ON THE QUASI-LIKELIHOOD ESTIMATION IN TIME SERIES MODELS

Abstract

The quasi-likelihood function was introduced by Wedderburn (1974) to be used for estimating the unknown parameters in generalized linear models. The idea of quasi-likelihood weakens the assumption that we know exactly the distribution of the random component in the model, and replaces it by an assumption about how the variance changes with the mean.

Wedderburn (1974) proved that the quasi-likelihood function has properties similar to those of log likelihood function, and that the log likelihood function is identical to the quasi-likelihood function if and only if the density of the random variable is a member of the one parameter exponential family.

The quasi-likelihood function could be used for estimation in the same way as the usual likelihood function. Wedderburn (1974) and McCullagh (1983) showed that the maximum quasi-likelihood estimates have many properties analogous to those of maximum likelihood estimates; in particular, the maximum quasi-likelihood estimate of the vector f3 (the vector of parameters in regression models) is asymptotically

normal with mean f3, and asymptotic covariance may be derived in the usual fashion from the second derivative matrix of the quasi-likelihood function. Also, if the underlying distribution comes from a natural exponential family, the maximum quasi­ likelihood estimate maximizes the likelihood function and so it has full asymptotic efficiency, which had been investigated by Firth (1987), and Hill and Tsai (1988). Wedderburn's original definition of quasi-likelihood for generalized linear models is extended by Neider and Pregibon (1987) to allow the comparison of variance functions as well as those of linear predictors and link functions. The thesis aims is to investigate the use of the generalized quasi-likelihood

function in estimation. The study includes different time series models. The study also includes the properties of the proposed method as well as a comparison between the method and the ordinary methods.

Chapter one presents some basic definitions and concept in order to make the thesis self contained.