Econometric Theory

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LIMIT THEORY FOR COINTEGRATED SYSTEMS WITH MODERATELY INTEGRATED AND MODERATELY EXPLOSIVE REGRESSORS

Tassos Magdalinosa1 and Peter C.B. Phillipsa2 c1

a1 Granger Centre for Time Series Econometrics University of Nottingham

a2 Yale University, University of Auckland, University of York, and Singapore Management University

Abstract

An asymptotic theory is developed for multivariate regression in cointegrated systems whose variables are moderately integrated or moderately explosive in the sense that they have autoregressive roots of the form ρni = 1 + ci/nα, involving moderate deviations from unity when α (0, 1) and ci are constant parameters. When the data are moderately integrated in the stationary direction (with ci < 0), it is shown that least squares regression is consistent and asymptotically normal but suffers from significant bias, related to simultaneous equations bias. In the moderately explosive case (where ci > 0) the limit theory is mixed normal with Cauchy-type tail behavior, and the rate of convergence is explosive, as in the case of a moderately explosive scalar autoregression (Phillips and Magdalinos, 2007, Journal of Econometrics 136, 115–130). Moreover, the limit theory applies without any distributional assumptions and for weakly dependent errors under conventional moment conditions, so an invariance principle holds, unlike the well-known case of an explosive autoregression. This theory validates inference in cointegrating regression with mildly explosive regressors. The special case in which the regressors themselves have a common explosive component is also considered.

Correspondence

c1 Address correspondence to Peter C.B. Phillips, Department of Economics, Yale University, P.O. Box 208268, New Haven, CT 06520-8268, USA; e-mail: peter.phillips@yale.edu.

Footnotes

Thanks go to two referees and a co-editor for comments on an earlier version of the paper. The original draft of the paper was written in April 2004. Phillips acknowledges partial research support from a Kelly Fellowship and the NSF under grants SES-04-142254 and SES 06-47086. Magdalinos thanks the EPSRC and the Onassis Foundation for scholarship support.

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