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Departure from normality of increasing-dimension martingales

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  • Arbus, Ignacio

Abstract

In this paper, we consider sequences of vector martingale differences of increasing dimension. We show that the Kantorovich distance from the distribution of the k(n)-dimensional average of n martingale differences to the corresponding Gaussian distribution satisfies certain inequalities. As a consequence, if the growth of k(n) is not too fast, then the Kantorovich distance converges to zero. Two applications of this result are presented. The first is a precise proof of the asymptotic distribution of the multivariate portmanteau statistic applied to the residuals of an autoregressive model and the second is a proof of the asymptotic normality of the estimates of a finite autoregressive model when the process is an AR([infinity]) and the order of the model grows with the length of the series.

Suggested Citation

  • Arbus, Ignacio, 2009. "Departure from normality of increasing-dimension martingales," Journal of Multivariate Analysis, Elsevier, vol. 100(6), pages 1304-1315, July.
    • Handle: RePEc:eee:jmvana:v:100:y:2009:i:6:p:1304-1315
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    References listed on IDEAS

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    1. Bruggemann, Ralf & Lutkepohl, Helmut & Saikkonen, Pentti, 2006. "Residual autocorrelation testing for vector error correction models," Journal of Econometrics, Elsevier, vol. 134(2), pages 579-604, October.
    2. Lütkepohl, Helmut, 2008. "Problems related to over-identifying restrictions for structural vector error correction models," Economics Letters, Elsevier, vol. 99(3), pages 512-515, June.
    3. Ignacio Arbués, 2008. "An Extended Portmanteau Test for VARMA Models With Mixing Nonlinear Constraints," Journal of Time Series Analysis, Wiley Blackwell, vol. 29(5), pages 741-761, September.
    4. Lewis, Richard & Reinsel, Gregory C., 1985. "Prediction of multivariate time series by autoregressive model fitting," Journal of Multivariate Analysis, Elsevier, vol. 16(3), pages 393-411, June.
    5. Helmut Lütkepohl, 2005. "Vector Error Correction Models," Springer Books, in: New Introduction to Multiple Time Series Analysis, chapter 6, pages 237-267, Springer.
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