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Expectation of Quadratic Forms in Normal and Nonnormal Variables with Econometric Applications

Author

Listed:
  • Yong Bao

    (Department of Economics, Purdue University)

  • Aman Ullah

    () (Department of Economics, University of California Riverside)

Abstract

We derive some new results on the expectation of quadratic forms in normal and nonnormal variables. Using a nonstochastic operator, we show that the expectation of the product of an arbitrary number of quadratic forms in noncentral normal variables follows a recurrence formula. This formula includes the existing result for central normal variables as a special case. For nonnormal variables, while the existing results are available only for quadratic forms of limited order (up to 3), we derive analytical results to a higher order 4. We use the nonnormal results to study the effects of nonnormality on the finite sample mean squared error of the OLS estimator in an AR(1) model and the QMLE in an MA(1) model.

Suggested Citation

  • Yong Bao & Aman Ullah, 2009. "Expectation of Quadratic Forms in Normal and Nonnormal Variables with Econometric Applications," Working Papers 200907, University of California at Riverside, Department of Economics, revised Jun 2009.
    • Handle: RePEc:ucr:wpaper:200907
    as

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    References listed on IDEAS

    as
    1. Ullah, A. & Srivastava, V. K. & Chandra, R., 1983. "Properties of shrinkage estimators in linear regression when disturbances are not normal," Journal of Econometrics, Elsevier, vol. 21(3), pages 389-402, April.
    2. Ivana Komunjer, 2007. "Asymmetric power distribution: Theory and applications to risk measurement," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 22(5), pages 891-921.
    3. Hoque, Asraul & Magnus, Jan R. & Pesaran, Bahram, 1988. "The exact multi-period mean-square forecast error for the first-order autoregressive model," Journal of Econometrics, Elsevier, vol. 39(3), pages 327-346, November.
    4. Hashem Pesaran, M. & Yamagata, Takashi, 2008. "Testing slope homogeneity in large panels," Journal of Econometrics, Elsevier, vol. 142(1), pages 50-93, January.
    5. Bao, Yong, 2007. "The Approximate Moments Of The Least Squares Estimator For The Stationary Autoregressive Model Under A General Error Distribution," Econometric Theory, Cambridge University Press, vol. 23(5), pages 1013-1021, October.
    6. Srivastava, V. K. & Maekawa, Koichi, 1995. "Efficiency properties of feasible generalized least squares estimators in SURE models under non-normal disturbances," Journal of Econometrics, Elsevier, vol. 66(1-2), pages 99-121.
    7. Magee, Lonnie, 1985. "Efficiency of iterative estimators in the regression model with AR(1) disturbances," Journal of Econometrics, Elsevier, vol. 29(3), pages 275-287, September.
    8. Ullah, Aman, 2004. "Finite Sample Econometrics," OUP Catalogue, Oxford University Press, number 9780198774488.
    9. Dufour, Jean-Marie, 1984. "Unbiasedness of Predictions from Estimated Autoregressions When the True Order Is Unknown," Econometrica, Econometric Society, vol. 52(1), pages 209-215, January.
    10. Kiviet, Jan F. & Phillips, Garry D.A., 1993. "Alternative Bias Approximations in Regressions with a Lagged-Dependent Variable," Econometric Theory, Cambridge University Press, vol. 9(1), pages 62-80, January.
    11. Ghazal, G. A., 1996. "Recurrence formula for expectations of products of quadratic forms," Statistics & Probability Letters, Elsevier, vol. 27(2), pages 101-109, April.
    12. Kadane, Joseph B, 1971. "Comparison of k-Class Estimators when the Disturbances are Small," Econometrica, Econometric Society, vol. 39(5), pages 723-737, September.
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    More about this item

    Keywords

    Expectation; Quadratic form; Nonnormality;
    All these keywords.

    JEL classification:

    • C10 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - General
    • C19 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Other

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