# ACTA UNIVERSITATIS LODZIENSIS Piotr Wdowiński*, Aneta

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ACTA UNIVERSITATIS LODZIENSIS Piotr Wdowiński*, Aneta

ACTA UNIVERSITATIS LODZIENSIS FOLIA OECONOMICA 192, 2005 P io tr W d o w iń ski* , A n e t a Z g l i ń s k a - P i e t r z a k ** TH E W ARSAW STOCK EXCHANGE INDEX WIG: M ODELING AND FORECASTING*** Abstract. In this paper we have assessed an influence of the NYSE Stock Exchange indexes (DJIA and NASDAQ) and European Stock indexes (D AX and FTSE) on the Warsaw Stock Exchange index WIG within a framework of a GARCH model. By applying a procedure of checking predictive quality of econometric models as proposed by Fair and Shiller (1990), we have found that the NYSE market has relatively more power than European market in explaining the WSE index WIG. Keywords: Warsaw Stock Exchange, stock index, GARCH model, forecasting. JEŁ Classification: C2, C5, C6, G l. 1. INTRODUCTION T h e problem o f searching for the influence o f large m ark ets on others has been well-know n from literature for years. H ow ever, this kind of relationship has been widely studied through analyzing correlations between individual indexes representing different m arkets (e.g. E rb et al. 1994; Bracker and K o ch 1999). In this paper we try to find such dependence between W IG index and foreign stock m ark et indexes (D JIA , N A S D A Q and D A X , F TSE) by estim ating param eters in regressions o f W IG . T h e estim ates show which foreign stock m ark et affects Polish stock exchange index m ore strongly. W e also test it by the forecasting approach using forecast errors o f W IG index obtained in individual regressions. We use also the idea o f com bined forecasts. * Dr (P h D ., Assistant Professor), Department of Econometrics, University of Łódź. ** Dr (Ph.D., Assistant Professor), Department of Econometrics, University of Łódź. *** We are grateful to our colleagues from the Department o f Econometrics, University of Łódź, and participants of FindEcon 2004 conference for helpful comments and suggestions on earlier draft of the paper. T here are m any m ethods used to com bine forecasts (e.g. C lem en 1989; G ran g er 1989). As show n by Clemen and W inkler (1986), simple com bination m ethods often work better than m ore com plex approach. The aim o f com bining forecasts is to investigate w hether the forecast com bin ation plays an im portant role in the im provem ent o f forecasting ac curacy. T he well-know n way o f com bining forecasts is to com pute linear com bination o f forecasts generated by alternative m odels or obtained by using different forecasting m ethods (e.g. Billio ct al. 2000, Claesscn and M ittnik 2002). D ependence between stock m arkets in different countries has been tested for years. M any analyzes deal with m easuring correlation between returns and diversified international portfolios (e.g. G rubel 1968, Levy and Sarnat 1970, A gm on 1972, Fiszeder 2003). In the 1990s there appeared research o f how changes to stock prices on one m ark et affect o th er m arkets (e.g. H am ao et al. 1990, K ing and W adhw ani 1990, Engle and Susmel 1993, Fiszeder 2001). T he focus o f this paper is to find the influence o f A m erican and E u ropean indexes on W arsaw Stock Exchange index W IG . T here is a lot o f research which prove th at this influence does exist. W e aim to examine which m ark et - A m erican o r E uropean - has a stronger im pact on W IG index. T h e p aper is structured as follows. In Section 2 we give a brief overview o f the G A R C H m ethodology. In Section 3 we test for influence o f foreign stock indexes on W IG index. Com bined forecasting o f W IG index is applied in Section 4, and finally we give concluding rem arks. 2. THE GARCH METHODOLOGY M any m odels have been proposed to describe volatility o f returns. N ow there is a com prehensive literature with several specifications o f autoregression m odels. M any em pirical analyzes, however, have show n th a t G A R C H ap p ro ach is the m ost