Online Publication Date: 19 April 2012 Publisher: Asian Economic and Social Society

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1 Online Publication Date: April Publisher: Asian Econoic and Social Society Using Entropy Working Correlation Matrix in Generalized Estiating Equation for Stock Price Change Model Serpilılıç (Faculty of Arts and Sciences, Departent of Statistics, Yıldız Technical University, Davutpasa Capus,, Esenler, İstanbul) Ahet Mete Çilingirtürk (Faculty of Econoics and Adinistrative Sciences, Marara University, Bahçelievler Capus,, Bahçelievler, İstanbul) Citation: Serpilılıç, Ahet Mete Çilingirtürk (): Using Entropy Working Correlation Matrix in Generalized Estiating Equation for Stock Price Change Model Journal of Asian Scientific Research Vol., No., pp.-.

2 Journal of Asian Scientific Research, Vol., No., pp.- Using Entropy Working Correlation Matrix in Generalized Estiating Equation for Stock Price Change Model Abstract Author (s) Serpilılıç Res. Assist., Faculty of Arts and Sciences, Departent of Statistics, Yıldız Technical University, Davutpasa Capus,, Esenler, İstanbul. E-ail: Ahet Mete Çilingirtürk Prof. Dr., Faculty of Econoics and Adinistrative Sciences, Marara University, Bahçelievler Capus,, Bahçelievler, İstanbul. E-ail: Longitudinal studies involving binary responses are widely applied in edical, health and econoic science research, have focused increasingly on how various independent variables affect responses over tie. These studies involve repeated observations on a subject and thus correlation within each subject is expected. Correct inferences can only be obtained by taking into account the correct specification of within-subject correlation structure between repeated observations. In recent years, non-noral longitudinal data is analyzed by Generalized Estiating Equations (GEE) ethod. Goodness-of-fit statistics have been suggested for selecting an appropriate working correlation structure in GEE with longitudinal binary data. The purpose of this article to provide an overview of the GEE approach for analyzing correlated binary data and to choose the structure of the correlation atrix between repeated observations for odel coparison, using data fro Istanbul Stock Exchange (ISE) to increase on the return. eywords: Working Correlation Structures; Generalized Estiating Equations; Longitudinal Binary Data; Entropy JEL Codes: C, G Introduction Generalized estiating equations (GEE) approach which extends generalized linear odels is a very popular for the situation of correlated data obtained longitudinal studies. Although GEE odels can be used for continuous responses, they have often becoe for analysis of categorical and count responses. GEE odels use quasi-likelihood estiation and full likelihood of the data is not necessary. It does not need ultivariate distributions, because GEE assues only a functional relationship for arginal distribution at

3 Using Entropy Working Correlation Matrix.. each tie point. (Hedeker and Gibbons,, ). This approach is coplex to interpret and ipleent fro classical analysis approach. Despite the any benefits of classical analysis, it has soe constraints. These are: () it does not odel the ean response changes across tie on each subject; () it has soe assuptions about the variance-covariance atrix. For exaple; in the classical regression odel, a single observation of the response variable is considered as the observational unit. Therefore, the statistical odeling assues independence between observations (Lee et al.,, ). But the assuption of independence is not usually used in longitudinal studies, because the relationship between repeated observations over tie on the sae subject can be correlated; () it does not consider about the structure of dependence between repeated observations obtained fro the sae subject. For these reasons, the classical approaches are insufficient. The structure of correlation is iportant to produce efficiency (i.e., statistical power) in the estiation of the regression paraeter. However, the loss of efficiency is lessened as the nuber of subjects gets large. If the correlation data is correctly identified, the inferences about hypothesis tests and confidence intervals will be valid and correct. In this study, we consider to correlated binary data and copare several criteria that can be obtained the final selection of working correlation structure. Generalized Estiating Equations A review of GEE ethod GEE odels are used for analyzing longitudinal binary studies involve binary responses for each subject and a set of covariates varying with or without tie. Consider a longitudinal binary data set coprising (X it,y it ) for i=,,n; t=,,n i. For the i th subject, there are n i repeated binary response variables. Define a n i x binary response vector as Y i =(Y i,,y ini )' and a n i xp covariate atrix as X i =(X i,,x ini )' with a p-diensional covariate vector X it. The binary response variable Y it = at tie t, if the subject i has response, success and Y it = if otherwise. It is assued that n i = for all i and N=n (Lin et al.,, ). The ost iportant proble in this ethod is to deterine the (co)variance structure. Even if the covariance structure has been isspecified in longitudinal studies, GEE ethod yields asyptotically noral and consistent for estiated paraeters. GEE specifications are siilar to generalized linear odel (GLM), but those of GLM with one addition are coprised by GEE approach. There are three specifications in this odel. First, the linear predictor is given as.() Then a link function is chosen in Equation...()

