Likelihood inference in Gaussian copula models for count time series via minimax exponential tilting

Abstract

Count time series arise in diverse contexts and may display a diversity of distributional features that may include overdispersion, zero–inflation, covariates’ effects and complex dependence structures. A class of models with the potential to account for this diversity is that of Gaussian copulas, which are computationally challenging to fit. A scalable and accurate likelihood approximation strategy is proposed that employs minimax exponential tilting (MET) to fit Gaussian copula models with arbitrary marginals and ARMA latent processes to count time series. The proposed method, called Time Series Minimax Exponential Tilting (TMET), exploits the exact conditional structure of causal and invertible ARMA processes to construct an optimized importance sampling density. Costly Cholesky decompositions are avoided by using a simplified Innovations algorithm to recursively compute conditional means and variances, and further accelerates computation through a sparse representation of the best linear prediction matrix. These innovations achieve linear computational complexity in the series length, while preserving key theoretical guarantees, including vanishing relative error in rare–event regimes. Simulation studies show that TMET outperforms widely used methods, including the Geweke–Hajivassiliou–Keane (GHK) simulator and the recent Vecchia–based MET (VMET) approach, especially in scenarios with low counts, strong dependence, and moving average latent processes. Beyond estimation, the copula framework is extended to include predictive inference and model diagnostics based on scoring rules and randomized quantile residuals. A real–world application to temperature data from the Kickapoo Downtown Airport in Texas demonstrates TMET’s advantages over the commonly used GHK simulator.