Parameter estimation in nonlinear AR - GARCH models

This paper develops an asymptotic estimation theory for nonlinear autoregressive models with conditionally heteroskedastic errors. We consider a general nonlinear autoregression of order p (AR(p)) with the conditional variance specified as a general nonlinear first order generalized autoregressive conditional heteroskedasticity (GARCH(1,1)) model. We do not require the rescaled errors to be independent, but instead only to form a stationary and ergodic martingale difference sequence. Strong consistency and asymptotic normality of the global Gaussian quasi maximum likelihood (QML) estimator are established under conditions comparable to those recently used in the corresponding linear case. To the best of our knowledge, this paper provides the first results on consistency and asymptotic normality of the QML estimator in nonlinear autoregressive models with GARCH errors. ∗We acknowledge financial support from the Academy of Finland (PS), the Finnish Foundation for the Advancement of Securities Markets (MM), OP-Pohjola Group Research Foundation (MM and PS), and the Yrjö Jahnsson Foundation (MM and PS). We thank a co-editor and two anonymous referees for helpful comments and suggestions. The first version of this paper was completed in May 2008 while the first author was a Post-Doctoral Research Fellow at University of Oxford’s Department of Economics. Parts of this research were also carried out while the first author was visiting the Center for Research in Econometric Analysis of Time Series (CREATES) at University of Aarhus (funded by the Danish National Research Foundation) and during the second author’s Fernand Braudel Fellowship at the European University Institute. Both institutions are thanked for their hospitality. Material from the paper has been presented at the Second Brussels-Waseda Seminar on Time Series and Financial Statistics, Brussels, June 2008; ESRC Econometric Study Group Annual Conference, Bristol, July 2008; Workshop on Nonparametric Function Estimation with Applications in Finance, Oulu, June 2009; Econometrics, Time Series Analysis and Systems Theory – A Conference in Honor of Manfred Deistler, Vienna, June 2009; 64th European Meeting of the Econometric Society, Barcelona, August 2009; and in seminars at Bilkent University, Graz University of Technology, Koç University, University of Aarhus, and University of Vienna. We thank the participants in these occasions for their comments. Address correspondence to: Mika Meitz, Department of Economics, Koç University, Rumelifeneri Yolu, 34450 Sariyer, Istanbul, Turkey; e-mail: [email protected]; or to: Pentti Saikkonen, Department of Mathematics and Statistics, University of Helsinki, P. O. Box 68, FIN–00014 University of Helsinki, Finland; e-mail: [email protected]. 1