DynamicAsymmetricGARCH

Transcription

1 DynamicAsymmetricGARCH Massimiliano Caporin Dipartimento di Scienze Economiche Università Ca Foscari di Venezia Michael McAleer School of Economics and Commerce University of Western Australia Revised: September 2004 Abstract This paper develops the Dynamic Asymmetric GARCH (or DA- GARCH) model that generalises asymmetric GARCH models such as the GJR model, making the asymmetric effect time dependent. We provide the stationarity conditions and show how GJR can be obtainedasaspecialcaseofdagarch.anapplicationtodailystock market indices is also presented to demonstrate the usefulness of the new model. Keywords: Asymmetric volatility, threshold GARCH, DAGARCH, stationarity conditions 1 Introduction The idea that positive and negative shocks have different impacts on dynamic volatility was incorporated in the GARCH model (see Engle (1982) ThefirstauthoraknowledgesfinancialsupportfromItalianMIURprojectCofin2002. The second author aknowledges the financial support of the Australian Research Council. 1

2 and Bollerslev(1986)) by Glosten, Jagannathan and Runkle(1993)(GJR), and generalised by Rabemananjara and Zakoian(1993) and Zakoian(1994) to the threshold(or TARCH) model. Several empirical studies have emphasized the relevance of the signs of the shocks on conditional volatility, but no particular attention has been given to the sizes of these shocks (for an exception, see the EGARCH model of Nelson(1991)). Inthispaperweprovideadifferentsolution,includingasizeeffectina special type of threshold GARCH model which generalises GJR to allow for multiple thresholds. Furthermore, we provide a second innovation by adding dynamics in the threshold structure to enforce variance persistence that is directlyrelatedtoboththesizeandsignofshocks. This paper develops the Dynamic Asymmetric GARCH (DAGARCH) model. We provide the stationarity conditions of the model and an empirical example for stock market indices to demonstrate the usefulness of the new model. 2 Dynamic Asymmetric GARCH: DAGARCH There is no theoretical or empirical reason to assume that the conditional distribution of asset returns is symmetric, with positive and negative shocks having the same impact on conditional variances. Furthermore, a similar assumptioncannotbesustainedforthesizeofshocks. IntheclassofGARCH models that deals explicitly with volatility asymmmetry, we present the Dynamic Asymmetric GARCH(DAGARCH) model which can be used to anal- 2

3 yse the size and sign effects within a GARCH structure. Furthermore, the DAGARCH model enables an analysis of the persistence of the size and sign effects of shocks by adding dynamics to volatility asymmetry. The conditional mean and conditional variance of DAGARCH(1,1) are given as follows: y t µ ( I t 1) =ε t =z t σ t (1) σ 2 t =ω+βσ 2 t 1+γ t 1 ε 2 t 1 (2) γ t 1 = γ 1 +φ 1 γ t 2 ε t 1 >d 1 γ 2 +φ 2 γ t 2 d 2 <ε t 1 <d 1.. γ m 1 +φ m 1 γ t 2 d m 1 <ε t 1 <d m 2 γ m +φ m γ t 2 ε t 1 <d m 1 (3) ω>0, β>0, d 1 >d 2 >...d m 2 >d m 1. (4) We do not make any assumptions regarding the dynamic evolution of the mean. Equation (1) defines the relation between the observed series, the conditional variances σ 2 t, and the standardised residuals z t. Equation (2) refers to the dynamic evolution of the conditional variances, while equation (3) focuses on the dynamic asymmetry. 3

4 Assumption 1: The standardised residuals are independently and identicallydistributed, withprobabilitydensityfunctionf(z t ), cumulativedistributionfunctionf(z t ), satisfyinge[z t ]=0, E[zt]=1,andwithanon- degenerate distribution. 2 Letc(z t )=β+γ t zt 1. 2 UsingtheresultsofLingandMcAleer(2002a,b), which extend and correct the results in He and Terasvirta(1999a,b), and the results in McAleer, Chan and Marinova(2003), we can state the following: Theorem1 Under Assumption 1, if E[c(z t )] δ < 1 for some δ (0,1], then there exists a unique, strictly stationary and ergodic solution of the DAGARCH model, with the following causal expansion: [ ] k σ 2 t = ω 1+ c(z t ), k=0 j=0 where the infinite sum converges almost surely. { Proof. DefinethefunctionΨ(δ)=E [c(z t )] δ}. UnderAssumption1, the standardised residuals have finite second moment, so that the function Ψ(δ) is at least twice differentiable. Furthermore, Ψ (δ) = d dδ [c(z t )] δ f(z t )dz t = d dδ [c(z t)] δ f(z t )dz t = E {ln[c(z t )][c(z t )] δ} 4

