Vector Equilibrium Correction Models with Non-Linear Discontinuous Adjustments

TL;DR: In this article, conditions for stationarity, geometric ergodicity as well as existence of moments are derived using a general multivariate Markov process and it is shown that imposing parametric restrictions on only one of the regimes of the non-linear vector autoregression is sufficient to ensure higher-order moments and linear cointegrating relations which are geometrically ergodic and hence also stationary.

Abstract: Cointegration is studied for a non-linear autoregressive process characterized by discontinuous and regime-dependent equilibrium or error correction. Here the disequilibrium, as measured by the norm of linear 'stable' or cointegrating relations, determines the regime and hence the equilibrium correction of the process. Importantly, switching between regimes is thereby allowed to be caused endogenously. The transition function may be either observable as in, e.g. threshold processes, or unobservable when transition probabilities are specified as in, e.g. autoregressive conditional root processes. Conditions for stationarity, geometric ergodicity as well as existence of moments are derived using a general multivariate Markov process. From these conditions it is shown that imposing parametric restrictions on only one of the regimes of the non-linear vector autoregression is sufficient to ensure higher-order moments and linear cointegrating relations which are geometrically ergodic and hence also stationary. Additionally, estimation is considered when the cointegrating relations are known and asymptotic theory is provided for this case. Based on many existing empirical analyses of, e.g. real exchange rates and interest rates spreads, the proposed dynamics appears to be desirable. This is also reflected in the included analysis of the German term structure where empirical evidence is found for discontinuous threshold error correction as opposed to classic linear error correction.