Hostname: page-component-7b7567c8c-tnzcf Total loading time: 0 Render date: 2026-11-15T02:56:27.796Z Has data issue: false hasContentIssue false
  • English
  • Français

MULTIVARIATE AUTOREGRESSION OF ORDER ONE WITH INFINITE VARIANCE INNOVATIONS

Published online by Cambridge University Press:  22 January 2008

M. Zarepour and
S.M. Roknossadati
M. Zarepour*
Affiliation:
University of Ottawa
S.M. Roknossadati
Affiliation:
University of Ottawa
*
Address correspondence to Mahmoud Zarepour, Department of Mathematics and Statistics, University of Ottawa, 585 King Edward Street, P.O. Box 450 STN A, Ottawa, Ontario K1N 6N5, Canada; e-mail: zarepour@uottawa.ca.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

We consider the limiting behavior of a vector autoregressive model of order one (VAR(1)) with independent and identically distributed (i.i.d.) innovations vector with dependent components in the domain of attraction of a multivariate stable law with possibly different indices of stability. It is shown that in some cases the ordinary least squares (OLS) estimates are inconsistent. This inconsistency basically originates from the fact that each coordinate of the partial sum processes of dependent i.i.d. vectors of innovations in the domain of attraction of stable laws needs a different normalizer to converge to a limiting process. It is also revealed that certain M-estimates, with some regularity conditions, as an appropriate alternative, not only resolve inconsistency of the OLS estimates but also give higher consistency rates in all cases.

Information

Type
Research Article
Information
Econometric Theory , Volume 24 , Issue 3 , June 2008, pp. 677 - 695
Copyright
Copyright © Cambridge University Press 2008

References

REFERENCES

Banjevic, D., Ishwaran, H., ’ Zarepour, M. (2002) A recursive method for functionals of Poisson processes. Bernoulli 8, 295–311.Google Scholar
Billingsley, P. (1968) Convergence of Probability Measures. Wiley.Google Scholar
Brockwell, P.J. ’ Davis, R.A. (1991) Time Series: Theory and Methods, 2nd ed. Springer Series in Statistics. Springer-Verlag.CrossRefGoogle Scholar
Caner, M. (1997) Weak convergence to a matrix stochastic integral with stable processes. Econometric Theory 13, 506–528.CrossRefGoogle Scholar
Chan, N.H. (1990) Inference for near-integrated time series with infinite variance. Journal of the American Statistical Association 85, 1069–1074.CrossRefGoogle Scholar
Chan, N.H. ’ Tran, L.T. (1989) On the first-order autoregressive process with infinite variance. Econometric Theory 5, 354–362.CrossRefGoogle Scholar
Cox, D.D. ’ Llatas, I. (1991) Maximum likelihood type estimation for nearly nonstationary autoregressive time series. Annals of Statistics 19, 1109–1128.CrossRefGoogle Scholar
Davis, R.A., Knight, K., ’ Liu, J. (1992) M-estimation for autoregressions with infinite variance. Stochastic Processes and Their Applications 40, 145–180.CrossRefGoogle Scholar
Feller, W. (1971) An Introduction to Probability Theory and Its Application, vol. 2, 2nd ed.Wiley.Google Scholar
Huber, P.J. (1981) Robust Statistics. Wiley Series in Probability and Mathematical Statistics. Wiley.CrossRefGoogle Scholar
Knight, K. (1989) Limit theory for autoregressive-parameter estimates in an infinite-variance random walk. Canadian Journal of Statistics 17, 261–278.CrossRefGoogle Scholar
Knight, K. (1991) Limit theory for M-estimates in an integrated infinite variance process. Econometric Theory 7, 200–212.CrossRefGoogle Scholar
Kurtz, T.G. ’ Protter, P. (1991) Weak limit theorems for stochastic integrals and stochastic differential equations. Annals of Probability 19, 1035–1070.CrossRefGoogle Scholar
LePage, R. (1980) Multidimensional infinitely divisible variables and processes, I. Stable case. Lecture Notes in Mathematics 1391, 153–163.CrossRefGoogle Scholar
Lucas, A. (1995) Unit root tests based on M-estimators. Econometric Theory 11, 331–346.CrossRefGoogle Scholar
Mittnik, S., Paulauskas, V., ’ Rachev, S.T. (2001) Statistical inference in regression with heavy-tailed integrated variables. Mathematical and Computer Modelling 34, 1145–1158.CrossRefGoogle Scholar
Paulauskas, V. ’ Rachev, S.T. (1998) Cointegrated processes with infinite variance innovations. Annals of Applied Probability 8, 775–792.CrossRefGoogle Scholar
Phillips, P.C.B. (1990) Time series regression with a unit root and infinite-variance errors. Econometric Theory 6, 44–62.CrossRefGoogle Scholar
Resnick, S. (1986) Point processes, regular variation and weak convergence. Advances in Applied Probability 18, 66–138.CrossRefGoogle Scholar
Resnick, S. ’ Greenwood, P. (1979) A bivariate stable characterization and domain of attraction. Journal of Multivariate Analysis 9, 206–221.CrossRefGoogle Scholar
Samorodnitsky, G. ’ Taqqu, M. (1994) Stable Non-Gaussian Random Processes. Chapman and Hall.Google Scholar