AN APPLICATION OF THE GARCH-t MODEL ON CENTRAL EUROPEAN STOCK RETURNS

Transcription

1 AN APPLICATION OF THE GARCH- MODEL ON CENTRAL EUROPEAN STOCK RETURNS Miloslav VOŠVRDA, Filip ŽIKEŠ * Absrac: The purpose of his paper is o invesigae he ime-series and disribuional properies of Cenral European sock reurns. We es he random walk hypohesis and hen consider an alernaive o random walk he ARIMA model for sock prices. The behavior of volailiy of reurns over ime is sudied using he GARCH- model which also allows us o learn more abou he disribuion properies of sock reurns. We employ he BDS es o assess he abiliy of he esimaed GARCH- model o capure all nonlineariies in sock reurns. Our empirical findings reveal ha he Czech and Hungarian sock marke indices are predicable from he ime series of hisorical prices, whereas ha of Poland is no. The reurns on all hree indices are condiionally heeroskedasic and non-normal. The esimaed number of degrees of freedom ranges from 18 o 4. Keywords: condiional heeroskedasiciy, GARCH, lepokurosis, marke efficiency JEL Classificaion: C32, G15 1. Inroducion One of he main principles of modern finance is ha expeced reurn is somehow relaed o risk. There are numerous equilibrium models ha specify his relaionship (see, for example, Cochrane, 2001 for deails). For insance, in he sandard Sharpe-Linner-Mossin CAPM, expeced reurn is proporional o sysemaic risk, i.e. o risk ha canno be diversified away. Alhough here is no a general agreemen in he lieraure as o wha model explains he realiy bes, i is widely acceped ha when agens are risk-averse expeced reurn is an increasing funcion of risk. A widely used measure of he risk of an asse is he sandard deviaion of reurns from he uncondiional mean. This measure can be loosely inerpreed as longrun volailiy, since i appears o be deermined by various economic fundamenal characerisics for ha paricular asse and is usually assumed consan for he pe- *) Academy of Sciences of he Czech Republic, Dep. of Economerics and Charles Universiy, Faculy of Social Sciences, Insiue of Economic Sudies, Oplealova 26, CZ Prague 1 ( vosvrda@uia.cas.cz; zikesf@yahoo.com). **) This research was suppored by he Gran Agency of Charles Universiy under gran GA UK 287/ 2003/A-EK/FSV, and by he Gran Agency of he Czech Republic under grans 402/01/0034 and 402/04/ l PRAGUE ECONOMIC PAPERS, 1, 2004

2 riod under sudy. I has been observed in numerous sudies, however, ha reurns are no homoskedasic and ha large price changes end o be followed by large changes of eiher sign and small changes end o be followed by small changes of eiher sign (Mandelbro, 1963, p. 418). Also i has been observed empirically ha he uncondiional disribuions of sock reurns exhibi faer ails (see, for example, Fama, 1965) han wha would be consisen wih he normal disribuion. Bu he disribuional properies of sock reurns have imporan implicaions for asse pricing: in he above menioned CAPM model, for example, he variances and covariances are used as a measure of he risk, bu as Bollerslev (1987, p. 542) noes, depending on he disribuion of he reurns, he variance may no be a valid or sufficien saisic o use. Anoher example of he imporance of he disribuional assumpions is he well-known Black-Scholes opion pricing model which ress on he hypohesis of normaliy of he underlying sock reurns and assumes consan volailiy of reurns over he life of he opion. Clearly, correcly specifying he disribuion of asse reurns is a necessary condiion for raional asse pricing. Alhough he lieraure on condiional heeroskedasiciy of sock reurns is fairly voluminous, only a few applicaions o emerging sock markes in Cenral Europe have been published o dae. Vošvrda e al. (1998) invesigae he ime series properies of he Czech PX-50 index and find significan condiional heeroskedasiciy in is daily reurns. Similarly, Gilmore and McManus (2001) find evidence of imevarying volailiy in weekly reurns on major Cenral European sock indices. Boh sudies, however, assume condiional normaliy of sock reurns raher han invesigaing he disribuions empirically. In his paper, we use Suden s -disribuion wih v degrees of freedom, where v is a parameer o be esimaed from he daa. As we will see below, his approach yields a beer fi and hence he resuling GARCH parameer esimaes are more efficien. This paper is organized as follows. Nex secion presens a brief descripion of he mehodology applied. Secion 3 conains he daa descripion and in Secion 4 he empirical resuls are repored. The paper concludes wih a shor summary and suggesions for fuure research in Secion Mehodology In his paper we focus on he Generalized Auoregressive Condiional Heeroskedasic Model (GARCH) developed in Bollerslev (1986). The vas majoriy of empirical sudies use he mehod of maximum likelihood o esimae he parameers of a GARCH model. The usual assumpion is made ha he error erm is condiionally normally disribued, e.g. ε ~ N (0, h ). Then he maximum likelihood esimaor is consisen and asympoically normally disribued even if he rue disribuion is nonnormal. The sandard errors of he esimaes, however, are inconsisen if he rue disribuion is no normal, bu Bollerslev and Wooldridge (1988) derive a robus esimaor of he covariance marix ha gives consisen esimaes of he sandard errors under such circumsances. This mehod is called he quasi-maximum likelihood esimaion (QMLE). Alernaively, one can assume a differen condiional disribuion for he error erm. Following Bollerslev (1987) we assume ha u is condiionally -disribued wih v degrees of freedom, i.e. v + 1 v Γ ( ε h ) ~ f ( ε h ) = Γ ( v 2) h ) v 1/ 2 1+ h ε 2 ( v 2) ( v + 1) /, v > 2 I is well-known propery of he Suden s -disribuion ha as he number of he degrees of freedom increases wihou bound, he -disribuion approaches a nor- PRAGUE ECONOMIC PAPERS, 1, 2004 l 27 2 (1)