5 and Ψ (δ) = E { (ln[c(z t )]) 2 [c(z t )] δ}. NotingthatΨ (δ)>0,sinceitscomponentstakeonlypositivevalues,and { that E ln[c(z t )][c(z t )] δ} collapses to E{ln[c(z t )]} when δ = 0, we can verify the following: i)byjensen sinequality,itfollowsthate{ln[c(z t )]}<0whenδ=0; ii) by the fact that Ψ(0) = 1, for small δ, the function Ψ(δ) must be less than 1(given convexity and first derivative), so that there exists some δ (0,1]thatsatisfiesE[c(z t )] δ <1; iii) finally, by applying Theorem 2.1 in Ling and McAleer (2002a), we obtain the stated result. Let us redefine the thresholds by using a compact formulation: S j =I(d j <ε t 1 <d j 1 ),wherei( )istheindicatorfunction. Remark1 Using d jσ t 1 = d j, where d j is an unobservable standardised threshold and rj a set defined over the d j values, the following equalities hold: S j =I(d j <ε t 1 <d j 1 )=I(d j <σ t 1 z t 1 <d j 1 ) =I ( ) ( d jσ t 1 <σ t 1 z t 1 <d j 1σ t 1 =I d j <z t 1 <dj 1) =S j and Sj =I ( z t 1 rj). 5

6 Remark 2 Whenever the model is re-cast in the standard GARCH representation, that makes explicit the ARCH parameter, a zero identification restrictionmustbeimposedononeofthe γ j coefficients. Remark 3 A set of sufficient conditions for the positivity of the conditional variances is the following: ω>0, β 0, γ j 0 and φ j 0 for j=1,2,...,m. Assumption 2: The thresholds d j are known (or fixed) a priori or, alternatively, it is assumed that the standardised residuals follow a given distribution such as a standardised normal, Student-t with k degrees of freedom, or GED. Thed j thresholdscanbesetinordertoidentifythetailsofthestandardised distribution, or to select positive and negative values. As an example, thelowerthresholdd m 1canbesettoF 1 (α)toidentifythelowerα-tailof the distribution. Remark4 ItfollowsfromTheorem2.1ofLingandMcAleer(2002a)that the necessary and sufficient condition for the existence of the second moment solutionise[c(z t )]<1. 6

7 Theorem 2 The necessary and sufficient condition for the existence of the second moment of DAGARCH(1,1) is m m β+ γ j e j +G φ j e j < 1 j=1 j=1 where G = m j=1 γ jm j 1 m j=1 φ jm j m j = E [ ] Sj e j = E [ z 2 ts j]. Note that the last two equations define scalars that depend on the assumptions for the standardised residual distribution and on the thresholds. Proof. Recallthatc(z t )=β+γ t zt. 2 Takingexpectationsofbothsides gives E[c(z t )] = β+e [ γ t z 2 t]. 7

8 Focusing on the expectation of the dynamic asymmetry term, we have E [ { ] m ] γ t zt 2 = E [ γj +φ j γ t 1 z 2 t I ( ) } z t rj j=1 { m } ] = E [ γj +φ j γ t 1 z 2 t I j (z t ) j=1 = m γ j E [ zti 2 j (z t ) ] m + φ j E [ γ t 1 zti 2 j (z t ) ]. j=1 j=1 Asγ t 1 isafunctionofz t 1,giventheassumptionoftemporalindependence ofthez t,itfollowsthat E [ ] m γ t zt 2 = γ j E [ zti 2 j (z t ) ] m + φ j E [ ] [ γ t 1 E z 2 t I j (z t ) ]. j=1 j=1 DefiningE[z 2 ti j (z t )]=e j (whichisobtainedbydirectcomputation,giventhe assumption on the distribution of the standardised residuals and knowledge ofthed j thresholds),uponsubstitutionyelds: E [ ] m m γ t zt 2 = γ j e j + φ j e j E [ ] γ t 1. j=1 j=1 We then compute the expectation of the Dynamic Asymmetry effect, which 8

