Forecasting Stocks with Multivariate Time Series Models.

International Journal of Mathematics and Statistics Invention (IJMSI) E-ISS: 3 4767 P-ISS: 3-4759 wwwijmsiorg Volume 4 Issue 9 ovember 06 PP-3-44 Forecasting Stocks with Multivariate Time Series Models I A Iwok and B C Okoro Deartment of Mathematics/Statistics, University of Port-Harcourt, PMB533, Port-Harcourt, Rivers State; igeria ABSTRACT: This work seeks to forecast stocks of the igerian banking sector using robability multivariate time series models The study involved the stocks from six different banks that were found to be analytically interrelated Stationarity of the six series were obtained by differencing Model selection criteria were emloyed and the best fitted model was selected to be a vector autoregressive model of order The model was subjected to diagnostic checks and was found to be adequate Consequently, forecasts of stocks were generated for the next two years KEYWORDS: Stationarity, VAR models, Stable rocess, White noise rocess and Cross correlation I ITRODUCTIO Most times, the issue of stock investment, stock market and stock trading is treated with less interest Both the learned and the illiterate world quickly switch to other channels when stock and its related discuss are going on television Most eole find it difficult to believe that stock is another viable area to invest in esecially at this eriod of economic doldrums The Stock and shares reresent ownershi interest in a business while stock market is a branch of the caital market The caital market is the backbone of any economy and is made u of the money market and the caital market The money market (commercial banks) is concerned with the trading of short-term instruments like bank deosits, treasury bills and treasury certificates while the caital market involves the long-term instruments The two markets comlement one another and thereby leading to a robust and balanced develoment of the financial system The caital market comrises markets and institutions that facilitate the issuance and secondary trading of long-term financial instruments Meanwhile, the money market functions basically to rovide short-term funds The caital market rovides funds to industries and governments to meet their long-term caital requirements such as financing of fixed investments like buildings, lants, machinery, bridges etc The caital market lays key roles in stimulating industries thereby enhancing robust economic growth and develoment One could imagine what the economy will be in the absence of a caital market Definitely, industrial growth would be deterred as the money market is not designed to rovide such funds The resence of a secondary market such as the stock exchange is a vital asect of the caital market Therefore, the stock market is at the heart of caital market develoment in any country Stock exchange A stock exchange is an arrangement (or lace) where large and small investors alike buy and sell (through stock brokers) securities (shares and bonds) of comanies and government agencies resectively This arrangement could be through comuters, telehone, fax, trading floor etc The stock exchange rovides the essential facilities for comanies and government to raise money for business exansion and develoment rojects through investors who own shares in cororations for the ultimate benefit of the economy Stocks and shares reresent ownershi interest in a business Peole invest in stock in other to share in the fortunes of comanies Some eole buy stocks with the hoe of seeing their caital grow; but keeing ace with inflation has always been the investor s rimary objective Before investing in stock as well as any business, certain stes are necessary The stock market is a financial game of winners and losers as is obtainable