3 mal disribuion wih zero mean and variance h, bu for finie v, he -disribuion has faer ails han he corresponding normal disribuion. Hence assuming he condiional -disribuion in connecion wih a GARCH model may beer accoun for he excess kurosis found in he sock marke reurns. Moreover, he esimaed number of degrees of freedom, v^ may indicae he source of he excess kurosis in he uncondiional sock reurns: according o Connolly (1992), if v^ < 10, boh non-normaliy and condiional heeroskedasiciy explain he excess kurosis in reurns, whereas if v^ > 30 condiional heeroskedasiciy is he only source of fa ails in he uncondiional disribuion of reurns. Since he applicaion of a GARCH model requires a properly specified equaion for he mean of reurns, we firs es he random walk hypohesis using a heeroskedasiciy-consisen variance raio es due o Lo and MacKinlay (1988). When he null hypohesis of random walk can be rejeced we consider an ARMA model as an alernaive for modelling he sock reurns. The usual Box-Jenkins mehodology is applied o idenify he model. The behavior of volailiy of reurns over ime is hen sudied wihin he conex of he appropriae mean equaion using he simple GARCH (1,1)- model, which is esimaed by maximum likelihood. Our model is hus summarized by he following equaions: r = µ + φ r ε = u 1 1 h, h = ω + δ h φ r + α ε p p + ε + θ ε θ ε, q q where r denoes he one-period rae of reurn on a marke index and {u } is a whie noise process wih uni variance. We also es for wo alernaives o he simple GARCH model: he hreshold ARCH (TARCH) developed by Zakoian (1994) and Glosen e al. (1993) and he ARCH-in-mean suggesed by Engle e al. (1987). Finally, we employ he BDS es (see Brock e al., 1987) o asses he abiliy of he esimaed GARCH- model o capure all nonlineariies in sock reurns. The lieraure on GARCH modelling of sock reurns from developed markes is voluminous (see Bollerslev e al., 1992, for an overview). The same, however, does no hold for emerging marke in Cenral Europe. In he majoriy of empirical sudies concerning emerging Cenral European sock markes, he focus is on marke efficiency and sock reurn predicabiliy using es procedures correced for general forms of heeroskedasiciy (e.g. Filer and Hanousek, 1996; Gilmore and McManus, 2001). Only in Vošvrda e al. (1998), he behavior of volailiy is sudied in greaer deail, bu he analysis is confined o he Czech sock marke index, PX-50. Hence, in his paper, we aemp o fill his gap and provide a more deailed and accurae picure of he behavior of volailiy of Cenral European sock index reurns. 3. Daa Descripion To perform he analysis of sock reurns on he Cenral European capial markes we focus on he 365-week period saring in January, 1996 and ending in December, We use weekly closing values of he value-weighed indices WIG, BUX and PX-50 of he Warsaw, Budapes and Prague sock exchanges, respecively. For an overview of he insiuional framework, rading sysem, number of lised companies and oher deails on he Cenral European sock markes see Filer and Hanousek (1996) and Gilmore and McManus (2001). For comparison, we also perform he analysis on he DAX index of Deusche Boerse. The daa were downloaded from Bloomberg. 28 l PRAGUE ECONOMIC PAPERS, 1, 2004

