- Email: [email protected]
Stationarity of stable power-GARCH processes
Journal of Econometrics 106 (2002) 97–107 www.elsevier.com/locate/econbase
Stationarity of stable power-GARCH processes Stefan Mittnika; ∗ , Marc S. Paolellaa , Svetlozar T. Rachevb a Institute
of Statistics and Econometrics, University of Kiel, Olshausenstr. 40, D-24098 Kiel, Germany b Institute of Statistics and Mathematical Economics, University of Karlsruhe, Kollegium am Schloss II, D-76128 Karlsruhe, Germany Received 15 July 1999; revised 27 March 2001; accepted 30 April 2001
Abstract We present conditions for strict stationarity of power-GARCH processes whose innovations are described by a heavy-tailed and possibly asymmetric stable Paretian distribution. The results generalize those of Bougerol and Picard (J. Econom. 52 (1992) 115), who derived analogous conditions for standard, i.e., power-two, GARCH processes with 5nite-variance innovations. ? 2002 Elsevier Science S.A. All rights reserved. JEL classi)cation: C22 Keywords: Asymmetry; Conditional heteroscedasticity; Financial modeling; Heavy tails; Integrated GARCH; State space representation; Stationarity
1. Introduction Autoregressive conditional heteroskedastic (ARCH) models introduced by Engle (1982) and the extension to generalized ARCH (GARCH) models by Bollerslev (1986) have become standard tools in econometrics—especially in empirical 5nance. They are capable of capturing two important features that characterize time series of returns on 5nancial assets: volatility clustering The research of S. Mittnik was supported by the Deutsche Forschungsgemeinschaft. Part of the research was conducted while S.T. Rachev was visiting the University of Kiel with support from the Alexander von Humboldt Foundation. ∗ Corresponding author. Tel.: +49-431-880-2166; fax: +49-431-880-2673. E-mail address: [email protected] (S. Mittnik).
0304-4076/02/$ - see front matter ? 2002 Elsevier Science S.A. All rights reserved. PII: S 0 3 0 4 - 4 0 7 6 ( 0 1 ) 0 0 0 8 9 - 6
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or conditional heteroskedasticity and excess leptokurtosis or heavy-tailedness, i.e., the tails of the unconditional distribution are heavier than implied by the normal distribution. Both of these phenomena were already observed in Mandelbrot (1963). However, in his work, he focused on the second property and proposed the stable Paretian distribution as a model for the unconditional asset returns. Unconditional heavy tails and GARCH phenomena are not unrelated. Diebold (1988) demonstrates that GARCH process driven by normally distributed innovations generate time series with heavy-tailed unconditional distributions; de Vries (1991) and Groenendijk et al. (1995) show that certain GARCH processes can give rise to unconditional stable Paretian distributions. When 5tting GARCH models to return series, it is often found that GARCH residuals still tend to be heavy tailed. To accommodate this, GARCH models with heavier conditional innovation distributions than those of the normal have been proposed—among them the Student’s t (Bollerslev, 1987) and the generalized error distribution, in short GED (Nelson, 1991). To allow for particularly heavy-tailed, conditional (and unconditional) return distributions, GARCH processes with non-normal stable Paretian error distributions have been considered (see McCulloch, 1985; Liu and Brorsen, 1995; Panorska et al., 1995; Mittnik et al., 1998). 1 The class of stable Paretian distributions contains the normal distribution as a special case, but also allows for heavy-tailedness, i.e., in5nite variance, and asymmetry. Neither the t nor the GED distribution share the latter property. The stable Paretian distribution also has the appealing property that it is the only distribution that arises as a limiting distribution of sums of independently, identically distributed (iid) random variables. This is required when error terms are assumed to be the sum of all external eIects that are not captured by the model. An occasional objection against the use of the non-normal stable Paretian distribution is that it has in5nite variance. This seems to contradict empirical studies suggesting the existence of third or fourth moments for various 5nancial return data (cf. Loretan and Phillips, 1994; Pagan, 1996). However, these 5ndings were almost exclusively arrived at by use of the Hill (1975) or related tail estimators, which are known to be highly unreliable for—even large—iid samples (cf. Mittnik and Rachev, 1993; Kratz and Resnick, 1996; Resnick, 1997; Adler, 1997; McCulloch, 1997; Mittnik et al., 1998; Paolella, 2001) and even worse for data with GARCH structures (Kearns and Pagan, 1 In fact, McCulloch (1985) presented a model which can be viewed as a precursor of Bollerslev’s (1986) GARCH model or, for that matter, Taylor’s (1986) power-one GARCH model as well as the integrated GARCH model of Engle and Bollerslev (1986). McCulloch’s model amounts to a power-one, integrated GARCH(1,1) process without a constant term driven by conditional symmetric stable Paretian innovations. Nelson (1991) considered the Cauchy distribution, which is a special case of the stable Paretian distribution. Engle and Bollerslev (1986) formulate integrated GARCH for 5nite-variance but not necessarily normal distributions.