in any business This necessitates why eole understudy businesses to know its nitty-gritty before investing in them If this is not well taken care of, the investor stands the risk of losing his investment while the oosite is the desired good news The challenge now is how do investors identify viable stocks and guide it towards making rofit? Is there a way to redict the stock market? An investor may see the rice of a certain stock advancing and choose to invest in it without taking into cognizance the rice movement and the likely movement in future Being oblivious of the future can sometimes be dangerous and disastrous in business as it may cause a huge loss of investment It is therefore ertinent that the rediction of the likelihood of the future stock trend is desirable Time series stand tall in addressing the challenge For one to better understand stock rices on the stock exchange, reference on the ast data is needed Hence, the work of most researchers have been recorded and documented for the urose of reference and historical reasons In other that data or observations be made relevant in determination and rediction of the future, it becomes necessary that regard is made to time wwwijmsiorg 3 Page

Forecasting Stocks with Multivariate Time Series Models The body of techniques available for analyzing series of deendent observations is called time series (Box et al, 008) Time series analysis investigates sequence of observations on a variable that have been measured at regularly recurring time oints The analysis of time series is based on the assumtion that successive values in the data file reresent consecutive measurements taken at equal time intervals There are two main goals of time series analysis which consists of identifying the nature of the henomenon reresented by the sequence of observations and forecasting (redicting future values of the time series variables) In data analysis, variables of interest can be univariate or multivariate In the case of univariate data analysis, the resonse variable is influenced by only one factor while that of multivariate case is influenced by multile factors In some situations, however, analyzing time series using multivariate methods is reasonable because univariate analysis could be limiting Multivariate time series analysis is emloyed when one wants to model and roffer exlanation on the interaction and counter interaction among a grou of time series variables This work intends to study these interrelationshis among variables and to build a multivariate time series model that can redict the future of stock values for six different banks in igeria II LITERATURE REVIEW Based on recent economic uncertainty, several discussions on the volatility of the stock market cannot be avoided In describing the current market, Robert Engle, a finance rofessor at ew York University stated, We have no idea where things are going (Merle, 008) Many a times, investors in stock have lost their investments as a result of oor market analysis Researchers over the years have researched on the ossible methods that could enable investors in stock to manage their investments gainfully Abdulsalam et al (0) used regression and a data mining technique to develoed tools for exloiting essentially time series data in financial institutions They built a rediction system that uses data mining technique in roducing eriodic forecasts of stock market rices Their technique was a comlement to roven numeric forecasting method using regression analysis The financial information obtained from the daily activity summary (equities) was taken as inut Regression analysis was adoted as a data mining technique to describe the trends of stock market rices of the igerian stock exchange Finally, redictions were made on the future stock