of acual hisorical daa (Y) arranged and indexed in a ime sequence (i). MA n n i 1 n Y i An example compuaion of a 3-monh single moving average is seen in Figure A.1.Here we see ha here are 39 monhs of acual hisorical daa and a 3-monh moving average is compued. Addiional columns of calculaions also exis in he example, calculaions ha are required o esimae he error of measuremens in using his moving-average approach. These errors are imporan as hey can be compared across muliple moving averages (i.e., 3-monh, 4-monh, 5- monh, ec.) as well as oher ime-series models (e.g., single moving average, seasonal addiive model, ec.) o find he bes fi ha minimizes hese errors. Figures A., A.3, and A.4 show he exac calculaions used in he moving-average model. Noice ha he forecas-fi value in period 4 of is a 3-monh average of he prior 3 periods (monhs 1 hrough 3). The forecasfi value for period 5 would hen be he 3-monh average of monhs hrough 4. This process is repeaed moving forward unil monh 40 (Figure A.3) where every monh afer ha, he forecas is fixed a Clearly, his approach is no suiable if here is a rend (upward or downward over ime) or if here is seasonaliy. Thus, error esimaion is imporan when choosing he opimal ime-series forecas model. Figure A. illusraes a few addiional columns of calculaions required for esimaing he forecas errors. The values from hese columns are used in Figure A.4 s error esimaion. ROV BizSas User Manual 10 Copyrigh Dr. Johnahan Mun

103 Monh Acual Forecas Fi Single Moving Average (3 Monhs) Error Y Y Yˆ Y Y Y1 Error Error E E Y 1 Y 1 Y % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % Forecas Forecas Forecas ˆ RMSE MSE MAD MAPE 0.80% Thiel's U 0.80 MA n n i 1 n Y i i 1,..., N Figure A.1 Single Moving Average (3 Monhs) Single Moving Average (3 Monhs) Y Yˆ Yˆ Monh Acual Forecas Fi Error Error Y Y Y 1 Error E E 1 Y Y1 Y % Abs Figure A. Calculaing Single Moving Average ROV BizSas User Manual 103 Copyrigh Dr. Johnahan Mun

104 Forecas Forecas Forecas Figure A.3 Forecasing wih a Single Moving Average Error Esimaion (RMSE, MSE, MAD, MAPE, and Theil s U) Several differen ypes of errors can be calculaed for ime-series forecas mehods, including he mean-squared error (MSE), roo mean-squared error (RMSE), mean absolue deviaion (MAD), and mean absolue percen error (MAPE). MSE is an absolue error measure ha squares he errors (he difference beween he acual hisorical daa and he forecas-fied daa prediced by he model) o keep he posiive and negaive errors from canceling each oher ou. This measure also ends o exaggerae large errors by weighing he large errors more heavily han smaller errors by squaring hem, which can help when comparing differen ime-series models. MSE is calculaed by simply aking he average of he Error column in Figure A.1. RMSE is he square roo of MSE and is he mos popular error measure, also known as he quadraic loss funcion. RMSE can be defined as he average of he absolue values of he forecas errors and is highly appropriae when he cos of he forecas errors is proporional o he absolue size of he forecas error. MAD is an error saisic ha averages he disance (absolue value of he difference beween he acual hisorical daa and he forecas-fied daa prediced by he model) beween each pair of acual and fied forecas daa poins. MAD is calculaed by aking he average of he Error column in Figure A.1, and is mos appropriae when he cos of forecas errors is proporional o he absolue size of he forecas errors. MAPE is a relaive error saisic measured as an average percen error of he hisorical daa poins and is mos appropriae when he cos of he forecas error is more closely relaed o he percenage error han he numerical size of he error. This error esimae is calculaed by aking Y Yˆ he average of he column in Figure A.1, where Y is he hisorical daa a ime, while Y Yˆ is he fied, or prediced, daa poin a ime using his ime-series mehod. Finally, an associaed measure is he Theil s U saisic, which measures he naivey of he model s forecas. Tha is, if he Theil s U saisic is less han 1.0, hen he forecas mehod used provides an esimae ha is saisically beer han guessing. Figure A.4 provides he mahemaical deails of each error esimae. ROV BizSas User Manual 104 Copyrigh Dr. Johnahan Mun

105 RMSE MSE MAD MAPE 0.80% Thiel's U 0.80 RMSE MSE MAD n i1 n i 1 MAPE n i 1 n i1 Error Error n Error n Y Yˆ Y n n i i i i MSE RMSE Theil' su i1 n n i1 Yˆ Y Y 1 Y Y Y 1 1 i i Figure A.4 Error Esimaion Single Exponenial Smoohing The second approach o use when no discernable rend or seasonaliy exiss in he ime-series daa is he single exponenial smoohing mehod. This mehod weighs pas daa wih exponenially decreasing weighs going ino he pas; ha is, he more recen he daa value, he greaer is weigh. This weighing largely overcomes he limiaions of moving averages or percenagechange models. The weigh used is ermed he alpha measure. The mehod is illusraed in Figures A.5 and A.6 and uses he following model: ESF Y 1 ( 1 ) ESF 1 where he exponenial smoohing forecas (ESF) a ime is a weighed average beween he acual value one period in he pas (Y -1 ) and las period s forecas (ESF -1 ), weighed by he alpha parameer (). ROV BizSas User Manual 105 Copyrigh Dr. Johnahan Mun