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99
1997). 2 Hence, the question of the maximally existing moment of 5nancial return data is yet an open one. In this paper, we present necessary and suJcient conditions for strict stationarity of GARCH processes driven by disturbances with possibly in5nitevariance and asymmetric stable Paretian distributions. Considering GARCH processes with 5nite-variance disturbances, Bollerslev (1986) established necessary and suJcient conditions for second-order stationarity; and Nelson (1990) derived analogous conditions for strict stationarity for GARCH(1,1) processes, which were subsequently extended by Bougerol and Picard (1992) to higher orders. In addition to allowing in5nite-variance disturbances, we generalize the results of Bougerol and Picard by also considering powerGARCH processes proposed, among others, by Ding et al. (1993) and Liu and Brorsen (1995), i.e., processes where the GARCH equation propagates not just conditional moments of order two, but, more generally, absolute moments of order ¿0. The results presented here also generalize those of Panorska et al. (1995). Their conditions are restricted to power-one processes (i.e., = 1) driven by symmetrically distributed innovations. Moreover, they fail to provide an analytic expression for the scaling factor entering the stationarity condition and resort to numerical approximations. A second and only indirectly related contribution of the paper is a new and more compact state space representation for GARCH processes. Compared to the state space representations presented in the literature, the new representation does not only simplify proving the suJcient conditions for stationarity given in Bougerol and Picard and that stated below, but also facilitates practical work with GARCH models such as volatility prediction or simulation. The paper is organized as follows. Section 2 brieLy introduces stable power-GARCH processes. Necessary and suJcient conditions for strict stationarity are established in Section 3. Section 4 presents the evaluation of a scaling factor, a particular expected value, which is required to operationalize the conditions, and considers various special cases. Concluding remarks are given in Section 5. 2. Stable Paretian power-GARCH processes Process yt is called a stable Paretian power-GARCH process, in short, an S; ; GARCH(r; s) process, if it is described by yt = t + ct t ; 2
iid
t ∼ S; ;
(1)
Adler (1997) went so far as to say that “[o]verall, it seems that the time may have come to relegate Hill-like estimators to the Annals of Not-Terribly-Useful Ideas”.
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and ct = 0 +
r
i |yt−i − t−i | +
i=1
s
j ct−j ;
(2)
j=1
where 0 ¿0, i ¿ 0, i = 1; : : : ; r, r ¿ 1, j ¿ 0, j = 1; : : : ; s, s ¿ 0, and S; denotes the standard asymmetric stable Paretian distribution 3 with stable index , skewness parameter ∈ [ − 1; 1], zero location parameter and unit scale parameter. The distribution is symmetric for = 0 and skewed to the right (left) for ¿0 (¡0). The stable index , which, in general, assumes values in the interval (0; 2], determines the tail-thickness of the distribution. The tails become thinner as approaches 2; and for = 2 the standard stable Paretian distribution coincides with the normal distribution N(0; 2). For ¡2, t does not possess moments of order or higher. Thus, only for ¿1, which will be assumed throughout the paper, does the mean exist. The power parameter, , satis5es 0¡¡. By letting the location parameter t in (1) be time-varying, we permit general mean equations, including, for example, regression and=or ARMA structures. 3. Stationarity of GARCH-stable processes To derive the necessary and suJcient conditions for strict stationarity we follow the strategy of Bougerol and Picard (1992) (hereafter BP) and cast the GARCH process de5ned by (1) and (2) in 5rst-order state space form. However, the state space representation we adopt below diIers from that of BP in that it is not only more compact but, more importantly, avoids the technical diJculties that arise for the empirically important GARCH(1,1) case. Setting m = max(r; s); the S; ; GARCH process given by (1) and (2) can be rewritten as Xt+1 = FXt + G |yt − t | + B; where F(m×m), G(m×1) 1 1 0 0 1 2 .. F = . m−1 0 0 0 0 m 3
(3)
and B(m×1) are de5ned by ··· 0 1 0 0 2 0 .. ; G = .. ; B = .. : .. . . . . 1 m−1 0 ··· 0 m 0
We are using the same parameterization as Samorodnitsky and Taqqu (1994) and Rachev and Mittnik (2000).
