ASYMPTOTIC NORMALITY OF THE QMLE ESTIMATOR OF ARCH IN THE NONSTATIONARY CASE

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1 Econometrica, Vol. 7, No. March, 004), 4 4 ASYMPOIC NORMALIY OF HE QMLE ESIMAOR OF ARCH IN HE NONSAIONARY CASE BY SØREN OLVER JENSEN AND ANDERS RAHBEK We establish consistency and asymptotic normality of the quasi-maximum likelihood estimator in the linear ARCH model. Contrary to the existing literature, we allow the parameters to be in the region where no stationary version of the process exists. his implies that the estimator is always asymptotically normal. KEYWORDS: ARCH, asymptotic normality, asymptotic theory, consistency, GARCH, nonstationarity, quasi-maximum likelihood estimation.. INRODUCION CONSIDER HE FIRS ORDER ARCH autoregressive conditional heteroscedastic) model introduced by Engle 98), as given by ) y t = σ t z t σ t = ω + αy t for t =, α>0, ω>0, and with z t an i.i.d.0 ) process with Vz t ) = Ez4 t ) = ζ< Asymptotic inference for the ARCH) and more general ARCH models, including GARCH models, has been studied in, e.g., Weiss 98), Lee and Hansen 994), Lumsdaine 99), and Kristensen and Rahbek 00). hese papers, as well as others in the econometric literature, all assume as a minimal requirement that the ARCH process y t is suitably ergodic or stationary such that laws of large numbers apply. We relax the condition on the stability of the y t process and allow it to be nonstationary and in particular not to have any moments. Our only condition is that Vz t )<. In the proofs we have aimed at a detailed presentation of the analysis of the score, information, and third derivatives of the likelihood function. We note that the arguments for the score and information in Lemma 3 and Lemma 4 are easily carried over to the stationary case by using the ergodic theorem similar to Lee and Hansen 994) and Lumsdaine 99). Moreover, we note that our result in Lemma 5 regarding the uniform boundedness of the third derivatives of the likelihood function does not use the nonstationary condition, and hence can be applied to the stationary case as well. As can be deduced from Remark, the established uniform boundedness corrects and significantly extends existing proofs of asymptotic normality for the ARCH) model. Anders Rahbek is grateful for support from the Danish Social Sciences Research Council, Centre for Analytical Finance CAF) as well as the EU network DYNSOCH. We thank David Lando for comments and discussions on our paper. We would also like to thank the two anonymous referees as well as the coeditor for constructive comments on our paper. 4

2 4 S. JENSEN AND A. RAHBEK. INFERENCE By Nelson 990) and Bougerol and Picard 99), y t is stationary a stationary version exists) and ergodic if and only if E logαz )<0. In particular, if z t t is Gaussian, then the if and only if condition is that α< exp Ψ )) 35, where Ψ ) is the Euler psi function; see Nelson 990). As mentioned our analysis is under the assumption that y t does not have a stationary version or equivalently, ) E logαz t ) 0 Consider the likelihood estimator based on maximizing the quasi likelihood l α) = ) log σ t + y t 3) σ t from which the QMLE quasi-maximum likelihood estimator) ˆα is found. Note that this is the true likelihood if z t is Gaussian and that we are conditioning on the initial value y 0. Our main result is the following: HEOREM : Assume that the ARCH process y t in ) does not allow a stationary version or equivalently ) holds. Assume further that the iid0 ) process z t is such that Vz t ) = ζ is finite, and the scale parameter ω is known. hen as the sequence of QMLE ˆα is consistent, and asymptotically normal, ˆα α) D N0σ ) where σ = ζα > 0 REMARK : Note that if z t is Gaussian, then σ = α in heorem. PROOF: ogether Lemmas 3, 4, and 5 in the next section establish the classical Cramér type conditions; see, e.g., Lehmann 999). 3. DERIVAION For exposition and without loss of generality we henceforth set ω = With the likelihood function given by 3), the score, information, and the third derivative of the log-likelihood with respect to α are found to be given by 4) 5) ) α l α) = α l α) = 3 α 3 l α) = ) y t y y t t ) y 4 t 4 ) 3 y t y t

3 ARCH MODEL 43 In the following we study the asymptotic behavior of these in order to establish consistency and asymptotic normality of the QMLE. First, consider the asymptotic behavior of y t : LEMMA : Assume that ) holds; then as t. y t as PROOF: his follows by heorem of Nelson 990). Next, consider the asymptotic behavior of the following type of averages: LEMMA : Assume that ) holds; then with m k positive integers, as t, y m t as if m = k, 7) α + αy t )k m 0 if m<k, and likewise, as, y m t as if m = k, 8) α + αy t )k m 0 if m<k. PROOF: he results follow by Lemma. Next turn to the score and the information: LEMMA 3: Under the assumptions of heorem, then with l α)/ α given by 4), ) ) / α l α) D ζ N 0 4α as. PROOF: Bydefinition/ ) l α)/ α = / ) s t where s t = ) y t y t he process s t is a Martingale difference sequence with respect to F t = σ{y t y t y 0 } as E s t E z t /α< and Es t F t ) = E z) y t t Next, using 8), = 0 Es t F t ) = y t + αy t ) ζ as ζ 4 4α > 0

