Hostname: page-component-76fb5796d-45l2p Total loading time: 0 Render date: 2024-04-30T06:36:26.569Z Has data issue: false hasContentIssue false
  • English
  • Français

Testing for Cointegration in a System of Equations

Published online by Cambridge University Press:  11 February 2009

In Choi  and
Byung Chul Ahn
In Choi
Affiliation:
Kookmin University
Byung Chul Ahn
Affiliation:
Youngnam University
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper introduces various consistent tests for the null of cointegration against the alternative of noncointegration that can be applied to a system of equations as well as to a single equation. The tests are analogs of Choi and Ahn's (1993, Testing the Null of Stationarity for Multiple Time Series, working paper, The Ohio State University) multivariate tests for the null of stationarity and use Park's (1992, Econometrica 60, 119–143) canonical cointegrating regression (CCR) residuals to make the tests free of nuisance parameters in the limit. The asymptotic distributions of the tests are complex but expressed in unified manner by using standard vector Brownian motion. These distributions are tabulated by simulation for some practical cases. Furthermore, the rates of divergence of the tests are reported. Because there are methods for estimating cointegrating matrices other than CCR, it is illustrated for a model without time trends that the tests we introduce work exactly the same way in the limit when Phillips and Hansen's (1990, Review of Economic Studies 57, 99–125) fully modified ordinary least-squares (OLS) procedure is used. Also, is shown that difficulties arise when OLS residuals are used to formulate the tests. Small-scale simulation results are reported to examine the finite sample performance of the tests. The tests are shown to work reasonably wellin finite samples. In particular, it is illustrated that using the multivariate tests introduced in this paper can be a better testing strategy in terms of the finite sample size and power than applying univariate tests several times to each equation in a system of equations.

Type
Articles
Information
Econometric Theory , Volume 11 , Issue 5 , October 1995 , pp. 952 - 983
Copyright
Copyright © Cambridge University Press 1995

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

Andrews, D.W.K. (1991) Heteroskedasticity and autocorrelation consistent covariance matrix estimation. Econometrica 59, 817–858.CrossRefGoogle Scholar
Andrews, D.W.K. & Monahan, J.C. (1992) An improved heteroskedasticity and autocorrelation consistent covariance matrix estimator. Econometrica 60, 953–966.CrossRefGoogle Scholar
Chan, N.H. & Wei, C.Z. (1988) Limiting distribution of least squares estimates of unstable autoregressive processes. Annals of Statistics 16, 367–401.CrossRefGoogle Scholar
Choi, I. (1994a) Durbin-Hausman tests for cointegration. Journal of Economic Dynamics and Control 18, 467–480.CrossRefGoogle Scholar
Choi, I. (1994b) Residual based tests for the null of stationarity with applications to U.S. macroeconomic time series. Econometric Theory 10, 720–746.CrossRefGoogle Scholar
Choi, I. (1994c) Spurious regressions and the residual based for cointegration when regressors are cointegrated. Journal of Econometrics 60, 313–320.Google Scholar
Choi, I. & Ahn, B.C. (1993) Testing the Null of Stationarity for Multiple Time Series. Working paper, The Ohio State University.Google Scholar
Choi, I. & Yu, B. (1993) A General Framework for Testing I(m) against I(m+k). Working paper, The Ohio State University.Google Scholar
Engle, R. & Granger, C.W.J. (1987) Co-integration and error correction: Representation, estimation and testing. Econometrica 55, 251–276.CrossRefGoogle Scholar
Granger, C.W.J. & Newbold, P. (1974) Spurious regressions in econometrics. Journal of Econometrics 2, 111–120.CrossRefGoogle Scholar
Hannan, E.J. & Heyde, C. (1972) On limit theorems for quadratic functions of discrete time series. Annals of Mathematical Statistics 43, 2058–2066.CrossRefGoogle Scholar
Hansen, B. (1992) Test for parameter instability in regressions with I(1) processes. Journal of Business and Economic Statistics 10, 321–336.Google Scholar
Kwiatkowski, D., Phillips, P.C.B., Schmidt, P., & Shin, Y. (1992) Testing the null hypothesis of stationarity against the alternative of a unit root: How sure are we that economic time series have a unit root? Journal of Econometrics 54, 159–178.CrossRefGoogle Scholar
Park, J.Y. (1990) Testing for unit roots and cointegration by variable addition. Advances in Econometrics 8, 107–133.Google Scholar
Park, J.Y. (1992) Canonical cointegrating regression. Econometrica 60, 119–143.CrossRefGoogle Scholar
Park, J.Y. & Ogaki, M. (1991). Seemingly Unrelated Canonical Cointegrating Regressions. Working paper, University of Rochester.Google 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. (1986) Understanding spurious regressions in econometrics. Journal of Econometrics 33, 311–340.CrossRefGoogle Scholar
Phillips, P.C.B. (1987) Time series regression with a unit root. Econometrica 55, 277–301.CrossRefGoogle Scholar
Phillips, P.C.B. (1990) Optimal inference in cointegrated system. Econometrica 59, 283–306.CrossRefGoogle Scholar
Phillips, P.C.B. & Durlauf, S. (1986) Multiple time series regression with integrated processes. Review of Economic Studies LIII, 473–495.CrossRefGoogle Scholar
Phillips, P.C.B. & Hansen, B. (1990) Statistical inference in instrumental variables regression with I(1) processes. Review of Economic Studies 57, 99–125.CrossRefGoogle Scholar
Phillips, P.C.B. & Ouliaris, S. (1988) Testing for cointegration using principal components methods. Journal of Economic Dynamics and Control 12, 205–230.CrossRefGoogle Scholar
Phillips, P.C.B. & Ouliaris, S. (1990) Asymptotic properties of residual based tests for cointegration. Econometrica 58, 165–193.CrossRefGoogle Scholar
Phillips, P.C.B. & Solo, V. (1992) Asymptotics for linear processes. Annals of Statistics 20, 971–1001.CrossRefGoogle Scholar
Quintos, C.E & Phillips, P.C.B. (1993) Parameter constancy in cointegrating regressions. Empirical Economics 18, 675–706.CrossRefGoogle Scholar
Saikkonen, P. (1991) Asymptotically efficient estimation of cointegration regressions. Econometric Theory 9, 1–21.CrossRefGoogle Scholar
Shin, Y. (1994) A residual based test of the null of cointegration against the alternative of no cointegration. Econometric Theory 10, 91–115.CrossRefGoogle Scholar
Stock, J. & Watson, M.W. (1988) Testing for common trends. Journal of the American Statistical Association 83, 1097–1107.CrossRefGoogle Scholar
Stock, J. & Watson, M.W. (1993) A simple estimator of cointegrating vectors in higher order integrated systems. Econometrica 61, 783–820.CrossRefGoogle Scholar
Tanaka, K. (1993) An alternative approach to the asymptotic theory of spurious regression, cointegration, and near cointegration. Econometric Theory 9, 36–61.CrossRefGoogle Scholar