On Construction and Estimation of Stationary Mixture Transition Distribution Models

Abstract

Mixture transition distribution (MTD) time series models build high-order dependence through a weighted combination of first-order transition densities for each one of a specified number of lags. We present a framework to construct stationary MTD models that extend beyond linear, Gaussian dynamics. We study conditions for first-order strict stationarity which allow for different constructions with either continuous or discrete families for the first-order transition densities given a prespecified family for the marginal density, and with general forms for the resulting conditional expectations. Inference and prediction are developed under the Bayesian framework with particular emphasis on flexible, structured priors for the mixture weights. Model properties are investigated both analytically and through synthetic data examples. Finally, Poisson and Lomax examples are illustrated through real data applications. Supplementary files for this article are available online.

KEYWORDS:

• Bayesian inference
• First-order strict stationarity
• Markov chain Monte Carlo
• Non-Gaussian time series

Supplementary Material

The supplementary material includes the proof for Proposition 2, sampling algorithm details, additional simulation results, model checking for the data examples, and R code to reproduce the results in Section 5.

Acknowledgments

This research was supported in part by the National Science Foundation under awards SES 1631963, and SES 2050012. The authors wish to thank three reviewers and an Associate Editor for useful comments.

Additional information

Funding

This research was supported in part by the National Science Foundation under awards SES 1631963, and SES 2050012. The authors wish to thank three reviewers and an Associate Editor for useful comments.