4 44 S. JENSEN AND A. RAHBEK where ζ = E z t ) = Vz t ).Furthermore,ass t is bounded by µ t = z t ) /4α we derive the Lindeberg type condition, E s { t s t > δ }) E µ { t µ t > δ }) 0 for some δ>0andas tends to using Vz t ) = ζ<. By the central limit theorem in Brown 97) the desired result follows. LEMMA 4: Under the assumptions of heorem, then with the observed information l α)/ α given by 5), ) α l as α) α > 0 as. PROOF: Rewrite minus the observed information as ) α l α) = κ t γ t with y 4 t κ t = z t and γ t = y4 t = 4 + αy t he strong law of large numbers implies as κ t κ t as κ< ) while 7) implies γ t as /α ) and hence the desired result follows. Finally, we turn to the uniform boundedness of the third derivative of the likelihood function. Note that the proof does not require nonstationarity of the process. LEMMA 5: Denote by Iαδ) the interval [α δ α + δ] 0 <δ<α. hen with 3 l α)/ α 3 given by ), it holds that sup 3 α l as 3 α) gα δ ) β< as α Iαδ)

5 ARCH MODEL 45 PROOF: Withα l = α δ 3 α l 3 α) = = 3 y t ) y t 3 y t 3 { + αy } ) t { + αy t }z t } ) 3 { + ααl z t + := gα δ ) and the results follows by the law of large numbers. α 3 l α 3 l ) α 3 l REMARK : When deriving consistency and asymptotic normality, the classical sufficient condition regarding bounds of the third derivatives of the likelihood function is that E sup 3 α Iαδ) α l 3 α) < In Basawa, Feigin, and Heyde 97, condition B.7)) this is incorrectly stated as sup α Iαδ) E 3 α l 3 α) < he mistake is reproduced in Weiss 98) and next in Lumsdaine 99), and their proofs may therefore not be complete. 4. CONCLUSION In this paper we have shown that for the ARCH) model the QMLE is always asymptotically Gaussian so long as the fourth order moment of the innovations z t is finite. his is somewhat surprising as most researchers have assumed one needs strict stationarity. For financial applications most often it is the GARCH ) as opposed to the ARCH) model that is applied and hence of much interest. In a forthcoming paper, which follows on from the developments given in this paper, Jensen and Rahbek 003) show that indeed the results hold for the GARCH ) model as well. hat is, whether or not the process is stationary, asymptotic normality holds and hence there are no knife edge results like [in] the unit root case as conjectured by Lumsdaine 99, p. 580). he derivations for the GARCH ) case are more involved and lengthy due to the added complexity of the lagged variance in the σ t specification. Department of Applied Mathematics and Statistics, University of Copenhagen, Denmark; soerent@math.ku.dk and Department of Applied Mathematics and Statistics, University of Copenhagen, Denmark; rahbek@math.ku.dk. Manuscript received October, 00; final revision received July, 003.

6 4 S. JENSEN AND A. RAHBEK REFERENCES BASAWA, I. V., P. D. FEIGIN, AND C. C. HEYDE 97): Asymptotic Properties of Maximum Likelihood Estimators for Stochastic Processes, Sankyā, Series A, 38, BOUGEROL, P., AND N. PICARD 99): Stationarity of GARCH Processes and of Some Nonnegative ime Series, Journal of Econometrics, 5, 5 7. BROWN, B. M. 97): Martingale Central Limit heorems, Annals of Mathematical Statistics, 4, 59. ENGLE, R. F. 98): Autoregressive Conditional Heteroscedasticity with Estimates of United Kingdom Inflation, Econometrica, 50, JENSEN, S.., AND A. RAHBEK 003): Asymptotic Normality for Non-Stationary, Explosive GARCH, Preprint No. 4, Department of Applied Mathematics and Statistics, University of Copenhagen. KRISENSEN, D.,AND A. RAHBEK 00): Asymptotics of the QMLE for a Class of ARCHq) Models, Preprint No. 0, Department of Applied Mathematics and Statistics, University of Copenhagen. LEE, S.-W.,AND B. HANSEN 994): Asymptotic heory for the GARCH ) Quasi-Maximum Likelihood Estimator, Econometric heory, 0, LEHMANN, E. L. 999): Elements of Large-Sample heory. New York: Springer-Verlag. LUMSDAINE, R. 99): Consistency and Asymptotic Normality of the Quasi-Maximum Likelihood Estimator in IGARCH ) and Covariance Stationary GARCH ) Models, Econometrica, 4, NELSON, D. B. 990): Stationarity and Persistence in the GARCH ) Model, Econometric heory,, WEISS, A. 98): Asymptotic heory for ARCH Models, Econometric heory,, 07 3.