106 Single Exponenial Smoohing Alpha RMSE Monh Acual Forecas Fi Forecas ESF Y -1 (1 ) ESF -1 Figure A.5 Single Exponenial Smoohing Figure A.6 shows an example of he compuaion. Noice ha he firs forecas-fied value in monh ( Y ˆ ) is always he previous monh s acual value (Y 1 ). The mahemaical equaion ges used only a monh 3 or saring from he second forecas-fied period. ROV BizSas User Manual 106 Copyrigh Dr. Johnahan Mun

107 Single Exponenial Smoohing Alpha 0.10 Monh Acual Forecas Fi Yˆ Y (146.64) (1 0.1)65. ESF Y -1 (1 ) ESF -1 Figure A.6 Calculaing Single Exponenial Smoohing Wih Trend and No Seasonaliy: Double Moving Average and Double Exponenial Smoohing For daa ha exhibi a rend bu no seasonaliy, he double moving average and double exponenial smoohing mehods work raher well. The following examples assume a quarerly seasonaliy (i.e., here are four quarerly periods in a year). Double Moving Average The double moving average mehod smoohes ou pas daa by performing a moving average on a subse of daa ha represens a moving average of an original se of daa. Tha is, a second moving average is performed on he firs moving average. The second moving-average applicaion capures he rending effec of he daa. Figures A.7 and A.8 illusrae he compuaion involved. The example shown is a 3-monh double moving average and he forecas value obained in period 40 is calculaed using he following: m 1 Forecas MA1, MA, MA1, MA, where he forecas value is wice he amoun of he firs moving average (MA 1 ) a ime, less he second moving average esimae (MA ) plus he difference beween he wo moving averages muliplied by a correcion facor ( divided ino he number of monhs in he moving average, m, less 1). ROV BizSas User Manual 107 Copyrigh Dr. Johnahan Mun

108 Double Moving Average (3 Monhs) RMSE Period Acual 3-monh MA 1 3-monh MA Forecas Fi Forecas Forecas MA1, MA, MA1, MA, m 1 Figure A.7 Double Moving Average (3 Monhs) ROV BizSas User Manual 108 Copyrigh Dr. Johnahan Mun

109 Double Moving Average (3 Monhs) Period Acual 3-monh MA 1 3-monh MA Forecas Fi (160.36) ( ) 3 1 m 1 Forecas MA1, MA, MA1, MA, Figure A.8 Calculaing Double Moving Average Double Exponenial Smoohing The second approach o use when he daa exhibis a rend bu no seasonaliy is he double exponenial smoohing mehod. Double exponenial smoohing applies single exponenial smoohing wice, once o he original daa and hen o he resuling single exponenial smoohing daa. An alpha () weighing parameer is used on he firs or single exponenial smoohing (SES) while a bea () weighing parameer is used on he second or double exponenial smoohing (DES). This approach is useful when he hisorical daa series is no saionary. Figure A.9 illusraes he double exponenial smoohing model, and Figure A.10 shows he compuaional deails. The forecas is calculaed using he following: DES SES ( SES SES 1) (1 ) DES Y (1 ) ( SES DES ) ROV BizSas User Manual 109 Copyrigh Dr. Johnahan Mun

110 Double Exponenial Smoohing Opimized Alpha Bea RMSE Period Acual SES DES Forecas Fi Forecas Forecas Forecas Forecas DES SES ( SES SES 1) (1 ) DES Y (1 )( SES DES ) Figure A.9 Double Exponenial Smoohing ROV BizSas User Manual 110 Copyrigh Dr. Johnahan Mun

111 Double Exponenial Smoohing Alpha Bea SES1 Y 65. DES 1 Y Y Period Acual SES DES Forecas Fi ( ) (1 0.5)( ) (146.64) (1 0.9) 65. ( ) DES SES ( SES SES 1) (1 ) DES Y (1 )( SES DES ) ( ) 1 Figure A.10 Calculaing Double Exponenial Smoohing No Trend and Wih Seasonaliy: Addiive Seasonaliy and Muliplicaive Seasonaliy Addiive Seasonaliy If he ime-series daa has no appreciable rend bu exhibis seasonaliy, hen he addiive seasonaliy and muliplicaive seasonaliy mehods apply. The addiive seasonaliy mehod is illusraed in Figures A.11 and A.1. ROV BizSas User Manual 111 Copyrigh Dr. Johnahan Mun

112 Addiive Seasonaliy Wih No Trend Level Seasonal RMSE Alpha Gamma Period Acual Level Seasonaliy Forecas Fi Level L Seasonaliy S Forecas F ( Y S s ) (1 )( L 1 m ) ( Y L ) (1 )( S L S ms s ) Figure A.11 Seasonal Addiive ROV BizSas User Manual 11 Copyrigh Dr. Johnahan Mun

113 The addiive seasonaliy model breaks he hisorical daa ino a level (L), or base-case, componen as measured by he alpha parameer (), and a seasonaliy (S) componen measured by he gamma parameer (). The resuling forecas value is simply he addiion of his base-case level o he seasonaliy value. (Please be aware ha all calculaions are rounded). Addiive Seasonaliy Wih No Trend Level Seasonal RMSE Alpha Gamma Period Acual Level Seasonaliy Forecas Fi Y ( ) (1 0.40)(87.00) 0.33( ) (1 0.33)(178.3) Level L Seasonaliy S Forecas F m ( Y Ss ) (1 )( L 1 ) ( Y L ) (1 )( S L S ms s ) Figure A.1 Calculaing Seasonal Addiive Muliplicaive Seasonaliy Similar o addiive seasonaliy, he muliplicaive seasonaliy model requires he alpha and gamma parameers. The difference is ha he model is muliplicaive, and so he forecas value is he muliplicaion of he base-case level by he seasonaliy facor. Figures A.13 and A.14 illusrae he compuaions required. (Please be aware ha all calculaions are rounded). ROV BizSas User Manual 113 Copyrigh Dr. Johnahan Mun