market rices of three banks in igerian banking sector Cambell (000) used multivariate system in modeling financial variables like stock returns This multivariate system allows stocks in one variable to roagate to the others Cambell (000) used vector auto regression as a mechanism to link vector stationary time series together It was discovered that rice formation was imacted by certain frictions like trading costs, short sale restrictions and circuit breaker Amihud (00) found that exected market illiquidity ositively affected ex ante stock excess return over time Additionally, stock returns were found to be negatively related over time to contemoraneous unexected illiquidity and the illiquidity in turn affected small firm stocks strongly Quintana and West (009) introduced new models for multile time series These new models were illustrated in an alication to international currency exchange rate data The models rovided a tractable and sequential rocedure for estimation of unknown covariance structure between series They carried out a rincial comonents analysis which enabled an easy model assessment Chordia et al (003) exlored liquidity movements in stock and treasury bond markets over a eriod of more than 800 trading days They found that a shock to quoted sreads in one market affected the sreads in other markets and the return volatility was an imortant drive of liquidity Innovations to stock and bond market liquidity and volatility was roved to be significantly correlated The correlation results confirmed that the common factors drive liquidity and volatility in the markets lay an imortant role in forecasting stock and bond liquidity Lendasse et al (000) develoed a method to redict non-linear tools This method was able to find non-linear relationshis in artificial and real-world financial series The method used several information as inut to comress the model to a state vector of limited size This facilitated the subsequent regression and the generalization ability of the forecasting algorithm of fitting a non-linear regression on the reduced vector An imroved result was obtained when the method was comared to linear and non-linear models that such comression was not used Larsen (00) develoed a stock rice rediction model He used a novel two layer reasoning aroach that emloyed domain knowledge from technical analysis in the first layer of reasoning This rocess guided a second layer of reasoning based on machine learning The model was sulemented by a money management strategy that used the historical success of redictions made by the model to determine the amount of caital to invest on future redictions When the method was tested based on a number of ortfolio simulations with trade signals generated, the model successfully outerformed the Oslo Benchmark Index (OSEBX) Lee et al (000) used the Auto-Regressive Integrated Moving Average (ARIMA) model to examine the imact of German hyerinflation in the 90s stock returns The result of the study showed that the wwwijmsiorg 33 Page

Forecasting Stocks with Multivariate Time Series Models hyerinflation in Germany in early 90s co-integrates with stock returns The fundamental relationshi between stock returns and the exected inflation was highly ositive They concluded that common stocks aear to be a hedge against inflation during this eriod Ibrahim and Agbaje (03) exloited the analytical technique of Auto regressive Distributed Lag (ARDL) bound test and examined the long-run relationshi and dynamic interactions between Stock Returns and Inflations in igeria using monthly Data of all Shares Price Index The results showed that there exist a long run relationshi between stock returns and inflation Ajayi and Mougoue (996) investigated the short-and long- run relationshi between stock rices and