A NON-LINEAR TIME SERIES APROACH TO MODELLING ASYMMETRY IN STOCK MARKET INDEXES

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1 A NON-LINEAR TIME SERIES APROACH TO MODELLING ASYMMETRY IN STOCK MARKET INDEXES Alessandra Amendola, Giseppe Sori Diparimeno di Scienze Economice Universià degli Sdi di Salerno Absrac: In is paper we propose an approac o modelling non-linear condiionally eeroscedasic ime series caracerised by asymmeries in bo e condiional mean and variance. Tis is acieved by combining a TAR model for e condiional mean wi a Canging Parameers Volailiy (CPV) model for e condiional variance. Empirical resls are given for e daily rerns of e S&P 500, NASDAQ composie and FTSE 00 sock marke indexes. Keywords: Canging Parameers Volailiy model, TAR, Kalman Filer, EM algorim.. Inrodcion De o e presence of asymmeric effecs in e mean and in e ime varying condiional variance, e complex beavior of financial ime series can be ardly capred by linear models. Hence, e las wo decades ave been caracerised by a growing ineres in e applicaion of non linear ime series modelling ecniqes o e analysis of financial daa. In order o simlaneosly capre differen aspecs of non-linear ime series beavior, Tong (990) firs proposed o combine e se of a non-linear model for e condiional mean wi a non linear model for e condiional variance. Tis idea was sccessively adoped in differen frameworks by varios aors (see Li and Li, 996, Li, Li and Li 997, Lndberg and Teräsvira, 998). In is paper we propose an alernaive modelling procedre in order o allow for asymmery in bo e condiional mean and variance. For modelling asymmery we combine a resold aoregressive srrcre (TAR, Tong 978) for e condiional mean wi a Canging Parameers Volailiy model for e condiional variance (CPV, Sori 999). Tis class of models can be considered as a sae-space generalisaion of e CHARMA models proposed by Tsay (987). Wi respec o convenional GARCH (Bollerslev, 986) ype models, e CPV model as wo main advanages. Firs, ineracion erms can be inclded in e condiional variance eqaion allowing o accon for asymmeric effecs. Second, i can incorporae ime-varying parameers in e volailiy model. Te performances of e proposed model in esimaing e condiional variances of ree differen sock marke indexes (S&P500, NASDAQ, FTSE) are assessed by means of a comparison wi e resls obained esimaing some differen condiional eeroscedasic srcres. In pariclar, we consider e classic GARCH model and wo alernaive specificaions, e TARCH (Rabemananjara and Zakoian, 993) and EGARCH (Nelson, 99) models, wic allow o capre asymmeric effecs in e condiional variance. Te paper is organised as follows: secion illsraes e eoreical backgrond nderlying or proposal; e resls of e empirical analysis are sown in secion 3 were some final commens are also given.

2 . Teoreical backgrond and modelling procedre.. Tresold models for e condiional mean Te Tresold AoRegressive (TAR) models were firs presened by Tong (978) and frer developed and applied in Tong and Lim (980) and Tong (983) (a more orog discssion can be fond in Tong,990). A TAR model can be regarded as a piecewise linear aoregressive srcre, wic allows o obain e decomposiion of a complex socasic sysem ino smaller sbsysems, based on e vales assmed by a resold variable, compared wi a se of predeermined vales, e resold vales. Le {Y } be a ime series, a TAR model for Y is given by: Y X = a d ( j) 0 R + a j p i= ( j) i Y i + ( j) () were { } is i.i.d. wi zero mean and finie variance, e resold vales, {w0, w,,..,wl}, are sc a w 0 <w <...<w l, w 0 = - and wl= + wi Rj = (w j-,w j], d is a posiive ineger. If we coose as resold variable a pas realisaion of Y, {Y -d }, e ime series {Y } follows a Self Exciing Tresold (SETAR) model. To invesigae e appropriaeness of a resold non-linear model insead of a simpler linear model, we perform e Tsay es for deecing resold non-lineariy. Tis esing procedre was presened in Tsay (989) and sccessively refined and generalised by Tsay (998). Assming a e aoregressive order p and e delay d are known, we firs perform an arranged regression based on e increasing order of e resold variable and en se e predicive residals calclaed by recrsive leas sqares o evalae e es saisic. Under e nll ypoesis of a linear model, e Tsay es is asympoically disribed as a cisqared random variable wi (p+) d.f.. In order o idenify e model we follow e ieraive procedre described in Tsay (998). Given e idenificaion of e resold variable, ofen sggesed by e nare of e specific problem, e resls of e non-lineariy es can be sed o firs selec a se of possible delays { d,..., d }. For eac combinaion of p, d and m, were m is e nmber of possible regimes (sally in e range of or 3), we en se a grid searc meod and e AIC (Akaike Informaion Crierion) o selec e resold vales and idenify e final model. Finally e model parameers in eac regime can be esimaed by e condiional leas sqares meod.. A sae space approac o e analysis of non-linear CH ime series Le be an nivariae series of predicion errors sc a ( ) ~ ( 0, ) and Cov(, -d ) = 0, d 0. Also assme a as finie momens p o e for order. A Canging Parameers Volailiy (CPV) model (Sori, 999) of order (r,s), wi r and s inegers, is defined as - N

