QUASI-LIKELIHOOD INFERENCE IN GARCH PROCESSES WHEN SOME COEFFICIENTS - PDF document

(24 January 2006) QUASI-LIKELIHOOD INFERENCE IN GARCH PROCESSES WHEN SOME COEFFICIENTS ARE EQUAL TO ZERO CHRISTIAN FRANCQ , ∗ GREMARS Université Lille 3 JEAN-MICHEL ZAKOIAN, ∗∗ GREMARS Université Lille 3 and CREST Abstract In this paper we establish the asymptotic distribution of the quasi-maximum likelihood (QML) estimator for generalized autoregressive conditional het- eroskedastic (GARCH) processes, when the true parameter may have zero coefficients. This asymptotic distribution is the projection of a normal vector distribution onto a convex cone. The results are derived under mild conditions which, for important subclasses of the general GARCH, coincide with those made in the recent literature when the true parameter is in the interior of the parameter space. Furthermore, the QML estimator is shown to converge to its asymptotic distribution locally uniformly. Using these results, we consider the problem of testing that one or several GARCH coefficients are equal to zero. The null distribution and the local asymptotic powers of the Wald, score and quasi-likelihood ratio tests are derived. The one-sided nature of the problem is exploited and asymptotic optimality issues are addressed. Keywords: Asymptotic efficiency of tests, Boundary, Chi-bar distribution, GARCH model, Quasi Maximum Likelihood Estimation, Local alternatives. JEL Codes: C12, C13, C22 ∗ Postal address: GREMARS, UFR MSES, Université Lille 3, Domaine du Pont de bois, BP 149, 59653 Villeneuve d’Ascq Cedex, France ∗∗ Postal address: CREST, 3 Avenue P. Larousse, 92245 Malakoff Cedex,France 1

2 1. Introduction Much attention has been given recently to the asymptotic properties of the quasi- maximum likelihood estimator (QMLE) in the context of GARCH( p, q ) processes. Whereas ARCH (AutoRegressive Conditionally Heteroskedastic) models have been introduced by Engle in 1982, and generalized by Bollerslev in 1986, it took about twenty years to see the emergence of consistency and asymptotic normality results for the general GARCH model under weak assumptions. Recent references dealing with the QML estimation of general GARCH( p, q ) are the dissertation by Boussama (1998), the monograph by Straumann (2005) and the papers by Berkes and Horváth (2003, 2004), Berkes, Horváth and Kokoszka (2003), Hall and Yao (2003) for GARCH models with heavy-tailed errors, and Francq and Zakoïan (2004) (hereafter FZ). It is benificial to use the QMLE in the GARCH framework because it is much less sensitive with respect to heavy tailed unconditional distributions than, for instance, the least- squares method. Other estimation procedures that are not demanding in terms of unconditional moments have recently been suggested by Liese (2004) and Ling (2005). See Giraitis, Leipus and Surgailis (2004) for a survey on GARCH modeling. The GARCH estimation theory however suffers the major weakness of excluding the presence of zero coefficients in the true parameter value. Indeed, one important difference between GARCH and other popular time series models, such as ARMA models, is that the admissible parameter space needs to be inequality restricted. The data generation mechanism requires the conditional variance to be always strictly positive, which is generally obtained by imposing a strictly positive intercept and non negative GARCH coefficients in the conditional variance equation (see however Nelson and Cao (1992) for weaker, but generally non explicit, conditions). A key regularity condition, imposed by the above cited papers, is that the true parameter must lie in the interior of the parameter space. This is essentially required for the asymptotic normality, not for the consistency of the QML estimator. For instance the asymptotic gaussian distribution of the QMLE does not obtain if, for instance, a GARCH( p, q ) model is estimated when the underlying process is a GARCH( p − 1 , q ), or a GARCH( p, q − 1 ) process. For hypothesis testing, it is crucial to be able to relax the assumption that the true

