Journal of Financial Econometrics Advance Access published August 23, 2006
Sample and Implied Volatility in GARCH Models Lajos Horva´th University of Utah Piotr Kokoszka Utah State University Ricˇardas Zitikis University of Western Ontario
abstract The unconditional variance of various GARCH-type models is a function hðqÞ of the parameter vector q which is estimated by b q. For most models used in practice, closed-form expressions of hðÞ have been found. On the contrary, the unconditional variance can be estimated by the sample variance ^2 . This article establishes the asymptotic distributions of the differences ^2 hðqÞ and ^2 hðb qÞ for broad classes of GARCH-type models. Even though both limit distributions are normal, the asymptotic variances are not equal. Potential practical consequences of these results are discussed.
keywords: GARCH processes, statistical hypothesis test, volatility The asymptotic theory for various types of GARCH processes has recently been advanced in several directions, see Horva´th, Kokoszka, and Teyssie`re (2001), Ling and McAleer (2002a,b), Li, Ling, and McAleer (2002), Berkes, Horva´th, and Kokoszka (2003, 2004), Francq and Zakoian (2004), and Berkes and Horva´th (2004), to name just a few references related to the subject of this article. However, some important and easy-to-formulate problems still remain unexplored.
We thank Professor Eric Renault for two extensive letters clarifying some issues discussed in Section 3 and directing us to relevant literature. L.H. was supported by NSF grants DMS-0604670 and INT-0223262 and NATO grant PST.EAP.CLG 980599. P.K. was supported by NSF grants DMS-0413653 and INT-0223262, and by NATO grant PST.EAP.CLG 980599. R.Z. was supported by a Discovery Research Grant from the Natural Sciences and Engineering Research Council (NSERC) of Canada. Address correspondence to Piotr Kokoszka, Department of Mathematics and Statistics, Utah State University, 3900 Old Main Hill, Logan, UT 84322-3900, or e-mail:
[email protected].
doi:10.1093/jjfinec/nbl002 ª The Author 2006. Published by Oxford University Press. All rights reserved. For permissions, please e-mail:
[email protected].
2
Journal of Financial Econometrics
To explain the contribution of this article, recall the GARCHðp, qÞ equations [cf. Bollerslev (1986)]: y k ¼ k e k ;
2k ¼ ! þ
X 1ip
i y2ki þ
X
j 2kj ;
ð1Þ
1jq
where fek , 1 < k < 1g is a sequence of zero mean independent and identically distributed random variables, and ! > 0, i 0, and j 0 are parameters. As it is always done in this context, we also assume that E e2k ¼ 1:
ð2Þ
Furthermore, we assume throughout this article that i > 0 and j > 0:
ð3Þ
Necessary and sufficient conditions under which the GARCHðp, qÞ equations have a unique, strictly stationary, and nonanticipative solution were found by Nelson (1991) when p ¼ 1 and q ¼ 1, and by Bougerol and Picard (1992a,b) for arbitrary p 1 and q 1. We assume throughout this article that those conditions hold and note here only that they imply 1 þ þ q < , for a constant < 1. Denote by q ¼ ð!, 1 , . . . , p , 1 , . . . , q Þ the vector of true GARCHðp, qÞ parameters. A generic element of the parameter space is denoted by u ¼ ðx, s1 , . . . , sp , t1 , . . . , tq Þ. Define the function h : R1þpþq ! ½0,1Þ by hðuÞ :¼
1
P
x P : si tj
i
j
Note that by assumption (2), the variance implied by the GARCH model is hðqÞ. On the contrary, the sample variance is X bn :¼ 1 y2 : Y n 1kn k bn hðqÞ tends to zero as n ! 1, but it It can be expected that the difference Y is not immediately clear at what rate and what the asymptotic distribution is. If we replace the unknown true parameter vector q by its estimator b q, we can then bn hðb qÞ. We will show that under ask the same question about the difference Y fairly general assumptions and for broad classes of models both differences are of the order n1=2 . In view of the above, the following questions arise: Do the statistics Tn ðq0 Þ :¼
pffiffiffi bn hðqÞÞ nðY
ð4Þ
Horva´th, Kokoszka & Zitikis j Sample and Implied Volatility
3
and Tn ðb qÞ :¼
pffiffiffi bn hðb nðY qÞÞ
ð5Þ
converge to the same distribution? Are the limiting distributions normal? Can the asymptotic variances be effectively computed? Do these statistics lead to useful tests? To answer these questions is the goal of this article. The remainder of this article is organized as follows. In Section 1, we state the relevant theoretical results for GARCHðp, qÞ processes. An extension to a broad class of augmented GARCH processes is developed in Section 2. Section 3 discusses some practical consequences of the theoretical results, whereas Section 4 contains their proofs.
1 MAIN RESULTS FOR THE GARCHðp, qÞ PROCESSES From now on, we denote by b q the commonly used pseudo maximum likelihood estimator (MLE) of q, see for example, Section 7.4.1 of Zivot and Wang (2003). To formulate our results, we must introduce additional notation. Under the assumptions stated in the introduction, we have 2k ¼ wk ðqÞ, where the function wk : R1þpþq ! ½0,1Þ is defined by wk ðuÞ :¼ c0 ðuÞ þ
X
ci ðuÞy2ki :
ð6Þ
1i 0 or log x. These two cases suggest imposing the following structural property on the function ðxÞ: ðxyÞ ¼ ðxÞaðyÞ þ bðyÞ;
ð12Þ
where aðyÞ and bðyÞ are some functions. Note that assumption (12) is, indeed, satisfied for both ðxÞ ¼ x and ðxÞ ¼ log x. Namely, in the first case we have Equation (12) with aðyÞ ¼ y and bðyÞ ¼ 0, whereas in the second case the equation holds with aðyÞ ¼ 1 and bðyÞ ¼ log y. In fact, these are essentially all the nontrivial solutions (in , a, and b) of the generalized Pexider’s equation (3.2), as seen from Theorem 4.9 on p. 65 of Castillo, Iglesias, and Ruiz-Cobo (2005). To formulate the main result of this section, Theorem 3 below, we need to introduce additional notation. First, let X b n ðÞ :¼ 1 ðy2k Þ: Y n 1kn bn of bn ðÞ becomes the sample second moment Y Note that when ðxÞ ¼ x, then Y the returns.
6
Journal of Financial Econometrics
bn ðÞ implied by the augmented We now must compute a counterpart of Y 2 GARCH model, that is, E½ðy0 Þ. For this, denote by q the parameter vector on which the functions cðÞ and dðÞ depend. Using Equations (11) and (12), it is easy to verify that E½ðy20 Þ ¼ HðqÞ; where HðqÞ :¼ E bðe20 Þ þ 0 E½dðe0 Þ; and where E aðe20 Þ 0 :¼ : 1 E½cðe0 Þ We assume of course that all the moments on the right-hand side of the above equations are finite, and that E½cðe0 Þ < 1: For the GARCH(1, 1) process, we have aðyÞ ¼ y; bðyÞ ¼ 0; cðyÞ ¼ y2 þ ; dðyÞ ¼ !:
ð13Þ
Hence, HðqÞ equals !=ð1 Þ, which is exactly the quantity hðqÞ introduced in the introduction. We now formulate an extension of Theorem 1 to the augmented GARCH processes defined by (11) with the function ðxÞ satisfying (12). Theorem 3 Assume that the random variables aðe20 Þ, bðe20 Þ, cðe0 Þ, dðe0 Þ, and ð20 Þ have finite second moments. Furthermore, assume that E½cðe0 Þ
