A C T A U N I V E R S I T A T I S L O D Z I E N S I S FOLIA OECONOMICA 192, 2005
P i o t r W d o w iń s k i* , A n e t a Z g l i ń s k a - P i e t r z a k **
T H E W ARSAW STOCK EXCHANGE INDEX WIG: M O DELING A N D FORECASTING***
Abstract. In this paper we have assessed an influence o f the NYSE Stock Exchange indexes (DJIA and NASDAQ ) and European Stock indexes (D A X and FTSE) on the Warsaw Stock Exchange index WIG within a framework of a GARCH model. By applying a procedure o f 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 a rk e ts o n others has been w ell-know n from literature for years. H ow ever, this kind of relationship has been widely studied thro u g h analyzing co rrelatio n s between individual indexes representing different m arkets (e.g. E rb et al. 1994; Bracker and K o c h 1999).
In this p ap er we try to find such dependence betw een W IG index and foreign stock m a rk e t indexes (D JIA , N A S D A Q and D A X , F T SE ) by estim ating param eters in regressions o f W IG .
T h e estim ates show which foreign stock m a rk e t affects Polish stock exchange index m ore strongly. W e also test it by th e forecasting app ro ach 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 o f Econometrics, University o f Łódź. ** Dr (Ph.D., Assistant Professor), Department of Econometrics, University o f Łódź. *** We are grateful to our colleagues from the Department o f Econometrics, University o f Łódź, and participants o f FindEcon 2004 conference for helpful comments and suggestions on earlier draft of the paper.
T here are m any m eth o d s used to com bine forecasts (e.g. C lem en 1989; G ra n g er 1989). As show n by Clem en and W inkler (1986), sim ple com bin atio n m eth o d s often w ork better th an m ore com plex ap p ro ach . T he aim o f com bining forecasts is to investigate w hether the forecast com b in atio n plays an im p o rtan t role in the im provem ent o f forecasting ac curacy. T h e w ell-know n way o f com bining forecasts is to co m pu te linear com b in atio n o f forecasts generated by alternative m odels or obtained by using different forecasting m eth o d s (e.g. Billio ct al. 2000, C laesscn and M ittn ik 2002).
D ependence betw een stock m arkets in different cou ntries has been tested for years. M any analyzes deal with m easuring co rrelatio n betw een returns and diversified in tern atio n al p ortfolios (e.g. G rubel 1968, Levy and S arnat 1970, A gm on 1972, Fiszeder 2003).
In the 1990s there app eared research o f how changes to stock prices on one m ark e t affect o th er m arkets (e.g. H a m a o et al. 1990, K in g and W adhw ani 1990, Engle and Susmel 1993, F iszeder 2001).
T h e focus o f this pap er is to find the influence o f A m erican and E u ro p ean indexes on W arsaw Stock E xchange index W IG . T here is a lot o f research which prove th a t this influence does exist. W e aim to exam ine which m a rk e t - A m erican o r E uro p ean - has a stron ger im p act on W IG index.
T h e p ap e r is stru ctu red as follows. In Section 2 we give a b rief 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. C om 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 retu rns. N ow there is a com prehensive literature with several specifications o f autoregression m odels. M any em pirical analyzes, how ever, have show n th a t G A R C H ap p ro ach is the m ost ap p ro p riate. We also apply G A R C H m odeling in this paper. T h is is the m ost p o p u la r class o f m odels used in m odeling the financial tim e series o f high frequency (e.g. A kgiray 1989, Schw ert and Seguin 1990, N elson 1991, A ndersen et al. 1999, Osiewalski and Pipień 1999 and 2004, Bollerslev and W right 2000, F iszeder 2001 and 2003, Brzeszczyński an d Kelm 2002, D o m an M . and D o m an R. 2003).
T h e G A R C H m odel has been proposed indep end en tly by Bollerslev (1986) and T ay lo r (1986) as a generalization o f A R C H m odel introduced by Engle (1982).
T h e m ain featu re o f A R C H m odel is to describe th e co n d itio n al variance as an auto regression process. H ow ever, m ost em pirical tim e series require using long-lag length A R C H m odels and a large n u m b er o f param eters m ust be estim ated. T h e solution o f the problem was G A R C H m odels which
gave b etter results (cf. Engle and Bollerslev 1986; N elson 1991).