appropriate. We also apply G A R C H m odeling in this paper. T his is the m ost po p u lar class o f m odels used in m odeling the financial tim e series o f high frequency (e.g. A kgiray 1989, Schwert and Seguin 1990, N elson 1991, A ndersen et al. 1999, Osiewalski and Pipień 1999 and 2004, Bollerslev and W right 2000, Fiszeder 2001 and 2003, Brzeszczyński and Kelm 2002, D om an M . and D om an R. 2003). T he G A R C H m odel has been proposed independently by Bollerslev (1986) and T ay lor (1986) as a generalization o f A R C H m odel introduced by Engle (1982). T h e m ain feature o f A R C H m odel is to describe the conditional variance as an autoregression process. H owever, m ost em pirical tim e series require using long-lag length A R C H m odels and a large num ber o f param eters m ust be estim ated. T he solution o f the problem was G A R C H m odels which gave better results (cf. Engle and Bollerslev 1986; N elson 1991). T he basic linear generalized autoregressive conditional heteroscedastic GARCHQ?, q) m odel is given as follows (e.g. Bollerslev 1986): (1) Уг = x (k)l®k + ev where: (2) e, = 9, %/h t, and h, is a function o f conditional variance represented as: (3) h, = y0 + X i=i YiŹ-i + Z Vj K - j, i where: 3, is i.i.d. with £ (9 ,) = 0 and v a r ( 9 , ) = l , yo >0* p ( p ^ 0. If я £ y i + Y j (Pj< 1* ^ e process h, is covariance stationary. i=i j= i Engle and Bollerslev (1986) considered also G A R C H processes with X ľ i + Y j V j — Ь which they denoted integrated G A R C H (IG A R C H ). In em pirical research the m ost frequently used is G A R C H (1, 1) m odel in w hich h, = y 0 + y iE ? - i + <Piht ~ i a n ^ Уо > Уi ^ 9 > i^ 0 . W hen У\ + (P\ < 1, th an unconditional variance o f et is given by var (e,) = . 1 1 i-Y i-V i T he coefficients o f the m odel are then easily interpreted, w ith the estim ate o f yt show ing the im pact o f current news on the conditional variance process and the estim ate o f <pt as the persistence o f volatility to a shock or, alternatively, the im pact o f “ old ” news on volatility. Recently a num ber o f new form ulations have been proposed, for example exponential G A R C H (p, q) m odel (E G A R C H , e.g. N elson 1991), G JR G A R C H m odel (e.g. G losten et al. 1993), G A R C H in the m ean (G A R C H M(p, q), e.g. Engle et al. 1987), power G A R C H (P G A R C H , e.g. D ing et al. 1993), and the fractionally integrated G A R C H (F IG A R C H , e.g. Baillie et al. 1996). Such m odels are com m only applied in financial tim e series research. The estim ation o f G A R C H m odels is bo th classic and non-classic, e.g. Bayesian ap p ro ach (e.g. Osiewalski and Pipień 1999, 2004). In Section 3 we test m odels for influence o f foreign stock m arkets on the Polish m arket. 3. TESTING FOR INFLUENCE OF FOREIGN STOCK INDEXES ON WIG INDEX In the p ap er we use a G A R C H m ethodology with G A R C H (1,1) models. We have found th at both A R C H and G A R C H effects appeared to be statistically significant in the dependence tested. T he focus o f the paper is to find the influence o f A m erican and European stock m ark et indexes on W IG index over the period 1.01.1995 to 29.12.2003 (2346 observations). A fter introducing suggested foreign indexes to the G A R C H (1, 1) equation describing W IG index, it was im possible to separate m utual relationships between index W IG and foreign indexes, m ainly D A X and F T SE , because the estim ates used to be statistically insignificant and had opposite signs which negated the assum ptions o f positive influence o f the biggest stock m ark ets on index W IG . Hence, we were unable to include both European indexes D A X and FT SE in one equation, and we decided to test for the influence o f E u ro p ean m arkets by two equations accordingly. In tu rn , we analyze the following models: • for A m erican m arket: • (4) • (5) • • wigt = a 0 -f a ^ j i a , - t + tx2nasdaq,^l + e(, for E u ro pean m arket: • • wig, = ot0 + tx ^a x , _ x + e(, or (6) • • wig, = a 0 + c tjts e t- i + e„ where