4 Journal of Asian Scientific Research, Vol., No., pp.- The coon choices for link function are identity, logit, and log for continuous, binary and count data, respectively. The variance is described as a function of the ean,..() where υ(μ it ) is a variance function and ϕ is a scale or distribution paraeter. When each subject is easured at all tie points, the working correlation atrix of the repeated observations is of size x. If a subject has been easured at n i tiepoints (n i <), each subject s correlation atrix R i will be of size n i xn i. α is a vector of association paraeters which are assued to be the sae for all subjects. (Hedeker and Gibbons,, ) Working Correlation Structures There are soe possible correlation structures to be appropriate to use in GEE. These structures are independent, exchangeable, autoregressive, -dependent and unstructured. The ost coonly used working correlation structures and estiators are given in the Table If data is balanced and there are clusters with sall nuber of observations, the unstructured correlation atrix is recoended. An exchangeable correlation atrix ay be ost appropriate for datasets with clustered observations, which ay not have a logical ordering at observations within a cluster. When the observations have been istied, it ay be appropriate to regard a odel where the M-dependent or autoregressive correlation is a function of the tie between observations. Any estiation of α is not perfored for both the independence and fixed working correlation structures (Horton and Lipsitz,, ). This paper presents the use of entropy for working correlation atrix, which supports an unstructured dependence within the tie points. Although the word entropy originated in the literature of therodynaics, its usage has penetrated alost all disciplines due to its association with the concept of inforation as envisaged by Claude Shannon. If the probabilities can be used rather than raw results, entropy can be calculated for one variable and can also be used for researching dependence between two or ore variables. Due to this future of entropy, it could be a possible alternative for correlation coefficient. Because of the iportance of working correlation atrix in GEE, it is crucial to use different working correlation structure in order to obtain efficient results. Therefore, entropy atrix could be the possible alternative for coon working correlation atrix. The entropy and entropy correlation coefficient( ) forulations are given in Equation -..().()

5 Using Entropy Working Correlation Matrix.. () values:. Residuals:.. (). These are used to estiate A i, R i and..(). Then the GEE s are solved again to obtain iproved estiates of β. GEE Estiation The working covariance atrix for Yi equals.. () where A i is x diagonal atrix with V(μ it ) as the t th diagonal eleent. ϕ is an overdispersion paraeter that can be estiated as follows:.() where N is the total nuber of observations and p is the nuber of regression paraeters. The square root of the overdispersion paraeter is called the scale paraeter. The GEE estiator of β is the solution of...() where Di is the atrix of derivatives () Iterative process for GEE s is given the following: Start with R i =independent (i.e., identity) and ϕ =: estiate β.. Use estiates to calculated fitted.(). Between step and are repeated to converge to a value of β (ılıç and Çilingirtürk,, ). Model Selection and Goodness of Fit Tests This paper exaines three odel selection criteria which are Marginal R, QIC and QIC U estiates. Repeated observations are correlated over tie points, therefore residuals are not independent. R in the ordinary least squares ethod cannot be used for GEE directly. An extension of R statistics in GLM is called as Marginal R for GEE (Zheng,, ). It can be calculated as shown below....() Marginal R is a statistical easure which is often interpreted as the proportion of response variation explained by the fitted odel. SAS PROC GENMOD could not be given Marginal R, so this easureent is calculated with the acro %SelectGEE.