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If r = s, we set, in G or F, i = 0, for i = s + 1; : : : ; m, or i = 0, for i = r + 1; : : : ; m. The kth component of state vector Xt ; denoted by Xk; t , is given by c if k = 1; t s r Xk; t = c + j |yt−j − t−j | if k = 2; : : : ; m: i t−i i=k
j=k
Because |yt − t | = ct | t | = e1; m Xt | t | , with e1; m = [1 0 : : : 0] denoting the 5rst m×1 unit vector, the state transition equation (3) can be expressed as Xt+1 = At Xt + B; where B is as in (3) and
(4)
| t | 1 + 1
| t | 2 + 2 .. At = F + Ge1; m | t | = . | t | m−1 + m−1
| t | m + m
1 0 0 0
0 ··· 0 1 0 . .. : . .. 0 1
0 ··· 0
Representation (4) has the advantage over that employed by BP in that it is also valid for GARCH(1,1) processes. BP’s general proof of their suJcient condition for strict stationarity does not apply to the GARCH(1,1) case, because matrix At of their representation has a zero row when p = q = 1. As a consequence, the results of Kesten and Spitzer (1984), which form the basis of their proof, are not applicable. This problem does not arise with representation (4), obviating a separate stationary proof for the GARCH(1,1) case. Apart from this mathematical “elegancy” argument, the above representation is appealing from a practical viewpoint, because it is more compact for higher-order processes. The state vector in (3) is of dimension max(r; s) rather than r + s − 1 with r ¿ 2, as required in BP, or max(r; s) + s, as in Baillie and Bollerslev (1992). Computationally intensive applications, such as back-testing procedures, Monte Carlo simulations or online computations, will bene5t from this reduction in complexity. The existence of solutions to (1) and (2) is equivalent to the existence of solutions to (4). It is determined by the largest Lyapunov exponent associated with the sequence of iid random matrices {At } and given by
1 1 a:s: log||At At−1 : : : At−n || = lim log||At At−1 : : : At−n ||; = inf E n→∞ n n→∞ n + 1 (5)
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when E(log+ ||At ||)¡∞ with log+ x = max(log x; 0) (see BP and references therein). In (5), || · || denotes any norm on Rn de5ning an operator norm on the set M (d) of d×d matrices by ||M || = sup{||Mx||= ||x||, x ∈ Rd ; x = 0}, for any M ∈ M (d). The Lyapunov exponent 6 E log||At ||¡∞ is well de5ned, because E| t | ¡∞. Then, is well de5ned for the sequence {An ; n ∈ Z}. Clearly, E(log+ ||B||)¡∞. Thus, by Theorem 3:2 of BP, we have: Proposition 1. The S; ; GARCH process de)ned by (1) and (2) has a stationary solution if and only if the largest Lyapunov exponent; ; associated with matrices {At } is strictly negative. The series Xt = B + ∞ i=1 At At−1 : : : At−i+1 B converges almost surely for all t; and process {Xt ; t in Z} is the unique strictly stationary and ergodic solution of (4). Proposition 2. The S; ; GARCH(r; s) process de)ned by (1) and (2) has a unique; strictly stationary solution if r s i + j 6 1; (6) "; ; i=1
i=1
where "; ; := E | t | . The proof of Proposition 2 follows the arguments given in BP. The only diIerence in our case is that the characteristic polynomial for EAt , s r i z −i − j z −j ; det(Iz − EAt ) = z r+s−1 1 − "; ; i=1
j=1
diIers slightly. BP consider a GARCH(r; s) process driven by 5nite-variance innovations and show that s r j 6 1 (7) i + i=1
j=1
is a suJcient condition for a unique strictly stationary solution. In the general stable case condition (7) needs to be modi5ed according to (6) in Proposition 2. The stationarity of an estimated model can be guaranteed by imposing (6) during estimation. To do so, the expected value "; ; = E| | has to be determined. 4. Determination of ; ; and special cases Factor "; ; depends on power as well as the stable index and skewness parameter of the standard stable Paretian distribution and, for ¡, is of
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the form (see Samorodnitsky and Taqqu, 1994, p. 18)
2 − 1 2 =2 arctan (; (1+(; ) cos ' 1− "; ; = ∞ 0 u− − 1 sin2 (u) du (8) with (; := tan )=2. To operationalize (8) for practical applications, the integral in the denominator needs to be evaluated. Use of identity cos 2u = 1 − ∞ 2 sin2(u) and substitution v = 2u allows us to express 0 u−−1 sin2 (u) du as ∞ 2+1 0 v−−1 (1 − cos v) dv or 4 −1 ∞ 2 ) if 0¡¡2; = 1; 2 '(1 − ) cos 2 −−1 u sin (u) du = (9) ) 0 if = 1 2 (see, for example, HoImann-JHrgensen, 1994, p. 218) yielding
−1 2 =2 arctan (; "; ; = + ' 1 − (1 + (; ) cos for ¡ and ¿1, where '(1 − ) cos ) 2 + = )=2
if = 1; if = 1:
(10)
(11)
Note that "; ; increases without bound as approaches . A plot of "; ; for the power-one case, i.e., "; ; 1 , is shown in Fig. 1. If, in addition to = 1, the distribution is symmetric, i.e., = 0, then
2 1 : "; 0; 1 = ' 1 − ) iid
For = 2, we have t ∼ N(0; 2) and, with Z ∼ N(0; 1),
2 +1 =2 "2; 0; = 2 E|Z | = √ ' ; 2 )
(12)
so that, as → 2, the limiting value in Fig. 1 for all ∈ [ − 1; 1] is 1.1284. If, instead, rescaled stable Paretian innovations, -t = 2−1=2 t , are used, then iid we have E|-t | = 2−=2 "; ; and, for = 2, -t ∼ N(0; 1) with
2=2 +1 E|-t | = √ ' : 2 ) 4
We use the fact that the gamma function is de5ned for all x ∈ R except for nonpositive integers.
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Fig. 1. Values of "; ; 1 as given by Eq. (10).
