Inference for a Special Bilinear Time Series Model

TL;DR

This paper introduces a novel GARCH-type maximum likelihood estimator for a special bilinear time series model, demonstrating its consistency, asymptotic normality, and good finite sample performance, addressing a challenging inference problem.

Contribution

It proposes a new nonstandard estimation method for a specific bilinear model, providing theoretical guarantees and practical validation.

Findings

Estimator is consistent and asymptotically normal.

Finite sample simulations show good performance.

Provides a simple estimator for asymptotic covariance.

Abstract

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.