Archimedean Copulas and Temporal Dependence

A central aspect of time series analysis is the modeling of dependence over time. Workhorse time series models such as the autoregressive moving average (ARMA) model popularized by Box and Jenkins (1970), the generalized autoregressive conditional heteroskedasticity (GARCH) model of Engle (1982) and Bollerslev (1986), or the autoregressive conditional duration (ACD) model of Engle and Russell (1998) impose explicit conditions on the way in which a process evolves stochastically over time. It is natural to wonder what kind of mixing or ergodic properties might be implied or excluded by such conditions. Conditions under which ARMA processes are geometrically ergodic have been provided by Pham and Tran (1985) and Mokkadem (1988), while conditions under which GARCH and ACD processes are geometrically ergodic have been provided by Carrasco and Chen (2002) and Meitz and Saikkonen (2008). Results such as these enhance our understanding of the models underlying much empirical work and can be used to justify the application of invariance principles to partial sums of functions of processes driven by those models. During the last five years, a new class of time series models has emerged in which copula functions are used to model dependence over time in a stationary Markov chain. The allure of this approach is that it facilitates the separate consideration of the dependence structure of the chain, specified using a copula, and the invariant distribution of the chain. This advantage was first emphasized by Darsow, Nguyen, and Olsen (1992). Chen and Fan (2006) introduced copulabased Markov models to the econometric literature, proposing a semiparametric estimation procedure in which the copula is specified parametrically while the