This is clearly unity for = 2, so that (6) reduces to (7), the condition associated with conventional normal-GARCH models. 5. Concluding remarks The stationarity condition for power-GARCH processes driven by stable Paretian innovations established in Proposition 2 gives rise to several remarks. The admissible parameter space for parameters in the conditional-volatility equation, i , i = 1; : : : ; r, and j , j = 1; : : : ; s, shrinks, under ceteris paribus conditions, as the tails of the innovations become heavier (i.e., ↓ 1), as the skewness of the innovation increases (i.e., as → ±1) and as the power parameter, , increases (i.e., ↑ ). Liu and Brorsen (1995) considered S; 0; GARCH(1; 1) models with = . In this case, the conditional mean of the quantity expressing the conditional volatility, ct , is in5nite. In simulation experiments we conducted, we found that setting = may lead to extremely erratic behavior of the conditionalvolatility process—a behavior we do not encounter in 5nancial data.
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Engle and Bollerslev (1986) refer to GARCH processes being driven by innovations with unit-variance and satisfying the borderline condition s r j = 1 (13) i + i=1
j=1
as an integrated GARCH or IGARCH process. Rewriting the GARCH process in form of an ARMA process for t2 (see Bollerslev, 1988), the condition implies that the resulting autoregressive polynomial, 1 − (1 + 1 )L − · · · − (m + m )Lm with m = max(r; s), has a unit root. In the S; ; GARCH(r; s) case, the analogue to (13) is r s i + j = 1; (14) "; ; i=1
j=1
giving rise to integrated S; ; GARCH(r; s) or S; ; IGARCH(r; s) processes. Under (14), the implied autoregressive polynomial for the ARMA representation of |yt − t | , given by 1−("; ; 1 +1 )L−· · ·−("; ; m +m )Lm , possesses a unit root and, thus, gives rise to persistence of conditional volatility. The estimation of S; ; GARCH(r; s) models as well as other stable Paretian models is discussed in Mittnik et al. (1999). Stationarity conditions and forecasting performance of asymmetric power GARCH models with generalized (asymmetric) Student’s t distributions are examined in Mittnik and Paolella (2000); and empirical comparisons between S; ; GARCH and Student’s t-GARCH models with value-at-risk applications are given in Mittnik and Paolella (2001). Acknowledgements We wish to thank Vygantas Paulauskas for helpful discussions, as well as two anonymous referees and the associate editor for constructive comments. References Adler, R.J., 1997. Discussion: heavy tail modeling and teletraJc data. The Annals of Statistics 25, 1849–1852. Baillie, R.T., Bollerslev, T., 1992. Prediction in dynamic models with time-dependent conditional variances. Journal of Econometrics 52, 91–113. Bollerslev, T., 1986. Generalized autoregressive conditional heteroskedasticity. Journal of Econometrics 31, 307–327. Bollerslev, T., 1987. A conditional heteroskedastic time series model for speculative prices and rates of return. Review of Economics and Statistics 69, 542–547. Bollerslev, T., 1988. On the correlation structure for the generalized autoregressive conditional heteroskedastic process. Journal of Financial Analysis 9, 121–131.
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Bougerol, P., Picard, N., 1992. Stationarity of GARCH processes and of some non-negative time series. Journal of Econometrics 52, 115–127. de Vries, C.G., 1991. On the relation between GARCH and stable processes. Journal of Econometrics 48, 313–324. Diebold, F.X., 1988. Empirical Modeling of Exchange Rate Dynamics. Springer, Berlin. Ding, Z., Granger, C.W., Engle, R.F., 1993. A long memory property of stock market returns and a new model. Journal of Empirical Finance 1, 83–106. Engle, R., 1982. Autoregressive conditional heteroskedasticity with estimates of the variance of United Kingdom inLation. Econometrica 50, 987–1007. Engle, R.F., Bollerslev, T., 1986. Modelling the persistence of conditional variances. Econometric Reviews 5, 1–50; 81–87. Groenendijk, P.A., Lucas, A., de Vries, C.G., 1995. A note on the relationship between GARCH and symmetric stable processes. Journal of Empirical Finance 2, 253–264. Hill, B.M., 1975. A simple general approach to inference about the tail of a distribution. Annals of Statistics 3, 1163–1174. HoImann-JHrgensen, J., 1994. Probability With a View Towards Statistics, Vol. I. Chapman & Hall, New York. Kearns, P., Pagan, A., 1997. Estimating the density tail index for 5nancial time series. The Review of Economics and Statistics 97, 