exchange rates in eight advanced economies Of interest, were the results on short-run effects in the US and UK markets They found that an increase in stock rices caused the currency to dereciate for both the US and the UK Ajayi and Mougoue (996) exlained that a rising stock market is an indicator of an exanding economy, which goes together with higher inflation exectations They added that foreign investors erceive higher inflation negatively and because of this, the demand for the currency dros and dereciates Owing to the above reviews, multivariate time series models are very crucial in modeling and identifying the joint structure on which decisions could deend The time varying techniques ossess the roerties of roviding an insight into the multivariate structure of several interrelated series The intent of this work is to identify the correlation structure of stock series of six Banks in igeria and ossibly build a multivariate time series model for the rediction of future stocks III METHODOLOGY 3 Stationarity A time series is said to be stationary if its statistical roerties remain constant through time These roerties are mean, variance etc A non stationary series Z t can be made stationary by differencing The differenced series is given as z t = Z t Z t = Z t () 3 Backward shift Oerator The Backward shift Oerator B is defined by B m Z t = Z t m () 33 The Backward Difference Oerator The backward difference oerator,, is define by = B (3) 34 Cross Correlation Given two time series variables X t and Y t, the cross correlation for lag k is given as r xy = C xy (4) S x S y where, c xy = n k x n t= t x y t+k y k = 0,, ; x and y are the samle means of x t and y t, s x and s y are the samle standard deviations resectively 35 White oise Process A white noise rocess a t = (a t,, a kt ) is a continuous random vector satisfying E a t = 0, E(a t a t ) = Σ a and E a t a s = 0 for s t (5) Σ a = covariance matrix which is assume to be non singular if not otherwise stated 36 Vector Autoregressive (VAR) Model The vector autoregressive (VAR) model gives an aroach in modeling dynamics among a set of time deendent variables It is an indeendent reduced form of dynamic model which entails constructing equation that makes each endogenous variable a function of their own ast values as well as ast values of all other endogenous variables The basic -lag Vector autoregressive VAR model has the form z t = c + Φ z t + Φ z t + + Φ z t + a t ; t = 0, ±, ±, (6) where z t = (z t,, z kt ) is a k vector of time series variable, Φ i are fixed (k k) coefficient matrices, c = (c,, c k ) is a fixed k vector of intercet terms a t = (a t,, a kt ) is an k white noise rocess The model can be exressed exlicitly in matrix form: wwwijmsiorg 34 Page

Forecasting Stocks with Multivariate Time Series Models z t z t = z kt φ φ φ k + + φ φ φ k φ φ φ k φ φ φ k φ k φ k φ kk z t z t z kt φ k φ k φ kk + z t z t z kt φ φ φ k + φ φ φ k a t a t a kt φ k φ k φ kk z t z t z kt (7) 37 Stable VAR () Processes The rocess (6) is stable if the roots of the auxiliary equation lie outside the unit circle That is if det (I n Φ z Φ z ) 0 for z (8) A stable VAR() rocess z t, t = 0, ±, ±,, is stationary 38 Autocovariances of a Stable VAR() Process Subtracting the mean μ from the VAR gives z t μ = φ z t μ + + φ z t μ + a t, (9) Post multilying both sides by z t l μ and taking exectation, we have for l = 0 using Γ z i = Γ z ( i) Γ z 0 = φ z t μ + + φ z t μ + Σ a = φ Γ z () + + φ Γ z + Σ a (0) If > 0 Γ z (l) = φ Γ z l + + φ Γ z l () From these equations, the autocovariance functions Γ z l for l can be obtained if φ,, φ and Γ z,,γl0 are known 39 Autocorrelation of a Stable VAR() Process The autocorrelations of a stable VAR () rocess are obtained from the matrix R z l = D Γ z l D () Where, D is a