GARCH Inference on a boundary 3 parameter value is an inner point of the parameter space. One typical situation where the positivity condition is violated is of course the case of conditional homoskedasticity. The problem of conditional homoskedasticity testing is particularly important in the finance literature. The model then reduces, under the null, to an independent white noise which legitimates the use of the so-called Black-Scholes formula for option pricing. Under the alternative of conditional heteroskedasticity, option pricing or Value-at-Risk calculation demand much more sophisticated methods. More generally, testing that some coefficients are null is an important subject in the GARCH framework. The non gaussianity of the QMLE may obviously have consequences on the asymptotic distribution of the standard tests statistics. The usual asymptotic χ 2 distribution of the Wald and Likelihood ratio is no longer valid. This problem is well-known (see Weiss (1986)), and has been investigated by Demos and Sentana (1998) among others. Our objective is to develop a complete asymptotic theory of estimation and testing in the context of GARCH processes, when the true parameter may be on the boundary of the parameter space. Fullfiling such an objective requires a series of steps: (i) deriving the asymptotic distribution of the QML estimator under, if possible, the same mild conditions as those employed when the parameter is in the interior of the parameter space. The main difficulty is that the standard equivalence in probability between the rescaled centered estimator and an asymptotically gaussian vector (the normalized score multiplied by the inverse of the Hessian matrix), does not hold. The asymptotic distribution of the QMLE will be obtained by approximating the quasi- likelihood by a quadratic function, and will be shown to be given by the projection of a normal vector onto a convex cone. (ii) deriving the asymptotic distributions of the commonly used tests, such as the Wald, Likelihood ratio and Rao-score tests, under the null of conditional homoskedas- ticity or, more generally, under the assumption that one or several GARCH coefficients are equal to zero. As is well-known, the asymptotic equivalence of the three tests does not hold when the parameter belongs to the boundary. (iii) establishing the regularity of the QML estimator over the whole parameter space. This step consists in studying the change of the asymptotic distribution derived in step (i) under a small change (of size n − 1 / 2 ) of the true parameter value. The third lemma of Le Cam being difficult to apply in our framework, we give a direct proof.

4 (iv) finally, comparing the asymptotic powers of the tests statistics (in the local and Bahadur senses) and studying their optimality properties. Two important examples are considered. In the first the nullity of only one coefficient is assumed under the null. In the second, the null hypothesis of no conditional heteroskedasticity is tested. In the latter case, a test exploiting the one-sided nature of the alternative will be compared to the previous ones. We will show that this test is locally asymptotically most stringent somewhere most powerful, a concept which has been introduced by Akharif and Hallin (2003). There are numerous antecedents in the literature to the results of the present paper. A systematic investigation of estimation with a parameter on a boundary, and a wide class of boundary hypothesis tests is in Andrews (1997, 1999, 2001). In particular, he considers the GARCH(1, q ) model under assumptions we will further discuss. Klüppelberg et al. (2002), and May and Szimayer (2001) consider testing for conditional heteroskedasticity in the AR(1)-GARCH(1,1) framework. To our knowl- edge, asymptotic results for the general GARCH( p, q ) when the parameter is on the boundary are not available in the literature. The use of a quadratic approximation to the objective function, and its optimization on a convex cone have been made by Chernoff (1954) and Andrews (2001) among many others (see the latter paper for a list of references). Testing problems in which, under the null hypothesis, the parameter is on the boundary of the maintained assumption have been considered e.g. by Chernoff (1954), Bartholomew (1959), Perlman (1969), Gouriéroux, Holly and Monfort (1982), Andrews (2001). Several papers consider one-sided alternatives. These include Wolak (1989), Rogers (1986), Silvapulle and Silvapulle (1995), King and Wu (1997); see the latter paper for further references. In particular, tests exploiting the one-sided nature of the ARCH alternative, against the null of no ARCH effect, have been proposed by Lee and King (1993), Hong (1997), Demos and Sentana (1998), Hong and Lee (2001), Andrews (2001), Dufour, Khalaf, Bernard and Genest (2004) among others. The article proceeds as follows. Section 2 describes the estimation problem of concern and recalls results available when θ 0 is not on the boundary. Section 3 establishes the asymptotic distribution of the QMLE when θ 0 is on the boundary. For a large class of GARCH models, the results are obtained without moments assumptions on the observed process. Section 4 establishes, without additional assumptions, the