THE UNIVERSITY OF TEXAS AT AUSTIN McCombs School of Business

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1 THE UNIVERSIT OF TEXAS AT AUSTIN McCombs School of Business STA 37.5 Tom Shively SIMPLE EXPONENTIAL SMOOTHING MODELS The statistical model for simple exponential smoothing is t = M t- + ε t ε t iid N(0, σ ) where M t- = α t- + ( - α)m t-.

2 Simulating data from a deterministic trend model The model is y t = α + β Time t + ε t ε t iid N(0, σ ) t = 0,, 9 where α = 0, β = 0. and σ = are the true data-generating parameter values

3 Simulating data from a deterministic trend model The model is y t = α + β Time t + ε t ε t iid N(0, σ ) t = 0,, 9 where α = 0, β = 0. and σ = are the true data-generating parameter values True_line Est_line The estimated regression line is yˆ = Time. 3

4 Simulating data from a simple exponential smoothing model with Alpha = (Random walk model) The model is t = M t- + ε t ε t iid N(0, σ ) where M t- = α t- + ( - α)m t-, α = and σ = are the true data-generating parameter values, and M 0 = 0 = Mean 4

5 Simulating data from a simple exponential smoothing model with Alpha = (Random walk model) The model is t = M t- + ε t ε t iid N(0, σ ) where M t- = α t- + ( - α)m t-, α = and σ = are the true data-generating parameter values, and M 0 = 0 = 0. 6 Simple exp. smoothing: Alpha=, Sigma= Mean 5

6 Excel spreadsheet used to generate data from a simple exponential smoothing model Row A B C D E Time t True_alpha 0 3 True_sigma 4 Epsilon.06 =NORMINV(RAND(),0,$B$4) t =NORMINV(RAND(),0,$B$4) Mean M t = t + ( - )M t M 0.06 = B$*E3 + (- B$)*D M = + ( - )M = B$*E4 + (- B$)*D3 M = + ( - )M t = M t- + t = D + C3 = M = D3 + C4 = M After the -values are generated in column C using the =NORMINV(RAND(),0,$B$4) formula, highlight the column, copy the values, and then use Paste > Values to past the values back into the same column. If you do not overwrite the -values in column C using Paste > Values, then they will change every time a change is made to the spreadsheet (which you do not want to happen). 6

7 Simulating data from a simple exponential smoothing model with Alpha = 0.5 The model is t = M t- + ε t ε t iid N(0, σ ) where M t- = α t- + ( - α)m t-, α = 0.5 and σ = are the true data-generating parameter values, and M 0 = 0 = Mean 7

8 Simulating data from a simple exponential smoothing model with Alpha = 0.5 The model is t = M t- + ε t ε t iid N(0, σ ) where M t- = α t- + ( - α)m t-, α = 0.5 and σ = are the true data-generating parameter values, and M 0 = 0 = 0. 6 Simple exp. smoothing: Alpha=0.5, Sigma= Mean 8

9 Comparing simple exponential smoothing models with Alpha = 0.5 and.0 6 Simple exp. smoothing: Alpha=0.5, Sigma= Mean 6 Simple exp. smoothing: Alpha=, Sigma= Mean 9

10 Simulating data from a simple exponential smoothing model with Alpha = 0 The model is t = M t- + ε t ε t iid N(0, σ ) where M t- = α t- + ( - α)m t-, α = 0 and σ = are the true data-generating parameter values, and M 0 = 0 = 0. 6 Simple exp. smoothing: Alpha=0, Sigma= Mean 0

11 Two examples each from simple exponential smoothing models with Alpha = 0, 0.5 and.0 _Alpha= _Alpha= _Alpha= _Alpha= _Alpha=0.5 _Alpha= _Alpha=0.5 _Alpha=0.5

12 Two examples each from simple exponential smoothing models with Alpha = 0, 0.5 and.0 _Alpha=0 _Alpha= _Alpha=0 _Alpha=0

13 Two realizations of Holt s exponential smoothing model for comparison purposes Holt's model: Alpha=0.8, Beta=0.7, Sigma= Mean Holt's model: Alpha=0.8, Beta=0.7, Sigma= Mean 3