114 Muliplicaive Seasonaliy Wih No Trend Level Seasonal RMSE Alpha Gamma Period Acual Level Seasonaliy Forecas Fi Level L Seasonaliy S Forecas F ( Y / Ss ) (1 )( L 1 m ) ( Y / L ) (1 )( S L S ms Figure A.13 Seasonal Muliplicaive s ) ROV BizSas User Manual 114 Copyrigh Dr. Johnahan Mun

115 Muliplicaive Seasonaliy Wih No Trend Level Seasonal RMSE Alpha Gamma Period Acual Level Seasonaliy Forecas Fi Y (1.49) ( /1.49) (1 0.)(178.3) 0.64( / ) (1 0.64)(1.49) Level L Seasonaliy S Forecas F ( Y / Ss ) (1 )( L 1 m ) ( Y / L ) (1 )( S L S ms s Figure A.14 Calculaing Seasonal Muliplicaive ) Wih Seasonaliy and Trend: Hol-Winer s Addiive and Hol-Winer s Muliplicaive When boh seasonaliy and rend exis, more advanced models are required o decompose he daa ino heir base elemens: a base-case level (L) weighed by he alpha parameer (); a rend componen (b) weighed by he bea parameer (); and a seasonaliy componen (S) weighed by he gamma parameer (). Several mehods exis, bu he wo mos common are he Hol-Winer s addiive seasonaliy and Hol-Winer s muliplicaive seasonaliy mehods. Hol-Winers Addiive Seasonaliy In he Hol-Winer s addiive model, he base-case level, seasonaliy, and rend are added ogeher o obain he forecas fi. Figures A.15 and A.16 illusrae he required compuaions for deermining a Hol-Winer s addiive forecas model. ROV BizSas User Manual 115 Copyrigh Dr. Johnahan Mun

116 Hol-Winers Addiive Seasonaliy Wih Trend Level Trend Seasonal RMSE Alpha Bea Gamma Period Acual Level Trend Seasonaliy Forecas Fi Level L ( Y S Trend b ( L L Seasonaliy S ( Y Forecas F m L s 1 ) (1 )( L 1 b 1 ) ) (1 )( b 1 ) L ) (1 )( S ) mb S m s s Figure A.15 Hol-Winer s Addiive ROV BizSas User Manual 116 Copyrigh Dr. Johnahan Mun

117 Hol-Winers Addiive Seasonaliy Wih Trend Level Trend Seasonal RMSE Alpha Bea Gamma Period Acual Level Trend Seasonaliy Forecas Fi Saring Trend Y ( ) (1 0.05)( ) 0.4( ) (1 0.4)(87.00) 1.00( ) (1 1.00)(0.00) Level L ( Y S s ) (1 )( L 1 b 1 ) Trend b ( L L 1 ) (1 )( b 1 ) Seasonaliy S ( Y L ) (1 )( S s ) Forecas F L mb S m m s Figure A.16 Calculaing Hol-Winer s Addiive Hol-Winers Muliplicaive Seasonaliy In he Hol-Winer s muliplicaive model, he base-case level and rend are added ogeher and muliplied by he seasonaliy facor o obain he forecas fi. Figures A.17 and A.18 show he required compuaion for deermining a Hol-Winer s muliplicaive forecas model when boh rend and seasonaliy exis. ROV BizSas User Manual 117 Copyrigh Dr. Johnahan Mun

118 Hol-Winers Muliplicaive Seasonaliy Wih Trend Level Trend Seasonal RMSE Alpha Bea Gamma Period Acual Level Trend Seasonaliy Forecas Fi Level L ( Y / S s ) (1 )( L 1 b 1) Trend b ( L L 1 ) (1 )( b 1) Seasonaliy S ( Y / L ) (1 )( S s ) Forecas F ( L mb ) S m ms Figure A.17 Hol-Winer s Muliplicaive ROV BizSas User Manual 118 Copyrigh Dr. Johnahan Mun

119 Hol-Winers Muliplicaive Seasonaliy Wih Trend Level Trend Seasonal Alpha Bea Gamma Period Acual Level Trend Seasonaliy Forecas Fi Saring Trend Y ( )(1.49) ( /1.49) (1 0.04)( ) 0.7( / 176.1) (1 0.7)(1.49) 1.00( ) (1 1.00)(0.00) Level L ( Y / S s ) (1 )( L 1 b 1) Trend b ( L L 1 ) (1 )( b 1) Seasonaliy S ( Y / L ) (1 )( Ss ) Forecas F ( L mb ) S m ms Figure A.18 Calculaing Hol-Winer s Muliplicaive ROV BizSas User Manual 119 Copyrigh Dr. Johnahan Mun

