Consistency of the asymptotic quasi-likelihood estimate on linear models

Consider a model y(t)=f(r)(theta) + M-t, 0 less than or equal to t less than or equal to T where theta is an element of Theta in an unknown parameter, f(t)(theta) is a linear predictable process, M-t is a martingale difference, and the nature of E(M-t(2)/Ft-1) is unknown. This paper presents an estimating procedure for theta based on the asymptotic quasi-likelihood methodology. Conditions under which the asymptotic quasi-likelihood estimate converges to the true parameter theta(0) are discussed. This method is applied to several simulated examples, and estimates of the unknown parameter are obtained by means of a two-stage technique. Comparison is made between the estimates obtained via this method and those obtained via the ordinary least squares method. Discussion is provided on the application of the model.