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Continuous Weak Convergence and Stochastic Equicontinuity Results for Integrated Processes with an Application to the Estimation of a Regression Model

Published online by Cambridge University Press:  11 February 2009

Pentti Saikkonen
Pentti Saikkonen
Affiliation:
University of Helsinki
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Abstract

The concepts of continuous and uniform weak convergence and versions of stochastic equicontinuity are discussed in the context of integrated processes of order one. The considered processes depend on a parameter vector in a specific fashion which is relevant for integrated and cointegrated systems with non-linearities in parameters. The results of the paper can be applied to obtain asymptotic distributions of estimators and test statistics in such systems. In a correctly specified cointegrated Gaussian system, this can be done in a very convenient way. Combining the results of this paper with available general maximum likelihood estimation theories readily shows that the maximum likelihood estimator is asymptotically optimal with a mixed normal limiting distribution. The usefulness of this approach is demonstrated by analyzing a regression model with autoregressive moving average errors and strictly exogenous regressors which may be either integrated of order one, asymptotically stationary, or nonstochastic and bounded.

Type
Articles
Information
Econometric Theory , Volume 9 , Issue 2 , April 1993 , pp. 155 - 188
Copyright
Copyright © Cambridge University Press 1993

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References

1.Andrews, D.W.K.Generic uniform convergence. Econometric Theory 8 (1992): 241–257.CrossRefGoogle Scholar
2.Basawa, I.V. & Scott, D.J.. Asymptotic Optimal Inference for Nonergodic Models. New York: Springer-Verlag, 1983.CrossRefGoogle Scholar
3.Billingsley, P.Convergence of Probability Measures. New York: Wiley, 1968.Google Scholar
4.Chan, N.H. & Wei, C.Z.. Limiting distributions of least squares estimates of unstable auto-regressive processes. The Annals of Statistics 16 (1988): 367–401.CrossRefGoogle Scholar
5.Dent, W.Computation of the exact likelihood function of an ARIMA process. Journal of Statistical Computation and Simulation 5 (1977): 193–206.CrossRefGoogle Scholar
6.Dudley, R.M.Real Analysis and Probability. Pacific Grove, California: Wadsworth & Brooks/Cole, 1989.Google Scholar
7.Engle, R.F. & Granger, C.W.J.. Cointegration and error correction: Representation, estimation and testing. Econometrica 55 (1987): 251–276.CrossRefGoogle Scholar
8.Hannan, E.J. & Rissanen, J.. Recursive estimation of mixed auto-regressive-moving average order. Biometrika 69 (1982): 81–94.CrossRefGoogle Scholar
9.Hansen, B.E.Convergence to stochastic integrals of general form. Manuscript. University of Rochester, 1990.Google Scholar
10.Johansen, S.Estimation and hypothesis testing of cointegration vectors in Gaussian vector autoregressive models. Econometrica 59 (1991): 1551–1581.CrossRefGoogle Scholar
11.Kohn, R.Asymptotic estimation and hypothesis testing results for vector linear time series models. Econometrica 47 (1979): 1005–1030.CrossRefGoogle Scholar
12.Newey, W.K.Uniform convergence in probability and stochastic equicontinuity. Econometrica 59 (1991): 1161–1167.CrossRefGoogle Scholar
13.Park, J.Y. & Phillips, P.C.B.. Statistical inference in regressions with integrated processes: Part II. Econometric Theory 5 (1989): 95–131.CrossRefGoogle Scholar
14.Phillips, P.C.B. Weak convergence to the matrix stochastic integral . Journal of Multivariate Analysis 24 (1988): 252–264.CrossRefGoogle Scholar
15.Phillips, P.C.B.Weak convergence of sample covariance matrices to stochastic integrals via martingale approximations. Econometric Theory 4 (1988): 528–533.CrossRefGoogle Scholar
16.Phillips, P.C.B.Optimal inference in cointegrated systems. Econometrica 59 (1991): 283–306.CrossRefGoogle Scholar
17.Phillips, P.C.B. & Durlauf, S.N.Multiple time series regression with integrated processes. Review of Economic Studies 53 (1986): 473–496.CrossRefGoogle Scholar
18.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
19.Phillips, P.C.B. & Solo, V.. Asymptotics for linear processes. The Annals of Statistics 20 (1992): 971–1001.CrossRefGoogle Scholar
20.Pierce, D.A.Least squares estimation in the regression model with autoregressive–moving average errors. Biometrika 58 (1971): 299–312.CrossRefGoogle Scholar
21.Pollard, D.Convergence of Stochastic Processes. New York: Springer–Verlag, 1984.CrossRefGoogle Scholar
22.Saikkonen, P.Asymptotic properties of some preliminary estimators for autoregressive moving average time series models. Journal of Time Series Analysis 7 (1986): 133–155.CrossRefGoogle Scholar
23.Solo, V.The order of differencing in ARIMA models. Journal of the American Statistical Association 79 (1984): 916–921.CrossRefGoogle Scholar
24.Stock, J.H.Asymptotic properties of least squares estimators of cointegrating vectors. Econometrica 56 (1987): 1035–1056.CrossRefGoogle Scholar
25.Sweeting, T.J.Uniform asymptotic normality of the maximum likelihood estimator. The Annals of Statistics 8 (1980): 1375–1381. Correction. The Annals of Statistics 10(1982): 320.CrossRefGoogle Scholar
26.Sweeting, T.J.On estimator efficiency in stochastic processes. Stochastic Processes and their Applications 15 (1983): 93–98.CrossRefGoogle Scholar