120 APPENDIX 3: A PRIMER ON REGRESSION ANALYSIS This appendix deals wih using regression analysis for forecasing purposes. I is assumed ha he reader is sufficienly knowledgeable abou he fundamenals of regression analysis. Therefore, insead of focusing on he deailed heoreical mechanics of he regression equaion, we here look a he basics of applying regression analysis and work hrough he various relaionships ha a regression analysis can capure, as well as he common pifalls in regression, including he problems of ouliers, nonlineariies, heeroskedasiciy, auocorrelaion, and srucural breaks. The Basics of Regression Analysis The general bivariae linear regression equaion akes he form of Y 0 1X, where 0 is he inercep, 1 is he slope, and is he error erm. I is bivariae as here are only wo variables: a Y, or dependen variable, and an X, or independen variable, where X is also known as he regressor (someimes a bivariae regression is also known as a univariae regression because here is only a single independen variable, X). The dependen variable is named as such because i depends on he independen variable; for example, sales revenue depends on he amoun of markeing coss expended on a produc s adverising and promoion, making he dependen variable sales and he independen variable markeing coss. An example of a bivariae regression is seen as simply insering he bes-fiing line hrough a se of daa poins in a wo-dimensional plane, as seen on he lef panel in Figure A3.1. In oher cases, a mulivariae regression can be performed, where here are muliple or n number of independen X variables, where he general regression equaion akes he form of Y X X X... X n n. In his case, he bes-fiing line will be wihin an n + 1 dimensional plane. Y Y Y X Y X Figure A3.1 Bivariae Regression However, fiing a line hrough a se of daa poins in a scaer plo as shown in Figure A3.1 may resul in numerous possible lines. The bes-fiing line is defined as he single unique line ha minimizes he oal verical errors, ha is, he sum of he absolue disances beween he acual daa poins (Y i ) and he esimaed line (Yˆ ) as shown on he righ panel of Figure A3.1. To find he bes-fiing line ha minimizes he errors, a more sophisicaed approach is required, ha is, ROV BizSas User Manual 10 Copyrigh Dr. Johnahan Mun

121 ROV BizSas User Manual 11 Copyrigh Dr. Johnahan Mun regression analysis. Regression analysis finds he unique bes-fiing line by requiring ha he oal errors be minimized, or by calculaing n i Y i Y i Min 1 ) ˆ ( where only one unique line minimizes his sum of squared errors. The errors (verical disance beween he acual daa and he prediced line) are squared o avoid he negaive errors from canceling ou he posiive errors. Solving his minimizaion problem wih respec o he slope and inercep requires calculaing firs derivaives and seing hem equal o zero: 0 ) ˆ ( and 0 ) ˆ ( n i i i n i i i Y Y d d Y Y d d which yields he Leas Squares Regression Equaions seen in Figure A3.. Figure A3. Leas Squares Regression Equaions Example: Given he following sales amouns ($millions) and adverising sizes (measured as linear inches by summing up all he sides of an ad) for a local newspaper, answer he following quesions. Adverising size (inch) Sales ($ millions) (a) Which is he dependen variable and which is he independen variable? The independen variable is adverising size, whereas he dependen variable is sales. (b) Manually calculae he slope ( 1 ) and he inercep ( 0 ) erms. X Y XY X Y (X)=10 (Y)=48.3 (XY)=1534. (X )=7308 (Y )=34.11 X Y n X X n Y X XY X X Y Y X X n i i n i i n i i n i i n i i i n i i n i i i and ) ( ) ( ) )( (

122 10(48.3) and (c) Wha is he esimaed regression equaion? Y = X, or Sales = (Size) (d) Wha would he level of sales be if we purchase a 8-inch ad in he paper? Y = (8) = $6.73 million dollars in sales Noe ha we only predic or forecas and canno say for cerain. This is only an expeced value or on average. Regression Oupu Using he daa in he previous example, a regression analysis can be performed using he Muliple Regression module in ROV BizSas. This module can run bivariae regressions such as he previous example, or muliple regression analysis (where more han one independen X variable exiss). Figure A3.3 shows he op par of ROV BizSas regression analysis oupu. Noice ha he coefficiens on he inercep and X variable confirm he resuls obained in he manual calculaion. Figure A3.3 Regression Oupu from ROV BizSas (Top Porion) The Muliple Regression analysis oupu also provides oher valuable saisics as seen in Figure A3.4. Mos of hese addiional saisical oupus perain o goodness-of-fi measures, ha is, a measure of how accurae and saisically reliable he model is. Analysis of Variance (ANOVA) resuls are also provided. ROV BizSas User Manual 1 Copyrigh Dr. Johnahan Mun

