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This paper studies the asymptotic properties of instrumental variable (IV) estimates of multivariate cointegrating regressions. The framework of study is based on earlier work by Phillips and Durlauf (1986) and Park and Phillips (1988, 1989). In particular, the results in these papers are extended to allow for IV regressions that accommodate deterministic and stochastic regressors as well as quite general deterministic processes in the data generating mechanism. It is found that IV regressions are consistent even when the instruments are stochastically independent of the regressors. This phenomenon, which contrasts with traditional theory for stationary time series, is a beneﬁcial artifact of spurious regression theory whereby stochastic trends in the instruments ensure their relevance asymptotically. Problems of inference are also addressed and some promising new theoretical results are reported. These involve a class of Wald tests which are modiﬁed by semiparametric corrections for serial correlation and for endogeneity. The resulting test statistics which we term fully modiﬁed Wald tests have limiting chi-squared distributions, thereby removing the obstacles to inference in cointegrated systems that were presented by the nuisance parameter dependencies in earlier work. Interestingly, IV methods themselves are insuﬀicient to achieve this end and an endogeneity correction is still generally required, again in contrast to traditional theory. Our results therefore provide strong support for the conclusion reached by Hendry (1986) that there is no free lunch in estimating cointegrated systems. Some simulation results are reported which seek to explore the sampling behavior of our suggested procedures. These simulations compare our fully modiﬁed (semiparametric) methods with the parametric error correction methodology that has been extensively used in recent empirical research and with conventional least squares regression. Both the fully modiﬁed and error correction methods work well in ﬁnite samples and the sampling performance of each procedure conﬁrms the relevance of asymptotic distribution theory, as distinct from superconsistency results, in discriminating between diﬀerent statistical methods.
Phillips, Peter C.B. and Hansen, Bruce E., "Statistical Inference in Instrumental Variables Regression with I(1) Processes" (1988). Cowles Foundation Discussion Papers. 1112.