14 Useful interpretation of simple exponential smoothing M t = α t + ( - α)m t- = α t + M t- - αm t- = M t- + α( t - M t- ) = M t- + α ε t (because t = M t- + ε t so ε t = t - M t- ) 4

15 Forecasting using a simple exponential smoothing One-period ahead forecast 3 = M 30 + ε 3 =.06 + ε 3 ε 3 ~ N(0, σ ) Mean 5

16 Excel spreadsheet to forecast 3 Row A B C D Time t t ExpSmoothAlpha=0.5 M t = 0.5 t + ( - 0.5)M t = B M 0 = = 0.5*B3 + (-0.5)*C M = ( - 0.5)M = 0.5*B4 + (-0.5)*C3 M = ( - 0.5)M ForecastAlpha=0.5 ˆ t = C 0.55 = C3 = M t Ŷ = M 0 Ŷ = M = 0.5*B3 + (-0.5)*C3 M 30 = ( - 0.5)M = C3 ˆ = M

17 Two-period ahead forecast The exponential smoothing model is t = M t- + t t iid N(0, ) where M t- = t- + ( - )M t-. Therefore, with t = 3, 3 = M M 3 = 3 + ( )M 30 For t = 3, = (M ) + ( )M 30 = M = M = M so 3 ~ N(M 30,( ) ). 7

18 Mean k-period ahead forecast In general, 30+k ~ N(M 30,[( k ) α + ] σ ) so the k-period ahead forecast of 30+k is M 30 =.06. 8

19 Important points regarding k-period ahead forecasts () There is uncertainty about what is going to be in the future so every future value (i.e. oneperiod ahead value, two-period ahead value, three-period ahead value, etc.) has a distribution to represent this uncertainty. () The long-run mean of the series standing at period t is the mean of each of these distributions and is therefore the forecast of each future value. (3) The uncertainty regarding values increases the further out in time you go. This is reflected in the fact that the variance of future values is increasing as k increases (i.e. the one-period ahead variance is (( )α +) σ = σ, the two-period ahead variance is (( ) α +) σ = (α +) σ, the three-period ahead variance is ((3 ) α +) σ = (α +) σ, etc.). 9

20 Updating forecasts given new data point =.76 3 = 3.6 M 3 = α 3 + ( α)m 30 =(0.5)(3.6)+(0.5)(.06) =.6 Forecasts for 3,, 36 given information through time period t = 3 = M 3 =.6 M 9 =.36 M 30 = α 30 + ( α)m 9 =(0.5)(.76)+(0.5)(.36) =.06 Forecasts for 3,, 36 given information through time period t = 30 = M 30 = Mean Please add these pages immediately following page 9 in STA37_SimpleExponentialSmoothing_PartsandCombined_UpdatedVersion.pdf

21 Excel spreadsheet for updated forecasts of 3 and 33 Row A B C D Time t t ExpSmoothAlpha=0.5 M t = 0.5 t + ( - 0.5)M t = B M 0 = = 0.5*B3 + (-0.5)*C M = ( - 0.5)M = 0.5*B4 + (-0.5)*C3 M = ( - 0.5)M ForecastAlpha=0.5 ˆ t = C 0.55 = C3 = M t Ŷ = M 0 Ŷ = M = 0.5*B33 + (-0.5)*C3 M 3 = ( - 0.5)M = C33 ˆ = M = C33 ˆ = M 3 33 Please add these pages immediately following page 9 in STA37_SimpleExponentialSmoothing_PartsandCombined_UpdatedVersion.pdf

22 Representing the current y t observation as a weighted average of past y observations The statistical model for simple exponential smoothing is t = M t- + t t iid N(0, ) where M t- = t- + ( - )M t-. Using some algebra, we can rewrite M t- as follows: M t- = t- + ( - )M t- = t- + ( - )[ t- + ( - )M t-3 ] (because M t- = t- + ( - )M t-3 ) = t- + ( - ) t- + ( - ) M t-3 = t- + ( - ) t- + ( - ) [ t-3 + ( - )M t-4 ] (because M t-3 = t-3 + ( - )M t-4 ) = t- + ( - ) t- + ( - ) t-3 + ( - ) 3 M t-4 = t- + ( - ) t- + ( - ) t ( - ) t- + ( - ) t- M 0 It can be shown that the weights, ( - ), ( - ),, ( - ) t- and ( - ) t- sum to one so it makes sense to think of M t- as a weighted average of t-, t-,, and M 0. Observation Weights Formula = 0 = 0. = 0.5 = 0.8 = t t- ( ) t-3 ( ) t-4 ( ) t-5 ( ) t-6 ( )

