Generalized least squares estimation for explosive AR(1) processes with conditionally heteroscedastic errors

This article is concerned with explosive AR(1) processes generated by conditionally heteroscedastic errors. Conditional least squares as well as generalized least squares estimation for autoregressive parameter are discussed and relevant limiting distributions are expressed as products of certain random variables. These results can be viewed as generalizations of classical results obtained for the standard explosive AR(1) model with i.i.d. errors (cf. [Fuller, W.A., 1996. Introduction to Statistical Time Series, second ed. Wiley, New York (Chapter 10)]). The model under consideration accommodates diverse conditionally heteroscedastic processes including Engle [1982. Autoregressive conditional heteroscedasticity with estimates of the variance of U.K. inflation. Econometrica 50, 987–1008]'s ARCH, threshold-ARCH and beta-ARCH processes. Based on residuals, least squares estimation for parameters appearing in the conditional variance is also discussed and is illustrated for various ARCH type processes.