Estimation and testing stationarity for double-autoregressive models

The paper considers the double-autoregressive model y(t) = phiy(t-1)+epsilon(t) with epsilon(t) = eta(t) root(omega + alphay(t-1)(2)). Consistency and asymptotic normality of the estimated parameters are proved under the condition E ln |phi +rootalphaeta(t)|<0, which includes the cases with |phi|=1 or |phi|>1 as well as E(epsilon(t)(2)) = infinity. It is well known that all kinds of estimators of phi in these cases are not normal when epsilon(t) 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.