123 Regression Analysis Repor Regression Saisics R-Squared (Coefficien of Deerminaion) Adjused R-Squared Muliple R (Muliple Correlaion Coefficien) Sandard Error of he Esimaes (SEy) nobservaions 7 The R-Squared or Coefficien of Deerminaion indicaes ha of he variaion in he dependen variable can be explained and accouned for by he independen variables in his regression analysis. However, in a muliple regression, he Adjused R-Squared akes ino accoun he exisence of addiional independen variables or regressors and adjuss his R-Squared value o a more accurae view of he regression's explanaory power. Hence, only of he variaion in he dependen variable can be explained by he regressors. The Muliple Correlaion Coefficien (Muliple R) measures he correlaion beween he acual dependen variable (Y) and he esimaed or fied (Y) based on he regression equaion. This is also he square roo of he Coefficien of Deerminaion (R-Squared). The Sandard Error of he Esimaes (SE y ) describes he dispersion of daa poins above and below he regression line or plane. This value is used as par of he calculaion o obain he confidence inerval of he esimaes laer. Regression Resuls Inercep Ad Size Coefficiens Sandard Error Saisic p-value Lower 5% Upper 95% Degrees of Freedom Hypohesis Tes Degrees of Freedom for Regression 1 Criical -Saisic (99% confidence wih df of 5) Degrees of Freedom for Residual 5 Criical -Saisic (95% confidence wih df of 5).5706 Toal Degrees of Freedom 6 Criical -Saisic (90% confidence wih df of 5).0150 The Coefficiens provide he esimaed regression inercep and slopes. For insance, he coefficiens are esimaes of he rue populaion values in he following regression equaion: Y = X 1 + X n X n. The Sandard Error measures how accurae he prediced Coefficiens are, and he - Saisics are he raios of each prediced Coefficien o is Sandard Error. The -Saisic is used in hypohesis esing, where we se he null hypohesis (Ho) such ha he real mean of he Coefficien = 0, and he alernae hypohesis (Ha) such ha he real mean of he Coefficien is no equal o 0. A -es is is performed and he calculaed -Saisic is compared o he criical values a he relevan Degrees of Freedom for Residual. The -es is very imporan as i calculaes if each of he coeffiens is saisisically significan in he presence of he oher regressors. This means ha he -es saisically verifies wheher a regressor or independen variable should remain in he regression or i should be dropped. The Coefficien is saisically significan if is calculaed -Saisic exceeds he Criical -Saisic a he relevan degrees of freedom (df). The hree main confidence levels used o es for significance are 90%, 95% and 99%. If a Coefficien's -Saisic exceeds he Criical level, i is considered saisically significan. Alernaively, he p-value calculaes each -Saisic's probabiliy of occurrence, which means ha he smaller he p-value, he more significan he Coefficien. The usual significan levels for he p-value are 0.01, 0.05, and 0.10, corresponding o he 99%, 95%, and 99% confidence levels. The Coefficiens wih heir p-values highlighed in blue indicae ha hey are saisically significan a he 95% confidence or 0.05 alpha level, while hose highlighed in red indicae ha hey are no saisically significan a any of he alpha levels. Analysis of Variance Sums of Squares Mean of Squares F- Saisic P-Value Hypohesis Tes Regression Criical F-saisic (99% confidence wih df of 4 and 3) Residual Criical F-saisic (95% confidence wih df of 4 and 3) Toal Criical F-saisic (90% confidence wih df of 4 and 3) The Analysis of Variance (ANOVA) able provides an F-es of he regression model's overall saisical significance. Insead of looking a individual regressors as in he -es, he F-es looks a all he esimaed Coefficiens' saisical properies. The F-saisic is calculaed as he raio of he Regression's Mean of Squares o he Residual's Mean of Squares. The numeraor measures how much of he regression is explained, while he denominaor measures how much is unexplained. Hence, he larger he F-saisic, he more significan he model. The corresponding p-value is calculaed o es he null hypohesis (Ho) where all he Coefficiens are simulaneously equal o zero, versus he alernae hypohesis (Ha) ha hey are all simulaneously differen from zero, indicaing a significan overall regression model. If he p-value is smaller han he 0.01, 0.05, or 0.10 alpha significance, hen he regression is significan. The same approach can be applied o he F-saisic by comparing he calculaed F-saisic wih he criical F values a various significance levels. Forecasing Period Acual (Y) Forecas (F) Error (E) (0.857) (0.899) (0.143) (0.014) Acual vs. Forecas Acual (Y) Forecas (F) Figure A3.4 Regression Oupu from Risk Simulaor ROV BizSas User Manual 13 Copyrigh Dr. Johnahan Mun

124 Goodness of Fi Goodness-of-fi saisics provide a glimpse ino he accuracy and reliabiliy of he esimaed regression model. They usually ake he form of a -saisic, F-saisic, R-squared saisic, adjused R-squared saisic, or Durbin-Wason saisic, and heir respecive probabiliies. (See he -saisic, F-saisic, and criical Durbin-Wason ables a he end of his manual for he corresponding criical values used laer in his chaper). The following paragraphs discuss some of he more common regression saisics and heir inerpreaion. The R-squared (R ), or coefficien of deerminaion, is an error measuremen ha looks a he percen variaion of he dependen variable ha can be explained by he variaion in he independen variable for a regression analysis. The coefficien of deerminaion can be calculaed by: R 1 n i1 n i1 ( Y Yˆ ) i ( Y Y ) i i 1 SSE TSS where he coefficien of deerminaion is one less he raio of he sums of squares of he errors (SSE) o he oal sums of squares (TSS). In oher words, he raio of SSE o TSS is he unexplained porion of he analysis, hus, one less he raio of SSE o TSS is he explained porion of he regression analysis. Figure A3.5 provides a graphical explanaion of he coefficien of deerminaion. The esimaed regression line is characerized by a series of prediced values (Yˆ ); he average value of he dependen variable s daa poins is denoed Y ; and he individual daa poins are characerized by Y i. Therefore, he oal sum of squares, ha is, he oal variaion in he daa or he oal variaion abou he average dependen value, is he oal of he difference beween he individual dependen Y values and is average (seen as he oal squared disance of i Y in Figure A3.5). The explained sum of squares, he porion ha is capured by he regression analysis, is he oal of he difference beween he regression s prediced value and he average dependen variable s daa se (seen as he oal squared disance of Yˆ Y in Figure A3.5). The difference beween he oal variaion (TSS) and he explained variaion (ESS) is he unexplained sums of squares, also known as he sums of squares of he errors (SSE). ROV BizSas User Manual 14 Copyrigh Dr. Johnahan Mun

