On the equivalence of marginal and approximate conditional likelihoods for correlation parameters under a normal model

Abstract

SUMMARY Marginal and approximate conditional likelihoods are obtained for the correlation parameters in a normal linear regression model with correlated errors. These likelihoods may be evaluated using the Kalman filter. It is shown that the marginal and conditional likelihoods are equivalent for any correlation matrix whose entries are continuous and differentiable functions of its parameters. The results are illustrated by the application of marginal and conditional likelihood methods to estimation of the correlation parameters in time series and correlated spatial processes. The concept of a marginal likelihood was introduced originally in the structural inference context by Fraser (1967), and later developed in the classical framework by Kalbfleisch & Sprott (1970) as a general method for eliminating nuisance parameters from the likelihood function. More recently, Cox & Reid (1987) introduced approximate conditional likelihoods which also address this problem. They argued that this conditional likelihood was preferable to the profile likelihood obtained by replacing the nuisance parameters in the likelihood by their maximum likelihood estimates when the parameters of interest are given. Following on the work of Cox & Reid (1987), Cruddas, Reid & Cox (1989) obtained an approximate conditional likelihood for the correlation parameter in several short series of autoregressive processes of order one with common variance and autocorrelation parameter. Based on a simulation study, Cruddas et al. (1989) showed that the estimate based on the approximate conditional likelihood had a much smaller bias and better coverage properties of the confidence interval than the maximum likelihood estimate from the profile likelihood. They also showed that the approximate conditional and the marginal likelihood were the same in this situation. The approximate conditional likelihood for the correlation parameters in a general normal regression model with correlated errors is obtained here, and is shown to be equivalent to the marginal likelihood for the same parameters.