Comparison Between the Exact Likelihood and Whittle Likelihood for Moving Average Processes

Abstract

For Gaussian stationary processes, the likelihood functions include the inverse and determinant of the covariance matrices, and Whittle likelihood is considered as a standard technique to avoid expensive matrix determinant and inversions under large sample size. In this paper, we investigate the difference between the exact likelihood and Whittle likelihood with finite sample size for moving average processes of order one. We elucidate the theoretical expressions of two likelihood functions and their expectations and evaluate the performance between exact likelihood and Whittle likelihood numerically. We find that the exact likelihood and Whittle likelihood perform similarly when the true value of parameter is close to zero, while the difference becomes large and Whittle estimator performs poorly when absolute value of parameter gets close to one. This is an important warning when we use the Whittle likelihood and estimator if the parameter of moving average process nears the boundary of parameter space.