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Computationally Efficient Recursions For Top-Order Invariant Polynomials With Applications

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  • Hillier, Grant
  • Kan, Raymond
  • Wang, Xiaolu

Abstract

The top-order zonal polynomials C ( A ), and top-order invariant polynomials C null ,…, null ( A 1 , …, A ) in which each of the partitions of k , i = 1, …, r , has only one part, occur frequently in multivariate distribution theory, and econometrics — see, for example, Phillips (1980, Econometrica 48, 861–878; 1984, Journal of Econometrics 26, 387–398; 1985, International Economic Review 26, 21–36; 1986, Econometrica 54, 881–896), Hillier (1985, Econometric Theory 1, 53–72; 2001, Econometric Theory 17, 1–28), Hillier and Satchell (1986, Econometric Theory 2, 66–74), and Smith (1989, Journal of Multivariate Analysis 31, 244–257; 1993, Australian Journal of Statistics 35, 271–282). However, even with the recursive algorithms of Ruben (1962, Annals of Mathematical Statistics 33, 542–570) and Chikuse (1987, Econometric Theory 3, 195–207), numerical evaluation of these invariant polynomials is extremely time consuming. As a result, the value of invariant polynomials has been largely confined to analytic work on distribution theory. In this paper we present new, very much more efficient, algorithms for computing both the top-order zonal and invariant polynomials. These results should make the theoretical results involving these functions much more valuable for direct practical study. We demonstrate the value of our results by providing fast and accurate algorithms for computing the moments of a ratio of quadratic forms in normal random variables.

Suggested Citation

  • Hillier, Grant & Kan, Raymond & Wang, Xiaolu, 2009. "Computationally Efficient Recursions For Top-Order Invariant Polynomials With Applications," Econometric Theory, Cambridge University Press, vol. 25(01), pages 211-242, February.
  • Handle: RePEc:cup:etheor:v:25:y:2009:i:01:p:211-242_09
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    References listed on IDEAS

