Statistica Sinica 17(2007), 161-175

A DOUBLE AR( $\mbox{\boldmath $p$}$) MODEL: STRUCTURE AND

ESTIMATION

Shiqing Ling

Hong Kong University of Science and Technology

Abstract: The paper considers the so-called double AR($p$) model,

$$y_{t}=\sum_{i=1}^{p}\phi_{i} y_{t-i}+\eta_{t}\sqrt{\omega+\sum_{i=1}^{p}\alpha_{i} y_{t-i}^{2}},$$

where $\eta_{t}\sim$ i.i.d. $N(0,1)$. It is shown that the necessary and sufficient condition for the existence of a strictly stationary solution to the model is that the top Lyapounov exponent $\gamma$, defined in the paper, be negative; the solution is then unique and geometrically ergodic. The necessary and sufficient condition for the existence of a strictly stationary solution to the model with $Ey_{t}^{2}<\infty$ is also obtained. The maximum likelihood estimator of the parameters in the model is shown to be asymptotically normal. The condition for this is again only that $\gamma$ is negative which includes the case with some roots of $1-\sum_{i=1}^{p}\phi_{i}z^{i}=0$ on or outside the unit circle, and the case with $Ey_{t}^{2}=\infty$. The result is novel because all kinds of estimated $\phi_{i}$'s in these cases are not asymptotically normal in the classical AR($p$) model with i.i.d. errors; it may provide new insights in this direction.

Key words and phrases: Asymptotic normality, double autoregressive model, maximum likelihood estimator, stationarity, geometric ergodicity.