171–175. Kesten, H., Spitzer, F., 1984. Convergence in distribution for products of random matrices. Zeitschrift fVur Wahrscheinlichkeitstheorie und Verwandte Gebiete 67, 363–386. Kratz, M., Resnick, S.I., 1996. The QQ-estimator and heavy tails. Communications in Statistics—Stochastic Models 12, 699–724. Liu, S., Brorsen, B.W., 1995. Maximum likelihood estimation of a GARCH-stable model. Journal of Applied Econometrics 10, 273–285. Loretan, M., Phillips, P.C.B., 1994. Testing the covariance stationarity of heavy-tailed time series. Journal of Empirical Finance 1, 211–248. Mandelbrot, B., 1963. The variation of certain speculative prices. Journal of Business 36, 394– 419. McCulloch, J.H., 1985. Interest-risk sensitive deposit insurance premia: stable ACH estimates. Journal of Banking and Finance 9, 137–156. McCulloch, J.H., 1997. Measuring tail thickness in order to estimate the stable index : a critique. Journal of Business and Economic Statistics 15, 74–81. Mittnik, S., Paolella, M.S., 2000. Conditional density and value-at-risk prediction of Asian currency exchange rates. Journal of Forecasting 19, 313–333. Mittnik, S., Paolella, M.S., 2001. Prediction of 5nancial downside-risk with heavy-tailed conditional distributions. In: Handbook of Heavy Tailed Distributions in Finance, North-Holland, to appear. Mittnik, S., Paolella, M.S., Rachev, S.T., 1998. A tail estimator for the index of the stable Paretian distribution. Communications in Statistics—Theory and Methods 27, 1239–1262. Mittnik, S., Rachev, S.T., 1993. Reply to comments on ‘Modeling asset returns with alternative stable models’ and some extensions. Econometric Reviews 12, 347–389. Mittnik, S., Rachev, S.T., Doganoglu, T., Chenyao, D., 1999. Maximum likelihood estimation of stable Paretian models. Mathematical and Computer Modelling 29, 275–293. Nelson, D., 1990. Stationarity and persistence in the GARCH(1,1) model. Econometric Theory 6, 318–344. Nelson, D., 1991. Conditional heteroskedasticity in asset returns: a new approach. Econometrica 59, 347–370. Pagan, A., 1996. The econometrics of 5nancial markets. Journal of Empirical Finance 3, 15–102. Panorska, A., Mittnik, S., Rachev, S.T., 1995. Stable GARCH Models for Financial Time Series. Applied Mathematics Letters 8, 33–37.
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Paolella, M.S., 2001. Testing the stable Paretian assumption. Mathematical and Computer Modeling, to appear. Rachev, S.T., Mittnik, S., 2000. Stable Paretian Models in Finance. Wiley, Chichester. Resnick, S., 1997. Heavy tail modeling and teletraJc data. The Annals of Statistics 25, 1805– 1869. Samorodnitsky, G., Taqqu, M., 1994. Stable Non-Gaussian Random Processes, Stochastic Models with In5nite Variance. Chapman & Hall, London. Taylor, S.J., 1986. Modelling Financial Time Series. Wiley, New York.
Stationarity of stable power-GARCH processes Stefan Mittnika; ∗ , Marc S. Paolellaa , Svetlozar T. Rachevb a Institute
of Statistics and Econometrics, University of Kiel, Olshausenstr. 40, D-24098 Kiel, Germany b Institute of Statistics and Mathematical Economics, University of Karlsruhe, Kollegium am Schloss II, D-76128 Karlsruhe, Germany Received 15 July 1999; revised 27 March 2001; accepted 30 April 2001
Abstract We present conditions for strict stationarity of power-GARCH processes whose innovations are described by a heavy-tailed and possibly asymmetric stable Paretian distribution. The results generalize those of Bougerol and Picard (J. Econom. 52 (1992) 115), who derived analogous conditions for standard, i.e., power-two, GARCH processes with 5nite-variance innovations. ? 2002 Elsevier Science S.A. All rights reserved. JEL classi)cation: C22 Keywords: Asymmetry; Conditional heteroscedasticity; Financial modeling; Heavy tails; Integrated GARCH; State space representation; Stationarity
1. Introduction Autoregressive conditional heteroskedastic (ARCH) models introduced by Engle (1982) and the extension to generalized ARCH (GARCH) models by Bollerslev (1986) have become standard tools in econometrics—especially in empirical 5nance. They are capable of capturing two important features that characterize time series of returns on 5nancial assets: volatility clustering The research of S. Mittnik was supported by the Deutsche Forschungsgemeinschaft. Part of the research was conducted while S.T. Rachev was visiting the University of Kiel with support from the Alexander von Humboldt Foundation. ∗ Corresponding author. Tel.: +49-431-880-2166; fax: +49-431-880-2673. E-mail address: [email protected] (S. Mittnik).
