A practical procedure for estimation of linear models via asymptotic quasi-likelihood (Q1302364)

Summary: This paper is concerned with the application of an asymptotic quasi-likelihood practical procedure to estimate the unknown parameters in linear stochastic models of the form \(y_t= f_t(\theta)+ M_t(\theta)\) \((t= 1,2,\dots, T)\), where \(f_t\) is a linear predictable process of \(\theta\) and \(M_t\) is an error term such that \(E(M_t|{\mathcal F}_{t- 1})= 0\) and \(E(M^2_t|{\mathcal F}_{t-1})< \infty\) and \({\mathcal F}_t\) is a \(\sigma\)-field generated from \(\{y_s\}_{s\leq t}\). For this model, to estimate the parameter \(\theta\in\Theta\), the ordinary least squares method is usually inappropriate (if there is only one observable path of \(\{y_t\}\) and if \(E(M^2_t|{\mathcal F}_{t- 1})\) is not a constant) and the maximum likelihood method either does not exist or is mathematically intractable. If the finite-dimensional distribution of \(M_t\) is unknown, to obtain a good estimate of \(\theta\) an appropriate predictable process \(g_t\) should be determined. In this paper, criteria for determining \(g_t\) are introduced which, if satisfied, provide more accurate estimates of the parameters via the asymptotic quasi-likelihood method.