the variables are first differences o f natural logarithm s, so they are the close-to-close returns on corresponding indexes. T h e estim ation was m ade by m axim um likelihood m e th o d 1 for daily d ata from 1.01.1995 to 29.12.2003. T he equations (4), (5) and (6) were recursively estim ated and helped to o b tain a series o f 250 one-period-ahead quasi ex-ante forecasts. T he estimates 1 We used Econometric Views package. o f individual equations were generally statistically significant. T he sum у1 + ф1 in the eq u atio n o f conditional variance was around 0.92, while the estim ates were ab o u t 0.12 and ф1 ab o u t 0.80. Hence, we can conclude th at the im pact o f cu rren t news on volatility in the conditional variance process is sm aller th a n the im pact o f “ old ” news. T he coefficient o f determ ination R -squared for corresponding equations (4), (5) and (6) was ab o u t 12%, 3% and 3% respectively. In T able 1 we give m easurem ent o f ex-post errors2 for forecasts obtained from respective equations o f index W IG . Table 1. Ex-post errors o f index WIG forecasts Specification DJIA-NASDAQ DAX FTSE MAE 0.00922 0.00926 0.00929 RMSE 0.01193 0.01197 0.01213 0.75195 0.82760 0.84897 n 0.005 0.004 0.005 n 0.411 0.575 0.582 n 0.588 0.425 0.417 THEIL T P\ (%) 19.69 13.39 15.75 TP2 (%) 50.21 43.57 47.72 In the analysis o f ex-post errors we used turn in g points test statistic - T P 1 - represented as: n\ T P I = N ( y * > 0 л r t-ir ,* -! > 0 |у , - 1 < 0) N (rtr , - i < 0 ) where: N (rtrt_ 1 < 0) - a num ber o f turning points in em pirical series; N (r tr* > 0 л r,_ t r*_! > 0 |r rr,_ i < 0) - a num ber o f po in ts, in which changes to direction in em pirical and forecasting series are the sam e in the period t and t — 1 under the condition th at the points are turning points in em pirical series. W e also used the following direction quality m easure (e.g. Welfe and Brzeszczyński 2000): 2 We used the following ex-post errors: MAE - mean absolute error, RMSE - root mean square error, THEIL - Theiľs inequality coefficient, /1, 12, 13 - decomposition of Theil’s inequality coefficient, TP 1 and TP2 - turning points test statistics. N (r ,r * > 0 ) (8) TP1 = N ш(r,r, * 0 ) where: N (rtr* > 0) - a num ber o f points in which direction changes in em pirical and forecasting series are the same. T he m ost accurate forecasts we obtained from the equation with American indexes. F o recasts basing on E uropean indexes, however, were close to each o th er as far as errors are concerned. In Section 4 we apply a m ethodology o f com bined forecasting. We test for relative im portance o f foreign stock m arkets in explaining W IG index. 4. COMBINED FORECASTS OF WIG INDEX In o rd er to assess the pow er o f influence o f foreign m arkets on the Polish m ark et we estim ated the following F air and Shiller (1990) equation (e.g. W dow iński 2004): (9) y , ~ y t-1 = + yi - i) + <*2(t-iý2t - yt- i ) + et where - denotes one-period-ahead forecasts o f y, generated by the m odel 1, i.e. the m odel w ith A m erican indexes based on inform ation available up to the m om ent t — 1 with the use o f recursive estim ation for each period t. T he predictor , - i ý 2t - denotes one-period-ahead forecasts g enerated accordingly by the m odel 2, i.e. th e m odel w ith E uropean indexes, while e is an erro r term , e ~ /N (0 , o f). If neither m odel 1, nor m odel 2 co ntain any relevant inform ation in term s o f forecasts quality for variable у in period t, the estim ates o f and a 2 will be statistically insignificant. If both m odels generate forecasts th at contain independent in form ation, the estim ates o f a.l and a 2 should both be statistically sig nificant. If b oth m odels contain inform ation but inform ation contained in forecasts generated by m odel 2 is completely contained in forecasts gene rated by m odel 1 and furtherm ore m odel 1 contains additional relevant in form ation, the estim ate o f ol1 will be statistically significant while the estim ate o f a 2 statistically