T h e basic linear generalized autoregressive con ditio n al 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 fu nction o f conditional variance represented as:
(3) h , = y0 + X Y i Ź - i + Z VjK -j,
i = i i
where: 3, is i.i.d. with £ (9 ,) = 0 and v a r ( 9 , ) = l , yo > 0* ( p ^ 0. If
p я
£ y i + Y j (P j< 1* ^ e process h, is covariance statio n ary . i = i j= i
Engle an d 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 i E ? - i + <Piht ~ i a n ^ У о > У i ^ 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 h e coefficients o f the m odel are then easily interp reted , w ith th e estim ate o f yt show ing the im pact o f curren t news on the co nditio n al 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 “ o ld ” news on volatility.
Recently a num ber o f new form ulations have been p ro p o sed , for exam ple 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), pow er G A R C H (P G A R C H , e.g. D ing et al. 1993), and th e 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. T he estim ation o f G A R C H m odels is b o th classic and non-classic, e.g. Bayesian a p p ro ach (e.g. Osiewalski and Pipień 1999, 2004).
In Section 3 we test m odels for influence o f foreign stock m ark ets on the P olish m arket.
3. TESTING FOR INFLUENCE OF FOREIGN STOCK INDEXES ON WIG INDEX
In the p ap e r we use a G A R C H m ethodology with G A R C H (1 ,1 ) m odels. W e have found th a t both A R C H and G A R C H effects app eared to be statistically significant in the dependence tested.
T h e focus o f the paper is to find the influence o f A m erican and E uropean stock m a rk e t 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 u tu al relationships betw een index W IG and foreign indexes, m ainly D A X and F T S E , because the estim ates used to be statistically insignificant and had op po site signs which negated the assum ptions o f positive influence o f the biggest stock m ark e ts on index W IG . H ence, we were unable to include b o th E urop ean indexes D A X and F T S E in one eq u atio n , and we decided to test for the influence o f E u ro p e a n m ark ets by two equ atio n s accordingly.
In tu rn , we analyze the follow ing m odels: • for A m erican m arket:
• • •
(4) wigt = a 0 -f a ^ j i a , - t + tx2nasdaq,^l + e(,
• fo r E u ro p e an m arket:
• •
(5) wig, = ot0 + tx ^ a x , _ x + e(,
o r
• •
(6) wig, = a 0 + c t j t s e t- i + e„
w here the variables are first differences o f n atu ral logarithm s, so they are the close-to-close retu rn s on correspond ing indexes.
T h e estim ation was m ad e by m axim um likelihood m e th o d 1 for daily d a ta from 1.01.1995 to 29.12.2003.
T h e 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 h e estim ates
o f individual equations were generally statistically significant. T h e sum у 1 + ф1 in the e q u a tio n o f co nditional variance was aro u n d 0.92, while th e estim ates were a b o u t 0.12 and ф1 a b o u t 0.80. Hence, we ca n conclude th a t the im pact o f c u rren t news on volatility in the co nd itional variance process is sm aller th a n the im pact o f “ o ld ” news.
T h e coefficient o f determ in atio n R -squared for co rresp o n d in g eq uatio ns (4), (5) and (6) was ab o u t 12% , 3% and 3% respectively.
In T ab le 1 we give m easurem ent o f ex-post errors2 for forecasts obtained from respective equ atio n s 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 THEIL 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 T P \ (%) 19.69 13.39 15.75 TP2 (%) 50.21 43.57 47.72
In th e analysis o f ex-post erro rs we used tu rn in g po ints test statistic - T P 1 - represented as:
n \ T P I = N ( y * > 0 л r t-ir ,* -! > 0 |у , - 1 < 0)
N (r tr , - i < 0 )
where:
N (rtrt_ 1 < 0) - a num ber o f tu rn in g points in em pirical series;
N ( r tr* > 0 л r,_ t r*_! > 0 |r rr,_ i < 0) - a n u m b er o f p o in ts, in w hich changes to d irection in em pirical and forecasting series are the sam e in the period t and t — 1 under the condition th a t the p o in ts are tu rn in g points in em pirical series.
W e also used the follow ing direction q uality m easure (e.g. W elfe 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 = шN (r,r, * 0 )
where:
N (r tr* > 0) - a nu m b er 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 Am erican indexes. F o rec asts basing on E u ro p ean indexes, how ever, were close to each o th e r as fa r as erro rs are concerned.
In Section 4 we apply a m ethodology o f com bined forecasting. We test for relative im p o rtan ce o f foreign stock m ark ets in explaining W IG index.