4 The choice of weekly daa is imporan for wo reasons. Firs, he effec of nonsynchronous rading induces spurious auocorrelaion ino he index reurns, bu he lower he daa frequency he less imporan his effec is (for deails see Lo and MacKinlay, 1990). Thus o miigae his effec we choose o use weekly raher han daily daa. Second, almos all saisical inferences ha will be drawn in he subsequen secion are based on asympoic heory and hence require a sufficien number of observaions o be reliable. Using monhly daa would herefore no be appropriae due o he shor hisory of he sock markes under sudy. We define he coninuously compounded rae of reurn in week on an marke index MI as r MI = lnmi lnmi 1 where MI = {WIG, BUX, PX50, DAX }. The descripive saisics for he weekly reurns on W, B, P and D are summarized in Table 1. Table 1 Descripive Saisics WIG BUX PX/50 DAX Mean * * * * Variance * * * * Maximum * * * * Minimum * * * * Skewness * * * * Kurosis * * * * Jarque-Bera ** ** ** ** Noe: **significan a he 1% level. The Jarque-Bera es indicaes significan deparures from normaliy for all indices. To learn more abou he shape of he uncondiional densiy funcion we employ a non-parameric kernel densiy esimaor raher han reporing he usual hisogram. Using Epanechnikov kernel and seing he bandwidh o he rule-of humb value proposed by Silverman (1986) we obain esimaes of he uncondiional densiies given in Figures 1 and 2. The esimaed densiy funcions of weekly reurns on he Cenral European sock marke indexes are similar o hose usually obained for indices of developed sock marke (e.g. Fama, 1965). The empirical densiies are lepokuric which has imporan implicaions for furher saisical ess and esimaion. As we noed in earlier, he fa ails of he uncondiional empirical disribuions can be aribued o condiional heeroskedasiciy and/or condiional non-normaliy. PRAGUE ECONOMIC PAPERS, 1, 2004 l 29

5 Figure 1 Kernel Densiy Esimaes 3 10 Y 5 Y X X*E 2 BUX WIG Figure 2 Kernel Densiy Esimaes Y 5 Y X*E X*E 2 PX-50 DAX 4. Empirical Resuls This secion repors he resuls of he analysis of sock reurns performed on he Cenral European sock marke indices. We sar wih he heeroskedasiciy-consisen random walk es developed in Lo and MacKinlay (1988). Nex come he resuls from ARIMA esimaion (when applicable) followed by he ARCH-LM es resuls for condiional heeroskedasiciy. Then we presen he esimaed GARCH-ype models and he likelihood raio ess for he null hypoheses ha a) he residuals from he ARIMA models are condiionally Gaussian and b) he residuals are homoskedasic and uncondiionally Suden s -disribued. We also examine he risk aversion of invesors using he likelihood raio es for he presence of he condiional volailiy erm in he mean equaion for reurns (ARCH-in-mean) and es for he asymmeric impac of shock o volailiy (TARCH). Finally, we asses he abiliy of he proposed GARCH models o remove all non-lineariy in he sandardized residuals using he Ljung-Box Q es and he BDS es. Table 2 summarizes he resuls of he variance raio es applied o he ime series of weekly closing log-values of WIG, BUX, PX-50 and DAX indices. Following 30 l PRAGUE ECONOMIC PAPERS, 1, 2004

