IDEAS home Printed from https://ideas.repec.org/p/ucr/wpaper/200907.html
   My bibliography  Save this paper

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

    Download full text from publisher

    File URL: https://economics.ucr.edu/repec/ucr/wpaper/09-07.pdf
    File Function: First version, 2009
    Download Restriction: no
    ---><---

    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. Hashem Pesaran, M. & Yamagata, Takashi, 2008. "Testing slope homogeneity in large panels," Journal of Econometrics, Elsevier, vol. 142(1), pages 50-93, January.
    3. 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.
    4. 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.
    5. 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.
    6. 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.
    7. Jan R. Magnus, 1979. "The expectation of products of quadratic forms in normal variables: the practice," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 33(3), pages 131-136, September.
    8. 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.
    9. 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.
    10. Ghazal, G. A., 1996. "Recurrence formula for expectations of products of quadratic forms," Statistics & Probability Letters, Elsevier, vol. 27(2), pages 101-109, April.
    11. Jan R. Magnus, 1978. "The moments of products of quadratic forms in normal variables," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 32(4), pages 201-210, December.
    12. 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.
    13. Kadane, Joseph B, 1971. "Comparison of k-Class Estimators when the Disturbances are Small," Econometrica, Econometric Society, vol. 39(5), pages 723-737, September.
    14. Ullah, Aman, 2004. "Finite Sample Econometrics," OUP Catalogue, Oxford University Press, number 9780198774488, November.
    15. Magnus, J.R., 1979. "The expectation of products of quadratic forms in normal variables : The practice Statistica Neerlandica," Other publications TiSEM fe936fc3-c7db-4806-80b3-6, Tilburg University, School of Economics and Management.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Kiviet, Jan F. & Phillips, Garry D.A., 2012. "Higher-order asymptotic expansions of the least-squares estimation bias in first-order dynamic regression models," Computational Statistics & Data Analysis, Elsevier, vol. 56(11), pages 3705-3729.
    2. Bao, Yong & Ullah, Aman, 2007. "The second-order bias and mean squared error of estimators in time-series models," Journal of Econometrics, Elsevier, vol. 140(2), pages 650-669, October.
    3. Hashem Pesaran, M. & Yamagata, Takashi, 2008. "Testing slope homogeneity in large panels," Journal of Econometrics, Elsevier, vol. 142(1), pages 50-93, January.
    4. Yong Bao & Aman Ullah, 2021. "Analytical Finite Sample Econometrics: From A. L. Nagar to Now," Journal of Quantitative Economics, Springer;The Indian Econometric Society (TIES), vol. 19(1), pages 17-37, December.
    5. Magnus, J.R. & Pesaran, B., 1990. "Evaluation Of Moment Of Quadratic Forms In Normal Variables," Papers 9021, Tilburg - Center for Economic Research.
    6. Aman Ullah & Yong Bao & Ru Zhang, 2014. "Moment Approximation for Unit Root Models with Nonnormal Errors," Working Papers 201401, University of California at Riverside, Department of Economics.
    7. Symeonides Spyridon D. & Karavias Yiannis & Tzavalis Elias, 2017. "Size corrected Significance Tests in Seemingly Unrelated Regressions with Autocorrelated Errors," Journal of Time Series Econometrics, De Gruyter, vol. 9(1), pages 1-41, January.
    8. van den Berg, G., 1986. "Small-sample properties of estimators of the autocorrelation coefficient," Other publications TiSEM 03414dd6-10fa-4f48-a742-e, Tilburg University, School of Economics and Management.
    9. Kiviet, Jan F. & Phillips, Garry D.A., 2014. "Improved variance estimation of maximum likelihood estimators in stable first-order dynamic regression models," Computational Statistics & Data Analysis, Elsevier, vol. 76(C), pages 424-448.
    10. Feng Xu & Zekai He, 2020. "Testing slope homogeneity in panel data models with a multifactor error structure," Statistical Papers, Springer, vol. 61(1), pages 201-224, February.
    11. Liu-Evans, Gareth, 2010. "An alternative approach to approximating the moments of least squares estimators," MPRA Paper 26550, University Library of Munich, Germany.
    12. Yu, Jun, 2012. "Bias in the estimation of the mean reversion parameter in continuous time models," Journal of Econometrics, Elsevier, vol. 169(1), pages 114-122.
    13. Qian, Chen & Giles, David E., 2007. "The bias of elasticity estimators in linear regression: Some analytic results," Economics Letters, Elsevier, vol. 94(2), pages 185-191, February.
    14. Magnus, J.R. & Pesaran, B., 1990. "Forecasting, Misspecification And Unit Roots: The Case Of Ar(1) Versus Arma (1,1)," Papers 9002, Tilburg - Center for Economic Research.
    15. Kan, Raymond, 2008. "From moments of sum to moments of product," Journal of Multivariate Analysis, Elsevier, vol. 99(3), pages 542-554, March.
    16. Hillier, Grant & Kan, Raymond & Wang, Xiaolu, 2009. "Computationally Efficient Recursions For Top-Order Invariant Polynomials With Applications," Econometric Theory, Cambridge University Press, vol. 25(1), pages 211-242, February.
    17. Mavroeidis, Sophocles & Sasaki, Yuya & Welch, Ivo, 2015. "Estimation of heterogeneous autoregressive parameters with short panel data," Journal of Econometrics, Elsevier, vol. 188(1), pages 219-235.
    18. Liu-Evans, Gareth, 2014. "A note on approximating moments of least squares estimators," MPRA Paper 57543, University Library of Munich, Germany.
    19. Vougas, Dimitrios V., 2007. "GLS detrending and unit root testing," Economics Letters, Elsevier, vol. 97(3), pages 222-229, December.
    20. Ghazal, G. A., 1996. "Recurrence formula for expectations of products of quadratic forms," Statistics & Probability Letters, Elsevier, vol. 27(2), pages 101-109, April.

    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

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:ucr:wpaper:200907. See general information about how to correct material in RePEc.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: . General contact details of provider: https://edirc.repec.org/data/deucrus.html .

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Kelvin Mac (email available below). General contact details of provider: https://edirc.repec.org/data/deucrus.html .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service hosted by the Research Division of the Federal Reserve Bank of St. Louis . RePEc uses bibliographic data supplied by the respective publishers.