Stability of nonlinear AR–GARCH models

This paper studies the stability of nonlinear autoregressive models with conditionally heteroskedastic errors. We consider a nonlinear autoregression of order p (AR(p)) with the conditional variance specified as a nonlinear first order generalized autoregressive conditional heteroskedasticity (GARCH(1,1)) model. Conditions under which the model is stable in the sense that its Markov chain representation is geometrically ergodic are provided. This implies the existence of an initial distribution such that the process is strictly stationary and β–mixing. Conditions under which the stationary distribution has finite moments are also given. The results cover several nonlinear specifications recently proposed for both the conditional mean and conditional variance. ∗The first author’s work was financially supported by the Jan Wallander’s and Tom Hedelius’ Foundation, Grant No. J03–41. The second author acknowledges financial support by the Yrjö Jahnsson Foundation and the Research Unit of Economic Structures and Growth (RUESG) in the University of Helsinki. Address correspondence to: Mika Meitz, Department of Economic Statistics, Stockholm School of Economics, P. O. Box 6501, SE–113 83 Stockholm, Sweden; 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