0304-4076/02/$ - see front matter ? 2002 Elsevier Science S.A. All rights reserved. PII: S 0 3 0 4 - 4 0 7 6 ( 0 1 ) 0 0 0 8 9 - 6
98
S. Mittnik et al. / Journal of Econometrics 106 (2002) 97–107
or conditional heteroskedasticity and excess leptokurtosis or heavy-tailedness, i.e., the tails of the unconditional distribution are heavier than implied by the normal distribution. Both of these phenomena were already observed in Mandelbrot (1963). However, in his work, he focused on the second property and proposed the stable Paretian distribution as a model for the unconditional asset returns. Unconditional heavy tails and GARCH phenomena are not unrelated. Diebold (1988) demonstrates that GARCH process driven by normally distributed innovations generate time series with heavy-tailed unconditional distributions; de Vries (1991) and Groenendijk et al. (1995) show that certain GARCH processes can give rise to unconditional stable Paretian distributions. When 5tting GARCH models to return series, it is often found that GARCH residuals still tend to be heavy tailed. To accommodate this, GARCH models with heavier conditional innovation distributions than those of the normal have been proposed—among them the Student’s t (Bollerslev, 1987) and the generalized error distribution, in short GED (Nelson, 1991). To allow for particularly heavy-tailed, conditional (and unconditional) return distributions, GARCH processes with non-normal stable Paretian error distributions have been considered (see McCulloch, 1985; Liu and Brorsen, 1995; Panorska et al., 1995; Mittnik et al., 1998). 1 The class of stable Paretian distributions contains the normal distribution as a special case, but also allows for heavy-tailedness, i.e., in5nite variance, and asymmetry. Neither the t nor the GED distribution share the latter property. The stable Paretian distribution also has the appealing property that it is the only distribution that arises as a limiting distribution of sums of independently, identically distributed (iid) random variables. This is required when error terms are assumed to be the sum of all external eIects that are not captured by the model. An occasional objection against the use of the non-normal stable Paretian distribution is that it has in5nite variance. This seems to contradict empirical studies suggesting the existence of third or fourth moments for various 5nancial return data (cf. Loretan and Phillips, 1994; Pagan, 1996). However, these 5ndings were almost exclusively arrived at by use of the Hill (1975) or related tail estimators, which are known to be highly unreliable for—even large—iid samples (cf. Mittnik and Rachev, 1993; Kratz and Resnick, 1996; Resnick, 1997; Adler, 1997; McCulloch, 1997; Mittnik et al., 1998; Paolella, 2001) and even worse for data with GARCH structures (Kearns and Pagan, 1 In fact, McCulloch (1985) presented a model which can be viewed as a precursor of Bollerslev’s (1986) GARCH model or, for that matter, Taylor’s (1986) power-one GARCH model as well as the integrated GARCH model of Engle and Bollerslev (1986). McCulloch’s model amounts to a power-one, integrated GARCH(1,1) process without a constant term driven by conditional symmetric stable Paretian innovations. Nelson (1991) considered the Cauchy distribution, which is a special case of the stable Paretian distribution. Engle and Bollerslev (1986) formulate integrated GARCH for 5nite-variance but not necessarily normal distributions.
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1997). 2 Hence, the question of the maximally existing moment of 5nancial return data is yet an open one. In this paper, we present necessary and suJcient conditions for strict stationarity of GARCH processes driven by disturbances with possibly in5nitevariance and asymmetric stable Paretian distributions. Considering GARCH processes with 5nite-variance disturbances, Bollerslev (1986) established necessary and suJcient conditions for second-order stationarity; and Nelson (1990) derived analogous conditions for strict stationarity for GARCH(1,1) processes, which were subsequently extended by Bougerol and Picard (1992) to higher orders. In addition to allowing in5nite-variance disturbances, we generalize the results of Bougerol and Picard by also considering powerGARCH processes proposed, among others, by Ding et al. (1993) and Liu and Brorsen (1995), i.e., processes where the GARCH equation propagates not just conditional moments of order two, but, more generally, absolute moments of order ¿0. The results presented here also generalize those of Panorska et al. (1995). Their conditions are restricted to power-one processes (i.e., = 1) driven by symmetrically distributed innovations. Moreover, they fail to provide an analytic expression for the scaling factor entering the stationarity condition and resort to numerical approximations. A second and only indirectly related contribution of the paper is a new and more compact state space representation for GARCH processes. Compared to the state space representations presented in the literature, the new representation does not only simplify proving the suJcient conditions for stationarity given in Bougerol and Picard and that stated below, but also facilitates practical work with GARCH models such as volatility prediction or simulation. The paper is organized as follows. Section 2 brieLy introduces stable power-GARCH processes. Necessary and suJcient conditions for strict stationarity are established in Section 3. Section 4 presents the evaluation of a scaling factor, a particular expected value, which is required to operationalize the conditions, and considers various special cases. Concluding remarks are given in Section 5. 2. Stable Paretian power-GARCH processes Process yt is called a stable Paretian power-GARCH process, in short, an S; ; GARCH(r; s) process, if it is described by yt = t + ct t ; 2
iid
t ∼ S; ;
(1)
Adler (1997) went so far as to say that “[o]verall, it seems that the time may have come to relegate Hill-like estimators to the Annals of Not-Terribly-Useful Ideas”.