9 isgivenby { m } ] E[γ t ] = E [ γj +φ j γ t 1 Ij (z t ) j=1 = = m E [ γ j I j (z t ) ] m + E [ ] φ j I j (z t )γ t 1 j=1 j=1 m m γ j E[I j (z t )]+ φ j E[I j (z t )]E [ ] γ t 1. j=1 j=1 Then,unconditionallyE[γ t ]=E [ γ t j ] and,usinge[ij (z t )]=m j (thatis, the probability of occurrence of each interval), gives E[γ t ] = m m γ j m j +E[γ t ] φ j m j, j=1 j=1 whichcanbesolvedas E[γ t ] = m j=1 γ jm j 1 m j=1 φ =G. jm j Collecting all terms gives E [ ] m m γ t zt 2 = γ j e j +G φ j e j j=1 j=1 9

10 and m m E[c(z t )] = β+ γ j e j +G φ j e j, j=1 j=1 which completes the proof. Remark 5 Note that when the dynamic asymmetric autoregressive terms are zero (that is, φ j =0for all j), this model collapses to a multiple threshold GJR-type representation. Furthermore, when the threshold is unique(that is, m=1),wecanderivethegjrmodel(d 0 =0)asaspecialcasebyimposing symmetry on the standardised residual. Claim 1 The ARMA representation of DAGARCH has a SETARMA structure. Proof. Settingv t =ε 2 t σ 2 t andsubstitutinggives m ] ε 2 t = ω+βε 2 t 1+ [ γj +φ j γ t 2 ε 2 t 1 S j +(1 βl)v t, j=1 ( ) wherethethresholdsareself-excitingasd j =d jσ 2 t 1=d j ε 2 t 1 v t. 10

11 Claim 2 The unconditional variance implied by DAGARCH is given by σ 2 = ω 1 β m j=1 γ je j G m j=1 φ. je j Proof. This follows from the stationarity condition and the SETARMA representation. Inthefollowing, forpracticalpurposes,wewilllabelthemodelasda- GARCH(q,m,d) where q represents the GARCH order (while the ARCH orderisconstrainedtobe1),misthenumberofasymmetricconstantterms, anddis thenumberof asymmetricdynamic coefficients(whichcanbeassumed constant over the thresholds, leading to d m). It is noted that DAGARCH(1,2,0) with the threshold set to zero is simply GJR. 3 Feasible estimation of DAGARCH IntheGJRmodel, thethresholdsarestableovertimeastheyarederived under the assumption of a symmetric distribution. In this case, we have E [ ] γ t zt 2 = E [ γ1 zti(z 2 t <0)+ γ 2 zti(z 2 t >0) ] = γ 1 E [ z 2 ti(z t <0) ] + γ 2 E [ z 2 ti(z t >0) ], 11

12 andgivenasymmetricz t densityfunction,wecanwrite E [ ] [ zt 2 = E z 2 t I(z t <0) ] +E [ zti(z 2 t >0) ] = 2 E [ z 2 ti(z t <0) ], E [ z 2 ti(z t <0) ] = E [ z 2 ti(z t >0) ] = 1 2 E[ z 2 t], which is then used to derive the standard GJR stationarity restriction. Furthermore,thesamederivationcanberecastusingε 2 t insteadofzt 2 sincethe zero threshold is not influenced by the value of the conditional variance, that is: E [ z 2 ti(z t <0) ] = E [ ε 2 ti(ε t <0) ]. In the DAGARCH model, the last inequality does not generally hold as the conditional variances are relevant in the definition of the thresholds. This can be observed directly in the d i thresholds, or indirectly in the d i thresholds through the derivation of the standardised residuals, which are not observable. Therefore,thethresholdsfortheobservableε t aretimevarying,creating a relevant computational problem in that the standard recursive estimation 12

13 approaches of GARCH-type models cannot be used. Furthermore, the d i thresholdsdependbothontheconditionalvariancesandonthed i thresholds. Infact,asshowninRemark1,d i =d iσ t 1,andthethresholdsonthe observablearetimedependent. Thesed i valuescanbedeterminedonlyon thebasisofadistributionalassumptiononthez t andshouldbefixed,considering the research purposes. Assuming that the interest is in comparing the effects of the tails and/or the positive and negative observable innovations, the various d i should be able to select innovations in the upper (orlower) α% tail. Consequently, we have a computational problem that is caused by two factors, namely an assumption on the standardised residuals which are not directly observed, and a set of time-varying thresholds which depends on the model parameter and specification. Inordertosolvebothoftheseproblems,weproposetwosolutionsinthe form of estimation algorithms. Algorithm A. Simplifying the DAGARCH thresholds: 1) make the d i thresholds dependent on the unconditional observable ( ) variance,thatis,d i =d i andassumeaprioriastudent-tora 1 N N j=1 ε2 j GED distribution for the standardised residuals; 2)computethed i thresholdsundertheassumeddistribution; 3) estimate DAGARCH; 4) test the assumed distribution for the obtained standardised residual and, if necessary, repeat the procedure from step 2). Algorithm B. Using a procedure that computes thresholds on the basis 13

