Hostname: page-component-7b9c58cd5d-9k27k Total loading time: 0 Render date: 2025-03-15T06:02:17.874Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Published online by Cambridge University Press:  01 April 2009

Tassos Magdalinos  and
Peter C.B. Phillips
Tassos Magdalinos
Affiliation:
Granger Centre for Time Series Econometrics University of Nottingham
Peter C.B. Phillips*
Affiliation:
Yale University, University of Auckland, University of York, and Singapore Management University
*
*Address correspondence to Peter C.B. Phillips, Department of Economics, Yale University, P.O. Box 208268, New Haven, CT 06520-8268, USA; e-mail: [email protected].
Get access Rights & Permissions [Opens in a new window]

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 ∈ ℝ 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.

Type
ARTICLES
Information
Econometric Theory , Volume 25 , Issue 2 , April 2009 , pp. 482 - 526
Copyright
Copyright © Cambridge University Press 2009

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Abadir, K.M. & Magnus, J.R. (2005) Matrix Algebra. Econometric Exercises, vol. 1. Cambridge University Press.CrossRefGoogle Scholar
Anderson, T.W. (1959) On asymptotic distributions of estimates of parameters of stochastic difference equations. Annals of Mathematical Statistics 30, 676–687.CrossRefGoogle Scholar
Andrews, D.W.K. (1987) Asymptotic results for generalized Wald tests. Econometric Theory 3, 348–358.CrossRefGoogle Scholar
Chan, N.H. & Wei, C.Z. (1987) Asymptotic inference for nearly nonstationary AR(1) processes. Annals of Statistics 15, 1050–1063.CrossRefGoogle Scholar
Elliott, G. (1998) On the robustness of cointegration methods when regressors almost have unit roots. Econometrica 66, 149–158.CrossRefGoogle Scholar
Hall, P. & Heyde, C.C. (1980) Martingale Limit Theory and Its Application. Academic Press.Google Scholar
Jacod, J. & Shiryaev, A.N. (1987) Limit Theorems for Stochastic Processes. Springer-Verlag.CrossRefGoogle Scholar
Johansen, S. (1988) Statistical analysis of cointegrating vectors. Journal of Economic Dynamics and Control 12, 231–254.CrossRefGoogle Scholar
Johansen, S. (1995) Identifying restrictions of linear equations with applications to simultaneous equations and cointegration. Journal of Econometrics 69, 111–132.CrossRefGoogle Scholar
Kitamura, Y. & Phillips, P.C.B. (1995) Efficient IV estimation in nonstationary regression: An overview and simulation study. Econometric Theory 12, 1095–1130.CrossRefGoogle Scholar
Kitamura, Y. & Phillips, P.C.B. (1997) Fully modified IV, GIVE and GMM estimation with possibly nonstationary regressors and instruments. Journal of Econometrics 80, 85–123.CrossRefGoogle Scholar
Park, J.Y. & Phillips, P.C.B. (1988) Statistical inference in regressions with integrated processes: Part 1. Econometric Theory 4, 468–497.CrossRefGoogle Scholar
Phillips, P.C.B. (1985) The distribution of matrix quotients. Journal of Multivariate Analysis 16, 157–161.CrossRefGoogle Scholar
Phillips, P.C.B. (1987) Towards a unified asymptotic theory for autoregression. Biometrika 74, 535–547.CrossRefGoogle Scholar
Phillips, P.C.B. (1988a) Regression theory for near-entegrated time series. Econometrica 56, 1021–1044.CrossRefGoogle Scholar
Phillips, P.C.B. (1988b) Multiple regression with integrated processes. In Prabhu, N.U.(ed.), Statistical Inference from Stochastic Processes, Contemporary Mathematics 80, 79–106.CrossRefGoogle Scholar
Phillips, P.C.B. (1991a) Optimal inference in cointegrated systems. Econometrica 59, 283–306.CrossRefGoogle Scholar
Phillips, P.C.B. (1991b) Error correction and long-run equilibrium in continuous time. Econometrica 59, 967–980.CrossRefGoogle Scholar
Phillips, P.C.B. (1994) Some exact distribution theory for maximum likelihood estimators of cointegration coefficients in error correction models. Econometrica 62, 73–94.CrossRefGoogle Scholar
Phillips, P.C.B. (1995) Fully modified least squares and vector autoregression. Econometrica 63, 1023–1078.CrossRefGoogle Scholar
Phillips, P.C.B. (2007) Unit root log periodogram regression. Journal of Econometrics 138, 104–124.CrossRefGoogle Scholar
Phillips, P.C.B. & Durlauf, S.N. (1986) multiple time series regression with integrated processes. Review of Economic Studies 53, 473–496.CrossRefGoogle Scholar
Phillips, P.C.B. & Hansen, B.E. (1990) Statistical inference in instrumental variables regression with I(1) processes. Review of Economic Studies 57, 99–125.CrossRefGoogle Scholar
Phillips, P.C.B. & Loretan, M. (1991) Estimating long-run economic equilibria. Review of Economic Studies 58, 407–436.CrossRefGoogle Scholar
Phillips, P.C.B. & Magdalinos, T. (2007a) Limit theory for moderate deviations from a unit root. Journal of Econometrics 136, 115–130.CrossRefGoogle Scholar
Phillips, P.C.B. & Magdalinos, T. (2007b) Limit theory for moderate deviations from a unit root under weak dependence. In Phillips, G.D.A. & Tzavalis, E. (eds.), The Refinement of Econometric Estimation and Test Procedures: Finite Sample and Asymptotic Analysis, pp. 123–162. Cambridge University Press.CrossRefGoogle Scholar
Phillips, P.C.B. & Magdalinos, T. (2008a) Inference in Cointegrated Systems with Mildly Integrated Regressors. Manuscript, Yale University.Google Scholar
Phillips, P.C.B. & Magdalinos, T. (2008b) Limit theory for explosively cointegrated systems. Econometric Theory 24, 865–887.CrossRefGoogle Scholar
Phillips, P.C.B. & Solo, V. (1992) Asymptotics for linear processes. Annals of Statistics 20, 971–1001.CrossRefGoogle Scholar
Saikkonen, P. (1991) Asymptotically efficient estimation of cointegration regressions. Econometric Theory 7, 1–21.CrossRefGoogle Scholar
Stock, J.H. & Watson, M.W. (1993) A simple estimator of cointegrating vectors in higher order integrated systems. Econometrica 61, 783–821.CrossRefGoogle Scholar