4. COMBINED FORECASTS OF WIG INDEX
In o rd e r to assess the pow er o f influence o f foreign m ark e ts on the Polish m a rk e t we estim ated the following F a ir and Shiller (1990) equation (e.g. W dow iński 2004):
(9) y , ~ yt-1 = + y i - i ) + <*2(t- iý2 t - yt- i ) + et
w here - 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 o n in fo rm atio n available up to the m om en t t — 1 w ith the use o f recursive estim ation for each period t. T h e p redictor , - i ý 2t - denotes on e-p eriod -ah ead forecasts g enerated accordingly by the m odel 2, i.e. th e m o d el w ith E u ro p e an indexes, while e is an e rro r term , e ~ /N (0 , o f). I f neither m odel 1, nor m odel 2 co n tain 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. I f both m odels generate forecasts th a t co n tain independent in fo rm atio n , the estim ates o f a.l and a 2 should b o th be statistically sig nificant. If b o th m odels con tain inform ation but in fo rm atio n contained in forecasts generated by m odel 2 is com pletely contained in forecasts gene rated by m odel 1 an d fu rth erm o re m odel 1 co ntains ad d itio n al relevant in fo rm atio n , the estim ate o f ol1 will be statistically significant while the estim ate o f a 2 statistically insignificant. If b o th forecasts co n tain the same in fo rm atio n , they are perfectly correlated and th e estim atio n o f param eters o f (9) is n o t possible.
B ecause th e influence o f E u ro p e a n m a rk e ts w as described by two equations, we estim ated m odel (9) for tw o cases:
• fo r 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 e q u a tio n (9).
Table 2. Estimation results o f Fair and Shiller equation
In te r c e p t D JIA --N A S D A Q D A X FT S E s. J-B D -W ARCH WHITE W ald (U S A ) W ald (E U R ) R2 (adj.) Obs. 0.0007 1.0018 0.5495 2.105 0.3989 1.4923 X J 0.0119 0.6407 (0.7258) 2.0719 0.2211 (0.6382) 0.3888 (0.4214) 4.4313 (0.0353) 2.227 (0.1356) 0.4429 250 0.0007 1.0226 0.6523 3.1112 J i 0.2915 1.3777 0.0119 0.8254 (0.6618) 2.0818 0.3223 (0.5702) 4.073 (0.3961) 9.6799 (0.0019) 1.8982 (0.1683) 0.4422 250
W ith italics we have denoted i-statistics with regard to estimates. Respective test p robab ilities w ith regard to test statistics are given in brackets. We applied the follow ing tests: Jarq u e-B era norm ality o f residuals test (J-B), conditional heteroscedasticity test (A R C H ), W hite’s test fo r heteroscedasticity (W hite), W ald coefficient restrictions test (W ald). T h e D -W stan ds for D u rb in -W atso n test statistic.
T h e test statistics and their probabilities in case o f J-B , D -W , A R C H and W hite’s tests den o te th a t errors in both m odels are n o rm al, with no a u to c o rre la tio n , no A R C H effects and no heteroskedasticity. T h u s we can test the pow er o f influence o f foreign indexes on W IG index using t- statistic.
It can be easily seen th a t the influence o f A m erican m a rk e t indexes D JIA and N A S D A Q turned o u t to be m ore relevant for W IG index than E u ro p e an indexes D A X and F T S E . It is su p p o rted by the significance of estim ates and th eir size. W e can conclude, th a t in fo rm atio n contained in forecasts generated by m odels (5) o r (6) is fully co n tain ed in forecasts by m odel (4). T herefore, A m erican indexes D JIA and N A S D A Q were m ore influential th a n E uro p ean 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 restriction s test, i.e. we should reject the null th a t the respective coefficient equals zero in case o f m odel (4) and should no t reject the null in case o f m odels (5) o r (6).
Sim ilar conclusions ab o u t the co-dependence o f m ark ets in case o f Poland were d raw n by e.g. F iszeder (2003).
Since the p aram eters in eq u atio n (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 th e n o n lin ear p rog ram m in g problem (N LP):
(10) m in /(co) = <o7 Vm, ю г1 = 1, to > 0, where: Г ( o . l
ю = - is the vector o f weights; L a>2\
V - the variance-covariance m atrix o f forecast errors.
In T ab le 3 we present covarianccs and co rrelatio ns o f forecasts errors.
Table 3. Covariance and correlation coefficients matrix o f forecast errors
Specification Covariance
DJIA-NASDAQ DAX FTSE
DJIA-NASDAQ 0.0001417 DAX 0.0001380 0.0001427 FTSE 0.0001375 0.0001427 0.0001427 Correlation DJIA-NASDAQ 1 DAX 0.9708 1 FTSE 0.9546 0.9875 1
A fter solving the problem o f n onlinear p rog ram m in g (10) we obtained the follow ing w eights which are given in T able 4.
