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Problems with the Asymptotic Theory of Maximum Likelihood Estimation in Integrated and Cointegrated Systems

Published online by Cambridge University Press:  11 February 2009

Pentti Saikkonen
Pentti Saikkonen
Affiliation:
University of Helsinki
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Abstract

Problems with the asymptotic theory of nonlinear maximum likelihood estimation in integrated and cointegrated systems are discussed in this paper. One problem is that standard proofs of consistency generally do not apply; another one is that, even if the consistency has been established, it can be difficult to deduce the limiting distribution of a maximum likelihood estimator from a conventional Taylor series expansion of the score vector. It is argued in this paper that the latter difficulty can generally be resolved if, in addition to consistency, an appropriate result of the order of consistency of the long-run parameter estimator of the model is available and the standardized sample information matrix satisfies a suitable extension of previous stochastic equicontinuity conditions. To make this idea applicable in particular cases, extensions of the author's recent stochastic equicontinuity results, relevant for many integrated and cointegrated systems with nonlinearities in parameters, are provided. As an illustration, a simple regression model with integrated and stationary regressors and nonlinearities in parameters is considered. In this model, the consistency and order of consistency of the long-run parameter estimator are obtained by employing extensions of well-known sufficient conditions for consistency. These conditions are applicable quite generally, and their verification in the special case of this paper suggests how to proceed in more complex models.

Type
Articles
Information
Econometric Theory , Volume 11 , Issue 5 , October 1995 , pp. 888 - 911
Copyright
Copyright © Cambridge University Press 1995

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References

REFERENCES

Andrews, D.W.K. (1992) Generic uniform convergence. Econometric Theory 8, 241–257.CrossRefGoogle Scholar
Basawa, I.V. & Scott, D.J. (1983) Asymptotic Optimal Inference for Nonergodic Models. New York: Springer-Verlag.CrossRefGoogle Scholar
Billingsley, P. (1968) Convergence of Probability Measures. New York: Wiley.Google Scholar
Boswijk, H.P. (1992) Cointegration, Identification and Exogeneity. Inference in Structural Error Correction Models. Amsterdam: Tinbergen Institute Research Series.Google Scholar
Heijmans, R.D.H. & Magnus, J.R. (1986) Consistent maximum-likelihood estimation with dependent observations. Journal of Econometrics 32, 253–285.CrossRefGoogle Scholar
Ibragimov, I.A. & Hasminskii, R.Z. (1981) Statistical Estimation, Asymptotic Theory. New York: Springer-Verlag.CrossRefGoogle Scholar
Jeganathan, P. (1988) Some Aspects of Asymptotic Theory with Applications to Time Series Models. Technical report, University of Michigan.Google Scholar
Johansen, S. (1991) Estimation and hypothesis testing of cointegration vectors in Gaussian vector autoregressive models. Econometrica 59, 1551–1581.CrossRefGoogle Scholar
Nagaraj, N.K. & Fuller, W.A. (1991) Estimation of the parameters of linear time series models subject to nonlinear restrictions. Annals of Statistics 19, 1143–1154.CrossRefGoogle Scholar
Newey, W.K. (1991) Uniform convergence in probability and stochastic equicontinuity. Econometrica 59, 1161–1167.CrossRefGoogle Scholar
Phillips, P.C.B. (1991) Optimal inference in cointegrated systems. Econometrica 59, 283–306.Google Scholar
Phillips, P.C.B. & Durlauf, S.N. (1986) Multiple time series regression with integrated processes. Review of Economic Studies 53, 473–496.Google Scholar
Pollard, D. (1985) New ways to prove central limit theorems. Econometric Theory 1, 295–313.CrossRefGoogle Scholar
Pötscher, B.M. & Prucha, I.R. (1991a) Basic structure of the asymptotic theory in dynamic nonlinear econometric models, part I: Consistency and approximation concepts. Econometric Reviews 10, 125–216.CrossRefGoogle Scholar
Pötscher, B.M. & Prucha, I.R. (1991b) Basic structure of the asymptotic theory in dynamic nonlinear econometric models, part II: Asymptotic normality. Econometric Reviews 10, 253–352.Google Scholar
Pötscher, B.M. & Prucha, I.R. (1994) Generic uniform convergence and equicontinuity concepts for random functions. An exploration of the basic structure. Journal of Econometrics 60, 23–63.CrossRefGoogle Scholar
Reinsel, G.C. & Ahn, S.K. (1992) Vector autoregressive models with unit roots and reduced rank structure: Estimation, likelihood ratio test, and forecasting. Journal of Time Series Analysis 13, 353–375.Google Scholar
Saikkonen, P. (1993) Continuous weak convergence and stochastic equicontinuity results for integrated processes with an application to the estimation of a regression model. Econometric Theory 9, 155–188.CrossRefGoogle Scholar
Stock, J.H. (1987) Asymptotic properties of least squares estimators of cointegrating vectors. Econometrica 56, 1035–1056.CrossRefGoogle Scholar
Sweeting, T.J. (1980) Uniform asymptotic normality of the maximum likelihood estimator. Annals of Statistics 8, 1375–1381. (Correction (1982) Annals of Statistics 10, 320.)Google Scholar
Wu, C.F. (1981) Asymptotic theory of nonlinear least squares estimation. Annals of Statistics 9, 501–513.CrossRefGoogle Scholar