IDEAS home Printed from https://ideas.repec.org/a/eee/stapro/v27y1996i2p101-109.html
   My bibliography  Save this article

Recurrence formula for expectations of products of quadratic forms

Author

Listed:
  • Ghazal, G. A.

Abstract

In this paper, we derive a recurrence formula for evaluating mathematical expectations of X'A1XX'A2X...X'AnX where X ~ Np(0, [Phi]) and Aj, J = 1, 2, ..., n are (p x p) nonstochastic symmetric matrices. Subsequently, this recurrence formula is applied to some specific problems. Various related results will also be reported.

Suggested Citation

  • Ghazal, G. A., 1996. "Recurrence formula for expectations of products of quadratic forms," Statistics & Probability Letters, Elsevier, vol. 27(2), pages 101-109, April.
    • Handle: RePEc:eee:stapro:v:27:y:1996:i:2:p:101-109
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/0167-7152(95)00050-X
    Download Restriction: Full text for ScienceDirect subscribers only
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Mikhail, William M. & Ghazal, G. A., 1991. "On a pooled estimator and its finite-sample moments," Journal of Econometrics, Elsevier, vol. 48(1-2), pages 195-214.
    2. F. J. H. Don, 1979. "The Expectation of Products of Quadratic Forms in Normal Variables," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 33(2), pages 73-79, June.
    3. Ghazal, G. A., 1994. "Moments of the ratio of two dependent quadratic forms," Statistics & Probability Letters, Elsevier, vol. 20(4), pages 313-319, July.
    4. H. Neudecker, 1968. "The Kronecker Matrix Product and Some of its Applications in Econometrics," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 22(1), pages 69-82, March.
    5. 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.
    6. Kadane, Joseph B, 1971. "Comparison of k-Class Estimators when the Disturbances are Small," Econometrica, Econometric Society, vol. 39(5), pages 723-737, September.
    7. Magnus, J.R. & Neudecker, H., 1979. "The commutation matrix : Some properties and applications," Other publications TiSEM d0b1e779-7795-4676-ac98-1, Tilburg University, School of Economics and Management.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Rasmus S. Pedersen & Anders Rahbek, 2014. "Multivariate variance targeting in the BEKK–GARCH model," Econometrics Journal, Royal Economic Society, vol. 17(1), pages 24-55, February.
    2. Peter M. Robinson & Francesca Rossi, 2014. "Improved Lagrange multiplier tests in spatial autoregressions," Econometrics Journal, Royal Economic Society, vol. 17(1), pages 139-164, February.
    3. Ghazal, G. A., 2000. "Recurrence formula for expectations of products of bilinear forms and expectations of bilinear forms and random matrices," Statistics & Probability Letters, Elsevier, vol. 48(1), pages 1-9, May.
    4. 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.
    5. repec:cep:stiecm:/2013/566 is not listed on IDEAS
    6. 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.
    7. Patrick Marsh, "undated". "Some Geometry for the Maximal Invariant in Linear Regression," Discussion Papers 04/07, Department of Economics, University of York.

    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. 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.
    3. 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.
    4. Sneek, J.M., 1982. "Some approximations to the exact distribution of sample autocorrelations for autoregressive moving average models," Serie Research Memoranda 0002, VU University Amsterdam, Faculty of Economics, Business Administration and Econometrics.
    5. Liu-Evans, Gareth, 2010. "An alternative approach to approximating the moments of least squares estimators," MPRA Paper 26550, University Library of Munich, Germany.
    6. Fryzlewicz, Piotr & Nason, Guy P., 2006. "Haar-Fisz estimation of evolutionary wavelet spectra," LSE Research Online Documents on Economics 25227, London School of Economics and Political Science, LSE Library.
    7. Paulo M. D. C. Parente & Richard J. Smith, 2021. "Quasi‐maximum likelihood and the kernel block bootstrap for nonlinear dynamic models," Journal of Time Series Analysis, Wiley Blackwell, vol. 42(4), pages 377-405, July.
    8. David E. Giles, 2005. "The Bias of Inequality Measures in Very Small Samples: Some Analytic Results," Econometrics Working Papers 0514, Department of Economics, University of Victoria.
    9. Joshua C. C. Chan & Liana Jacobi & Dan Zhu, 2019. "How Sensitive Are VAR Forecasts to Prior Hyperparameters? An Automated Sensitivity Analysis," Advances in Econometrics, in: Topics in Identification, Limited Dependent Variables, Partial Observability, Experimentation, and Flexible Modeling: Part A, volume 40, pages 229-248, Emerald Group Publishing Limited.
    10. Campos, Julia & Ericsson, Neil R. & Hendry, David F., 1996. "Cointegration tests in the presence of structural breaks," Journal of Econometrics, Elsevier, vol. 70(1), pages 187-220, January.
    11. Naoto Kunitomo, 1979. "Asymptotic Optimality of the Limited Information Maximum Likelihood Estimator in Large Econometric Models," Discussion Papers 503, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
    12. Kirill S. Evdokimov & Andrei Zeleneev, 2023. "Simple Estimation of Semiparametric Models with Measurement Errors," Papers 2306.14311, arXiv.org, revised Mar 2024.
    13. O. J. Boxma & E. J. Cahen & D. Koops & M. Mandjes, 2019. "Linear Stochastic Fluid Networks: Rare-Event Simulation and Markov Modulation," Methodology and Computing in Applied Probability, Springer, vol. 21(1), pages 125-153, March.
    14. Loperfido, Nicola, 2014. "Linear transformations to symmetry," Journal of Multivariate Analysis, Elsevier, vol. 129(C), pages 186-192.
    15. Liu, Shuangzhe & Leiva, Víctor & Zhuang, Dan & Ma, Tiefeng & Figueroa-Zúñiga, Jorge I., 2022. "Matrix differential calculus with applications in the multivariate linear model and its diagnostics," Journal of Multivariate Analysis, Elsevier, vol. 188(C).
    16. Christian Gourieroux & Joann Jasiak, 2022. "Long Run Risk in Stationary Structural Vector Autoregressive Models," Papers 2202.09473, arXiv.org.
    17. Magnus, J.R. & Pesaran, B., 1990. "Evaluation Of Moment Of Quadratic Forms In Normal Variables," Papers 9021, Tilburg - Center for Economic Research.
    18. Arismendi, J.C., 2013. "Multivariate truncated moments," Journal of Multivariate Analysis, Elsevier, vol. 117(C), pages 41-75.
    19. Kiviet, Jan F., 2020. "Testing the impossible: Identifying exclusion restrictions," Journal of Econometrics, Elsevier, vol. 218(2), pages 294-316.
    20. Reinaldo B. Arellano-Valle & Adelchi Azzalini, 2022. "Some properties of the unified skew-normal distribution," Statistical Papers, Springer, vol. 63(2), pages 461-487, April.

    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:eee:stapro:v:27:y:1996:i:2:p:101-109. See general information about how to correct material in RePEc.

    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: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/wps/find/journaldescription.cws_home/622892/description#description .

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

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.