23 Estimating Alpha in a simple exponential smoothing model The estimate of α will be the value that gives the best in-sample forecasts in the sense that the RMSE for the one-step ahead forecast errors is minimized. where This value is obtained using Solver. The true data-generating model is t = M t- + ε t ε t iid N(0, σ ) M t- = α t- + ( - α)m t-, α = 0.5 and σ = are the true parameter values, and M 0 = 0 = 0.

24 Excel Spreadsheet Row A B C D E F G Time t 0 t ExpSmoothAlpha=0. M t = 0. t + ( - 0.)M t = B ForecastAlpha=0. ˆ t + = M t ForecastErrorAlpha=0. e t = t - Ŷ t ExpSmoothAlpha=0.5 M t = 0.5 t + ( - 0.5)M t = B ForecastAlpha=0.5 M 0 = 0 M 0 = = 0.*B3 + (-0.)*C = C = B3 - D3 = 0.5*B3 + (-0.5)*F = F M = 0. + ( - 0.)M 0 Ŷ = M 0 e = - M = ( - 0.5)M 0 Ŷ = M = 0.*B4 + (-0.)*C3 = C3 = B4 - D4 = 0.5*B4 + (-0.5)*F3 = F3 M = 0. + ( - 0.)M Ŷ = M e = - Ŷ M = ( - 0.5)M Ŷ = M = B3 D = 0.*B3 + (-0.)*C3 M 30 = ( - 0.)M = C3 ˆ = M 30 3 e 30 = 30 - ˆ = 0.5*B3 + (-0.5)*F3 M 30 = ( - 0.5)M 9 ˆ t = F3 3 = M t ˆ = M 30

25 Choice of α Choose the α that makes the one-step ahead forecast errors as small as possible Zero ForecastError_Alpha=0. Zero ForecastError_OptimalAlpha Use root mean square error (RMSE) to summarize the size of the one-step ahead forecast errors. RMSE for various values of α α RMSE

26 RMSE. R M S E Alpha 4

27 Using Solver to estimate Alpha Excel commands to use Solver Click on Data and click on Solver. Type $H$ in the box next to Set Target Cell. Click on Min. Type $C$ into the box under By Changing Cells. To add the constraints, click on Add and type $C$ into the box under Cell Reference, click on <= and type.00 into the box under Constraint, and then click on Add. Repeat the process except click on >=, and type 0.00 into the box under Constraint. Then click on OK, click on Solve, and click on OK. Row A B C D E F G H Time t t Est_alpha M M t Forecast ˆ t M t e t Error ˆ t t Error_Sq =B =C$*B3+( C$)*D =D =B3 D3 =F3^ =C$*B4+( C$)*D3 =D3 =B4 D4 =F4^ e t RMSE n e t n t.005 =SQRT(AVERAGE(G3:G3)) 5

28 4 Simple exp. smoothing: True_alpha=0.5, Sigma= Estimated Mean True Mean 6

29 Forecasting using the estimated value of Alpha in a simple exponential smoothing model True_Mean Est_Mean 7

30 Checking whether a simple exponential smoothing model using the estimated value of Alpha is the appropriate model Zero ForecastError_OptimalAlpha 0 Frequency More Bin 8

31 Application to monthly Dow Jones Industrial Average The simple exponential smoothing model is t = M t- + ε t ε t iid N(0, σ ) where M t- = α t- + ( - α)m t Monthly Dow Jones Industrial Average Dow Jones 9