125 Y Y SSE = Y- Y 1 ESS = - Y Explained Y Y i - Y TSS is he proporion of explained o oal 1 (SSE/TSS) R X Figure A3.5 Explaining he Coefficien of Deerminaion Anoher relaed saisic, he adjused coefficien of deerminaion, or he adjused R-squared ( R ), correcs for he number of independen variables (k) in a mulivariae regression hrough a degrees of freedom correcion o provide a more conservaive esimae: R 1 n i1 n i1 ( Y i ( Y i Yˆ ) i Y ) /( k ) /( k 1) SSE /( k ) 1 TSS /( k 1) The adjused R-squared should be used insead of he regular R-squared in mulivariae regressions because every ime an independen variable is added ino he regression analysis, he R-squared will increase, which indicaes ha he percen variaion explained has increased. This increase occurs even when nonsensical regressors are added. The adjused R-squared akes he added regressors ino accoun and penalizes he regression accordingly, providing a much beer esimae of a regression model s goodness of fi. Oher goodness-of-fi saisics include he -saisic and he F-saisic. The former is used o es if each of he esimaed slope and inercep(s) is saisically significan, ha is, if i is saisically significanly differen from zero (herefore making sure ha he inercep and slope esimaes are saisically valid). The laer applies he same conceps bu simulaneously for he enire regression equaion including he inercep and slope(s). Using he previous example, he following illusraes how he -saisic and F-saisic can be used in a regression analysis. (You can use ROV BizSas o generae he corresponding able of criical values by going o ROV BizSas Probabiliies Saisical Tables Suden T Table). I is assumed ha he reader is somewha familiar wih hypohesis esing and ess of significance in basic saisics. ROV BizSas User Manual 15 Copyrigh Dr. Johnahan Mun

126 Figure A3.6 ANOVA and Goodness-of-Fi Table Example: Given he informaion from he muliple regression oupu in Figure A3.6, inerpre he following: a) Perform a hypohesis es on he slope and he inercep o see if hey are each significan a a wo-ailed alpha ( of The null hypohesis, H 0, is such ha he slope 1 = 0 and he alernae hypohesis, H a, is such ha 1 0. The -saisic calculaed is , which exceeds he -criical (.9687 obained from he -saisic able a he end of his manual) for a wo-ailed alpha of 0.05 and oal degrees of freedom n k = 7 1 = 6. Therefore, he null hypohesis is rejeced and one can sae ha he slope is saisically significanly differen from 0, indicaing ha he regression s esimae of he slope is saisically significan. This hypohesis es can also be performed by looking a he - saisic s corresponding p-value (0.0054), which is less han he alpha of 0.05, which means he null hypohesis is rejeced. The hypohesis es is hen applied o he inercep, where he null hypohesis is such ha he inercep 0 = 0 and he alernae hypohesis is such ha 0 0. The - saisic calculaed is , which exceeds he criical -value of.9687 for n k (7 1 = 6) degrees of freedom. Therefore, he null hypohesis is rejeced indicaing ha he inercep is saisically significanly differen from 0, which means ha he regression s esimae of he slope if saisically significan. The calculaed p-value (0.0007) is also less han he alpha level, which means ha he null hypohesis is also rejeced. b) Perform a hypohesis es o see if boh he slope and inercep are significan as a whole, in oher words, if he esimaed model is saisically significan a an alpha ( of The simulaneous null hypohesis, H 0, is such ha 0 = 1 = 0 and he alernae hypohesis, H a, is The calculaed F-value is , which exceeds he criical F-value (5.99 obained from he able a he end of his manual) for k(1) degrees of freedom in he numeraor and n k (7 1 = 6) degrees of freedom for he denominaor. Therefore, he null hypohesis is rejeced indicaing ha boh he slope and inercep are simulaneously significanly differen from 0 and ROV BizSas User Manual 16 Copyrigh Dr. Johnahan Mun

127 ha he model as a whole is saisically significan. This resul is confirmed by he p-value of (significance of F), which is less han he alpha value, hereby rejecing he null hypohesis and confirming ha he regression as a whole is saisically significan. c) Using he regression oupu in Figure A3.7, inerpre he R value. How is i relaed o he correlaion coefficien? Figure A3.7 Addiional Regression Oupu from Risk Simulaor The calculaed R is 0.815, meaning ha 81.5% of he variaion in he dependen variable can be explained by he variaion in he independen variable. The R is simply he square of he correlaion coefficien; ha is, he correlaion coefficien beween he independen and dependen variable is Assumpions in Regression Analysis The following six assumpions are he requiremens for a regression analysis o work: 1. The relaionship beween he dependen and independen variables is linear.. The expeced value of he errors or residuals is zero. 3. The errors are independenly and normally disribued. 4. The variance of he errors is consan, or homoskedasic, and no varying over ime. 5. The errors are independen and uncorrelaed wih he explanaory variables. 6. The independen variables are uncorrelaed o each oher, meaning ha no mulicollineariy exiss. One very simple mehod o verify some of hese assumpions is o use a scaer plo. This approach is simple o use in a bivariae regression scenario. If he assumpion of he linear model is valid, he plo of he observed dependen variable values agains he independen variable values should sugges a linear band across he graph wih no obvious deparures from lineariy. Ouliers may appear as anomalous poins in he graph, ofen in he upper righ-hand or lower lefhand corner of he graph. However, a poin may be an oulier in eiher an independen or dependen variable wihou necessarily being far from he general rend of he daa. If he linear model is no correc, he shape of he general rend of he X-Y plo may sugges he appropriae funcion o fi (e.g., a polynomial, exponenial, or logisic funcion). Alernaively, he plo may sugges a reasonable ransformaion o apply. For example, if he X-Y plo arcs from lower lef o upper righ so ha daa poins eiher very low or very high in he independen variable lie below he sraigh line suggesed by he daa, while he middle daa poins of he independen variable lie on or above ha sraigh line, aking square roos or logarihms of he independen variable values may promoe lineariy. If he assumpion of equal variances, or homoskedasiciy, for he dependen variable is correc, he plo of he observed dependen variable values agains he independen variable should sugges a band across he graph wih roughly equal verical widh for all values of he ROV BizSas User Manual 17 Copyrigh Dr. Johnahan Mun