Table 4. The weights in a NLP problem
DJIA-NASDAQ D AX FTSE
0.5664 0.4336 X
0.6734 X 0.3266
A s we can notice the weights are close to estim ated param eters in equation (9). T hus we confirmed earlier conclusions abo ut the strong influence o f A m erican m ark et on the Polish stock m ark et. As show n in T ab le 5, we could n o t sub stantially reduce the variance o f com bined forecast errors, because o f the high and positive correlation betw een forecast erro rs from individual m odels (4), (5) an d (6).
Table 5. Variance of forecast errors Individual forecast DJIA-NASDAQ 0.0001417 DAX 0.0001427 FTSE 0.0001427 Combined forecast DJIA-NASDAQ-DAX 0.0001406 DJ1A-NASDAQ-FTSE 0.0001409
A fter calculating optim al w eights, we calculated com bined forecasts and assessed th eir accuracy. T h e results are given in T a b le 6.
Table 6. Ex-post errors o f combined forecasts of WIG index
Specification DJIA-NASDAQ DJIA-NASDAQ-FTSE
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 T P\ (%) 15 17 TP2 (%) 48.55 48.96
It is clearly show n th a t com bined forecasts are n o t sup erior to forecasts obtained from each m odel separately.
F inally, we tested the forecasting quality o f the eq u atio n s including indexes o f b oth foreign m arkets at the sam e time. We estim ated the following equations:
• • • •
(11) wig, = a 0 + ctydjia,-1 + &2nasd a q ,-! + a 3dax,_ i + e,
• • • •
(12) wig, = ß 0 + ß l d ji a ,- 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 e a n m ark et were usually insignificant. Also, as before, we obtain ed o pp osite signs o f estim ates w hich does n o t com ply w ith the p o stu late o f positive influence o f the biggest stock m ark e ts on W IG index.
By applying the all-index ap p ro ach itself we were n o t able to determ ine the pow er o f influence o f individual m arkets on the Polish m ark e t due to high co rrelatio n coefficients betw een indexes and difficulties in estim ating the p aram eters. T herefore, the com bined forecasts ap p ro ach suggested in the p ap er can be considered correct.
D espite questio n ab le statistics in results o f regressions (11) an d (12) and their w eak econom ic properties we used them to fo recast W IG index. T he errors o f these forecasts are show n in T ab le 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 n o t o f b etter qu ality than forecasts ob tain ed from individual m odels separately.
5. CONCLUSIONS
In this p ap e r we attem pted to assess the influence o f A m erican and E u ro p e an stock m ark ets o n the Polish stock m ark et. In o u r analysis we used the G A R C H m ethodology which is p o p u lar to describe the financial phenom ena o f high frequency. We have found th a t the situ atio n on Am erican m a rk e t had m o re influence on the Polish m ark e t th an the situ atio n on E u ro p e an m ark e ts in the analysed period d u rin g 1.01.1995 - 29.12.2003. W e based these conclusions on the analyzis o f forecasting equ atio n s for W IG index and on the analysis o f com bined forecasting. O u r results are consistent w ith those obtained by o th er au th o rs for the P olish stock m ark et (e.g. F iszeder 2001 and 2003; Brzeszczyński and K elm 2002).
W e should notice, how ever, a few aspects which can affect conclusions d raw n from this kind o f research.
F irstly, it m u st be rem em bered th a t there is a stro n g co-dependence am o n g th e biggest stock m arkets so it is difficult to show th e im pact o f one individual m ark e t on an o th er one in a sim ple ap p ro ach .
Secondly, it is very im portant to choose the proper indexes as determ inants o f m a rk e t developm ent. It is necessary, then, to test also the influence o f o th er indexes and stock m ark ets - including em erging m ark e ts - on the Polish m arket.
T hirdly, the Polish stock m ark et m ay be treated by foreign investors as one o f secondary im portance, and consequently dom estic investm ent funds m ay influence the variability o f retu rn s m ore th an situ atio n o n foreign m arkets.
In th e p ap e r we have used only series o f 250 one-perio d-ahead forecasts. In fu tu re research we will attem p t to show how the p aram eters reflecting the influence o f A m erican stock m a rk e t and o th er m ark e ts on the Polish m a rk e t change over time.
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P io tr W dow iński, A n eta Z g liń sk a -P ie trza k
MODELOWANIE I PROGNOZWANIE INDEKSU WIG (Streszczenie)
W artykule podjęliśmy próbę oceny wpływu indeksów rynku amerykańskiego DJIA i NASDAQ oraz indeksów rynku europejskiego D A X 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.