32 Excel spreadsheet before Solver Row A B C D E F G H I ear Time t DJIA t Jun , Jul-99 0, Aug-99 0,89.8 ExpSmooth M t = α t + ( - α)m t- 0, = C = E$*C3 + (- E$)*D = E$*C4 + (- E$)*D3 Parameter α 0.0 Forecast ˆ t + = M t 0, = D 0, = D3 Error e t = t - Ŷ t = C3 - F = C4 - F4 ErrorSquare e t 99,634.9 =G3^ 6,44.99 =G4^ RMSE n e t n t = =SQRT(AVERAGE(H3:H86)) 5 Sep , , , Apr-06 8, , , May-06 83, , , Jun-06 84, , ,

33 Excel spreadsheet before Solver Row A B C D E F G H I ear Time t DJIA t Jun , Jul-99 0, Aug-99 0,89.8 ExpSmooth M t = α t + ( - α)m t- 0, = C = E$*C3 + (- E$)*D = E$*C4 + (- E$)*D3 Parameter α 0.0 Forecast ˆ t + = M t 0, = D 0, = D3 Error e t = t - Ŷ t = C3 - F = C4 - F4 ErrorSquare e t 99,634.9 =G3^ 6,44.99 =G4^ RMSE n e t n t = =SQRT(AVERAGE(H3:H86)) 5 Sep , , , Apr-06 8, , , May-06 83, , , Jun-06 84, , ,998.0 Excel spreadsheet after Solver Row A B C D E F G H I ear Time t DJIA t ExpSmooth M t = α t + ( - α)m t- Parameter α Forecast ˆ t+ = M t Error e t = t - Ŷ t ErrorSquare e t RMSE n e t n t = Jun , , Jul-99 0, , , Aug-99 0, , , Sep , , , May-06 83, , , Jun-06 84, , ,08.4 3

34 ,000 In-sample forecasts using optimal value of alpha,000 0,000 9,000 8,000 7, Dow Jones Forecast, , Error 5 0 Frequency Bin 3

35 Simple exponential smoothing for the Dow Jones data using StatTools StatTools instructions for simple exponential smoothing To use StatTools in Excel, you must first open it outside Excel by clicking on its icon. Then inside Excel, click on StatTools in the menu at the top of the Excel screen. To run an analysis using StatTools, you must first create a StatTools data set containing the variable(s) you want to analyze. To do this, click on Data Set Manager in the top left hand corner of the StatTools screen. In the Data Set Manager dialog box, click on New, click on the Select the range icon immediately to the right of the Excel Range box, highlight the column in the Excel worksheet containing Dow Jones, click OK, and then click OK again. To forecast using a simple exponential smoothing model, click Time Series and Forecasting at the top of the StatTools screen, and then click on Forecast. In the StatTools-Forecast dialog box, click the box next to Dow Jones, type the number of periods you want to forecast in the Number of Forecasts box, click the Optimize Parameters box, and click Exponential Smoothing (Simple). Then click Time Scale, click Monthly, type 999 for the Starting ear and 6 for the Starting Month (since the starting month is June), and then click OK. The output (including the estimate of and the forecasts) will be put into a new worksheet labelled Forecast. The output is: Simple Exponential Smoothing Forecasts for Dow Jones Forecasting Constant (Optimized) Level (Alpha) 0.84 Simple Exponential Mean Abs Err Root Mean Sq Err Mean Abs Per% Err 3.4% Forecasting Data Dow Jones Level Forecast Error Jun Jul Aug May Jun Jul

36 Simple exponential smoothing for the Dow Jones data using R R script ######################################################################################################### # # ou must set the working directory properly # If the R package ggplot is not installed then it must be installed by typing the command (at the "> prompt"): # install.packages("ggplot") # If the R package forecast is not installed then it must be installed by typing the command (at the "> prompt"): # install.packages("forecast") # ou must type (at the "> prompt"): library (ggplot) # ou must type (at the "> prompt"): library (forecast) # To run, type (at the "> prompt"): source("simple_exp_smooth_djia_monthly_june999-june006_usingses.r") # (where Simple_Exp_Smooth_DJIA_Monthly_June999-June006_UsingSES.R is the name of file containing the # R script given below) # ######################################################################################################### # # Open file for output # sink ("C:/Users/shivelyt/Box Sync/Courses/37_Spring05/R_Scripts/Simple_Exp_Smoothing/SES_Output.txt", append=false, split=true) # # Read data # file <- "DJIA_Monthly_June999-June006.dat" DJIA_table <- read.table(file, header = FALSE, sep = "") colnames(djia_table) <- c("time", "DJIA") print(head(djia_table)) # # Plot DJIA vs. ear # g = ggplot() g <- g + geom_line(data=djia_table, aes(time,djia), color="black", lty=) 34

