Forecasting Stocks with Multivariate Time Series Models.

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1 International Journal of Mathematics and Statistics Invention (IJMSI) E-ISS: P-ISS: 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

2 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

3 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

4 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

5 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 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

6 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 DFirst bank * DAccess bank * DUBA * DUnion bank * DWema bank * DWema bank * 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 Z t(corr ) = 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 = z t z t z 3t z 4t z 5t z 6t (5) This can be exressed exlicitly as follows: First Bank : z t = z t z 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 = z 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

7 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 = 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 (ii) Lag (iii) Lag wwwijmsiorg 38 Page

8 Forecasting Stocks with Multivariate Time Series Models (iv) Lag (v) Lag Aendix B Table : Values of the Selected Criteria with their Resective lags S AIC BIC HQC M() - Value Aendix C: Residual Autocorrelation Matrices (lag -3) of the Fitted Model R = R = R 3 = wwwijmsiorg 39 Page

9 GTB Wemabank Unionbank 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 time time time time Aendix E Table : Forecasts of the Multivariate Model Year Month First bank Access bank UBA Union bank Wema bank GTB 06 JA FEB MAR APR MAY JU JUL AUG SEP OCT OV DEC JA FEB MAR APR MAY JU JUL AUG SEP OCT OV DEC wwwijmsiorg 40 Page

10 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 FEB MAR APR MAY JU JUL AUG SEP OCT OV DEC JA FEB MAR APR MAY JU JUL AUG SEP OCT OV DEC JA FEB MAR APR MAY JU JUL AUG SEP OCT OV DEC JA FEB MAR APR MAY JU JUL AUG SEP OCT OV DEC JA FEB MAR APR MAY JU JUL AUG SEP OCT OV DEC JA FEB MAR APR MAY JU JUL AUG wwwijmsiorg 4 Page

11 Forecasting Stocks with Multivariate Time Series Models SEP OCT OV DEC JA FEB MAR APR MAY JU JUL AUG SEP OCT OV DEC JA FEB MAR APR MAY JU JUL AUG SEP OCT OV DEC JA FEB MAR APR MAY JU JUL AUG SEP OCT OV DEC JA FEB MAR APR MAY JU JUL AUG SEP OCT OV DEC JA FEB MAR APR MAY JU JUL AUG SEP OCT OV DEC JA FEB MAR APR MAY JU , wwwijmsiorg 4 Page

12 Forecasting Stocks with Multivariate Time Series Models JUL AUG SEP OCT OV DEC JA FEB MAR APR MAY JU JUL AUG SEP OCT OV DEC JA FEB MAR APR MAY JU JUL AUG SEP OCT OV DEC JA FEB MAR APR MAY JU JUL AUG SEP OCT OV DEC JA FEB MAR APR MAY JU JUL AUG SEP OCT OV DEC JA FEB MAR APR MAY JU JUL AUG SEP OCT OV DEC wwwijmsiorg 43 Page

13 Forecasting Stocks with Multivariate Time Series Models References [] Abdulsalam, G, Jobby, D and Tsay, K (0) Regression and a data mining technique in financial Institutions Journal of Statistics and Economics Vol 5, o 9, [] Amihud, R (00) Illiquidity in Stock Market Journal of Financial Research 9: : [3] Ajayi, R A and Mougoue, M (996): On the Dynamic Relation Between Stock Prices and exchange rates Journal of Econometric Methods Vol 8, o 6, [4] Box, GEP, Jenkins, GM and Reinsel, GC (008) Time Series Analysis, Forecasting and Control John Willey and Sons, Inc Hoboken, ew Jersey ISB (cloth) QA80B _5 dc [5] Cambell (000) Multivariate system of modeling financial variables Euroean Journal of Economic and Social Systems, 4 o(000) 8-9 [6] Chordia, M Jude, P and Pete, C (003) An Emirical Analysis of Stock and Bond Market Liquidity Economics and Finance Vol 40, [7] Ibrahim, K and Agbaje, R (03) The Relationshi between Stock Return and Inflation in igeria Journal of Econometrics Vol 9, o 6, [8] Larsen, J I (00) Predicting Stock Prices Using Technical Analysis and Machine Learning Journal of Economics and Comuter Science Vol 7, o 5, 35-4 [9] Lee, F, Teddy, J and Moon,S (000) Stock Returns during German Hyerinflation The Quarterly Review of Economic Policy, Germany QREP, Vol 5, o 8, [0] Lendasse, G, Hannan, C and Tuti, J (000) on-linear Financial Time Series Forecasting and Alications Perry Pub Ltd ISB: [] Lutkeol H(005) : ew introduction to multile Time Series Analysis Sringer Berlin Heidebelg ew York ISB SPI [] Merle, K, P (008) A Dwindling Economy Equador Economic Submit Vol3, Issue 8, 5-64 [3] Quintana and West (009) An alternative model for multile time series Euroean Scientific Journal; Vol 9, o 4 ISS: (rint)e-ISS: wwwijmsiorg 44 Page