3 r i= i, -i s = a + b + e = C x + e x A x + q = j= j, -j (a) (b) were e is a Gassian wie noise observaion error e ~ N(0, σ e ), q (n ), wi n=r+s, is a Gassian serially ncorrelaed sysem error, q ~ N ( 0, Q), and x (n ) is an n- dimensional sae vecor wi sae variables given by e socasically varying parameers a i, (i=,...,r) and b j, (j=,...,s). Te observaion marix is C = [,..., r,..., s ] wile A is an (n n) ransiion marix of nknown coefficiens. Te specificaion of e model is compleed by e sal assmpions E ( q ) = 0, {,z } e z x N ( m, ) wi E[ q ( x0 ) ] = 0 and E ( x e ) = 0, 0 ~ 0 P Under e above assmpions e model is condiionally Gassian and e Kalman filer can be sed o obain a MMSE esimae of e sae vecor. Te condiional variance is recrsively esimaed as 0 ) ' e = C P - ( C + σ (3) were = Var( x ). Te condiional variance eqaion (3) can be also wrien as = σ e + p + r i= s i, i( ) s i= r + j= r + i p + i, j( ) r+ s j= r + p ( i r ) j, j( ) ( j r ) ( j r ) r r + p + i= j= s i= j= r + r p i, j( ) i, j( ) i i j ( j r ) (4) P - wi p i, j( ) being e elemen of place (i,j) in P -. Compared o convenional approaces, e CPV model as wo main advanages. Firs, i allows for ime varying parameers in e condiional variance specificaion (4). Second, ineracion erms beween pas innovaions and volailiies are easily inclded in e model. Te coice A = 0 yields a more parsimonios random coefficien version of e CPV model a we will call e consrained CPV model or, abbreviaed, CPV-C. In a CPV-C model e condiional variance parameers are consan b e ineracion erms are sill presen. I can be sown (Sori, 999) a, if A = 0 and e covariance marix Q is diagonal, e resling CPV-C model will ave e same condiional variance as a Generalized Aoregressive Condiionally Heeroskedasic (GARCH) model (Bollerslev, 986) of e same order. Similarly, for s=0 e model is a random coefficien

4 aoregressive model of order r. A generalizaion of model () wic allows for simlaneos modelling of condiional mean and variance is given by e regression CPV model y = M β + = M β + C x + e x Ax + = q (5a) (5b) were y is an observed ime series, M is a vecor ( g) of endogenos or exogenos regressors and β a vecor (g ) of nknown parameers. Te model parameers A, Q, and β can be esimaed maximizing a Gassian log-likeliood fncion expressed in e classical predicion error decomposiion form. Te formlaion of model (5) is qie general and e regression erm inclded ino e observaion eqaion can incorporae an ARMA ype srcre wi exogenos explanaory variables. Also, replacing e consan parameer vecor β by a ime variable vecor β, we can easily accommodae for some common nonlinear srcres sc as Tresold Aoregressive models. Under e assmpion of condiional normaliy, e log-likeliood fncion of a regression CPV model can be wrien as σ e were T T e T ( ) "( y; A, σ e, Q) = log( π ) log[ ( ) ] (6) = = ( ) e y = C x M β and ' ( C ) σ ( ) = C P e wi x ( - = E x y ) and P ( - = Var x y ). Te nknown parameers in { A, e, Q } can be esimaed by maximising e Gassian log-likeliood (6) sing a version of e EM algorim ailored for sae space models by W e al. (996). Alernaively scoring or qasi- Newon meods cold also be sed (see Wason and Engle, 983, for a discssion on e applicaion of e meod of scoring in e conex of sae space models). Finally, i is wor noing a, wen forecasing from a CPV model, e condiional variance affecs e esimae of e condiional mean E( ) in wo differen ways. Firs, eners e sae pdaing eqaion, second, e esimaed E( x ) does no necessarily ave o be eqal o zero. 3. Asymmeric effecs in e CPV model If e model order is sc a s>0, asymmeric effecs are inrodced ino e CPV-C condiional variance specificaion by means of e ineracion erms beween pas socks and volailiies. I follows a, in CPV-C ype models, e effec of a pas sock on e presen

