Estimating Long Run Economic Equilibria
Our subject is econometric estimation and inference concerning long-run economic equilibria in models with stochastic trends. Our interest is focused on single equation speciﬁcations such as those employed in the Error Correction Model (ECM) methodology of David Hendry (1987, 1989 inter alia) and the semiparametric modiﬁed least squares method of Phillips and Hansen (1989). We start by reviewing the prescriptions for empirical time series research that are presently available. We argue that the diversity of choices is confusing to practitioners and obscures the fact that statistical theory is clear about optimal inference procedures. Part of the diﬀiculty arises from the many alternative time series representations of cointegrated systems. We present a detailed analysis of these various representations, the links between them, and the estimator choices to which they lead. An asymptotic theory is provided for a wide menu of econometric estimators and system speciﬁcations, accommodating diﬀerent levels of prior information about the presence of unit roots and the nature of short-run dynamic adjustments. The single equation ECM approach is studied in detail and our results lead to certain recommendations. Weak exogeneity and data coherence are generally insuﬀicient for valid conditioning on the regressors in this approach. Strong exogeneity and data coherency are suﬀicient to validate conditioning. But the requirement of strong exogeneity rules out most cases of interest because long-run economic equilibrium typically relates interdependent variables for which there is substantial time series feedback. One antidote for this problem in practice is the inclusion of leads as well as lags in the diﬀerences of the regressors. The simulations that we report, as well as the asymptotic theory support the use of this procedure in practice. Our results also support the use of dynamic speciﬁcations that involve lagged long-run equilibrium relations rather than lagged diﬀerences in the dependent variable. Finally, our simulations point to problems of overﬁtting in single equation ECM’s. These appear to have important implications for empirical research in terms of size distortions that are produced in signiﬁcance tests that utilize nominal critical values delivered by conventional asymptotic theory. In sum, our results indicate that the single equation ECM methodology has good potential for further development and improvement. But in comparison with the semi parametric modiﬁed least squares method of Phillips and Hansen (1989) the latter method seems superior for inferential purposes in most cases.
Phillips, Peter C.B. and Loretan, Mico, "Estimating Long Run Economic Equilibria" (1989). Cowles Foundation Discussion Papers. 1171.