insignificant. If both forecasts contain the same inform ation, they are perfectly correlated and the estim ation o f param eters o f (9) is n o t possible. Because th e influence o f E u ro p e an m ark e ts was described by two equations, we estim ated m odel (9) for tw o cases: • for em pirical W IG index and forecasts generated by (4) and (5). • for em pirical W IG index and forecasts generated by (4) and (6). Below in T ab le 2 we present estim ation results o f equation (9). Table 2. Estimation results of Fair and Shiller equation In te r cept D JIA -N A S D A Q DAX FT S E 0.0007 1.0018 0.5495 2.105 0.3989 1.4923 X J 0.0007 1.0226 0.6523 3.1112 J i 0.2915 1.3777 W ald (U SA ) W ald (E U R ) 0.0119 0.2211 0.3888 4.4313 0.6407 2.0719 (0.6382) (0.4214) (0.0353) (0.7258) (0.1356) 0.0119 0.3223 4.073 0.8254 9.6799 2.0818 (0.5702) (0.3961) (0.0019) (0.6618) 1.8982 (0.1683) s. J-B D-W ARCH WHITE 2.227 R2 (adj.) Obs. 0.4429 250 0.4422 250 W ith italics we have denoted i-statistics with regard to estimates. Respective test probabilities with regard to test statistics are given in brackets. We applied the following tests: Jarque-B era norm ality o f residuals test (J-B), conditional heteroscedasticity test (A R C H ), W hite’s test for heteroscedasticity (W hite), W ald coefficient restrictions test (W ald). T he D -W stands for D u rb in -W atso n test statistic. T he test statistics and their probabilities in case o f J-B, D -W , A R C H and W hite’s tests denote th at errors in both m odels are n orm al, with no au to co rrelatio n , no A R C H effects and no heteroskedasticity. T hus we can test the pow er o f influence o f foreign indexes on W IG index using tstatistic. It can be easily seen th at the influence o f A m erican m ark e t indexes D JIA and N A S D A Q turned ou t to be m ore relevant for W IG index than E u ro p ean indexes D A X and F T SE . It is supported by the significance of estim ates and their size. W e can conclude, th at inform ation contained in forecasts generated by m odels (5) o r (6) is fully contained in forecasts by m odel (4). Therefore, A m erican indexes D JIA and N A S D A Q were m ore influential th an E uropean indexes (D A X or F T S E ) as regards W IG index. T his conclusion is also confirm ed by W ald coefficient restrictions test, i.e. we should reject the null th at the respective coefficient equals zero in case o f m odel (4) and should not reject the null in case o f m odels (5) or (6). Sim ilar conclusions about the co-dependence o f m arkets in case o f Poland were d raw n by e.g. Fiszeder (2003). Since the p aram eters in equation (9) do n o t sum up to one, we should n o t tre a t them as weights in calculating com bined forecasts. T herefore, to calculate the weights, we used the nonlinear program m ing problem (NLP): m in /(co) = <o7 Vm, ( 10) ю г1 = 1, to > 0, where: ю = Г( o . l - is the vector o f weights; La>2\ V - the variance-covariance m atrix o f forecast errors. In T able 3 we present covarianccs and correlations o f forecasts errors. Table 3. Covariance and correlation coefficients matrix of forecast errors Covariance Specification DJIA-NASDAQ DAX FTSE DJIA-NASDAQ DAX FTSE 0.0001417 0.0001380 0.0001375 0.0001427 0.0001427 0.0001427 Correlation DJIA-NASDAQ DAX FTSE 1 0.9708 0.9546 1 0.9875 1 A fter solving the problem o f nonlinear program m ing (10) we obtained the follow ing weights which are given in T able 4. Table 4. The weights in a NLP problem DJIA-NASDAQ DAX FTSE 0.5664 0.4336 X 0.6734 X 0.3266 As we can notice the weights are close to estim ated param eters in equation (9). Thus we confirmed earlier conclusions about the strong influence o f A m erican m arket on the Polish stock m arket. As show n in T able 5, we could n o t substantially reduce the variance o f com bined forecast errors, because o f the high and positive correlation between forecast errors from individual m odels (4), (5) and (6). Table 5. Variance of forecast errors Individual forecast 0.0001417 0.0001427 0.0001427 DJIA-NASDAQ DAX FTSE Combined forecast 