diagonal matrix with the standard deviation of the comonent of z t on the main diagonal Thus, 0 D = γ 0 0 and the correlation between z i,t and z j,t l is ρ ij l = γ ij (l) γ ii (0) γ jj (0) which is just the ij th element of R z l γ kk 0 30 VAR Order Selection The three selection criteria that will be used to determine the order of the VAR rocess are: (i) Akaike Information Criterion (AIC) given as AIC = In Σ ε + number of estimated arameter = In Σ a + k (ii) Hannan-Quin Criterion (HQC) given as HQC = In Σ a + Ink freely estimated arameters = In Σ a + In n (iii) Bayesian Information Criterion (BIC) given as BIC = In Σ a + In freely estimated arameters (3) (4) wwwijmsiorg 35 Page

0 9 8 37 46 55 64 73 8 9 00 09 8 7 36 45 54 63 7 8 90 99 Forecasting Stocks with Multivariate Time Series Models = In Σ a + In k where is the VAR order, Σ a is the estimate of white noise covariance matrix Σ ε k is the number of time series comonents of the vector time series is the samle size In all the criteria above, each estimate is chosen so as to minimize the value of the criterion IV DIAGOSTIC CHECKS There is always a need to diagnose a model after being fitted to a data This is rimarily done to examine whether the model is adequate or not A general way of achieving this is to examine the behaviour of the residuals matrices Under the assumtion of model adequacy, the residuals are exected to follow a white noise rocess According to Lutkeohl (005); if ρ uv (i) is the true correlation coefficients corresonding to the r uv (i), then we have the following hyothesis test at 5% level to check whether or not a given multivariate series follows a white noise rocess or not The hyothesis states: H 0 : ρ uv i = 0 versus H : ρ uv i 0 Decision: Reject H 0 if r uv,i > To test for H 0, the autocorrelations of the residuals are comuted and their absolute values are comared with If these absolute values are all less than ; it is concluded that the multivariate model is adequately fitted V FORECASTIG Suose the fitted model in (6) is found to be adequate, then it can used to generate forecasts The forecasts are generated by obtaining the estimates: z t = c + Φ z t + Φ z t + + Φ z t ; t = 0, ±, ±, Given the forecast origin t, the forecasts so obtained are the minimum mean square error forecasts (Lutkeohl, 005) VI DATA AALYSIS AD RESULTS The multivariate data used for this work are the monthly recorded stocks from first Bank Z t, Access Bank Z t, UBA Z 3t, Union Bank Z 4t, GTB Z 5t and Wema Bank Z 6t Thus, the multivariate time series can be reresented as the random vector Z t = Z t, Z t, Z 3t, Z 4t, Z 5t, Z 6t The data which sans between 999 to 05 dislayed in table of aendix E The R-software was used in the analysis 6 Time Series Plot The time series lot of the six time series data are dislayed as multile grahs in figure below 70 60 50 40 30 0 First bank Access bank UBA 0 0 Figure : Time series lot of the six raw series It is quite clear from the above lots that none of the series is stationary Differencing is therefore required to achieve stationary wwwijmsiorg 36 Page

0 9 8 37 46 55 64 73 8 9 00 09 8 7 36 45 54 63 7 8 90 99 Forecasting Stocks with Multivariate Time Series Models 6 Differenced Series Plots Using the first difference in equation (), the lot of the differenced series is shown in figure below 40 30 0 0 0-0 -0 DFirst bank * DAccess bank * DUBA * DUnion bank * DWema bank * DWema bank * -30-40 Figure : Time series lot of the six differenced series The lots in figure clearly show that the six series are stationary and the multivariate technique can now be alied 63 The Raw data Correlation Matrix The correlation matrix of the six random variables Z t(corr ) below shows that the variables are highly ositively correlated Hence, the multivariate technique can