Comments on Pairwise Likelihood in Time Series Models

This note is concerned with the asymptotic properties of pairwise likelihood estimation procedures for linear time series models. The latter includes ARMA as well as fractionally integrated ARMA processes, where the fractional integration parameter d < 0.5. In some cases, including AR(1) processes and longmemory processes with d < 0.25, the loss in efficiency in using pairwise likelihood methods is slight. On the other hand, for some models such as the MA(1), the loss in efficiency can be large, and for long-memory models with d > 0.25, the pairwise likelihood estimator is not even asymptotically normal. A comparison between using all pairs and consecutive pairs of observations in defining the likelihood is given. We also explore the application of pairwise likelihood to a popular nonlinear model for time series of counts. In this case, the likelihood based on the entire data set cannot be computed without resorting to simulation-based procedures. On the other hand, it is possible to numerically compute the pairwise likelihood precisely. We illustrate the good performance of pairwise likelihood in this case.