Residual Log-Periodogram Inference for Long-Run-Relationships

Abstract

We assume that some consistent estimator of an equilibrium relation between non-stationary series integrated of order d E (0:5; 1:5) is used to compute residuals ˆut = yt - xt (or differences there of). We propose to apply the semiparametric log-periodogram regression to the (differenced) residuals in order to estimate or test the degree of persistence ± of the equilibrium deviation ut. Provided converges fast enough, we describe simple semiparametric conditions around zero frequency that guarantee consistent estimation of ±. At the same time limiting normality is derived, which allows to construct approximate confidence intervals to test hypotheses on ±. This requires that d ¡ ± > 0:5 for superconsistent b¯, so the residuals can be good proxies of true cointegrating errors. Our assumptions allow for stationary deviations with long memory, 0 · ± < 0:5, as well as for non-stationary but transitory equilibrium errors, 0:5 < ± < 1. In particular, if xt contains several series we consider the joint estimation of d and ±. Wald statistics to test for parameter restrictions of the system have a limiting Â2 distribution. We also analyze the benefits of a pooled version of the estimate. The empirical applicability of our general cointegration test is investigated by means of Monte Carlo experiments and illustrated with a study of exchange rate dynamics.