Hostname: page-component-586b7cd67f-rdxmf Total loading time: 0 Render date: 2024-11-20T10:23:22.228Z Has data issue: false hasContentIssue false
  • English
  • Français

Statistical Inference in Regressions with Integrated Processes: Part 1

Published online by Cambridge University Press:  18 October 2010

Joon Y. Park  and
Peter C.B. Phillips
Joon Y. Park
Affiliation:
Cowles Foundation, Yale University
Peter C.B. Phillips
Affiliation:
Cowles Foundation, Yale University
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper develops a multivariate regression theory for integrated processes which simplifies and extends much earlier work. Our framework allows for both stochastic and certain deterministic regressors, vector autoregressions, and regressors with drift. The main focus of the paper is statistical inference. The presence of nuisance parameters in the asymptotic distributions of regression F tests is explored and new transformations are introduced to deal with these dependencies. Some specializations of our theory are considered in detail. In models with strictly exogenous regressors, we demonstrate the validity of conventional asymptotic theory for appropriately constructed Wald tests. These tests provide a simple and convenient basis for specification robust inferences in this context. Single equation regression tests are also studied in detail. Here it is shown that the asymptotic distribution of the Wald test is a mixture of the chi square of conventional regression theory and the standard unit-root theory. The new result accommodates both extremes and intermediate cases.

Type
Research Article
Information
Econometric Theory , Volume 4 , Issue 3 , December 1988 , pp. 468 - 497
Copyright
Copyright © Cambridge University Press 1988 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

1. Andrews, D.W.K. Least-squares regression with integrated or dynamic regressors under weak error assumptions. Econometric Theory 3 (1987): 98–116.10.1017/S026646660000414XGoogle Scholar
2. Billingsley, P. Convergence of probability measures. New York: John Wiley, 1968.Google Scholar
3. Box, G.E.P. & Jenkins, G.M.. Time series analysis: forecasting and control. San Francisco, California: Holden Day, 1976.Google Scholar
4. Chan, N.H. & Wei, C.Z.. Limiting distributions of least-square estimates of unstable autoregressive processes. Annals of Statistics 16 (1988): 367–401.10.1214/aos/1176350711Google Scholar
5. Dickey, D.A. & Fuller, W.A.. Distribution of the estimators for autoregressive time series with a unit root. Journal of American Statistical Association 74 (1979): 427–431.Google Scholar
6. Dickey, D.A. & Fuller, W.A.. Likelihood ratio statistics for autoregressive time series with a unit root. Econometrica 49 (1981): 1057–1072.10.2307/1912517Google Scholar
7. Durlauf, S.N. & Phillips, P.C.B.. Trends versus random walks in time series analysis. Cowles Foundation Discussion Paper No. 788, Yale University, 1986.Google Scholar
8. Eberlain, E. On strong invariance principles under dependence assumptions. Annals of Probability 14 (1986): 260–270.Google Scholar
9. Engle, R.F. & Granger, C.W.J.. Cointegration and error correction: representation, estimation, and testing. Econometrica 55 (1987): 251–276.10.2307/1913236Google Scholar
10. Fuller, W.A. Introduction to statistical time series. New York: John Wiley, 1976.Google Scholar
11. Hall, P. & Heyde, C.C.. Martingale limit theory and its applications. New York: Academic Press, 1980.Google Scholar
12. Krämer, W. Least-squares regression when the independent variable follows an ARIMA process. Journal of the American Statistical Association 81 (1986): 150–154.Google Scholar
13. Muirhead, R.J. Aspects of multivariate statistical theory. New York: John Wiley, 1982.Google Scholar
14. Ouliaris, S. Testing for unit roots and cointegration in multiple time series. Yale Doctoral Dissertation, 1987.Google Scholar
15. Phillips, P.C.B. Understanding spurious regressions in econometrics. Journal of Econometrics 33 (1986): 311–340.10.1016/0304-4076(86)90001-1Google Scholar
16. Phillips, P.C.B. Time-series regression with a unit root. Econometrica 55 (1987): 277–301.10.2307/1913237Google Scholar
17. Phillips, P.C.B. Weak convergence to the matrix stochastic integral ∫ 0 1 BdB′ . Journal of Multivariate Analysis 24 (1988): 252–264.10.1016/0047-259X(88)90039-5Google Scholar
18. Phillips, P.C.B. Weak convergence of sample covariance matrix to stochastic integrals via martingale approximations. Econometric Theory 4 (1988): 528–533.10.1017/S026646660001344XGoogle Scholar
19. Phillips, P.C.B. & Durlauf, S.N.. Multiple time-series regression with integrated processes. Review of Economic Studies 53 (1986): 473–496.10.2307/2297602Google Scholar
20. Phillips, P.C.B. & Ouliaris, S.. Testing for cointegration using principal components methods. Journal of Dynamics and Control (1988): 1–26.Google Scholar
21. Phillips, P.C.B. & Park, J.Y.. Asymptotic equivalence of OLS and GLS in regressions with integrated regressors. Journal of American Statistical Association 83 (1988): 111–115.Google Scholar
22. Phillips, P.C.B. & Perron, Pierre. Testing for a unit root in time-series regression. Biometrika 75 (1988): 335–346.10.1093/biomet/75.2.335Google Scholar
23. Stock, J. Asymptotic properties of least-squares estimators of cointegrating vectors. Econometrica 56 (1988): 1035–1056.Google Scholar
24. Stock, J. & Watson, M.. Testing for common trends. Mimeo, Harvard University, 1986.Google Scholar
25. Strasser, H. Martingale difference arrays and stochastic integrals. Probability Theory and Related Fields 72 (1986): 83–98.10.1007/BF00343897Google Scholar
26. Tsay, R.S. & Tiao, G.C.. Asymptotic properties of multivariate nonstationary processes with applications to autoregressions. University of Chicago, Technical Report #54, 1986.Google Scholar
27. West, K.D. Asymptotic normality, when regressors have a unit root. Princeton Discussion Paper in Economics #110, Princeton University, 1986.Google Scholar
28. White, H. Asymptotic theory for econometricians. New York: Academic Press, 1984.Google Scholar
29. White, H. & Domowitz, I.. Nonlinear regression with dependent observations. Econometrica 52 (1984): 143–162.10.2307/1911465Google Scholar