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and ct = 0 +
r
i |yt−i − t−i | +
i=1
s
j ct−j ;
(2)
j=1
where 0 ¿0, i ¿ 0, i = 1; : : : ; r, r ¿ 1, j ¿ 0, j = 1; : : : ; s, s ¿ 0, and S; denotes the standard asymmetric stable Paretian distribution 3 with stable index , skewness parameter ∈ [ − 1; 1], zero location parameter and unit scale parameter. The distribution is symmetric for = 0 and skewed to the right (left) for ¿0 (¡0). The stable index , which, in general, assumes values in the interval (0; 2], determines the tail-thickness of the distribution. The tails become thinner as approaches 2; and for = 2 the standard stable Paretian distribution coincides with the normal distribution N(0; 2). For ¡2, t does not possess moments of order or higher. Thus, only for ¿1, which will be assumed throughout the paper, does the mean exist. The power parameter, , satis5es 0¡¡. By letting the location parameter t in (1) be time-varying, we permit general mean equations, including, for example, regression and=or ARMA structures. 3. Stationarity of GARCH-stable processes To derive the necessary and suJcient conditions for strict stationarity we follow the strategy of Bougerol and Picard (1992) (hereafter BP) and cast the GARCH process de5ned by (1) and (2) in 5rst-order state space form. However, the state space representation we adopt below diIers from that of BP in that it is not only more compact but, more importantly, avoids the technical diJculties that arise for the empirically important GARCH(1,1) case. Setting m = max(r; s); the S; ; GARCH process given by (1) and (2) can be rewritten as Xt+1 = FXt + G |yt − t | + B; where F(m×m), G(m×1) 1 1 0 0 1 2 .. F = . m−1 0 0 0 0 m 3
(3)
and B(m×1) are de5ned by ··· 0 1 0 0 2 0 .. ; G = .. ; B = .. : .. . . . . 1 m−1 0 ··· 0 m 0
We are using the same parameterization as Samorodnitsky and Taqqu (1994) and Rachev and Mittnik (2000).
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If r = s, we set, in G or F, i = 0, for i = s + 1; : : : ; m, or i = 0, for i = r + 1; : : : ; m. The kth component of state vector Xt ; denoted by Xk; t , is given by c if k = 1; t s r Xk; t = c + j |yt−j − t−j | if k = 2; : : : ; m: i t−i i=k
j=k
Because |yt − t | = ct | t | = e1; m Xt | t | , with e1; m = [1 0 : : : 0] denoting the 5rst m×1 unit vector, the state transition equation (3) can be expressed as Xt+1 = At Xt + B; where B is as in (3) and
(4)
| t | 1 + 1
| t | 2 + 2 .. At = F + Ge1; m | t | = . | t | m−1 + m−1
| t | m + m
1 0 0 0
0 ··· 0 1 0 . .. : . .. 0 1
0 ··· 0
Representation (4) has the advantage over that employed by BP in that it is also valid for GARCH(1,1) processes. BP’s general proof of their suJcient condition for strict stationarity does not apply to the GARCH(1,1) case, because matrix At of their representation has a zero row when p = q = 1. As a consequence, the results of Kesten and Spitzer (1984), which form the basis of their proof, are not applicable. This problem does not arise with representation (4), obviating a separate stationary proof for the GARCH(1,1) case. Apart from this mathematical “elegancy” argument, the above representation is appealing from a practical viewpoint, because it is more compact for higher-order processes. The state vector in (3) is of dimension max(r; s) rather than r + s − 1 with r ¿ 2, as required in BP, or max(r; s) + s, as in Baillie and Bollerslev (1992). Computationally intensive applications, such as back-testing procedures, Monte Carlo simulations or online computations, will bene5t from this reduction in complexity. The existence of solutions to (1) and (2) is equivalent to the existence of solutions to (4). It is determined by the largest Lyapunov exponent associated with the sequence of iid random matrices {At } and given by
1 1 a:s: log||At At−1 : : : At−n || = lim log||At At−1 : : : At−n ||; = inf E n→∞ n n→∞ n + 1 (5)
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when E(log+ ||At ||)¡∞ with log+ x = max(log x; 0) (see BP and references therein). In (5), || · || denotes any norm on Rn de5ning an operator norm on the set M (d) of d×d matrices by ||M || = sup{||Mx||= ||x||, x ∈ Rd ; x = 0}, for any M ∈ M (d). The Lyapunov exponent 6 E log||At ||¡∞ is well de5ned, because E| t | ¡∞. Then, is well de5ned for the sequence {An ; n ∈ Z}. Clearly, E(log+ ||B||)¡∞. Thus, by Theorem 3:2 of BP, we have: Proposition 1. The S; ; GARCH process de)ned by (1) and (2) has a stationary solution if and only if the largest Lyapunov exponent; ; associated with matrices {At } is strictly negative. The series Xt = B + ∞ i=1 At At−1 : : : At−i+1 B converges almost surely for all t; and process {Xt ; t in Z} is the unique strictly stationary and ergodic solution of (4). Proposition 2. The S; ; GARCH(r; s) process de)ned by (1) and (2) has a unique; strictly stationary solution if r s i + j 6 1; (6) "; ; i=1
i=1
where "; ; := E | t | . The proof of Proposition 2 follows the arguments given in BP. The only diIerence in our case is that the characteristic polynomial for EAt , s r i z −i − j z −j ; det(Iz − EAt ) = z r+s−1 1 − "; ; i=1
j=1
diIers slightly. BP consider a GARCH(r; s) process driven by 5nite-variance innovations and show that s r j 6 1 (7) i + i=1
j=1
