Quasi-maximum likelihood estimators in generalized linear models with autoregressive processes

Abstract

The paper studies a generalized linear model (GLM) $y_t = h(x_t^T \beta ) + \varepsilon _t ,t = 1,2,...,n,$ where $\varepsilon _1 = \eta _1 ,\varepsilon _t = \rho \varepsilon _{t - 1} + \eta _t ,t = 2,3,...,n,$ h is a continuous differentiable function, ηt’s are independent and identically distributed random errors with zero mean and finite variance σ 2. Firstly, the quasi-maximum likelihood (QML) estimators of β, ρ and σ 2 are given. Secondly, under mild conditions, the asymptotic properties (including the existence, weak consistency and asymptotic distribution) of the QML estimators are investigated. Lastly, the validity of method is illuminated by a simulation example.