5 volailiy derives from e sm of wo componens. Of ese, e firs, as in GARCH models, is given by a linear fncions of e pas sqared sock wi no regard for is sign. Differenly, e second adds a frer conribion o e vale of e condiional variance depending on e sign of e sock. Tis erm is, in modle, proporional o e magnide of e sock rescaled by pas condiional sandard deviaions. In order o clarify is poin, we consider, as an example, e simple CPV-C(,) model wi condiional variance eqaion given by 0 + β + = ν + α δ (7) 0 σ e were ν =, α = Q,, β = Q, and δ = Q,. As i is eviden from eqaion (7), e asymmery in e relaion beween e condiional variance and pas residals comes from e cross-erm δ. In pariclar, for δ <0, a posiive penaly erm will be added o if <0 wile a posiive qaniy will be sbraced from if >0. A similar reasoning applies o e case in wic δ >0. Tis sggess a simple esing procedre for verifying e presence of asymmeric effecs in e relaionsip beween condiional variance and pas socks by esing for δ significanly differen from 0. Te presence of leverage effecs can be esed by e ypoesis a δ <0. Fig. sows e simlaed news impac crve for a pariclar CPV-C (,) model incorporaing leverage effecs. Fig. Simlaed news impac crve for a CPV-C(,) model wi parameers ν 0 =0.05 α =0.5, β =0.65 and δ = Also, an aracive feare of e model is a, wen and δ <0, e magnide of e penaly erm added o e condiional variance will no depend only on e magnide of e sock iself b i will be weiged, or, more properly, rescaled, by e condiional sandard

6 deviaion a ime (-). So e greaer will be e ncerainy associaed wi e negaive sock, and e greaer will be e impac of on. Te same argmen applies o iger order models even if, in ig dimensional srcres, care sold be aken in e idenificaion of e relevan lags of ineracion in order o avoid o incr e so called crse of dimensionaliy. 3. Empirical resls In is secion we presen e resls of an applicaion of e proposed modelling approac o e analysis of some sock marke indexes. In pariclar we consider e daily U.S. NASDAQ Composie, e U.S. Sandard and Poor's 500 and e U.K. FTSE 00. Te sample covers e period from Janary 996 o 6 November 999. Rerns are calclaed as logarimic firs differences, R = (log X ). Te plos of e original daa and rerns are sown in Fig.. Even if e algorim for e maximisaion of e likeliood fncion is able o narally deal wi e simlaneos esimaion of e condiional mean and variance model parameers, in order o garanee fll comparabiliy of e performances of e models considered in esimaing e condiional variance of e series, we rever o a wo sage modelling procedre. Firs, in order o invesigae e presence of a resold ype non-linear srcre in e daa we ave performed Tsay's es for non-lineariy. Te resls obained for differen vales of e delay parameer are sown in Tab.. Te es leads o rejec e nll ypoesis of lineariy for all of e series considered. Teoreical and empirical evidence sgges a e beavior of sock marke rerns is inflenced by wa as appened in e previos days. Terefore we coose as possible resold variables lagged vales of e rerns. Tis leads o e idenificaion of a SETAR model for e condiional mean. Te es also indicaes, as e bes coice for e delay d, e one wic corresponds o e iges vale of e es saisic. For all e series we ave cosen e aoregressive order p o lie in e range [, 5], and we ave considered m=,3 as e possible nmber of regimes. For eac combinaion of d, p and s we ave cosen e SETAR model a gives e minimm AIC. Tab.: Tsay's resold non-lineariy es F SP500 7HVW GI FTSE 7HVW GI Nasdaq 7HVW GI

7 Fig. : From op o boom: original daa (lef) and logarimic firs differences (rig) of S & P 500, NASDAQ and FTSE 00. Te final models idenified, afer refining e resolds and e orders, are a SETAR(3,5) wi delay d=4 and w= for e S&P 500, a SETAR(,) wi d=4 and w= for e FTSE 00, and a SETAR (,) wi d=4 and w=0.00 for e NASDAQ series. Te leas sqares esimae of e models parameers and e corresponding sandard errors (in pareneses) are sown in Tab.. In order o assess e fiing accracy of e SETAR models we calclae some widely sed loss fncions (Tab.3).