128 independen variable. Tha is, he shape of he graph should sugges a iled cigar and no a wedge or a megaphone. A fan paern like he profile of a megaphone, wih a noiceable flare eiher o he righ or o he lef in he scaer plo, suggess ha he variance in he values increases in he direcion where he fan paern widens (usually as he sample mean increases), and his, in urn, suggess ha a ransformaion of he dependen variable values may be needed. As an example, Figure A3.8 shows a scaer plo of wo variables: sales revenue (dependen variable) and markeing coss (independen variable). Clearly, here is a posiive relaionship beween he wo variables, as is eviden from he regression resuls in Figure A3.9, where he slope of he regression equaion is a posiive value (0.7447). The relaionship is also saisically significan a 0.05 alpha, and he coefficien of deerminaion is 0.43, indicaing a somewha weak bu saisically significan relaionship. Figure A3.8 Scaer Plo Showing a Posiive Relaionship ROV BizSas User Manual 18 Copyrigh Dr. Johnahan Mun

129 Figure A3.9 Bivariae Regression Resuls for Posiive Relaionship Compare ha o a muliple linear regression as shown in Figure A3.10, where anoher independen variable, pricing srucure of he produc, is added. The regression s adjused coefficien of deerminaion (adjused R-squared) is now 0.6, indicaing a much sronger regression model. The pricing variable shows a negaive relaionship o he sales revenue, a very much expeced resul, as according o he law of demand in economics, a higher price poin necessiaes a lower quaniy demanded and, hence, lower sales revenues. The -saisics and corresponding probabiliies (p-values) also indicae a saisically significan relaionship. ROV BizSas User Manual 19 Copyrigh Dr. Johnahan Mun

130 Figure A3.10 Muliple Linear Regression Resuls for Posiive and Negaive Relaionships In conras, Figure A3.11 shows a scaer plo of wo variables wih lile o no relaionship, which is confirmed by he regression resul in Figure A3.1, where he coefficien of deerminaion is 0.066, close o being negligible. In addiion, he calculaed -saisic and corresponding probabiliy (p-value) indicae ha he markeing expenses variable is saisically insignifican a he 0.05 alpha level. Therefore, he regression equaion is no significan (a fac ha is also confirmed by he low F-saisic). ROV BizSas User Manual 130 Copyrigh Dr. Johnahan Mun

131 Figure A3.11 Scaer Plo Showing No Relaionship Figure A3.1 Muliple Regression Resuls Showing No Relaionship ROV BizSas User Manual 131 Copyrigh Dr. Johnahan Mun

132 The Pifalls of Forecasing: Ouliers, Nonlineariy, Mulicollineariy, Heeroskedasiciy, Auocorrelaion, and Srucural Breaks Oher han i being good modeling pracice o creae scaer plos prior o performing regression analysis, he scaer plo can also someimes, on a fundamenal basis, provide significan amouns of informaion regarding he behavior of he daa series. Blaan violaions of he regression assumpions can be spoed easily and efforlessly, wihou he need for more deailed and fancy economeric specificaion ess. For insance, Figure A3.13 shows he exisence of ouliers. Figure A3.14 s regression resuls, which include he ouliers, indicae ha he coefficien of deerminaion is only 0.5 as compared o in Figure A3.15 when he ouliers are removed. Values may no be idenically disribued because of he presence of ouliers. Ouliers are anomalous values in he daa. They may have a srong influence over he fied slope and inercep, giving a poor fi o he bulk of he daa poins. Ouliers end o increase he esimae of residual variance, lowering he chance of rejecing he null hypohesis (H 0 ). They may be due o recording errors, which may be correcable, or hey may be due o he dependen-variable values no all being sampled from he same populaion. Apparen ouliers may also be due o he dependen-variable values being from he same, bu non-normal, populaion. Ouliers may show up clearly in an X-Y scaer plo of he daa, as poins ha do no lie near he general linear rend of he daa. A poin may be an unusual value in eiher an independen or dependen variable wihou necessarily being an oulier in he scaer plo. The mehod of leas squares involves minimizing he sum of he squared verical disances beween each daa poin and he fied line. Because of his, he fied line can be highly sensiive o ouliers. In oher words, leas squares regression is no resisan o ouliers, hus, neiher is he fied-slope esimae. A poin verically removed from he oher poins can cause he fied line o pass close o i, insead of following he general linear rend of he res of he daa, especially if he poin is relaively far horizonally from he cener of he daa (he poin represened by he mean of he independen variable and he mean of he dependen variable). Such poins are said o have high leverage: he cener acs as a fulcrum, and he fied line pivos oward high-leverage poins, perhaps fiing he main body of he daa poorly. A daa poin ha is exreme in dependen variables bu lies near he cener of he daa horizonally will no have much effec on he fied slope, bu by changing he esimae of he mean of he dependen variable, i may affec he fied esimae of he inercep. ROV BizSas User Manual 13 Copyrigh Dr. Johnahan Mun

133 Figure A3.13 Scaer Plo Showing Ouliers Figure A3.14 Regression Resuls wih Ouliers ROV BizSas User Manual 133 Copyrigh Dr. Johnahan Mun

