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Testing for Cointegration When Some of the Cointegrating Vectors are Prespecified

Published online by Cambridge University Press:  11 February 2009

Michael T.K. Horvath  and
Mark W. Watson
Michael T.K. Horvath
Affiliation:
Stanford University
Mark W. Watson
Affiliation:
Princeton University
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Abstract

Many economic models imply that ratios, simple differences, or “spreads” of variables are I(0). In these models, cointegrating vectors are composed of 1's, 0's, and —1's and contain no unknown parameters. In this paper, we develop tests for cointegration that can be applied when some of the cointegrating vectors are prespecified under the null or under the alternative hypotheses. These tests are constructed in a vector error correction model and are motivated as Wald tests from a Gaussian version of the model. When all of the cointegrating vectors are prespecified under the alternative, the tests correspond to the standard Wald tests for the inclusion of error correction terms in the VAR. Modifications of this basic test are developed when a subset of the cointegrating vectors contain unknown parameters. The asymptotic null distributions of the statistics are derived, critical values are determined, and the local power properties of the test are studied. Finally, the test is applied to data on foreign exchange future and spot prices to test the stability of the forward–spot premium.

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Type
Articles
Information
Econometric Theory , Volume 11 , Issue 5 , October 1995 , pp. 984 - 1014
Copyright
Copyright © Cambridge University Press 1995

