Inference for a Special Bilinear Time Series Model

It is well known that estimating bilinear models is quite challenging. Many different ideas have been proposed to solve this problem. However, there is not a simple way to do inference even for its simple cases. This paper studies the special bilinear model $$Y_t=\mu+\phi Y_{t-2}+ bY_{t-2}\varepsilon_{t-1}+ \varepsilon_t,$$ where $\{\varepsilon_t\}$ is a sequence of i.i.d. random variables with mean zero. We first give a sufficient condition for the existence of a unique stationary solution for the model and then propose a GARCH-type maximum likelihood estimator for estimating the unknown parameters. It is shown that the GMLE is consistent and asymptotically normal under only finite fourth moment of errors. Also a simple consistent estimator for the asymptotic covariance is provided. A simulation study confirms the good finite sample performance. Our estimation approach is novel and nonstandard and it may provide a new insight for future research in this direction.