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The Joint Moment Generating Function of Quadratic Forms in Multivariate Autoregressive Series

Published online by Cambridge University Press:  11 February 2009

Karim M. Abadir  and
Rolf Larsson
Karim M. Abadir
Affiliation:
University of York
Rolf Larsson
Affiliation:
Stockholm University
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Abstract

Let (X1) be a discrete multivariate Gaussian autoregressive process of order 1. The paper derives the exact finite-sample joint moment generating function (m.g.f.) of the three quadratic forms constituting the sufficient statistic of the process. The formula is then specialized to some cases of interest, including the m.g.f. of functional of multivariate Ornstein-Uhlenbeck processes that arise asymptotically from more general (X1) processes as well.

Type
Research Article
Information
Econometric Theory , Volume 12 , Issue 4 , October 1996 , pp. 682 - 704
Copyright
Copyright © Cambridge University Press 1996

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References

REFERENCES

Abadir, K.M. (1993a) The limiting distribution of the autocorrelation coefficient under a unit root. Annals of Statistics 21, 1058–1070.CrossRefGoogle Scholar
Abadir, K.M. (1993b) OLS bias in a nonstationary autorcgression. Econometric Theory 9, 81–93.CrossRefGoogle Scholar
Abadir, K.M. (1993c) On the asymptotic power of unit root tests. Econometric Theory 9, 189–221.CrossRefGoogle Scholar
Abadir, K.M. (1995a) The joint density of two functionals of a Brownian motion. Mathematical Methods of Statistics 4, 449–462. (Correction, 1996, 5, 124.)Google Scholar
Abadir, K.M. (1995b). The limiting distribution of the t ratio under a unit root. Econometric Theory 11, 775–793.CrossRefGoogle Scholar
Abadir, K.M. & Rockinger, M. (1995) The Devil's Horns: A Problem with the Densities of AR Statistics. HEC CR 541, Paris.Google Scholar
Aiuppa, T.A. & Bargmann, R.E. (1977) Distribution of a linear combination of quadratic forms, a four-moment approximation. Proceedings of the Business and Economics Statistics Section, American Statistical Association 706–710.Google Scholar
Anderson, T.W. (1951). Estimating linear restrictions on regression coefficients for multivariate normal distributions. Annals of Mathematical Statistics 22, 327–351. (Correction, 1980, Annals of Statistics 8, 1400.)Google Scholar
Anderson, T.W. & Kunitomo, N. (1992) Asymptotic distributions of regression and autoregres-sion coefficients with martingale difference disturbances. Journal of Multivariate Analysis 40, 221–243.CrossRefGoogle Scholar
Box, G.E.P. & Tiao, G.C. (1977) A canonical analysis of multiple time series. Biometrika 64, 355–365.CrossRefGoogle Scholar
Chan, N. & Wei, C.Z. (1987) Asymptotic inference for nearly nonstationary AR(1) processes. Annals of Statistics 15, 1050–1063.CrossRefGoogle Scholar
Chan, N. & Wei, C.Z. (1988) Limiting distributions of least squares estimates of unstable auto-regressive processes. Annals of Statistics 16, 367–401.CrossRefGoogle Scholar
Cramer, H. (1946) Mathematical Methods of Statistics. Princeton, New Jersey: Princeton University Press.Google Scholar
Gurland, J. (1948) Inversion formulae for the distribution of ratios. Annals of Mathematical Statistics 19, 228–237.CrossRefGoogle Scholar
Jensen, S.T. & Nielsen, B. (1995) The Joint Laplace Transform of a Non-symmetric Quadratic Functions and Function of a Brownian Motion. Preprint 7, Institute of Mathematical Statistics, University of Copenhagen.Google Scholar
Johansen, S. (1988) Statistical analysis of cointegration vectors. Journal of Economic Dynamics and Control 12, 231–254.CrossRefGoogle Scholar
Larsson, R. (1994) Bartlett Corrections for Unit Root Test Statistics. Preprint 2, Institute of Mathematical Statistics, University of Copenhagen.Google Scholar
Larsson, R. (1995) The asymptotic distributions of some test statistics in near-integrated AR processes. Econometric Theory 11, 306–330.CrossRefGoogle Scholar
Magnus, J.R. (1988) Linear Structures. London: Charles Griffin & Co.Google Scholar
Muirhead, R.J. (1982) Aspects of Multivariate Statistical Theory. New York: John Wiley & Sons.CrossRefGoogle Scholar
Nabeya, S. & Tanaka, K. (1990) A general approach to the limiting distribution for estimators in time series regression with nonstable autoregressive errors. Econometrica 58, 145–163.Google Scholar
Perron, P. (1991) A continuous time approximation to the unstable first-order autoregressive process: The case without an intercept. Econometrica 59, 211–236.CrossRefGoogle Scholar
Phillips, P.C.B. (1985) The distribution of matrix quotients. Journal of Multivariate Analysis 16, 157–161.CrossRefGoogle Scholar
Phillips, P.C.B. (1987a) Asymptotic expansions in nonstationary vector autoregressions. Econometric Theory 3, 45–68.CrossRefGoogle Scholar
Phillips, P.C.B. (1987b) Towards a unified asymptotic theory for autoregression. Biometrika 74, 535–547.CrossRefGoogle Scholar
Phillips, P.C.B. (1988) Regression theory for near-integrated time series. Econometrica 56, 1021–1043.CrossRefGoogle Scholar
Phillips, P.C.B. & Durlauf, S.N. (1986) Multiple time series regression with integrated processes. Review of Economic Studies 53, 473–495.CrossRefGoogle Scholar
Solo, V. (1984) The order of differencing in ARIMA models. Journal of the American Statistical Association 79, 916–921.CrossRefGoogle Scholar
Tanaka, K. (1993) An alternative approach to the asymptotic theory of spurious regression, cointegration and near cointegration. Econometric Theory 9, 36–61.CrossRefGoogle Scholar
Tsay, R.S. & Tiao, G.C. (1990) Asymptotic properties of multivariate nonstationary processes with applications to autoregressions. Annals of Statistics 18, 220–250.CrossRefGoogle Scholar
White, J.S. (1958) The limiting distribution of the serial correlation coefficient in the explosive case. Annals of Mathematical Statistics 29, 1188–1197.CrossRefGoogle Scholar
White, J.S. (1959) The limiting distribution of the serial correlation coefficient in the explosive case II. Annals of Mathematical Statistics 30, 831–834.CrossRefGoogle Scholar