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Statistical inference in instrumental variables regression with $$I(1)$$ processes. (English) Zbl 0703.62098
Summary: This paper studies the asymptotic properties of instrumental variable (IV) estimates of multivariate cointegration regressions and allows for 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 $$\chi^2$$ distributions, thereby removing the obstacles to inference in cointegrated systems that were presented by the nuisance parameter dependencies in earlier work.
Some simulation results are reported which seek to explore the sampling behaviour 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 super-consistency results, in discriminating between statistical methods.

##### MSC:
 62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH) 62P20 Applications of statistics to economics 62H15 Hypothesis testing in multivariate analysis 62E20 Asymptotic distribution theory in statistics 62H12 Estimation in multivariate analysis
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