134 Figure A3.15 Regression Resuls wih Ouliers Deleed However, grea care should be aken when deciding if he ouliers should be removed. Alhough in mos cases when ouliers are removed, he regression resuls look beer, a priori jusificaion mus firs exis. For insance, if one is regressing he performance of a paricular firm s sock reurns, ouliers caused by downurns in he sock marke should be included; hese are no ruly ouliers as hey are ineviabiliies in he business cycle. Forgoing hese ouliers and using he regression equaion o forecas one s reiremen fund based on he firm s socks will yield incorrec resuls a bes. In conras, suppose he ouliers are caused by a single nonrecurring business condiion (e.g., merger and acquisiion) and such business srucural changes are no forecas o recur. Under such condiions, hese ouliers should be removed and he daa cleansed prior o running a regression analysis. Figure A3.16 shows a scaer plo wih a nonlinear relaionship beween he dependen and independen variables. In a siuaion such as his one, a linear regression will no be opimal. A nonlinear ransformaion should firs be applied o he daa before running a regression. One simple approach is o ake he naural logarihm of he independen variable (oher approaches include aking he square roo or raising he independen variable o he second or hird power) and regress he sales revenue on his ransformed markeing-cos daa series. Figure A3.17 shows he regression resuls wih a coefficien of deerminaion a 0.938, as compared o in Figure A3.18 when a simple linear regression is applied o he original daa series wihou he nonlinear ransformaion. If he linear model is no he correc one for he daa, hen he slope and inercep esimaes and he fied values from he linear regression will be biased, and he fied slope and inercep esimaes will no be meaningful. Over a resriced range of independen or dependen variables, nonlinear models may be well approximaed by linear models (his is, in fac, he basis of linear inerpolaion), bu for accurae predicion a model appropriae o he daa should be seleced. An examinaion of he X-Y scaer plo may reveal wheher he linear model is appropriae. If here is a grea deal of variaion in he dependen variable, i may be difficul o decide wha he ROV BizSas User Manual 134 Copyrigh Dr. Johnahan Mun

135 appropriae model is; in his case, he linear model may do as well as any oher, and has he virue of simpliciy. Figure A3.16 Scaer Plo Showing a Nonlinear Relaionship Figure A3.17 Regression Resuls Using a Nonlinear Transformaion ROV BizSas User Manual 135 Copyrigh Dr. Johnahan Mun

136 Figure A3.18 Regression Resuls Using Linear Daa However, grea care should be aken here as boh he original linear daa series of markeing coss and he nonlinearly ransformed markeing coss should no be added in he regression analysis. Oherwise, mulicollineariy occurs. Tha is, markeing coss are highly correlaed o he naural logarihm of markeing coss, and if boh are used as independen variables in a mulivariae regression analysis, he assumpion of no mulicollineariy is violaed and he regression analysis breaks down. Figure A3.19 illusraes wha happens when mulicollineariy srikes. Noice ha he coefficien of deerminaion (0.938) is he same as he nonlinear ransformed regression (Figure A3.17). However, he adjused coefficien of deerminaion wen down from (Figure A3.17) o (Figure A3.19). In addiion, he previously saisically significan markeing-coss variable in Figure A3.18 now becomes insignifican (Figure A3.19) wih a probabiliy value increasing from close o zero o A basic sympom of mulicollineariy is low -saisics coupled wih a high R-squared (Figure A3.19). ROV BizSas User Manual 136 Copyrigh Dr. Johnahan Mun

137 Figure A3.19 Regression Resuls Using Boh Linear and Nonlinear Transformaions Anoher common violaion is heeroskedasiciy, ha is, he variance of he errors increases over ime. Figure A3.0 illusraes his case, where he widh of he verical daa flucuaions increases, or fans ou, over ime. In his example, he daa poins have been changed o exaggerae he effec. However, in mos ime-series analyses, checking for heeroskedasiciy is a much more difficul ask. Noice in Figure A3.1 ha he coefficien of deerminaion dropped significanly when heeroskedasiciy exiss. As is, he curren regression model is insufficien and incomplee. If he variance of he dependen variable is no consan, hen he error s variance will no be consan. The mos common form of such heeroskedasiciy in he dependen variable is ha he variance of he dependen variable may increase as he mean of he dependen variable increases for daa wih posiive independen and dependen variables. Unless he heeroskedasiciy of he dependen variable is pronounced, is effec will no be severe: he leas-squares esimaes will sill be unbiased, and he esimaes of he slope and inercep will eiher be normally disribued if he errors are normally disribued or a leas normally disribued asympoically (as he number of daa poins becomes large) if he errors are no normally disribued. The esimae for he variance of he slope and overall variance will be inaccurae, bu he inaccuracy is no likely o be subsanial if he independen variable values are symmeric abou heir mean. Heeroskedasiciy of he dependen variable is usually deeced informally by examining he X-Y scaer plo of he daa before performing he regression. If boh nonlineariy and unequal variances are presen, employing a ransformaion of he dependen variable may have he effec of simulaneously improving he lineariy and promoing equaliy of he variances. Oherwise, a weighed leas-squares linear regression may be he preferred mehod of dealing wih nonconsan variance of he dependen variable. ROV BizSas User Manual 137 Copyrigh Dr. Johnahan Mun

138 Figure A3.0 Scaer Plo Showing Heeroskedasiciy wih Nonconsan Variance Figure A3.1 Regression Resuls wih Heeroskedasiciy ROV BizSas User Manual 138 Copyrigh Dr. Johnahan Mun