Skip to main content Accessibility help

We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.

Close cookie message
×
Home
  • Home
Article
  • Cited by 48

  • Get access
    Add to cart USD25.00
    Check if you have access via personal or institutional login
    Log in Register

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

  • Tassos Magdalinos (a1) and Peter C.B. Phillips (a2)

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.

Copyright

COPYRIGHT: © Cambridge University Press 2009

Corresponding author

*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.

References

Hide All
Abadir, K.M. & Magnus, J.R. (2005) Matrix Algebra. Econometric Exercises, vol. 1. Cambridge University Press.
Anderson, T.W. (1959) On asymptotic distributions of estimates of parameters of stochastic difference equations. Annals of Mathematical Statistics 30, 676–687.
Andrews, D.W.K. (1987) Asymptotic results for generalized Wald tests. Econometric Theory 3, 348–358.
Chan, N.H. & Wei, C.Z. (1987) Asymptotic inference for nearly nonstationary AR(1) processes. Annals of Statistics 15, 1050–1063.
Elliott, G. (1998) On the robustness of cointegration methods when regressors almost have unit roots. Econometrica 66, 149–158.
Hall, P. & Heyde, C.C. (1980) Martingale Limit Theory and Its Application. Academic Press.
Jacod, J. & Shiryaev, A.N. (1987) Limit Theorems for Stochastic Processes. Springer-Verlag.
Johansen, S. (1988) Statistical analysis of cointegrating vectors. Journal of Economic Dynamics and Control 12, 231–254.
Johansen, S. (1995) Identifying restrictions of linear equations with applications to simultaneous equations and cointegration. Journal of Econometrics 69, 111–132.
Kitamura, Y. & Phillips, P.C.B. (1995) Efficient IV estimation in nonstationary regression: An overview and simulation study. Econometric Theory 12, 1095–1130.
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.
Park, J.Y. & Phillips, P.C.B. (1988) Statistical inference in regressions with integrated processes: Part 1. Econometric Theory 4, 468–497.
Phillips, P.C.B. (1985) The distribution of matrix quotients. Journal of Multivariate Analysis 16, 157–161.
Phillips, P.C.B. (1987) Towards a unified asymptotic theory for autoregression. Biometrika 74, 535–547.
Phillips, P.C.B. (1988a) Regression theory for near-entegrated time series. Econometrica 56, 1021–1044.
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.
Phillips, P.C.B. (1991a) Optimal inference in cointegrated systems. Econometrica 59, 283–306.
Phillips, P.C.B. (1991b) Error correction and long-run equilibrium in continuous time. Econometrica 59, 967–980.
Phillips, P.C.B. (1994) Some exact distribution theory for maximum likelihood estimators of cointegration coefficients in error correction models. Econometrica 62, 73–94.
Phillips, P.C.B. (1995) Fully modified least squares and vector autoregression. Econometrica 63, 1023–1078.
Phillips, P.C.B. (2007) Unit root log periodogram regression. Journal of Econometrics 138, 104–124.
Phillips, P.C.B. & Durlauf, S.N. (1986) multiple time series regression with integrated processes. Review of Economic Studies 53, 473–496.
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.
Phillips, P.C.B. & Loretan, M. (1991) Estimating long-run economic equilibria. Review of Economic Studies 58, 407–436.
Phillips, P.C.B. & Magdalinos, T. (2007a) Limit theory for moderate deviations from a unit root. Journal of Econometrics 136, 115–130.
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.
Phillips, P.C.B. & Magdalinos, T. (2008a) Inference in Cointegrated Systems with Mildly Integrated Regressors. Manuscript, Yale University.
Phillips, P.C.B. & Magdalinos, T. (2008b) Limit theory for explosively cointegrated systems. Econometric Theory 24, 865–887.
Phillips, P.C.B. & Solo, V. (1992) Asymptotics for linear processes. Annals of Statistics 20, 971–1001.
Saikkonen, P. (1991) Asymptotically efficient estimation of cointegration regressions. Econometric Theory 7, 1–21.
Stock, J.H. & Watson, M.W. (1993) A simple estimator of cointegrating vectors in higher order integrated systems. Econometrica 61, 783–821.
  • Cited by 48

  • Get access
    Add to cart USD25.00
    Check if you have access via personal or institutional login
    Log in Register

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

  • Tassos Magdalinos (a1) and Peter C.B. Phillips (a2)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.