14 of a simpler previously estimated model: 1) assume a Student-t or a GED distribution for the standardised residualsandcomputethed i thresholds; 2a) estimate a standard GARCH model under the distributional assumptionofstep1)andsavetheestimatedconditionalvariancesas GARCH σ 2 t; 2b) test the distributional assumption of the standardised residuals and, ifnecessary,updatethed i thresholds; 2c)computethed i thresholdsusing GARCH σ 2 t andd i; 3a) estimate DAGARCH with a zero restriction on the dynamic asymmetricautoregressivetermsandsavetheconditionalvariancesas DAGARCH,1 σ 2 t; 3b) test the distributional assumption of standardised residuals and, if necessary,updatethed i thresholds; 3c)computethed i thresholdsusing DAGARCH,1 σ 2 t andd i; 4a) estimate DAGARCH with an equality restriction on the dynamic asymmetricautoregressivetermsandsavetheconditionalvariancesas DAGARCH,2 σ 2 t; 4b) test the distributional assumption of standardised residuals and, if necessary,updatethed i thresholds; 4c)computethed i thresholdsusing DAGARCH,2 σ 2 t andd i; 5a) estimate a full DAGARCH and save the conditional variances as DAGARCH,3σ 2 t; 5b) test the distributional assumption of standardised residuals and, if necessary,updatethed i thresholds; 5c)computethed i thresholdsusing DAGARCH,3 σ 2 t andd i; 6) if necessary, repeat step 5) until convergence is achieved. 14

15 Clearly, the full estimation of the model is possible, even though computationally demanding. The second solution is computationally more intensive but it allows a comparison of the likelihoods obtained by the different approaches. Furthermore, the error included in the time-varying threshold will belowerthaninthefirstcasewhichusesstaticthresholds. Finally,theresultofstep6)canbecomparedwiththeestimationoutputofanunrestricted DAGARCH model with endogenous determination of the thresholds(which canbeviewedasapossiblestep7)withparameterstartingvaluesgivenby the step 6) results). 4 Dynamic asymmetry in daily stock indices This section provides an empirical analysis of the variance asymmetry of stock market indices. We consider the following daily indices: Dow Jones industrials(dj), Nikkei 225(NK), Eurostoxx 50(EX) and FTSE 100. Table 1 reports the sample period, the number of observations(obs.) and the sample moments of the logarithmic returns. The means and standard deviations (SD) of all four series are similar; the skewness of DJ is similar to that of FTSE,andthatofEXissimilartoNK;andthekurtosisofFTSEissimilar tothatofnk. [Insert Table 1 here] Using these logarithmic returns series, we estimated several specifications of the DAGARCH model and compared them with the standard GARCH(1,1) 15

16 and GJR(1,1) models. In particular, we considered GARCH, GJR(labelled DAGARCH(1,2,0)), GJR with common or specific dynamic asymmetry(da- GARCH(1,2,1) and DAGARCH(1,2,2), respectively), and two DAGARCH models with three thresholds (DAGARCH(1,4,0) and DAGARCH(1,4,1)). The DAGARCH(1,4,4) model was also considered but is not reported here as the dynamics appeared to be stable over the different thresholds. The following tables report the various estimated models for the indices givenintable1. Notalltheestimatedmodelsarereported. Insomecases, DAGARCH with three thesholds is preferred to the GJR representation, while in other cases the generalisation of the GJR with asymmetric dynamic is the preferred model. Figure 1 reports a comparison of the asymmetric impacts with and without dynamics. The models are preferred in terms of likelihoods using the standard LR test, as all models are nested. An interesting result concerns the asymmetric dynamics which are present for the negative returns values. [Insert Figure 1 here] [Insert Tables 2-5 here] 5 Concluding remarks This paper developed the Dynamic Asymmetric GARCH (or DAGARCH) model which generalises the GJR model to allow for multiple thresholds and dynamics in the asymmetric conditional variance. The DAGARCH model 16