6 Journal of Asian Scientific Research, Vol., No., pp.- One of the goodness-of-fit statistics, Akaike s Inforation Criterion (AIC), can be used for coparing copetitive odels. But this criteria could not be used for GEE ethod. Because GEE is not a likelihood-based ethod. In this reason, Pan () introduced a selection ethod which naed as Ouasilikelihood under the Independence Model Criterion (QIC). QIC is siilar to AIC. The forulas AIC and QIC are given as follows: () where L is the log likelihood and p is the diension of β..() Is quasi likelihood coputed using R, Ω _I is the inverse of the variance atrix by fitting an independence odel and is odified sandwich estiate of variance fro the odel with R in Equation. When approxiates p, Pan () also proposed QIC U which could be useful in variable selection, but it is not used for odel coparison. QIC U s forula is given in Equation (Hardin and Hilbe,, ). () Marginal R, QIC and QIC U are the criteria of the evaluation of choosing the best odel. In this process, the odel with lower value of QIC and QIC U and higher value of Marginal R should be taken into account. These criteria are obtained by special acro software in the SAS. progra (support.sas.co/resources/papers/proceedi ngs/-.pdf,, ). Stock Price Change Model The significance and purpose of this study The purpose of this study is to odel the ISE Stock Price Change and to present the entropy working correlation atrix. The odel will have a technical analyses approach, because it takes just the increase signal in a quarterly base. According to goodness-of-fit criteria, appropriate working correlation structures in the GEE analysis of longitudinal studies with binary responses is deterined. Furtherore, it is showed that which variables is the ost effective on stock prices. One of the ost thought in investors is to predict the future direction of stock prices. Stock prices have ore volatility than the other investents. The ost coonly used analyses in evaluating of the stock prices are fundaental and technical analysis. Fundaental analysis is a ethod of evaluating securities by attepting to easure the intrinsic value of a stock. Fundaental analysts study everything fro the overall econoy and industry conditions to the financial condition and anageent of copanies. Technical analysis is the evaluation of securities by eans of studying statistics generated by arket activity, such as past

7 Using Entropy Working Correlation Matrix.. prices and volue. Technical analysts do not attept to easure a security's intrinsic value but instead use stock charts to identify patterns and trends that ay suggest what a stock will do in the future ( asp, ). Data Sources Using transactions data chosen ISE between / and / quarterly, we estiate paraeter estiates and the corresponding standard errors under exchangeable, AR(), M() dependent, unstructured and entropy correlation structures via GEE ethod. We copare the criteria for choosing between these structures. In this study, response variable is stock price. According to the technical analysis, it is coded if it increases according to the previous -onth period, otherwise Covariate variables are transaction volue, stock dividend, cash dividend, increase of capital, price index, exchange rate of dollar, Noenclature Generale des Activites Econoiques dans I'Union Europeenne, NACE, (General Nae for Econoic Activities in the European Union) Codes and tie. Transaction volue and dividend paid in cash are coded if it increases according to the previous -onth period, otherwise The effects of NACE, increase of capital and stock dividend of these factors on stock return are not statistically significant. We take the transaction volue and dividend paid in cash as the covariate variables. Findings and Results Let μ it denote the ean, the probability of increasing stock prices for i=,, stocks and t= (baseline),,,,,,, onths. logit link function for binary responses can be shown as follows: () where β, β, β, β are the regression coefficient paraeters for intercept, the size of transaction volue, dividend paid in cash and tie respectively. Table presents the results of the GEE odels using several various working correlations. The analysis results are siilar in the estiated paraeters for all structures. The negative sign of the regression coefficient of tie variable indicates that decreasing on stock prices is stronger at the beginning of the follow-up period. Fro the results, high the size of transaction volue and dividend paid in cash have significantly positive effect on increasing stock prices. However, as shown in Table, p-values iply that there are statistically significant effects on these variables. Table suarizes the results of the analysis with different working correlation structures. Although QIC and Marginal R have selected the best fitting odel, these criteria are very close for all structures. The M() dependent structure is found to have the sallest QIC in all other structures and thus is selected as the preferred working correlation structures. For without tie dependent data set, this structure is used to consider as a function of tie between observations, M() dependent

8 Journal of Asian Scientific Research, Vol., No., pp.- is preferred. Conclusions We have discussed selecting the working correlation structure in GEE with longitudinal binary response. An application of longitudinal studies to data on stock price is used as an exaple. GEE is relatively new ethod for the analysis of longitudinal studies on stock price. GEE ethod yields the estiates of regression coefficients and their variances fro different correlation structures that can be sensitivity to incorrect specification. will be estiated asyptotically noral and consistently, even when the working correlation structure is isspecified. The choice of R i will influence the efficiency for estiates of paraeters and variances. It is ore efficient to use R i that is chosen correct specification. For this study, the results of the correlation structure M-dependent and AR() are ore siilar. If the repeated observations across subjects are easured at equally spaced in tie, AR() structure is preferred in longitudinal data (Shults et al.,, ). M-dependent structure is used to consider as a function of tie between observations for without tie dependent data set. According to Marginal R, QIC and QIC U of selection criteria, M() dependent structure is preferred in this study. However, one of the ain points of this study is to copare the efficiency of the entropy atrix as a working correlation structure to other structures. According to Marginal R, QIC and QIC U of selection criteria (lower QIC and OIC U values and higher Marginal R value), entropy atrix is preferred instead of unstructured correlation atrix. This result showed that entropy atrix could be used as a working correlation structure instead of unstructured correlation atrix in this study. In order to deterine the status of the stock price, financial ratios are calculated by balance sheets, financial and incoe stateents of copanies. Further studies are needed to investigate how to affect these ratios as covariate variables when ore coon working correlation structures are used.

