Hostname: page-component-848d4c4894-p2v8j Total loading time: 0 Render date: 2024-05-05T06:42:50.628Z Has data issue: false hasContentIssue false
  • English
  • Français

Some Invariance Principles and Central Limit Theorems for Dependent Heterogeneous Processes

Published online by Cambridge University Press:  18 October 2010

Jeffrey M. Wooldridge  and
Halbert White
Jeffrey M. Wooldridge
Affiliation:
Massachusetts Institute of Technology and University of California, San Diego
Halbert White
Affiliation:
Massachusetts Institute of Technology and University of California, San Diego
Get access Rights & Permissions [Opens in a new window]

Abstract

Building on work of McLeish, we present a number of invariance principles for doubly indexed arrays of stochastic processes which may exhibit considerable dependence, heterogeneity, and/or trending moments. In particular, we consider possibly time-varying functions of infinite histories of heterogeneous mixing processes and obtain general invariance results, with central limit theorems following as corollaries. These results are formulated so as to apply to economic time series, which may exhibit some or all of the features allowed in our theorems. Results are given for the case of both scalar and vector stochastic processes. Using an approach recently pioneered by Phillips, and Phillips and Durlauf, we apply our results to least squares estimation of unit root models.

Type
Research Article
Information
Econometric Theory , Volume 4 , Issue 2 , August 1988 , pp. 210 - 230
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. Non-strong mixing autoregressive processes. Journal of Applied Probability 21 (1983): 930–934.10.2307/3213710Google Scholar
2. Andrews, D.W.K. A near independent, but non-strong mixing triangular array. Journal of Applied Probability 22 (1984): 729–731.10.2307/3213877Google Scholar
3. Andrews, D.W.K. Least squares regression with integrated or dynamic regressors under weak error assumptions. Econometric Theory 3 (1987): forthcoming.10.1017/S026646660000414XGoogle Scholar
4. Bierens, H., Uniform, J. consistency of kernel estimators of a regression function under generalized conditions. Journal of the American Statistical Association 11 (1983): 699–707.10.1080/01621459.1983.10478031Google Scholar
5. Billingsley, P. Convergence of probability measures. New York: Wiley, 1968.Google Scholar
6. Billingsley, P. Probability and measure. New York: Wiley, 1979.Google Scholar
7. Durrett, R. & Resnick, S.. Functional limit theorems for dependent variables. Annals of Probability 6 (1978): 829–846.10.1214/aop/1176995431Google Scholar
8. Gallant, A.R. Nonlinear statistical models. New York: Wiley, 1987.10.1002/9780470316719Google Scholar
9. Gallant, A.R. & White, H.. A unified theory of estimation and inference for nonlinear dynamic models. Oxford: Basil Blackwell, 1988.Google Scholar
10. Granger, C.W.J. Models that generate trends. UCSD Department of Economics Discussion Paper, 1985.Google Scholar
11. Granger, C.W.J. & Joyeaux, R.. An introduction to long-memory time series models and fractional differencing. Journal of Time Series Analysis 1 (1980): 15–29.10.1111/j.1467-9892.1980.tb00297.xGoogle Scholar
12. Hall, P. & Heyde, C.C.. Martingale limit theory and its application. New York: Academic Press, 1980.Google Scholar
13. McLeish, D.L. A maximal inequality and dependent strong laws. Annals of Probability 3 (1975): 826–836.10.1214/aop/1176996269Google Scholar
14. McLeish, D.L. Invariance principles for dependent variables. Zeitschrift für Wahrschein-lichskeitstheorie und Verwundete Gebiete 32 (1975): 165–178.10.1007/BF00532611Google Scholar
15. McLeish, D.L. On the invariance principle for nonstationary mixingales. Annals of Probability 5 (1977): 616–621.10.1214/aop/1176995772Google Scholar
16. Park, J.Y. & Phillips, P.C.B.. Statistical inference in regressions with integrated processes: part 1. Cowles Foundation Discussion Paper 811, 1986.Google Scholar
17. Park, J.Y. & Phillips, P.C.B.. Statistical inference in regressions with integrated processes: part 2. Cowles Foundation Discussion Paper 819, 1987.Google Scholar
18. Phillips, P.C.B. Weak convergence to the matrix stochastic integral ∫ BdB'. Cowles Foundation Discussion Paper 796, 1986.Google Scholar
19. Phillips, P.C.B. Understanding spurious regressions in econometrics. Journal of Econometrics 33 (1986): 311–340.10.1016/0304-4076(86)90001-1Google Scholar
20. Phillips, P.C.B. Time series regression with a unit root. Econometrica 55 (1987): 277–302.10.2307/1913237Google Scholar
21. Phillips, P.C.B. Asymptotic expansions in nonstationary vector autoregressions. Econometric Theory 3 (1987): forthcoming.10.1017/S0266466600004126Google Scholar
22. Phillips, P.C.B. Weak convergence of sample covariance matrices to stochastic integrals via martingale approximations. Cowles Foundation Discussion Paper 846, 1987.Google Scholar
23. 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
24. Serfling, R.J. Contributions to central limit theory for dependent variables. Annals of Mathematical Statistics 39 (1968): 1158–1175.10.1214/aoms/1177698240Google Scholar
25. White, H. Asymptotic theory for econometricians. New York: Academic Press, 1984.Google Scholar
26. White, H. & Domowitz, I.. Nonlinear regression with dependent observations. Econometrica 52 (1984): 143–162.10.2307/1911465Google Scholar
27. Wooldridge, J.M. Asymptotic properties of econometric estimators. Ph.D. Dissertation, University of California, San Diego, 1986.Google Scholar
28. Wooldridge, J.M. & White, H.. Some invariance and central limit theorems for dependent heterogeneous processes. UCSD Department of Economics Discussion Paper 86–18, 1986.Google Scholar