6 Lo and MacKinlay (1988), we compue he variance raio s (1 + M^ r (q)) and he corresponding robus es saisics for q = 2, 4, 8, 16. The null hypohesis of random walk is decisively rejeced for he Czech index PX-50. The variance raio s for PX- 50 is larger han 1 for q = 2 implying posiive firs-order auocorrelaion in weekly reurns on his index. The firs-order auocorrelaion coefficien is approximaely equal o 11 %. In case of he Hungarian BUX index, he variance raio for q = 8 is saisically differen from one a he 5 % significance level, hereby rejecing he random walk null hypohesis. For WIG and DAX indices, he null hypohesis of random walk canno be rejeced. Table 2 Random Walk Tes Resuls q = 2 q = 4 q = 8 q = M^ r(q) WIG z*(q) Prob M^ r(q) BUX z*(q) Prob M^ r(q) PX-50 z*(q) Prob M^ r(q) DAX z*(q) Prob The resuls of he random walk es indicae ha he Czech and Hungarian sock conain predicable componens. These findings are in conradicion o an earlier sudy due o Filer and Hanousek (1996) who were no able o rejec he null hypohesis of random walk for PX-50, BUX and WIG using weekly daa for he period saring a he daa each index was firs calculaed and ending in June, Since our sample and ha used by Filer and Hanousek are almos non-overlapping (hey have only 24 common observaions) i appears ha predicabiliy in he Czech and Hungarian sock reurns has increased over he las decade. I may be ineresing o divide our sample ino wo non-overlapping subsamples and run he random walk es independenly in he subsamples. Our sample, however, conains only 365 weekly observaions and hus he size of he subsamples would be small resuling ino he variance raio es having low power. For more on sock reurn predicabiliy on Cenral European sock markes see Žikeš, Since he random-walk model was rejeced for he Czech and Hungarian socks we will now search for a parameric model ha would give us more insigh ino he naure and degree of predicabiliy. We do his by idenifying and esimaing an ARIMA (p,1,q) model using he Box-Jenkins mehodology. For he PX-50 index we find an ARIMA (1,1,1) model o bes describe he daa generaing process and for BUX index, ARIMA (2,1,0) appears o be he bes alernaive. We do no repor he esimaion oupu here since boh models will be re-esimaed allowing for ARCH effecs in he residuals in he subsequen paragraph. 1) Nex we run he ARCH-LM 1) The esimaion oupus are available from he auhors upon reques. PRAGUE ECONOMIC PAPERS, 1, 2004 l 31

7 es for he null hypohesis of no condiional heeroskedasiciy in he residuals from he ARIMA models. For WIG and DAX indices, we apply he es o he residuals from a regression of reurns on a consan only, since for hese indices he null hypohesis of random walk could no be rejeced (e.g. he residuals should approximae a whie noise sequence). We se he order of he es, p = 8. Also we run he Ljung- Box Q-es of order eigh on he esimaed squared residuals. Table 3 summarizes he resuls. Table 3 ARCH-LM Tes Resuls WIG BUX PX-50 DAX ARCH-LM Tes Sa Prob Ljung-Box Q-sa Prob Excep for he Hungarian BUX index, boh ess clearly indicae presence of condiional heeroskedasiciy in he esimaed residuals. In case of he BUX index he null hypohesis of homoskedasiciy is rejeced a he 10% significance level using he Ljung-Box Q-es bu he ARCH-LM es fails o rejec he null a he convenional significance levels. Hence he presence of condiional heeroskedasiciy in he residuals from he ARIMA (2,1,0) model for BUX index has o be furher invesigaed. We now urn o modelling condiional heeroskedasiciy in he index reurns. In paricular, we assume he GARCH (1,1) model for he residuals from he corresponding mean regression equaions. I urns ou ha his model has superior performance in comparison o oher GARCH specificaions. Thus for he Czech PX-50 index we esimae an ARIMA (1,1,1)-GARCH (1,1) model, for he BUX index an ARI- MA (2,1,0)-GARCH (1,1) model and for WIG and DAX indices we assume a random walk model whose incremens obey a GARCH (1,1) process. We esimae hese models by he mehod of maximum likelihood under he assumpion ha he residuals are Suden- disribued wih v degrees of freedom. 2) The esimaed coefficiens are asympoically normally disribued. Table 4a repors he esimaed random walk model wih condiionally heeroskedasic incremens for he Polish WIG index. Table 4a GARCH (1,1)- Model for WIG Coefficien Sd. error z-sa. Probabiliy µ ϖ δ α v n.m. n.m. 2) The BHHH algorihm is employed o maximize he likelihood funcion. 32 l PRAGUE ECONOMIC PAPERS, 1, 2004