9 Using Entropy Working Correlation Matrix.. Table Coonly used working correlation structures and estiators Working Correlation Structures Definition Exaple The nuber of paraeters Estiators Independent Corr Y, Y ij ik j=k j In this case, working correlation is not estiated. Exchangeab le Corr Y, Y ij ik j=k j ˆ N * p * N nini i ij ik i j k ee Unstructure d Corr Y, Y ij ik j=k jk j.. t.. t t t.. tt p i ˆ jk ij ik ee Autoregressi ve of first order [AR()] Corr Y, Y ij t=,,...,n -j i, jt i t t.. t t t.. ˆ p i n i ee ij i, j i jni M-dependen t t=.. t.. Corr Yij, Yik t t=,,... t t> t t.. M t ˆ t ij i, t p i j ni t t i n t i ee Fixed Corr Y, Y r ij ik jk r.. r t r.. r t rt rt.. (User specified) In this case, working correlation is not estiated.

10 Journal of Asian Scientific Research, Vol., No., pp.- Paraeter Intercept The size of transaction volue, (TV) Dividend paid in cash, (DC) Table.Analysis of the GEE paraeter and standard error (SE) estiates, using various working correlation structures Exchangeabl M() Unstructured AR() Entropy Entropy e Dependent Esti Esti Esti Esti SE SE SE SE ate ate ate ate * * * * * * - - * * * * * * * * * * ** * * * Estia Estia SE SE te te * * ** - * Marginal R QIC QIC U *p<, ** p<, Table.Estiated working correlation atrices for various structures Excha ngeab le e e e e e e e e Unstr uctur ed e e e e e e e e

11 Using Entropy Working Correlation Matrix AR() e e e e e e e e Entro py e e e e e e e e

12 Journal of Asian Scientific Research, Vol., No., pp M() Depen dent e e e e e e e e Entro py e e e e e e e e

13 Using Entropy Working Correlation Matrix.. References Hardin, J. W.,&Hilbe, J. M. ().Generalized Estiating Equations Chapan and Hall/ CRC Press. Boca Raton. Hedeker, D., & Gibbons, R. D. ().Longitudinal Data Analysis John Willey & Sons, Inc.New Jersey: Hoboken. Horton, N. J., &Lipsitz, S. R. ().Review of Software to Fit Generalized Estiating Equation Regression Models The Aerican Statistician., -. ılıç, S., &Çilingirtürk, A. M. ().GenelleştirilişTahinDenkleleri ndeorelasyonyapılarınınincelenesi th International Econoetrics, Operation Research and Statistics Syposiu.Denizli, Turkey, May -,, -. Lee, J-H., Herzog, T. A., Meade, C. D., Webb, M. S. and Brandon, T. H. ().The use of GEE for analyzing longitudinal binoial data: A prier using data fro a tobacco intervention Addictive Behaviors., -. Lin, -C., Chen, Y-J.andShyr, Y. ().A nonparaetric soothing ethod for assessing GEE odels with longitudinal binary data.statistics in Medicine., -. Shults, J., Sun, W.,Tu, X., i, H., Asterda, J.,Hilbe, J. M. and Ten-Have, T. ().A coparison of several approaches for choosing between working correlation structures in generalized estiating equation analysis of longitudinal binary data. Statistics in Medicine., -. Using GEE to odel student s satisfaction: A SAS Macro Approach.(t.y.) [Online] Available: support.sas.co/resources/papers/proceedin gs/-.pdf (April, ). What is the difference between fundaental and technical analysis? (t.y.) [Online] Available: asp (June, ). Zheng, B. ().Suarizing the goodness of fit on generalized linear odels for longitudinal data.statistics in Medicine., -.