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    1. Phillips, Peter C B, 1985. "The Exact Distribution of LIML: II," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 26(1), pages 21-36, February.
    2. Magnus, J.R., 1978. "The moments of products of quadratic forms in normal variables," Other publications TiSEM 17c77a44-1789-4cf4-a382-a, Tilburg University, School of Economics and Management.
    3. Phillips, P. C. B., 1984. "The exact distribution of exogenous variable coefficient estimators," Journal of Econometrics, Elsevier, vol. 26(3), pages 387-398, December.
    4. Sargan, J D, 1976. "Econometric Estimators and the Edgeworth Approximation," Econometrica, Econometric Society, vol. 44(3), pages 421-448, May.
    5. Forchini, G., 2002. "The Exact Cumulative Distribution Function Of A Ratio Of Quadratic Forms In Normal Variables, With Application To The Ar(1) Model," Econometric Theory, Cambridge University Press, vol. 18(04), pages 823-852, August.
    6. Phillips, P C B, 1986. "The Exact Distribution of the Wald Statistic," Econometrica, Econometric Society, vol. 54(4), pages 881-895, July.
    7. Phillips, P C B, 1980. "The Exact Distribution of Instrumental Variable Estimators in an Equation Containing n + 1 Endogenous Variables," Econometrica, Econometric Society, vol. 48(4), pages 861-878, May.
    8. Smith, Murray D., 1989. "On the expectation of a ratio of quadratic forms in normal variables," Journal of Multivariate Analysis, Elsevier, vol. 31(2), pages 244-257, November.
    9. Hillier, Grant H., 1985. "On the Joint and Marginal Densities of Instrumental Variable Estimators in a General Structural Equation," Econometric Theory, Cambridge University Press, vol. 1(01), pages 53-72, April.
    10. Chikuse, Yasuko, 1987. "Methods for Constructing Top Order Invariant Polynomials," Econometric Theory, Cambridge University Press, vol. 3(02), pages 195-207, April.
    11. repec:adr:anecst:y:1986:i:4:p:05 is not listed on IDEAS
    12. repec:adr:anecst:y:1986:i:4 is not listed on IDEAS
    13. Sawa, Takamitsu, 1972. "Finite-Sample Properties of the k-Class Estimators," Econometrica, Econometric Society, vol. 40(4), pages 653-680, July.
    14. Hillier, Grant H & Kinal, Terrence W & Srivastava, V K, 1984. "On the Moments of Ordinary Least Squares and Instrumental Variables Estimators in a General Structural Equation," Econometrica, Econometric Society, vol. 52(1), pages 185-202, January.
    15. Jan R. Magnus, 1986. "The Exact Moments of a Ratio of Quadratic Forms in Normal Variables," Annals of Economics and Statistics, GENES, issue 4, pages 95-109.
    16. De Gooijer, Jan G., 1980. "Exact moments of the sample autocorrelations from series generated by general arima processes of order (p, d, q), d=0 or 1," Journal of Econometrics, Elsevier, vol. 14(3), pages 365-379, December.
    17. Magnus, J.R., 1986. "The exact moments of a ratio of quadratic forms in normal variables," Other publications TiSEM c6725407-ac3c-44fd-b6d1-5, Tilburg University, School of Economics and Management.
    18. Hillier, G.H., 1999. "The density of a quadratic form in a vector uniformly distributed on the n-sphere," Discussion Paper Series In Economics And Econometrics 9902, Economics Division, School of Social Sciences, University of Southampton.
    19. Ghazal, G. A., 1996. "Recurrence formula for expectations of products of quadratic forms," Statistics & Probability Letters, Elsevier, vol. 27(2), pages 101-109, April.
    20. Hillier, Grant, 2001. "THE DENSITY OF A QUADRATIC FORM IN A VECTOR UNIFORMLY DISTRIBUTED ON THE n-SPHERE," Econometric Theory, Cambridge University Press, vol. 17(01), pages 1-28, February.
    21. Hillier, G. H. & Satchell, S. E., 1986. "Finite-Sample Properties of a Two-Stage Single Equation Estimator in the SUR Model," Econometric Theory, Cambridge University Press, vol. 2(01), pages 66-74, April.
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    Cited by:

    1. Bao, Yong & Kan, Raymond, 2013. "On the moments of ratios of quadratic forms in normal random variables," Journal of Multivariate Analysis, Elsevier, vol. 117(C), pages 229-245.
    2. Christopher L. Skeels & Frank Windmeijer, 2016. "On the Stock-Yogo Tables," Bristol Economics Discussion Papers 16/679, Department of Economics, University of Bristol, UK, revised 25 Nov 2016.
    3. Grant Hillier & Federico Martellosio, 2013. "Properties of the maximum likelihood estimator in spatial autoregressive models," CeMMAP working papers CWP44/13, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    4. Grant Hillier & Raymond Kan & Xiaolu Wang, 2008. "Generating functions and short recursions, with applications to the moments of quadratic forms in noncentral normal vectors," CeMMAP working papers CWP14/08, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    5. Hillier, Grant & Kan, Raymond & Wang, Xiaoulu, 2009. "Generating functions and short recursions, with applications to the moments of quadratic forms in noncentral normal vectors," Discussion Paper Series In Economics And Econometrics 918, Economics Division, School of Social Sciences, University of Southampton.
    6. Hillier, Grant & Kan, Raymond & Wang, Xiaoulu, 2009. "Generating functions and short recursions, with applications to the moments of quadratic forms in noncentral normal vectors," Discussion Paper Series In Economics And Econometrics 0918, Economics Division, School of Social Sciences, University of Southampton.

    More about this item

    JEL classification:

    • C16 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Econometric and Statistical Methods; Specific Distributions
    • C46 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods: Special Topics - - - Specific Distributions
    • C63 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Computational Techniques

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