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Estimation and Testing of Cointegrated Systems by an Autoregressive Approximation

Published online by Cambridge University Press:  18 October 2010

Pentti Saikkonen
Pentti Saikkonen
Affiliation:
University of Helsinki
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Abstract

This paper studies the estimation and testing of general cointegrated systems by using an autoregressive approximation. Simple estimators for both the cointegration vectors and their weight matrix in the autoregressive error correction model representation of the system are developed. Since these estimators assume that the number of cointegration vectors and their normalization are fixed in advance, convenient specification tests for checking the validity of these assumptions are also provided. The asymptotic distributions of the estimators and test statistics are derived by assuming that the order of the auto-regressive approximation increases with the sample size at a suitable rate. This generalizes some previous results derived for finite-order autoregressions as no assumption of a finite-parameter data-generating process is imposed. The estimators and tests of the paper are interpreted in terms of autoregressive spectral density estimators at the zero frequency and, in the special case of a finite-order Gaussian autoregression, their relation to maximum likelihood procedures is discussed. All estimators of the paper can be applied with simple least-squares techniques and used to construct conventional Wald tests with asymptotic chi-square distributions under the null hypothesis. The limit theory of the specification tests is nonstandard, similar to that in univariate unit root tests.

Type
Articles
Information
Econometric Theory , Volume 8 , Issue 1 , March 1992 , pp. 1 - 27
Copyright
Copyright © Cambridge University Press 1992

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References

REFERENCES

1.Ann, S.K. & Reinsel, G.C.. Estimation for partially nonstationary multivariate autoregressive models. Journal of the American Statistical Association 85 (1990): 813–823.Google Scholar
2.Banerjee, A., Dolado, J.J., Hendry, D.F. & Smith, G.W.. Exploring equilibrium relationships i n econometrics through static models: Some Monte Carlo evidence. Oxford Bulletin ofEco nomics and Statistics 48 (1986): 253–277.CrossRefGoogle Scholar
3.Berk, K.N.Consistent autoregressive spectral estimates. The Annals of Statistics 2 (1974): 489–502.Google Scholar
4.Engle, R.F. & Granger, C.W.J.. Cointegration and error correction: Representation, Estimation and Testing. Econometrica 55 (1987): 251–276.Google Scholar
5.Engle, R.F., Hendry, D. F. & Richard, J.-F.. Exogeneity. Econometrica 51 (1983): 277–304.CrossRefGoogle Scholar
6.Engle, R.F. & Yoo, B.S.. Cointegrated economic time series: A survey with new results. University of California, San Diego, Discussion Paper 87–26R, 1989.Google Scholar
7.Hannan, E.J.Multiple Time Series. New York: Wiley, 1970.CrossRefGoogle Scholar
8.Hannan, E. J. & Deistler, M.. The Statistical Theory of Linear Systems. New York: Wiley, 1988.Google Scholar
9.Hansen, B.E. & Phillips, P.C.B.. Estimation and inference in models of cointegration: A simulation study. Advances in Econometrics (1989): 8 (1990): 225–248.Google Scholar
10.Johansen, S.Statistical analysis of cointegration vectors. Journal of Economic Dynamics and Control 12 (1988): 231–254.CrossRefGoogle Scholar
11.Johansen, S. Estimation and hypothesis testing of cointegration vectors in Gaussian vector autoregressive models. Preprint, University of Copenhagen, 1989.Google Scholar
12.Johansen, S. & Juselius, K.. Maximum likelihood estimation and inference on cointegration with applications to the demand for money. Oxford Bulletin of Economics and Statistics 52 (1990): 169–210.CrossRefGoogle Scholar
13.Lewis, R. & Reinsel, G.C.. Prediction of multivariate time series by autoregressive model fitting. Journal of Multivariate Analysis 16 (1985): 393–411.CrossRefGoogle Scholar
14.Oberhofer, W. & Kmenta, J.. A general procedure for obtaining maximum likelihood estimates in generalized regression models. Econometrica 42 (1974): 579–590.CrossRefGoogle Scholar
15.Park, J.Y. & Phillips, P.C.B.. Statistical inference in regressions with integrated processes: Part I. Econometric Theory 4 (1988): 468–497.CrossRefGoogle Scholar
16.Park, J.Y. & Phillips, P.C.B.. Statistical inference in regressions with integrated processes: Part II. Econometric Theory 5 (1989): 95–131.Google Scholar
17.Paulsen, J.Order determination of multivariate autoregressive time series with unit roots. Journal of Time Series Analysis 5 (1984): 115–127.CrossRefGoogle Scholar
18.Phillips, P.C.B.Weak convergence of sample covariance matrices to stochastic integrals via martingale approximations. Econometric Theory 4 (1988): 528–533.CrossRefGoogle Scholar
19.Phillips, P.C.B.Optimal inference in cointegrated systems. Econometrica 59 (1991): 283–306.CrossRefGoogle Scholar
20.Phillips, P.C.B. Spectral regression for cointegrated time series. Forthcoming in W. Barnett (ed.), Nonparametric and Semiparametric Methods in Economics and Statistics, Cambridge: Cambridge University Press, 1991,Google Scholar
21.Phillips, P.C.B. & Durlauf, S.N.. Multiple time series regression with integrated processes. Review of Economic Studies 53 (1986): 473–496.CrossRefGoogle Scholar
22.Phillips, P.C.B. & Hansen, B.E.. Statistical inference in instrumental variables regression with 1(1) processes. Review of Economic Studies 57 (1990): 99–125.CrossRefGoogle Scholar
23.Phillips, P.C.B. & Loretan, M.. Estimating long run economic equilibria. Review of Economic Studies 58 (1991): 407–436.CrossRefGoogle Scholar
24.Pötscher, B.M.Model selection under nonstationarity: autoregressive models and stochastic regression models.The Annals of Statistics 17 (1989): 1257–1274.CrossRefGoogle Scholar
25.Said, S.E. & Dickey, D.A.. Testing for unit roots in autoregressive-moving average models of unknow n order. Biometrika 71 (1984): 599–607.CrossRefGoogle Scholar
26.Saikkonen, P.Asymptotically efficient estimation of cointegration regressions. Econometric Theory 1 (1991): 1–21.Google Scholar
27.Sims, C.A., Stock, J.H. & Watson, M.W.. Inference in linear time series models with some unit roots. Econometrica 58 (1990): 113–144.CrossRefGoogle Scholar
28.Stock, J.H.Asymptotic properties of least squares estimators of cointegrating vectors. Econometrica 5 (1987): 1035–1056.CrossRefGoogle Scholar
29.Stock, J.H. & Watson, M.. Testing for common trends. Journal of the American Statistical Association 83 (1988): 1097–1107.Google Scholar
30.Stock, J.H. & Watson, M.. A simple MLE of cointegrating vectors in higher order integrated systems. Technical working paper No. 83, National Bureau of Economic Research, Inc., 1989.CrossRefGoogle Scholar
31.Tsay, R.S.Order selection in nonstationary autoregressive models. The Annals of Statistics 12 (1984): 1425–1433.Google Scholar