Estimation and testing stationarity for double-autoregressive models
The paper considers the double-autoregressive model "y"<sub>"t"</sub> = "φ""y"<sub>"t" - 1</sub>+"ϵ"<sub>"t"</sub> with "ϵ"<sub>"t"</sub> =<formula format="inline"><file name="rssb_432_mu1.gif" type="gif" /></formula>. Consistency and asymptotic normality of the estimated parameters are proved under the condition "E" ln |"φ" +√"&agr;""η"<sub>"t"</ sub>|>0, which includes the cases with |"φ"|=1 or |"φ"|>1 as well as <formula format="inline"><file name="rssb_432_mu2.gif" type="gif" /></formula>. It is well known that all kinds of estimators of "φ" in these cases are not normal when "ϵ"<sub>"t"</sub> are independent and identically distributed. Our result is novel and surprising. Two tests are proposed for testing stationarity of the model and their asymptotic distributions are shown to be a function of bivariate Brownian motions. Critical values of the tests are tabulated and some simulation results are reported. An application to the US 90-day treasury bill rate series is given. Copyright 2004 Royal Statistical Society.