is a suJcient condition for a unique strictly stationary solution. In the general stable case condition (7) needs to be modi5ed according to (6) in Proposition 2. The stationarity of an estimated model can be guaranteed by imposing (6) during estimation. To do so, the expected value "; ; = E| | has to be determined. 4. Determination of ; ; and special cases Factor "; ; depends on power as well as the stable index and skewness parameter of the standard stable Paretian distribution and, for ¡, is of
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the form (see Samorodnitsky and Taqqu, 1994, p. 18)
2 − 1 2 =2 arctan (; (1+(; ) cos ' 1− "; ; = ∞ 0 u− − 1 sin2 (u) du (8) with (; := tan )=2. To operationalize (8) for practical applications, the integral in the denominator needs to be evaluated. Use of identity cos 2u = 1 − ∞ 2 sin2(u) and substitution v = 2u allows us to express 0 u−−1 sin2 (u) du as ∞ 2+1 0 v−−1 (1 − cos v) dv or 4 −1 ∞ 2 ) if 0¡¡2; = 1; 2 '(1 − ) cos 2 −−1 u sin (u) du = (9) ) 0 if = 1 2 (see, for example, HoImann-JHrgensen, 1994, p. 218) yielding
−1 2 =2 arctan (; "; ; = + ' 1 − (1 + (; ) cos for ¡ and ¿1, where '(1 − ) cos ) 2 + = )=2
if = 1; if = 1:
(10)
(11)
Note that "; ; increases without bound as approaches . A plot of "; ; for the power-one case, i.e., "; ; 1 , is shown in Fig. 1. If, in addition to = 1, the distribution is symmetric, i.e., = 0, then
2 1 : "; 0; 1 = ' 1 − ) iid
For = 2, we have t ∼ N(0; 2) and, with Z ∼ N(0; 1),
2 +1 =2 "2; 0; = 2 E|Z | = √ ' ; 2 )
(12)
so that, as → 2, the limiting value in Fig. 1 for all ∈ [ − 1; 1] is 1.1284. If, instead, rescaled stable Paretian innovations, -t = 2−1=2 t , are used, then iid we have E|-t | = 2−=2 "; ; and, for = 2, -t ∼ N(0; 1) with
2=2 +1 E|-t | = √ ' : 2 ) 4
We use the fact that the gamma function is de5ned for all x ∈ R except for nonpositive integers.
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Fig. 1. Values of "; ; 1 as given by Eq. (10).
This is clearly unity for = 2, so that (6) reduces to (7), the condition associated with conventional normal-GARCH models. 5. Concluding remarks The stationarity condition for power-GARCH processes driven by stable Paretian innovations established in Proposition 2 gives rise to several remarks. The admissible parameter space for parameters in the conditional-volatility equation, i , i = 1; : : : ; r, and j , j = 1; : : : ; s, shrinks, under ceteris paribus conditions, as the tails of the innovations become heavier (i.e., ↓ 1), as the skewness of the innovation increases (i.e., as → ±1) and as the power parameter, , increases (i.e., ↑ ). Liu and Brorsen (1995) considered S; 0; GARCH(1; 1) models with = . In this case, the conditional mean of the quantity expressing the conditional volatility, ct , is in5nite. In simulation experiments we conducted, we found that setting = may lead to extremely erratic behavior of the conditionalvolatility process—a behavior we do not encounter in 5nancial data.
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Engle and Bollerslev (1986) refer to GARCH processes being driven by innovations with unit-variance and satisfying the borderline condition s r j = 1 (13) i + i=1
j=1
as an integrated GARCH or IGARCH process. Rewriting the GARCH process in form of an ARMA process for t2 (see Bollerslev, 1988), the condition implies that the resulting autoregressive polynomial, 1 − (1 + 1 )L − · · · − (m + m )Lm with m = max(r; s), has a unit root. In the S; ; GARCH(r; s) case, the analogue to (13) is r s i + j = 1; (14) "; ; i=1
j=1
giving rise to integrated S; ; GARCH(r; s) or S; ; IGARCH(r; s) processes. Under (14), the implied autoregressive polynomial for the ARMA representation of |yt − t | , given by 1−("; ; 1 +1 )L−· · ·−("; ; m +m )Lm , possesses a unit root and, thus, gives rise to persistence of conditional volatility. The estimation of S; ; GARCH(r; s) models as well as other stable Paretian models is discussed in Mittnik et al. (1999). Stationarity conditions and forecasting performance of asymmetric power GARCH models with generalized (asymmetric) Student’s t distributions are examined in Mittnik and Paolella (2000); and empirical comparisons between S; ; GARCH and Student’s t-GARCH models with value-at-risk applications are given in Mittnik and Paolella (2001). Acknowledgements We wish to thank Vygantas Paulauskas for helpful discussions, as well as two anonymous referees and the associate editor for constructive comments. References Adler, R.J., 1997. Discussion: heavy tail modeling and teletraJc data. The Annals of Statistics 25, 1849–1852. Baillie, R.T., Bollerslev, T., 1992. Prediction in dynamic models with time-dependent conditional variances. Journal of Econometrics 52, 91–113. Bollerslev, T., 1986. Generalized autoregressive conditional heteroskedasticity. Journal of Econometrics 31, 307–327. Bollerslev, T., 1987. A conditional heteroskedastic time series model for speculative prices and rates of return. Review of Economics and Statistics 69, 542–547. Bollerslev, T., 1988. On the correlation structure for the generalized autoregressive conditional heteroskedastic process. Journal of Financial Analysis 9, 121–131.
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