Abstract
Autoregressive moving-average (ARMA) difference equations are ubiquitous models for short memory time series and have parsimoniously described many stationary series. Variants of ARMA models have been proposed to describe more exotic series features such as long memory autocovariances, periodic autocovariances, and count support set structures. This review paper enumerates, compares, and contrasts the common variants of ARMA models in today’s literature. After the basic properties of ARMA models are reviewed, we tour ARMA variants that describe seasonal features, long memory behavior, multivariate series, changing variances (stochastic volatility) and integer counts. A list of ARMA variant acronyms is provided.
Highlights
Autoregressive moving-average (ARMA) models are fundamental stationary time series models
The ARMA class is dense in all short memory stationary series; the class is parsimonious in that it flexibly generates a variety of different stationary autocovariance shapes from a few parameters
Likelihood methods are preferred when the ARMA model has a movingaverage component as solutions to moment equations based on sample autocovariances may not exist
Summary
Autoregressive moving-average (ARMA) models are fundamental stationary time series models (stationary here refers to covariance or second order stationary and not strict stationarity). Many extensions and variants of ARMA models have been developed to describe departures from short memory stationary characteristics, such as long memory autocovariances, periodicities, stochastic volatility (changing variances), multivariate series and discrete counts. We enumerate many of the ARMA variants, discussing what they intend to achieve, and compare and contrast their probabilistic and statistical structures. The time series literature is extensive and many general and specialized texts exist. Brockwell and Davis (2002), Chatfield (2003), Shumway and Stoffer (2006), Box, Jenkins and Reinsel (2008) and Cryer and Chan (2008) are comprehensive course texts for introductory material, emphasizing mainly the univariate case.
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