address the interrelationshi amongst the variables 000 0655 0655 000 0636 0800 080 0740 0055 0803 0750 0605 Z t(corr ) = 0636 0800 000 0704 078 0649 080 0740 0704 000 069 0789 0556 0750 0803 0605 078 0649 069 0789 000 0806 0806 000 64 The Cross Correlations The cross correlation matrices at different lags (lags 5) are dislayed in aendix A The high values further confirm the interrelationshi among the variables and the aroriateness of fitting multivariate model to the series 65 Model Selection The AIC, BIC and HQC at the different lags are dislayed in table of aendix B The three selection criteria attains minimum (the bolded values) at lag Hence the selected model is VAR() 66 Model Presentation Using equation (7) in the methodology, the VAR() model with significant arameters is resented in matrix form as shown: z t z t z 3t z 4t z 5t z 6t = 0357 0000 0000 0000 0348 0000 + 078 0000 0000 0000 0000 0000 0000 088 0000 0000 09 0000 0000 0000 0940 0000 007 0000 0000 0000 0000 0998 006 0000 0000 0000 004 0000 084 0000 0058 030 069 0000 0000 0975 z t z t z 3t z 4t z 5t z 6t (5) This can be exressed exlicitly as follows: First Bank : z t = 0357 + 078z t + 0058z 6t + a t (6) Access Bank : z t = 088z t + 030z 6t + a t (7) UBA : z 3t = 0940z 3t 004z 5t + 069z 6t + a 3t (8) Union Bank : z 4t = 0998z 4t + a 4t (9) Wema Bank : z 5t = 0348 09z t + 007z 3t + 006z 4t + 084z 5t + a 5t (0) GTB : z 6t = 0975z 6t + a 6t () + a t a t a 3t a 4t a 5t a 6t wwwijmsiorg 37 Page

Forecasting Stocks with Multivariate Time Series Models VII DIAGOSTIC CHECKS After obtaining the multivariate model (5), the next ste is to ascertain whether the model is adequately fitted or not The following diagnoses are carried out to this effect 7 Residual Autocorrelation Function As stated in section 4 of the methodology, the following hyothesis are alied: H 0 : ρ uv i = 0 versus H ρ uv i 0 In the series, we had n = 04 This gives the boundary condition for residual autocorrelation function to be = 0400 04 and H 0 is rejected if r uv,i > = 0400 n Comaring the values of autocorrelations in the residual correlation matrices at different lags (lags 3) in aendix A with r uv,i, we find that none of the residual autocorrelations is greater than 0400 This means that the residuals follow a white noise rocess In other words, the fitted model is adequate 7 Residual Plots The residual lots obtained by fitting the model (5) are lotted in aendix D The lots show that all the six residual series: a t, a t, a 3t, a 4t, a 5t, a 6t of the residual vector a t resemble a white noise rocess which is art of the assumtions needed to be satisfied for model adequacy VIII FORECASTS Since the constructed model has satisfied the basic assumtion of model adequacy, it can be used for generating forecasts The forecasts for the years 06 and 07 generated by multivariate model are dislayed in table of aendix E IX DISCUSSIO AD COCLUSIO Forecasting stocks is very imortant in the banking sector esecially in the resent day devastating economy in igeria Since the gains and losses in stocks are highly robable, there is need to be guided by robability models that can redict the future stocks Such models are time series models which could either be univariate or multivariate Univariate models can only handle a series that is indeendent of any other series However, many time series that arise in nature are interrelated The interrelationshi of such series can easily be revealed by the correlation structure exhibited by the series The stocks of the different banks considered in this work were found to be highly correlated; and the alied multivariate methods took care of the correlation structure of the comonent series of the vector Since the constructed multivariate model was found to be adequate, the generated forecasts are reliable The forecasts can serve as a guide to banks that may wish to involve in stocks in 07 Aendix A: Raw Data Cross Correlations at different Lags (i) Lag 0656 0636 080 0556 0653 080 074 0805 0606 0704 078 0657 0690 0597 074 (ii) Lag 0656 0636 080 0557 0657 0803 0745 0808 0608 0703 078 0666 0689 0506 054 (iii) Lag 3 0656 0636 0803 0557 066 0808 075 080 0609 070 077 0673 0689 053 063 wwwijmsiorg 38 Page

