Approximating Gaussian Copula models for count time series: connecting the distributional transform and a continuous extension

Gaussian copulas are versatile models for the analysis of time series data as they allow for the separate modeling of their marginal and association structures. However, likelihood–based inference for count time series is computationally intensive for large samples. This is so because the likelihood is a multivariate normal probability that lacks a closed–form expression, making its computation a challenging numerical problem. We study a likelihood approximation method based on a continuous extension that avoids the need for approximating high–dimensional integrals, and show that the previously proposed distributional transform likelihood approximation is a particular case. We also obtain a novel expression for this approximate likelihood that can be efficiently evaluated using the innovations algorithm. Through simulation experiments we identify scenarios where the proposed method achieves similar approximation accuracy as the Geweke–Hajivassiliou–Keane (GHK) method, but with far superior computational efficiency, as well as scenarios where this is not the case. We illustrate the efficacy of the method by fitting a Gaussian copula model to the number of weekly campylobacteriosis infections in Hamburg, Germany.