Hostname: page-component-76fb5796d-x4r87 Total loading time: 0 Render date: 2024-04-26T10:04:04.990Z Has data issue: false hasContentIssue false
  • English
  • Français

Weak Convergence of Sample Covariance Matrices to Stochastic Integrals Via Martingale Approximations

Published online by Cambridge University Press:  18 October 2010

P.C.B. Phillips
P.C.B. Phillips*
Affiliation:
Cowles Foundation, Yale University
Get access Rights & Permissions [Opens in a new window]

Abstract

Under general conditions the sample covariance matrix of a vector martingale and its differences converges weakly to the matrix stochastic integral ∫01BdB′, where B is vector Brownian motion. For strictly stationary and ergodic sequences, rather than martingale differences, a similar result obtains. In this case, the limit is ∫01BdB′ + Λ and involves a constant matrix Λ of bias terms whose magnitude depends on the serial correlation properties of the sequence. This note gives a simple proof of the result using martingale approximations.

Type
Brief Report
Information
Econometric Theory , Volume 4 , Issue 3 , December 1988 , pp. 528 - 533
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. Brillinger, D.R. Time series: data analysis and theory. San Francisco, California: Holden-Day, 1981.Google Scholar
2. Chan, N.H. & Wei, C.Z.. Limiting distributions of least-squares estimates of unstable autoregressive processes. Annals of Statistics 16 (1988): 367–401.10.1214/aos/1176350711Google Scholar
3. Hall, P. & Heyde, C.C.. Martingale limit theory and its application. New York: Academic Press, 1980.Google Scholar
4. Park, J.Y. & Phillips, P.C.B.. Statistical inference in regressions with integrated processes: Part 1, Econometric Theory 4 (1988): 468–497.10.1017/S0266466600013402Google Scholar
5. Park, J.Y. & Phillips, P.C.B.. Statistical inference in regressions with integrated processes: Part 2. Econometric Theory 5 (1989): forthcoming.Google Scholar
6. Phillips, P.C.B. Understanding spurious regressions in econometrics. Journal of Econometrics 33 (1986): 311–340.Google Scholar
7. Phillips, P.C.B. Time-series regression with a unit root. Econometrica 55 (1987): 277–301.10.2307/1913237Google Scholar
8. Phillips, P.C.B. Asymptotic expansions in nonstationary vector autoregressions. Econometric Theory 3 (1987): 45–68.10.1017/S0266466600004126Google Scholar
9. 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
10. Phillips, P.C.B. Towards a unified asymptotic theory of autoregression. Biometrika 74 (1987): 535–547.10.1093/biomet/74.3.535Google Scholar
11. Phillips, P.C.B. Regression theory for near integrated time series. Econometrica (1988): forthcoming.10.2307/1911357Google Scholar
12. Phillips, P.C.B. Partially identified econometric models. Cowles Foundation Discussion Paper No. 845, Yale University, July 1987.Google Scholar
13. Phillips, P.C.B. Multiple regression with integrated time series. AMS/IMS/SIAMConference on Stochostic Processes (1988): forthcoming.10.1090/conm/080/999009Google Scholar
14. 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
15. Sims, C.A., Stock, J.H. & Watson, M.W.. Inference in linear time-series models with some unit roots. Mimeographed, Minnesota University, 1986.Google Scholar
16. Solo, V. The order of differencing in ARMA models. Journal of the American Statistical Association 79 (1984): 916–921.10.1080/01621459.1984.10477111Google Scholar
17. Stock, J.H. Asymptotic properties of least-squares estimators of cointegrating vectors. Econometrica 55 (1987): 1035–1057.10.2307/1911260Google Scholar
18. Stock, J.H. & Watson, M.W.. Testing for common trends. Hoover Institution Working Paper No. E-87–2, 1987.Google Scholar