8 The esimaed GARCH coefficiens δ 1 and α 1 are significan a he 1% and 10% level, respecively and heir sum is less hen one implying ha he GARCH model is saionary, hough he volailiy is fairly persisen since (δ 1 + α 1 ) is close o one. The esimaed number of degrees of freedom of he condiional -disribuion is equal o 6.5 which suggess ha he reurns on he WIG index are condiionally non-normally disribued. To summarize, boh heeroskedasiciy and non-normaliy accoun for he fa ails in he empirical disribuion of he reurns on WIG. Table 4b GARCH (1,1)- Model for BUX Recall ha boh he Ljung-Box Q es for serial correlaion in he squared residuals and he ARCH-LM es failed o rejec he null hypohesis of no ARCH effecs in he residuals from he ARIMA (2,1,0) model for he Hungarian BUX index. From he esimaion oupu, repored in Table 4b we see, however, ha he GARCH coefficiens δ 1 and α 1 are saisically significan a he 1% and 10% level, respecively. The esimaed number of degrees of freedom is equal o 4.2 which is a very small value given he fac ha for v 4 he kurosis of a Suden- random variable wih v degrees of freedom is infinie. Bu he low value of v^ is no surprising since he empirical kurosis of he BUX reurns is quie high, equal o Thus, as wih he WIG index, he lepokurosis in he empirical disribuion funcion of he reurns on he BUX index is induced by boh heeroskedasiciy and non-normaliy. Table 4c GARCH (1,1)- Model for PX-50 Coefficien Sd. error z-sa. Probabiliy µ φ θ ϖ δ α v n.m. n.m. R 2 = Coefficien Sd. error z-sa. Probabiliy µ φ ϖ δ α v n.m. n.m. R 2 = PRAGUE ECONOMIC PAPERS, 1, 2004 l 33

9 In Table 4c, here is summarized he esimaed ARIMA (1,1,1)-GARCH (1,1) model for he Czech index, PX-50. We obain very similar resuls o hose for WIG and BUX in ha he GARCH parameers δ 1 and α 1 are saisically significan a he 1% and 10% level, respecively, and v^ is much less han en. Noe ha alhough he GARCH (1,1) model is saionary, δ 1 + α 1 = 0.983, i.e. he volailiy of he reurns on PX-50 is very persisen. Table 4d GARCH (1,1)- Model for DAX Coefficien Sd. error z-sa. Probabiliy µ ϖ δ α v n.m. n.m. Finally, we examine he German index DAX (see Table 4d.). Surprisingly, he esimaed number of degrees of freedom of he condiional Suden- disribuion for reurns is quie large compared o hose for he marke indices of he Cenral European sock markes. I falls ino he inconclusive region defined in Connolly (1992) and hence i is difficul o commen on he disribuional properies of reurns wihou furher esing. Inuiively, since he GARCH parameers are highly significan and he volailiy of reurns is very persisen (δ 1 + α 1 = 0.993) i may be he case ha he excess kurosis in he DAX reurns is induced primarily by condiional heeroskedasiciy. To es his hypohesis rigorously, we employ a likelihood raio es. The Table 5 presens he likelihood raios for a) he null hypohesis ha he reurns on he corresponding marke indices are condiionally normally disribued (1/v = 0) and b) he null hypohesis of uncondiionally Suden- disribued reurns wih v degrees of freedom (α 1 = δ 1 = 0). The likelihood raios are asympoically χ 2 -disribued wih a) one, and b) wo degrees of freedom. The 5% criical values are hus 3.84 and 5.99, respecively. Noe, however, ha according o Bollerslev (1987) he disribuion of he es saisic LR 1/v=0 in samples of moderae sizes ends o be more concenraed owards he origin han he χ 12 -disribuion and hence a es based on he 3.84 criical value has lower size (is more conservaive). Bollerslev(1987) provides he appropriae 5% criical value, Table 5 LR Tes Resuls v^ LR 1/v=0 LR α1 = δ 1 = 0 WIG * 16.04* BUX * 16.52* PX * 13.76* DAX * 61.37* Noe: *significan a he 1% level. 34 l PRAGUE ECONOMIC PAPERS, 1, 2004

