Efficiency analysis of ten estimation procedures for quantitative linear models with autocorrelated errors

Many estimation procedures for quantitative linear models with autocorrelated errors have been proposed in the literature. A number of these procedures have been compared in various ways for different sample sizes and autocorrelation parameters values and for structured or random explanatory vaiables. In this paper, we revisit three situations that were considered to some extent in previous studies, by comparing ten estimation procedures: Ordinary Least Squares (OLS), Generalized Least Squares (GLS), estimated Generalized Least Squares (six procedures), Maximum Likelihood (ML), and First Differences (FD). The six estimated GLS procedures and the ML procedure differ in the way the error autocovariance matrix is estimated. The three situations can be defined as follows: Case 1, the explanatory variable x in the simple linear regression is fixed; Case 2,x is purely random; and Case 3x is first-order autoregressive. Following a theoretical presentation, the ten estimation procedures are compared in a Monte Carlo study conducted in the time domain, where the errors are first-order autoregressive in Cases 1-3. The measure of comparison for the estimation procedures is their efficiency relative to OLS. It is evaluated as a function of the time series length and the magnitude and sign of the error autocorrelation parameter. Overall, knowledge of the model of the time series process generating the errors enhances efficiency in estimated GLS. Differences in the efficiency of estimation procedures between Case 1 and Cases 2 and 3 as well as differences in efficiency among procedures in a given situation are observed and discussed.