Vector equilibrium correction models with non‐linear discontinuous adjustments

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.innovations of the VEC process have finite moments of order n. In addition, a characterization is given for the process X t as a non-linear I(1) type process. Asymptotic normality is shown to hold for the parameters of the nonlinear VEC model for the case of known cointegrating vectors. As mentioned before the results are used in an illustration with the German term structure, where quite clear-cut evidence is found for a TAR VEC model. The results clearly support the existence of an area with no disequilibrium adjustment: the small disequilibria (namely the interest rate spreads) are not corrected, either by the long-term or by the short-term interest rate. On the contrary, large yield spreads induce error correction but from the short-term rate only, which is consistent with the expectations hypothesis of the term structure of interest rates.Non-linearity and non-stationarity have been studied mainly in a univariate framework, with some recent exceptions such as Tsay regarding stability in non-linear-type error correction models, the theory in Escribano and Mira (2002) and Corradi et al. (2000) applies to continuous and suitably smooth transition functions. Saikkonen (2002, Theorem 5) establishes geometric ergodicity for a general class of processes with observable transition functions. 1 The paper is organized as follows. Section 2 recalls results concerning linear I(1) cointegrated processes which will prove useful for the subsequent non-linear analysis. It also introduces the non-linear vector error correction model under study. Section...

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