Forecasting Stocks with Multivariate Time Series Models (iv) Lag 4 0658 0636 0803 0557 0769 0808 075 087 060 070 077 0688 0687 056 0549 (v) Lag 5 0658 0636 0803 0557 0669 0808 075 087 060 070 077 0688 0687 056 0549 Aendix B Table : Values of the Selected Criteria with their Resective lags S AIC BIC HQC M() - Value 0-8966 -8966-8966 0 0-034 -579-8666 484553 0 3-9974 -0863-537 43879 0735 4 3-8943 -0377-837 48496 0009 5 4-7537 -945-0806 3530 05098 6 5-669 -8743-04847 47995 003 7 6-4904 -7977-0069 67579 08684 8 7-3 -73-9664 79 08538 9 8-09 -6565-936 3449 05537 0 9-358 -58659-9004 37655 03935 0-0089 -5534-8640 9669 0779-0945 -45004-83359 3565 05035 3-098 -3993-8774 506697 00533 4 3-0906 -3939-7868 66779 08709 Aendix C: Residual Autocorrelation Matrices (lag -3) of the Fitted Model 000 000 005 003 009 008 000 003 0050 0 0078 000 R = 00 030 007 007 00 000 000 0003 0006 0067 0004 0005 004 00 006 000 0007 0000 00 0035 009 0077 0084 0003 00 00 0 0066 00 0055 0033 0033 0 0044 008 03 R = 00 004 0 007 0033 00 004 0069 0034 04 0097 006 00 0005 00 0054 0 0005 00 00 003 0033 0 007 004 007 000 00 003 00 0055 0056 00 005 0000 00 R 3 = 0038 0 005 00 005 0077 00 0055 005 0069 00 006 00 00 005 003 007 00 00 007 0033 0045 003 004 wwwijmsiorg 39 Page

-04-0 00 0 04 GTB -08-06 -04-0 00 0-05 00 05 0 5 Wemabank -06-04 -0 00 0 04-5 -0-05 00 05 0 Unionbank -05 00 05 Forecasting Stocks with Multivariate Time Series Models Aendix D: Time Series Plots of the Residuals a t, a t, a 3t, a 4t, a 5t, a 6t Resectively 0 50 00 50 00 time 0 50 00 50 00 time 0 50 00 50 00 time 0 50 00 50 00 time 0 50 00 50 00 0 50 00 50 00 Aendix E Table : Forecasts of the Multivariate Model Year Month First bank Access bank UBA Union bank Wema bank GTB 06 JA 35 33 946 8 5 88 FEB 386 30 3 44 77 7 MAR 6 37 5 93 46 74 APR 556 34 8 344 95 MAY 635 47 4 0 36 44 JU 76 56 90 693 44 54 JUL 36 53 35 856 89 43 AUG 64 57 36 399 34 383 SEP 574 734 96 087 398 OCT 5 745 88 65 57 333 OV 533 73 6 74 56 40 DEC 74 86 3 736 67 406 07 JA 63 3 70 544 3 FEB 38 736 46 967 587 50 MAR 73 53 33 789 647 67 APR 9 645 37 9 63 68 MAY 36 69 08 377 73 JU 87 66 6 3956 57 0 JUL 753 76 44 39 567 3 AUG 05 677 055 633 499 4 SEP 34 638 705 565 36 55 OCT 366 69 977 534 38 9 OV 37 65 88 355 36 3 DEC 30 678 935 44 337 wwwijmsiorg 40 Page

Forecasting Stocks with Multivariate Time Series Models Table : Stock Data of the Six Banks Year Month First bank Access bank UBA Union bank Wema bank GTB 999 JA 6 3 65 33 3 405 FEB 38 55 0 9 59 995 MAR 8 85 35 55 0 APR 05 0 545 67 4 5 MAY 565 9 38 63 47 45 JU 38 647 3 08 345 5 JUL 6 383 3 9 407 555 AUG 6 46 09 39 47 SEP 6 435 65 8 3 5 OCT 6 36 69 0 3 455 OV 6 36 67 0 3 44 DEC 6 3 65 33 3 405 000 JA 0 35 78 093 3 4 FEB 065 343 8 4 4 4 MAR 96 43 775 074 53 39 APR 0 345 755 07 3 305 MAY 98 345 73 07 5 305 JU 97 305 84 07 95 36 JUL 87 3 8 07 9 35 AUG 8 3 69 067 83 35 SEP 74 9 74 07 7 73 OCT 789 7 8 07 6 OV 69 53 8 96 63 3 DEC 0 4 808 99 7 85 00 JA 99 345 7 69 9 9 FEB 05 335 65 735 4 9 MAR 06 303 64 8 5 75 APR 533 86 66 78 95 94 MAY 34 7 68 978 8 7 JU 54 38 95 0 3 6 JUL 93 55 8 855 9 6 AUG 93 56 675 86 4 SEP 95 4 67 3 75 35 OCT 95 334 74 05 05 35 OV 0 375 75 09 48 4 DEC 69 3 84 05 48 5 00 JA 3 343 9 45 48 45 FEB 486 366 59 04 48 57 MAR 4 38 03 9 48 4 APR 534 3 5 0 348 35 MAY 607 367 444 0 36 57 JU 696 546 89 6 98 35 JUL 43 575 45 806 30 AUG 68 689 35 63 9 36 SEP 55 6 96 06 374 OCT 56 706 88 085 368 OV 576 604 6 76 34 DEC 84 745 3 84 8 4 003 JA 974 7 3 93 87 56 FEB 75 8 544 3 9 57 MAR 677 8 9 35 05 55 APR 96 785 46 3485 34 57 MAY 3456 8 5 380 77 57 JU 335 899 75 39 85 57 JUL 5 904 39 449 8 57 AUG 5 9 35 39 97 57 SEP 95 9 49 9 57 OCT 7 998 5 49 384 537 OV 9 93 49 4 59 DEC 9 9 49 398 6 004 JA 65 877 055 49 49 638 FEB 4 9 079 49 399 64 MAR 344 87 0 49 44 687 APR 475 885 96 49 485 67 MAY 5 886 0 49 49 603 JU 55 79 0 6 55 634 JUL 03 9 744 86 54 58 AUG 075 66 687 875 537 557 wwwijmsiorg 4 Page

