An Application of the Functional Central Limit Theorem in the Unit Root Test in Autoregressive Models

Document Type : Scientific Research

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Abstract

To study the limiting distribution of the test statistics used in the unit root problems, we usually need to use the known theorem Donsker (Functional central limit theorem). In this paper, we study the limiting behavior of the unit root test statistics in the AR (1) model without and with a constant term by an indirect use of the Donsker theorem where the error terms are  with noise. We also consider the case when the error terms are nonwhite noise stationary and then generalize our results to the AR (p) models. Several examples are provided to clarify the issue.
To study the limiting distribution of the test statistics used in the unit root problems, we usually need to use the known theorem Donsker (Functional central limit theorem). In this paper, we study the limiting behavior of the unit root test statistics in the AR (1) model without and with a constant term by an indirect use of the Donsker theorem where the error terms are  with noise. We also consider the case when the error terms are nonwhite noise stationary and then generalize our results to the AR (p) models. Several examples are provided to clarify the issue.

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References

Billingsley, P. (1999). Convergence of Probability Measures, 2nd Edition. Wiley, New York.
Brockwell, P.J and Davis, R.A. (2002). Introductionto Time Series and Forecasting, Second Edition, 2nd Edition, New York.
Dickey, D.A. and Fuller, W.A. (1979). Distribution of the estimators for autoregressive time series with a unit root. J. Amer. Statist. Assoc. 74, 427–431.
Donsker, M.D. (1951). An invariance principle for certain probabilitylimit theorems. Mem. Amer. Math. Soc. 6, 1–12.
Gould, J.P. and C.R. Nelson. (1974). The Stochastic Structure of the Velocity of Money, American Economic Review, 64, 405-418.
Wei, William W.S. (2006) Time series analysis: univariate and multivariate methods, 2nd Edition, Pearson, Addison Wesley.

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