10 The null hypohesis of uncondiionally Suden- disribued is decisively rejeced for all four marke indices. These findings are in line wih our esimaes of he GARCH parameers: hey all were saisically significan. Hence, condiional normaliy is rejeced for he Cenral European sock markes indices. The likelihood raio es fails o rejec he null for he German DAX. Again, hese resuls are in accordance wih our esimaes of v: he smaller he v^, he larger he likelihood raio for he null hypohesis of condiional normaliy. To summarize, on he basis of he GARCH- model and he likelihood raios we have found ha in case of he reurns on he Cenral European sock marke indices, boh condiional heeroskedasiciy and non-normaliy accouns for excess kurosis in he empirical disribuion funcions of index reurns. For he German index DAX, he picure is less clear cu, hough i appears ha he primary source of he fa ails in he empirical disribuion is imevarying volailiy. Nex we es he simple GARCH model agains wo alernaive specificaions. Firs, we considered an asymmeric impac of posiive and negaive shocks o reurns on heir expeced volailiy (TARCH). Second, we included he expeced volailiy erm, h, ino he mean equaion for reurns (ARCH-in-mean). Boh alernaive specificaions can be esed using he usual likelihood raio. Table 6 repors he likelihood raios for a) he null hypohesis ha he shocks o reurns have a symmeric effec on fuure volailiy (λ = 0) and b) he null hypohesis of risk-neuraliy of invesors (π = 0). Under boh null hypoheses he likelihood raios are asympoically χ 2 disribued wih one degree of freedom, hence he 5% criical value is equal o Table 6 LR Tes Resuls LR λ=0 LR π=0 WIG * BUX n.m. 2.21* PX * DAX n.m. 0.01* Noe: *significan a he 5% level. Noe ha we do no repor he values of LR λ=0 for BUX and DAX indices because he esimaes of he consan erm ϖ in he TARCH variance equaion are negaive which violaes he assumpion ha all GARCH coefficiens are posiive (a negaive Table 7 GARCH-M- Model for WIG Coefficien Sd. error z-sa. Probabiliy µ π ϖ δ α v n.m. n.m. PRAGUE ECONOMIC PAPERS, 1, 2004 l 35

11 ϖ may cause h o be negaive for some which is no consisen wih he non-negaiviy propery of variance). For WIG and PX-50 he null hypohesis ha he shocks o reurns have symmeric impac on volailiy canno be rejeced on he convenional significance levels. Turning o he null hypohesis of risk-neuraliy, he likelihood raio es fails o rejec he null for BUX, PX-50 and DAX, bu does rejec i a he 1% significance level for he Polish WIG index. Since he null hypohesis of risk neuraliy was rejeced in case of he WIG index, we now esimae he ARCH-in-mean model assuming Suden- disribuion of he error erms. Table 7 summarizes he esimaion oupu. The esimae of π, which measures he degree of risk aversion, is posiive and significan a he 5% level. I implies ha he invesors rading on he Polish sock marke are risk-averse: hey require higher reurns in periods when hey expec higher volailiy (risk). 3) The fac ha expeced volailiy, h, is significan in he mean equaion has also imporan implicaions for reurn predicabiliy. To see his, recall ha h is a funcion of pas squared residuals from he mean equaion. The expeced reurn on he WIG index for period +1 based on he informaion available in period, can be hus wrien as E [r + 1 Φ ] = µ + πh + 1, (2) = µ + π(ϖ + δ 1 h + α 1 ε 2 ), (3) = µ + π(ζ + ψ(l)ε 2 ), (4) where ψ(l) is an infinie-order polynomial in he lag operaor. 4) Now because boh h and ε 2 are known a ime he reurn for period + 1 is predicable from he ime Table 8 Diagnosic Tess WIG BUX PX-50 DAX Q(4) * 6.39 Q(8) * Q(12) * Q 2 (4) * 1.60 Q 2 (8) * 4.67 Q 2 (12) * 4.93 BDS(2) * 0.58 BDS(4) * 0.09 BDS(6) * 0.57 Noe: *significan a he 5% level. 3) There has been evidence in he lieraure on he ARCH-in-mean models of he sensiiviy of he parameer esimaes o he disribuional assumpions made o obain he esimaes by MLE (see Bollerslev e.a., 1992, for an overview). I has been repored, for example, ha by changing he condiional disribuion from normal o Suden-, he esimaed π dropped subsanially and was no-longer significan. When we use normal disribuion o esimae he ARCH-in-mean model for he WIG index, we obain π^ = 7.73 wih (Bollerslev and Woodridge, 1991) robus p-value equal o Thus he sensiiviy observed on developed markes carries over o he Polish sock marke. 4) We assume ha δ 1 < 1 and hence he GARCH (1,1) process can be expressed as an infinie order ARCH. 36 l PRAGUE ECONOMIC PAPERS, 1, 2004