8 Tab. Leas sqares esimaes and sandard error of SETAR models Regime a 0 a a a 3 a 4 a 5 SP500 I ( ) II ( ) FTSE I 0.00 (0.0005) II (0.0004) Nasdaq I ( ) (0.0507) (0.0434) (0.0439) II (0.0474) ( ) ( ) ( ) ( ) (0.0474) ( ) ( ) ( ) Tab.3: Vales of differen loss fncions for e esimaion of e condiional mean RMSE MAE MAPE ( 0-6 ) THEIL ( 0-6 ) S&P FTSE Nasdaq For eac series we ave en performed an ARCH-LM es (Engle, 98) on e residals of e resold model esimaed for e condiional mean. Te resls of e es and e analysis of e aocorrelaion fncions of e sqared residals sgges e presence of aoregressive condiional eeroskedasiciy in e daa. In order o deec any possible asymmery in e condiional variance componen, we look a e cross correlaion beween e sqared sandardized residals and lagged sandardized residals (Fig. 3). Tese cross correlaions sold be zero if asymmeric effecs are no presen and be negaive in presence of asymmery. Te esimaed correlaions sow evidence in favor of e ypoesis of asymmery for e S&P 500 and NASDAQ series wile, for e FTSE 00, e vales of e aocorrelaion fncion lie inside e ± s.e. confidence bands excep for lag 5. Te nex sep is o idenify and esimae a siable CPV model for e condiional variance. In pariclar, we consider e parsimonios CPV-C specificaion. Te order of e model o be fied as been cosen o minimise e vale of e Scwarz Crierion (SC). Tab. 4 repors e vales of e SC for differen model specificaions ogeer wi e AIC and log-likeliood vales. Te searc as been resriced wiin e inervals r and 0 s. For all e series a CPV-C (,) model is idenified.

9 Fig 3: From op o boom, cross-correlaions beween and i for S&P 500, NASDAQ and FTSE00.

10 Tab.4: AIC, SC and log-likeliood vales for differen CPV models Orders AIC SC LL SP FTSE Nasdaq Te maximm likeliood esimaes and relaive sandard errors of e CPV-C model parameers are sown in Tab. 5. Tab. 5: ML parameer esimaes and asympoic s.e. (in pareneses) SP ( ) FTSE ( ) Nasdaq ( ) σ Q Q Q e (0.00) 0.6 (0.03) 0.8 (0.094) (0.078) (0.096) (0.068) 0.86 (0.053) (0.064) (0.04) Te negaive sign of e Q parameer confirms e presence of a leverage effec as sggesed by e cross-correlaion analysis. Te resls ave been compared wi ose obained sing some differen condiional variance specificaion. Namely we ave considered e classical GARCH(p,q) model (Bollerslev, 986), given by: q p = α0 + αi + β j i= j= ; e Exponenial GARCH model (EGARCH, Nelson 99), wi condiional variance specificaion given by:

11 ln( p 0 ) ) = α + β j ln( j + γ + λ ; j= e TARCH model (Rabemananjara and Zakoian, 993) wic, as e EGARCH, incldes asymmeric effecs in e condiional variance: q p = α + i i + 0 α γ d + β j i= j= wi d = if <0 and d =0 oerwise. Te esimaed parameers for e GARCH, E-GARCH and TARCH models fied for e condiional variance of eac of e ree series are given in ables 6, 7 and 8, respecively. Tab. 6 Parameers esimaes and sandard error of GARCH models α 0 α α β β SP (0.0000) (0.0303) (0.009) FTSE (0.0000) Nasdaq (0.0000) ( ) (0.040) (0.0654) j (0.075) (0.095) Tab. 7 Parameers esimaes and sandard error of EGARCH models α 0 γ γ λ λ β SP (0.0950) ( ) ( ) (0.0437) ( ) FTSE ( ) Nasdaq (0.549) (0.0534) ( ) (0.0579) (0.083) (0.0098) (0.0035) (0.0568) Tab. 8 Parameers esimaes and sandard error of TGARCH models α 0 α α θ β SP (0.0000) (0.0673) (0.0866) ( ) (0.0798) FTSE (0.0000) Nasdaq ( ) (0.070) (0.0579) (0.0347) (0.004) ( ) For eac model, e performance in e esimaion of e condiional variance as been assessed on e basis of five differen loss fncions, namely:

12 T MSE v = T [ ĥ ] LSEv = T [ln( ) ln( ĥ )] = T v = MAE = T ĥ LAE = T ln( ) ln( ĥ ). T = T v = HMSE v = T T = ˆ Bollerslev e al. (994) poin o a, in many cases, RMSE can be inappropriae, as a measre of goodness of fi, since i penalizes condiional variance esimaes wic are differen from e realized sqared residals in a flly symmerically fasion. Alernaive loss fncions wic penalize condiional variance esimaes asymmerically are e logarimic loss fncions (LSE v and LAE v ) and e Heeroscedasiciy Adjsed MSE (HMSE v ). Te resls obained ave been repored in Table 9. Te las wo colmn give e vales of e maximised log-likeliood and e nmber of esimaed parameers for eac model. Tab.9: Vales of differen loss fncions for e esimaion of e condiional variance RMSE v MAE v LSE v LAE v HMSE v LL NP S&P500 CPV-C(,) GARCH(,) E-GARCH(,) TARCH(,) Nasdaq CPV-C(,) GARCH(,) E-GARCH(,) TARCH(,) FTSE CPV-C(,) GARCH(,) E-GARCH(,) TARCH(,) Te performances of e CPV-C model resls o be beer for e series caracerised by a sbsanial asymmeric componen (S&P500, NASDAQ) an for e FTSE00, for wic e cross-correlaion analysis (Fig.) does no sow a so srong evidence in favor of e ypoesis of asymmery. In pariclar for e NASDAQ series e CPV-C performs beer an all e oer models on e basis of all e loss fncions sed. For e S&P 500, again, Sricly, e maximised log-likeliood vale of e CPV model is no comparable o ose obained for e oer models (GARCH, EGARCH, TARCH) since, wile e likeliood fncion for eac of e laer is defined on e residals of e model esimaed for e condiional mean, e vale repored for e CPV model refers o e classical predicion error decomposiion form of e likeliood wic is defined on e series of e one sepaead predicion errors made in forecasing ese residals.

13 e minimm RMSE v, MAE v and HMSE v vales are obained for e CPV-C model wile a worse performance is acieved on e basis of e logarimic loss fncions LSE v and LAE v. wic exaggerae e ineres in predicing wen residals are close o 0. Acknowledgemens: Tis paper is sppored by MURST98 "Modelli Saisici per l'analisi delle Serie Temporali" References Bollerslev T. (986): Generalized aoregressive condiional eeroskedasiciy, Jornal of Economerics, vol. 3, Engle R.F., (98): Aoregressive condiional eeroskedasiciy wi esimaes of e variance of Unied Kingdom Inflaion, Economerica, vol.50, Li C.W., Li W.K. (996): On a Doble resold Aoregresssive Heeroscedasic Time Series Model, Jornal of Applied Economerics, vol., Li J., Li W.K., Li C.W. (997): On a resold aoregression wi condiional eeroscedasic variances, Jornal of Saisical Planning and Inference, vol.6, Li J., Li W.K., Li C.W. (997): On a resold aoregression wi condiional eeroscedasic variances, Jornal of Saisical Planning and inference, vol.6, Lndberg S., Teräsvira T., (998), Modelling economic ig-freqencies ime series wi STAR-STGARCH models, Working Paper n.9, Sockolm Scool of Economics Nelson D.B., (99), Condiional eeroskedasiciy in asse rerns; a new approac, Economerica, vol.59. Rabemananjara R. Zakoian J.M, (993), Tresold ARCH Models and asymmeries in volailiy, Jornal of Applied Economerics, vol.8. Sori G. (999): Te CPV model: a sae space generalizaion of GARCH processes, Proceedings of e conference SCO 99, "Modelli Complessi e Meodi Compazionali Inensivi per la Sima e la Previsione", Tong H., (978), On a Tresold models, in Paern Recogniion and Signal Processing, Cen C.H.(ed.) Sijoff & Noordoff, Amserdam. Tong H., (983), Lecre noes in Saisics Tresold models in non-linear ime series analysis, Springe-Verlag. Tong H., (990): Non-Linear Time Series: A Dynamical Sysem Approac, Oxford Pblicaions, Oxford Universiy Press. Tong H., Lim K.S, (980), Tresold aoregression, limi cycles and cyclical daa, Jornal of Royal Saisical Sociey, B, vol.4. Tsay R. (987): Condiional eeroskedasic ime series models, Jornal of e American Saisical Associaion, vol. 8, Tsay R. (998) Tesing and Modelling Mlivariae Tresold Models, Jornal of e American Saisical Associaion, 93, pp Tsay R.S., (989), Tesing and modelling resold aoregressive processes, Jornal of e American Saisical Associaion vol.84, n.405. Wason M. W., Engle R. F. (983) Alernaive algorims for e esimaion of dynamic facor, MIMIC and varying coefficien regression models, Jornal of Economerics, 3, W L. S., Pai J. S., Hosking J. R. M. (996) An algorim for esimaing parameers of saespace models, Saisics & Probabiliy Leers, 8,