0.0001406 0.0001409 DJIA-NASDAQ-DAX DJ1A-NASDAQ-FTSE A fter calculating optim al weights, we calculated com bined forecasts and assessed th eir accuracy. T he results are given in T ab le 6. Table 6. Ex-post errors of combined forecasts of WIG index Specification DJIA-NASDAQ MAE 0.00918 0.00916 RMSE 0.01186 0.01188 THEIL 0.78937 0.79123 n 0.005 0.005 4 0.514 0.515 4 0.537 0.534 DJIA-NASDAQ-FTSE TP\ (%) 15 17 TP2 (%) 48.55 48.96 It is clearly obtained from Finally, we indexes o f both equations: show n th a t com bined forecasts are n o t superior to forecasts each m odel separately. tested the forecasting quality o f the equations including foreign m arkets at the same time. We estim ated the following (11) • • • • wig, = a 0 + ctydjia,-1 + &2nasdaq,-! + a 3dax,_ i + e, (12) wig, = ß 0 + ß l d jia ,-l + ß 2nasdaqt- i + ß 1f t s e t- 1 + Zt. • • • • T h e estim ates o f a 3 and ß 3 describing the influence o f E u ro p ean m arket were usually insignificant. Also, as before, we obtained opposite signs o f estim ates which does not com ply with the postulate o f positive influence o f the biggest stock m arkets on W IG index. By applying the all-index approach itself we were no t able to determ ine the pow er o f influence o f individual m arkets on the Polish m arket due to high correlatio n coefficients between indexes and difficulties in estim ating the param eters. Therefore, the com bined forecasts approach suggested in the paper can be considered correct. D espite questionable statistics in results o f regressions (11) and (12) and their weak econom ic properties we used them to forecast W IG index. The errors o f these forecasts are shown in T able 7. Table 7. Ex-post errors of forecasts obtained in all-index equations Specification DJIA-NASDAQ-DAX DJIA-NASDAQ-FTSE MAE 0.00918 0.00932 RMSE 0.01191 0.0123 THEIL 0.75361 0.7531 n 0.005 0.005 n 0.42 0.394 4 0.579 0.606 TP 1 (%) 21.26 19.69 TP2 (%) 51.04 50.21 A s in the case o f com bined forecasts they are no t o f better quality than forecasts obtained from individual m odels separately. 5. CONCLUSIONS In this p ap er we attem pted to assess the influence o f A m erican and E u ro p ean stock m arkets on the Polish stock m arket. In o u r analysis we used the G A R C H m ethodology which is popular to describe the financial phenom ena o f high frequency. We have found th at the situation on Am erican m ark et had m ore influence on the Polish m arket than the situation on E u ro p ean m ark ets in the analysed period during 1.01.1995 - 29.12.2003. We based these conclusions on the analyzis o f forecasting equations for W IG index and on the analysis o f com bined forecasting. O u r results are consistent w ith those obtained by other authors for the Polish stock m arket (e.g. Fiszeder 2001 and 2003; Brzeszczyński and K elm 2002). W e should notice, however, a few aspects which can affect conclusions d raw n from this kind o f research. F irstly, it m ust be rem em bered th at there is a strong co-dependence am ong the biggest stock m arkets so it is difficult to show the im pact of one individual m ark et on another one in a sim ple approach. Secondly, it is very im portant to choose the proper indexes as determ inants o f m ark et developm ent. It is necessary, then, to test also the influence of o th er indexes and stock m arkets - including em erging m arkets - on the Polish m arket. 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P io tr W dow iński, A n eta Z g liń sk a -P ietrza k MODELOWANIE I PROGNOZWANIE INDEKSU WIG (Streszczenie) W artykule podjęliśmy próbę oceny wpływu indeksów rynku amerykańskiego DJIA i NASDAQ oraz indeksów rynku europejskiego DAX i FTSE na indeks WIG z giełdy w Warszawie. D o modelowania tego wpływu wykorzystaliśmy metodologię GARCH. Stosując metodologię łączenia prognoz oraz metodologię oceny jakości prognostycznej modeli ekono metrycznych, zaproponowane w pracy Fair i Shiller (1990), pokazaliśmy, że rynek NYSE ma względną przewagę nad rynkiem europejskim w wyjaśnieniu zmian indeksu WIG.