Translate ergodicity conditions to copula-parameter restrictions for MLE consistency

Derive verifiable sufficient conditions stated directly on the parameters of the autoregressive copula C and the moving-aggregate (MAG) copula K in the copula-based time series model U_t = h(ε_t, …, ε_{t−q+1}, W_{t−q}) and W_t = g(ε_t, W_{t−1}, …, W_{t−p}), where g and h are conditional quantile functions and {ε_t} are i.i.d. U(0,1), that guarantee stationarity and ergodicity of the estimated latent processes {Ŵ_t} and {ê_t} for all feasible parameter values, thereby enabling consistency of the maximum-likelihood estimator.

Background

To establish consistency of the maximum-likelihood estimator for the proposed copula-based ARMA generalization, the authors need stationarity and ergodicity of the estimated latent sequences {Ŵ_t} and {ê_t}. They note that these properties can follow from general results on stochastic recurrence equations (Douc et al., 2014), and that, under additional assumptions, M-estimation results (Wooldridge, 1994) yield uniform laws of large numbers and consistency.

However, the paper points out that the current conditions are high-level and not yet expressed in terms of concrete constraints on the AR and MAG copula families and their parameters. Turning these theoretical requirements into explicit parameter restrictions would make the consistency theory practically verifiable for applied modeling.