Abstract
For an ARCH model, we propose a multistage weighted least squares (WLS) estimate which consists of repeated WLS procedures until the corresponding asymptotic variance equals that of the quasi-maximum likelihood estimate (QMLE). At every stage, the current estimate is of a WLS type weighted by the squared conditional variance evaluated at the estimate of the previous stage. Initially, the weighting parameter is any fixed and known value in the parameter space. The procedure provides, without any moment requirement, an asymptotically Gaussian estimate having the same asymptotic distribution as the QMLE even in the unstable case. Apart from the initialization stage, two additional stages are required in the stable case to obtain the same asymptotic distribution as the QMLE, while in the unstable case only one stage is enough. So in all, the proposed procedure involves three stages WLS in the stable case and two stages WLS in the unstable case.
Similar content being viewed by others
References
Aknouche A, Al-Eid E (2012) Asymptotic inference for unstable periodic ARCH processes. Stat Inf Stoch Process 15(1):61–79
Aknouche A, Al-Eid E, Hmeid AM (2011) Offline and online weighted least squares estimation of nonstationary power ARCH processes. Stat Prob Lett 81: 1535–1540
Aue A, Horváth L (2011) Quasi-likelihood estimation in stationary and nonstationary autoregressive models with random coefficients. Statistica Sinica 21:973–999
Berkes I, Horváth L, Kokoskza P (2003) GARCH processes: structure and estimation. Bernoulli 9: 201–227
Berkes I, Horváth L, Ling S (2009) Estimation in nonstationary random coefficient autoregressive models. J Time Ser Anal 30: 395–416
Billingsley P (1968) Convergence of probability measure. Wiley, New York
Billingsley P (1995) Probability and measure. Wiley, New York
Bose A, Mukherjee K (2003) Estimating the ARCH parameters by solving linear equations. J Time Ser Anal 24: 127–136
Bougerol P, Picard N (1992) Stationarity of GARCH processes and some nonnegative time series. J Econom 52: 115–127
Boussama F (2000) Normalité asymptotique de l’estimateur du pseudo-maximum de vraisemblance d’un modéle GARCH. C R Acad Sci Paris 331: 81–84
Engle RF (1982) Autoregressive conditional heteroskedasticity with estimates of variance of UK inflation. Econometrica 50: 987–1008
Francq C, Zakoïan JM (2004) Maximum likelihood estimation of pure GARCH and ARMA-GARCH processes. Bernoulli 10: 605–637
Francq C, Zakoïan JM (2007) Quasi-maximum likelihood estimation in GARCH processes when some coefficients are equal to zero. Stoch Process Appl 117: 1265–1284
Francq C, Zakoïan JM (2010a) Inconsistency of the MLE and inference based on weighted LS for LARCH models. J Econom 159: 151–165
Francq C, Zakoïan JM (2010b) Strict stationarity testing and estimation of explosive ARCH models. Preprint MPRA, 22414
Francq C, Zakoïan JM (2012) Strict stationarity testing and estimation of stationary and explosive GARCH models. Econometrica 80(2):821–861
Goldie C, Maller R (2000) Stability of perpetuities. Ann Prob 28: 1195–1218
Hall P, Heyde CC (1980) Martingale limit theory and its applications. Academic Press, New York
Jensen ST, Rahbek A (2004a) Asymptotic normality of the QML estimator of ARCH in the nonstationary case. Econometrica 72: 641–646
Jensen ST, Rahbek A (2004b) Asymptotic inference for nonstationary GARCH. Econom Theory 20: 1203–1226
Klüppelberg C, Lindner A, Maller R (2004) A continuous time GARCH process driven by a Lévy process: stationarity and second order behavior. J Appl Prob 41: 601–622
Lee SW, Hansen BE (1994) Asymptotic theory for the GARCH(1,1) quasi-maximum likelihood estimator. Econom Theory 10: 29–52
Ling S (2007) Self-weighted and local quasi-maximum likelihood estimators for ARMA-GARCH/IGARCH models. J Econom 140: 849–873
Ling S, Li D (2008) Asymptotic inference for a nonstationary double AR(1) model. Biometrika 95: 257–263
Linton O, Pan J, Wang H (2010) Estimation for a non-stationary semi-strong GARCH(1,1) model with heavy-tailed errors. Econom Theory 26: 1–28
Lumsdaine RL (1996) Consistency and asymptotic normality of the quasi-maximum likelihood estimator in IGARCH(1,1) and covariance stationary GARCH(1,1) models. Econometrica 64: 575–596
Nelson DB (1990) Stationarity and persistence in the GARCH(1,1) model. Econom Theory 6: 318–334
Pantula SG (1988) Estimation of autoregressive models with ARCH errors. Sankhya 50: 119–138
Schick A (1996) \({\sqrt{n}}\) -consistent estimation in a random coefficient autoregressive model. Aust J Stat 38: 155–60
Weiss AA (1986) Asymptotic theory for ARCH models: estimation and testing. Econom Theory 2: 107–131
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Aknouche, A. Multistage weighted least squares estimation of ARCH processes in the stable and unstable cases. Stat Inference Stoch Process 15, 241–256 (2012). https://doi.org/10.1007/s11203-012-9073-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11203-012-9073-7