12 series of hisorical volailiy and pas squared residuals. The conclusion ha WIG s reurns are predicable is in conradicion o he resul of he robus variance raio es for he null hypohesis of random walk. Bu his should no be surprising. The variance raio es, by consrucion, has power only agains linear dependencies in he ime series of reurns. In an ARCH-in-mean model here is a correlaion beween curren reurn and lagged squared disurbances. I follows ha he variance raio es canno deec such non-lineariies and hence fails o rejec he null hypohesis of random walk. 5) The ARCH-in-mean model herefore provides us wih an imporan insigh ino he ime-series predicabiliy of sock reurns. Finally, having esimaed he corresponding ARIMA-GARCH models for he sudied sock marke indices, we should perform diagnosic esing o asses wheher he models are correcly specified. We sar by esing he sandardized residuals for serial correlaion. This may seem superfluous because he ARIMA models were specified in a way ha would remove possibly all auocorrelaion in he residuals. Bu because we did no repor he esimaes and diagnosic checks before for he sake of breviy, we should do i now. The firs hree rows of Table 8 herefore repor he Ljung-Box Q saisics applied o he sandardize residuals from he models of Tables 4b, c, d and 7. Clearly, he null hypohesis of no serial correlaion canno be rejeced on he convenional significance levels and hence he mean equaions appear o be correcly specified. Nex we run he Ljung-Box Q es on he squared sandardized residuals (we label he es saisic Q 2 o emphasize ha he es is run on squared sandardized residuals). The null hypohesis of no serial correlaion in squared sandardized residuals can be rejeced only for he PX-50 index. Hence he GARCH (1,1) model in his case fails o appropriaely fi he volailiy process. No reasonable alernaive specificaions, however, were found o perform beer. 6) Finally, we apply he BDS es on he sandardized residuals o deermine, wheher he GARCH model removes all non-lineariies in he ime-series of reurns. We se he proximiy parameer o 0.5 and run he es for embedding dimensions 2, 4, and 6. The null hypohesis ha he sandardized residuals are independenly idenically disribued can be rejeced a 5 % for PX-50 index only, which is consisen wih he findings of he Ljung-Box Q es. To summarize, he diagnosic checks performed on he sandardized residuals provide convincing evidence ha, excep for he PX-50 index, he ARIMA-GARCH models were appropriaely specified and can hus be used for raional predicions of boh index reurns and heir volailiy. 5. Conclusion The aim of his paper has been o invesigae he behavior of volailiy and he disribuional properies of he Cenral European sock index reurns. Using weekly daa for he period 1996:1 hrough 2002:12, we focused on he Czech, Hungarian and Polish sock markes and considered he German marke as a benchmark. We exended he previous empirical work by explicily modelling he fa ails in he un- 5) To deec non-lineariies in sock reurns, we could also apply he BDS es o he reurns. Bu since we have seen ha he sock reurns are heeroskedasic he resul of he BDS es would be necessarily ambiguous: we would no know wheher we should aribue he rejecion of he iid null hypohesis o nonlineariies in he mean or non-lineariies in he variance. 6) The correlogram of he squared sandardized residuals reveals ha here is a significan spike in boh he auocorrelaion and parial auocorrelaion funcions a lag 7. This indicaes ha a GARCH (1,8) may beer fi he daa. Some of he esimaed coefficiens in his model are, however, negaive, which is ruled ou by assumpion. The esimaes are available from he auhor upon reques. PRAGUE ECONOMIC PAPERS, 1, 2004 l 37