17 accommodates both the size and sign effects of shocks and induces some dynamics in the variance asymmetry. The stationarity conditions are derived, as well as two algorithms for model estimation. An empirical application is also provided, with several DAGARCH specifications being estimated using the daily returns of the Dow Jones, Nikkei225, FTSE100 and EuroStoxx50 indicestoshowtheusefulnessofthenewmodel. References [1] Bollerslev, T., 1986, Generalized autoregressive conditional heteroskedasticity, Journal of Econometrics 31, [2] Engle, R.F., 1982, Autoregressive conditional heteroskedasticity with estimates of the variance of United Kingdom inflation, Econometrica 50, [3] Glosten L., R. Jagannathan and D. Runkle, 1993, Relationship between the expected value and the volatility of the nominal excess returns on stocks, Journal of Finance 48, [4] He,C.andT.Terasvirta,1999a,Propertiesofmomentsofafamilyof GARCH processes, Journal of Econometrics 92, [5] He, C. and T. Terasvirta, 1999b, Fourth moment structure of the GARCH(p,q) process, Econometric Theory 15,

18 [6] Ling, S. and M. McAleer, 2002a, Stationarity and the existence of moments of a family of GARCH processes, Journal of Econometrics 106, [7] Ling, S. and M. McAleer, 2002b, Necessary and sufficient moment conditionas for the GARCH(r,s,) and asymmetric power GARCH(r,s) models, Econometric Theory 18, [8] McAleer, M., F. Chan and D. Marinova, 2003, An econometric analysis of asymmetric volatility: theory and application to patents, Journal of Econometrics, forthcoming [9] Nelson, D.B., 1991, Conditional heteroskedasticity in asset returns: a new approach, Econometrica 59, [10] Rabemananjara R. and J.M. Zakoian, 1993, Threshold ARCH models and asymmetries in volatility, Journal of Applied Econometrics 8, [11] Zakoian, J.M., 1994, Threshold heteroskedastic functions, Journal of Economic Dynamics and Control 18,

19 Figure1: FTSEIndex-DynamicEvolutionofγ t fortheperiodnovember 2002-March

20 Table 1: Descriptive Statistics of Logarithmic Returns Index Sample Obs. Mean SD Skewness Kurtosis DJ 02/01/30-29/04/ FTSE 02/04/84-08/04/ EX 01/01/92-08/04/ NK 04/01/84-11/05/

21 Table 2: Nikkei estimates (insignificant estimates are in bold) Parameters GARCH(1,1) DAGARCH(1,2,0) DAGARCH(1,2,1) DAGARCH(1,2,2) ω β γ 1 (α) γ φ φ LogLik γ 1 (α) reports the ARCH(1) parameter in GARCH(1,1), the constant asymmetry parameter for negative returns values in DAGARCH(1,2,j), and the constant asymmetry parameter for large negative returns in DAGARCH(1,4,j). γ 2 reports the constant asymmetry parameter for small negative returns values in DAGARCH(1,4,j). γ 3 reports the constant asymmetry parameter for positive returns values in DAGARCH(1,2,j), and the constant asymmetry parameter for small positive returns values in DAGARCH(1,4,j) estimates. γ 4 reports the constant asymmetry parameter for large positive returns values in DAGARCH(1,4,j). φ 1 reports the dynamic asymmetry parameter in DAGARCH(1,j,1), and the dynamic asymmetry parameter for negative returns values in DAGARCH(1,2,2). φ 2 reports the dynamic asymmetry parameter for positive returns values in DAGARCH(1,2,2). 21

22 Table 3: Dow Jones estimates (insignificant estimates are in bold) Parameters GARCH(1,1) DAGARCH(1,2,0) DAGARCH(1,2,1) DAGARCH(1,2,2) ω β γ 1 (α) γ φ φ LogLik See the footnotes for Table 2. 22

23 Table 4: FTSE estimates (insignificant estimates are in bold) Parameters GARCH(1,1) DAGARCH(1,2,0) DAGARCH(1,2,1) DAGARCH(1,2,2) DAGARCH(1,4,0) DAGARCH(1,4,1) ω β γ 1 (α) γ γ γ φ φ LogLik See the footnotes for Table 2. 23

24 Table 5: Eurostoxx estimates (insignificant estimates are in bold) Parameters GARCH(1,1) DAGARCH(1,2,0) DAGARCH(1,2,1) DAGARCH(1,2,2) DAGARCH(1,4,0) DAGARCH(1,4,1) ω β γ 1 (α) γ γ γ φ φ LogLik See the footnotes for Table 2. 24