Forecasting Stocks with Multivariate Time Series Models SEP 996 584 596 84 576 53 OCT 957 698 543 954 576 49 OV 049 698 58 9 576 49 DEC 05 698 579 33 576 5 005 JA 4 698 745 6 576 63 FEB 483 698 8 794 576 603 MAR 6 698 799 65 448 595 APR 6 698 74 78 99 69 MAY 86 698 75 9 98 5 JU 797 698 668 9 99 508 JUL 996 698 608 98 3 5 AUG 0 698 638 68 308 554 SEP 0 683 69 65 3 63 OCT 0 683 0 4 358 85 OV 0 683 05 395 4 88 DEC 0 683 039 50 433 099 006 JA 0 683 99 797 437 35 FEB 3098 683 36 9 559 496 MAR 8 683 49 739 6 6 APR 999 683 9 845 608 68 MAY 305 683 6 3055 584 94 JU 83 683 05 3599 56 94 JUL 898 683 46 38 5 94 AUG 95 683 00 537 47 94 SEP 35 683 794 47 3 94 OCT 5 683 9 5 37 69 OV 4 663 88 3 393 69 DEC 36 663 905 393 69 007 JA 34 663 0 034 393 FEB 335 663 0 9 393 0 MAR 46 663 0 35 393 9 APR 67 663 0 459 393 998 MAY 94 663 0 459 393 906 JU 30 663 94 459 393 04 JUL 3099 663 94 7 393 0 AUG 3 663 306 553 393 06 SEP 3 9 493 596 393 53 OCT 3 9 498 5 393 95 OV 3 9 344 548 374 DEC 3 9 3 548 374 4 008 JA 33 9 9 548 374 3 FEB 375 9 6 548 374 37 MAR 37 9 8 548 374 509 APR 4474 9 75 548 374 66 MAY 4899 75 348 548 374 7 JU 535 69 499 3055 374 43 JUL 69 66 484 78 79 399 AUG 477 793 05 935 8 834 SEP 46 83 3 856 35 867 OCT 43 07 7 646 3 79 OV 37 456 53 33 6 DEC 359 5 66 4 35 806 009 JA 3689 5 393 76 45 86 FEB 38 5 3799 994 486 99 MAR 38 5 3799 30 56 33 APR 404 5 3799 335 605 377 MAY 83 5 4396 3949 845 39 JU 57 5 505 44 93 346 JUL 554 68 5695 4609 095 33 AUG 374 3049 506 36 0 9 SEP 399 3049 5 4078 898 989 OCT 344 3049 5 43 957 953 OV 435 3049 505 40 5 305 DEC 447 3049 495 4306 5 3463 00 JA 405 6 489 47 5 33 FEB 434 567 489 43 5 393 MAR 4445 58 4899 44 5 3596 APR 4 594 547 3698 5 334 MAY 434 558 65 385 5 334 JU 434 548 3,7 3463 5 336 wwwijmsiorg 4 Page

Forecasting Stocks with Multivariate Time Series Models JUL 4 497 377 4 5 565 AUG 77 48 4 5 0 SEP 957 3098 77 4 5 88 OCT 6 395 96 33 49 67 OV 37 96 6 83 49 53 DEC 986 63 49 309 0 JA 43 58 738 738 49 835 FEB 765 84 0 0 95 8 MAR 558 59 8 8 47 988 APR 896 577 93 93 3 03 MAY 999 705 579 579 35 64 JU 607 739 3 66 34 7 JUL 485 6 3 373 7 4 AUG 485 5 4 6 78 397 SEP 43 33 4 69 6 35 OCT 485 66 73 3 55 OV 40 0 63 605 55 DEC 45 5 83 7 4 56 0 JA 47 7 86 655 08 78 FEB 44 6 33 593 7 807 MAR 639 7 5 605 05 98 APR 54 3 8 555 5 754 MAY 4 03 53 70 JU 35 97 07 485 099 666 JUL 34 08 05 563 08 68 AUG 9 94 93 495 089 54 SEP 9 90 47 0 598 OCT 74 0 90 487 03 598 OV 34 8 9 46 66 DEC 373 8 95 4 9 776 03 JA 476 9 976 4 7 83 FEB 57 999 36 7 905 MAR 395 85 76 346 33 997 APR 343 78 59 30 8 6 MAY 348 6 65 77 08 63 JU 5 3 564 7 558 JUL 44 073 55 8 089 44 AUG 066 064 4 09 076 54 SEP 96 095 369 09 068 67 OCT 00 085 36 09 064 46 OV 9 55 09 055 4 DEC 967 4 67 459 054 408 04 JA 956 67 587 054 445 FEB 94 7 67 05 406 MAR 94 6 7 05 349 APR 978 4 374 395 054 60 MAY 00 7 408 38 05 5 JU 09 3 366 373 05 5 JUL 73 76 475 44 05 7 AUG 74 76 483 465 05 8 SEP 485 86 45 75 05 95 OCT 6 89 458 784 054 98 OV 64 896 45 73 05 935 DEC 68 805 456 735 05 3 05 JA 63 687 84 086 457 FEB 648 7 806 0 08 46 MAR 7 0 8 05 7 6 APR 708 9 689 9 36 555 MAY 8 37 85 03 999 JU 803 09 8 44 43 JUL 855 06 78 5 58 AUG 903 767 74 08 578 SEP 9 035 755 05 5 OCT 94 969 78 0 5 53 OV 0 0 75 0 4 73 DEC 0 96 89 963 70 wwwijmsiorg 43 Page

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