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References

REFERENCES

Ahn, S.K. & Reinsel, G.C. (1990) Estimation for partially nonstationary autoregressive models. Journal of the American Statistical Association 85, 813–823.CrossRefGoogle Scholar
Anderson, T.W. (1951) Estimating linear restrictions on regression coefficients for multivariate normal distributions. Annals of Mathematical Statistics 22, 327–351.CrossRefGoogle Scholar
Bekaert, G. & Hodrick, R.J. (1993) On biases in the measurement of foreign exchange risk premiums. Journal of International Money and Finance 12 (2), 115–138.CrossRefGoogle Scholar
Bobkowsky, M.J. (1983) Hypothesis Testing in Nonstationary Time Series. Ph.D. Thesis, Department of Statistics, University of Wisconsin.Google Scholar
Brillinger, D.R. (1980) Time Series, Data Analysis and Theory, expanded ed. San Francisco: Holden-Day.Google Scholar
Cavanagh, C.L. (1985) Roots Local to Unity. Manuscript, Department of Economics, Harvard University.Google Scholar
Chan, N.H. (1988) On parameter inference for nearly nonstationary time series. Journal of the American Statistical Association 83, 857–862.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
Chan, N.H. & Wei, C.Z. (1988) Limiting distributions of least squares estimates of unstable autoregressive processes. Annals of Statistics 16(1), 367–401.CrossRefGoogle Scholar
Davies, R.B. (1977) Hypothesis testing when a parameter is present only under the alternative. Biometrika 64, 247–254.CrossRefGoogle Scholar
Davies, R.B. (1987) Hypothesis testing when a parameter is present only under the alternative. Biometrika 74, 33–43.Google Scholar
Elliott, G. (1993) Efficient Tests for a Unit Root When the Initial Observation Is Drawn from Its Unconditional Distribution. Manuscript, Harvard University.Google Scholar
Elliott, G., Rothenberg, T.J., & Stock, J.H. (1995) Efficient tests of an autoregressive unit root. Econometrica, forthcoming.Google Scholar
Engle, R.F. & Granger, C.W.J. (1987) Cointegration and error correction: Representation, estimation, and testing. Econometrica 55, 251–276. Reprinted in Engle, R.F. & Granger, C.W.J. (eds.), Long-Run Economic Relations: Readings in Cointegration. New York: Oxford University Press, 1991.CrossRefGoogle Scholar
Evans, M.D.D. & Lewis, K.K. (1992) Are Foreign Exchange Rates Subject to Permanent Shocks. Manuscript, University of Pennsylvania.Google Scholar
Hansen, B.E. (1990) Inference When a Nuisance Parameter Is Not Identified Under the Null Hypothesis. Manuscript, University of Rochester.Google Scholar
Hansen, B.E. (1993) Testing for Unit Roots Using Covariates. Manuscript, University of Rochester.Google Scholar
Horvath, M. & Watson, M. (1995) Critical Values for Likelihood Based Tests for Cointegration When Some of the Cointegrating Vectors Are Prespecified. Manuscript, Stanford University.Google Scholar
Johansen, S. (1988) Statistical analysis of cointegrating vectors. Journal of Economic Dynamics and Control 12, 231–254. Reprinted in Engle, R.F. & Granger, C.W.J. (eds.), Long-Run Economic Relations: Readings in Cointegration. New York: Oxford University Press, 1991.CrossRefGoogle Scholar
Johansen, S. (1991) Estimation and hypothesis testing of cointegrating vectors in Gaussian vector autoregression models. Econometrica 59, 1551–1580.Google Scholar
Johansen, S. (1992a) Determination of cointegration rank in the presence of a linear trend. Oxford Bulletin of Economics and Statistics 54, 383–397.CrossRefGoogle Scholar
Johansen, S. (1992b) The Role of the Constant Term in Cointegration Analysis of Nonstationary Variables. Preprint 1, Institute of Mathematical Statistics, University of Copenhagen.Google Scholar
Johansen, S. & Juselius, K. (1990) Maximum likelihood estimation and inference on cointegration—With applications to the demand for money. Oxford Bulletin of Economics and Statistics 52(2), 169–210.CrossRefGoogle Scholar
Johansen, S. & Juselius, K. (1992) Testing structural hypotheses in a multivariate cointegration analysis of the PPP and UIP of UK. Journal of Econometrics 53, 211–244.Google Scholar
Kremers, J.J.M., Ericsson, N.R., & Dolado, J.J. (1992) The power of cointegration tests. Oxford Bulletin of Economics and Statistics 54(3), 325–348.CrossRefGoogle Scholar
Ng, S. & Perron, P. (1993) Unit Root Tests in ARM A Models with Data Dependent Methods for the Selection of the Truncation Lag. Manuscript, University of Montreal.Google Scholar
Osterwald-Lenum, M. (1992) A note with quantiles of the asymptotic distribution of the maximum likelihood cointegration rank test statistics. Oxford Bulletin of Economics and Statistics 54, 461–471.CrossRefGoogle Scholar
Park, J.Y. (1990) Maximum Likelihood Estimation of Simultaneous Cointegrating Models. Manuscript, Institute of Economics, Aarhus University.Google Scholar
Park, J.Y. & Phillips, P.C.B. (1988) Statistical inference in regressions with integrated regressors I. Econometric Theory 4, 468–497.CrossRefGoogle Scholar
Phillips, P.C.B. (1987) Toward a unified asymptotic theory for autoregression. Biometrika 74, 535–547.CrossRefGoogle Scholar
Phillips, P.C.B. (1988) Multiple regression with integrated regressors. Contemporary Mathematics 80, 79–105.CrossRefGoogle Scholar
Phillips, P.C.B. (1991) Optimal inference in cointegrated systems. Econometrica 59(2), 283–306.CrossRefGoogle Scholar
Phillips, P.C.B. (1993) Robust Nonstationary Regression. Manuscript, Cowles Foundation, Yale University.Google 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. & Solo, V. (1992) Asymptotics for linear processes. Annals of Statistics 20, 971–1001.Google Scholar
Sims, C.A., Stock, J.H., & Watson, M.W. (1990) Inference in linear time series models with some unit roots. Econometrica 58(1), 113–144.Google Scholar
Stock, J.H. (1991) Confidence intervals of the largest autoregressive root in U.S. macroeconomic time series. Journal of Monetary Economics 28, 435–460.CrossRefGoogle Scholar
Tsay, R.S. & Tiao, G.C. (1990) Asymptotic properties of multivariate nonstationary processes with applications to autoregressions. Annals of Statistics 18, 220–250.CrossRefGoogle Scholar