37 # # Save data as a time series object # DJIA_time_series <- ts(djia_table[]) colnames(djia_time_series) <- "DJIA" # # Use SES to obtain smoothed series using alpha=0. # fit <- ses(djia_time_series, alpha=0., initial="simple", h=3) print (fit$model) print (fit$mean) print (fit$fitted) plot.data <- data.frame(djia_table[:85,], fit$fitted) colnames(plot.data) <- c("time", "Smooth_Pt") g <- g + geom_line(data=plot.data, aes(time, Smooth_Pt), color="red", lty=) rm(plot.data) # # Use SES to obtain smoothed series using alpha=0.6 # fit <- ses(djia_time_series, alpha=0.6, initial="simple", h=3) print (fit$model) print (fit$mean) print (fit$fitted) plot.data <- data.frame(djia_table[:85,], fit$fitted) colnames(plot.data) <- c("time", "Smooth_Pt6") g <- g + geom_line(data=plot.data, aes(time, Smooth_Pt6), color="blue", lty=) g <- g + xlab("time") + ylab("dow Jones") + ggtitle("time series plot of Dow Jones (Solid black line) and in-sample forecasts for alpha=0. and 0.6") rm(plot.data) # # Use SES to obtain smoothed series using optimal value of alpha # fit3 <- ses(djia_time_series, initial="simple", h=3) print (fit3$model) print (fit3$mean) print (fit3$fitted) plot.data <- data.frame(djia_table[:85,], fit3$fitted) colnames(plot.data) <- c("time", "Smooth_Optimal") h = ggplot() h <- h + geom_line(data=djia_table, aes(time,djia), color="black", lty=) h <- h + geom_line(data=plot.data, aes(time, Smooth_Optimal), color="blue", lty=) h <- h + xlab("time") + ylab("dow Jones") + ggtitle("time series plot of Dow Jones (Solid black) and in-sample forecasts (dashed blue)") rm(plot.data) 35

38 # # Save plot to a pdf file and then print # ggsave('plot.pdf', g) shell.exec(file.path(getwd(), "plot.pdf")) ggsave('plot.pdf', h) shell.exec(file.path(getwd(), "plot.pdf")) dev.off() # # Close file for output # closeallconnections() 36

39 Time DJIA R output Call: ses(x = DJIA_time_series, h = 3, initial = "simple", alpha = 0.) Smoothing parameters: alpha = 0. Initial states: l = sigma: Time Series: Start = 86 End = 88 Frequency = [] Time Series: Start = End = 85 Frequency = [] [3] [5] [37] [49] [6] [73] [85]

40 Call: ses(x = DJIA_time_series, h = 3, initial = "simple", alpha = 0.6) Smoothing parameters: alpha = 0.6 Initial states: l = sigma: Time Series: Start = 86 End = 88 Frequency = [] Time Series: Start = End = 85 Frequency = [] [3] [5] [37] [49] [6] [73] [85]

41 Call: ses(x = DJIA_time_series, h = 3, initial = "simple") Smoothing parameters: alpha = Initial states: l = sigma: Time Series: Start = 86 End = 88 Frequency = [] Time Series: Start = End = 85 Frequency = [] [3] [5] [37] [49] [6] [73] [85]

42 Time series plot of Dow Jones (Solid black line) and in sample forecasts for alpha=0. and Dow Jones Time 40

43 Time series plot of Dow Jones (Solid black line) and in sample forecasts (dashed blue line) Dow Jones Time 4

University of Texas at Dallas School of Management. Investment Management Spring Estimation of Systematic and Factor Risks (Due April 1)

University of Texas at Dallas School of Management Finance 6310 Professor Day Investment Management Spring 2008 Estimation of Systematic and Factor Risks (Due April 1) This assignment requires you to perform