13 condiional disribuions of sock reurns using a GARCH-ype model wih Suden- disribued disurbances. To briefly summarize our analysis, we found ha: he Cenral European sock index prices do no follow random walk, sock index reurns are predicable from he ime-series of hisorical reurns using a linear (ARMA) or non-linear (ARCH-in-mean) parameric model, sock reurn volailiy changes over ime and can be well described by a GARCH-ype model, empirical densiy funcions of sock reurns exhibi excess kurosis ha can be aribued o boh condiional heeroskedasiciy and non-normaliy. We hasen o emphasize ha he predicabiliy of sock reurns need no be a sympom of marke inefficiency. Even in an efficien marke sock reurns may be correlaed simply because he expeced reurn sysemaically changes over ime. Bu our analysis did no sudy he way expeced reurn is deermined a all and hence any conclusion regarding he EMH is impossible. Our empirical findings have also imporan implicaions for asse pricing. Consider, for simpliciy, he well-known Black-Scholes model for pricing European syle call opions on a non-dividend paying sock. This model is derived under hree crucial assumpions: a) log-sock prices follow random walk, b) reurns are normally disribued, and c) volailiy is consan hrough ime. Bu none of hese assumpions is saisfied empirically, so he Black-Scholes canno be used for raional pricing of opions on he Cenral European sock marke indices. References Bollerslev, T. P. (1986), Generalized Auoregressive Condiional Heeroskedasiciy. Journal of Economerics, 31, pp (1987), A Condiionally Heeroskedasic Time Series Model for Speculaive Prices and Raes of Reurn. The Review of Economics and Saisics, 69, pp Bollerslev, T., Chou, R., Kroner, K. (1992), ARCH Modelling in Finance. Journal of Economerics, 52, pp Bollerslev, T. P., Wooldridge, J. M. (1992), Quasi-Maximum Likelihood Esimaion and Inference in Dynamic Models wih Time Varying Covariances. Economeric Reviews, 11, pp Brock, W., Decher, W., Scheinkman, J. (1987), A Tes for Independence Based on he Correlaion Dimension. Universiy of Wisconsin a Madison, Deparmen of Economics, Working Paper. Cochrane, J. H. (2001), Asse Pricing. Princeon: Princeon Universiy Press. Connolly, R. A. (1989), An Examinaion of he Robusness of he Weekend Effec. Journal of Financial and Quaniaive Analysis, 24, pp Engle, R., Lilien, D. M., Robins, R. P. (1987), Esimaing Time-Varying Risk Premia in he Term Srucure: The ARCH-M Model. Economerica, 55, pp Fama, E. F. (1965), The Behavior of Sock Marke Prices. Journal of Business, 38, pp Filer, R. K., Hanousek, J. (1996), The Exen of Efficiency in Cenral European Equiy Markes. Cener for Economic Research and Graduae Educaion, Charles Universiy, Prague, Working Paper. Gilmore, C. G., McManus, G. M. (2001), Random-Walk and Efficiency Tess of Cenral European Equiy Markes. European Financial Managemen Associaion Conference, Lugano. Glosen, L. R., Jaganahan, R., Runkle, D. (1993), On he Relaion beween he Expeced Value and Volailiy of he Normal Excess Reurn on Socks. Journal of Finance, 48, pp Lo, A. W., MacKinlay, A. C. (1988), Sock Prices Do No Follow Random Walk: Evidence from a Simple Specificaion Tes. Review of Financial Sudies, 1, pp (1990), An Economeric Analysis of Nonsynchronous Trading. Journal of Economerics, 45, pp Mandelbro, B. (1963), The Variaion of Cerain Speculaive Prices. Journal of Business, 36, pp l PRAGUE ECONOMIC PAPERS, 1, 2004

14 Silverman, B. W. (1986), Densiy Esimaion for Saisics and Daa Analysis. New York: Chapman and Hall. Vošvrda, M., Filáček, J., Kapička, M. (1998), The Efficien Marke Hypohesis and Volailiy: he Case of Prague Sock Exchange. Bullein of he Czech Economeric Sociey, 7, pp Zakoian, J. M. (1994), Threshold Heeroskedasic Models. Journal of Economic Dynamics and Conrol, 18, pp Žikeš, F. (2003), The Predicabiliy of Asse Reurns: An Empirical Analysis of Cenral European Sock Markes. Charles Universiy, Prague, Maser Thesis (hp://ies.fsv.cuni.cz/knih/papers/zikes.pdf). PRAGUE ECONOMIC PAPERS, 1, 2004 l 39