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Simulating Normal Rectangle Probabilities and Their Derivatives: The Effects of Vectorization

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  • Vassilis Argyrou Hajivassiliou

    (Yale University)

Abstract

An extensive literature in econometrics and in numerical analysis has considered the computationally difficult problem of evaluating the multiple integral representing the probability of a multivariate normal random vector constrained to lie in a rectangular region. A leading case of such an integral is the negative orthant probability, implied by the multinomial probit (MNP) model used in econometrics and biometrics. Classical parametric estimation of this model requires, for each trial parameter vector and each observation in a sample, evaluation of a normal orthant probability and its derivatives with respect to the mean vector and the variance-covariance matrix. Several Monte Carlo simulators have been developed to approximate the orthant probability integral and its linear and logarithmic derivatives that limit computation while possessing properties that facilitate their use in iterative calculations for statistical inference. In this paper, I discuss Gauss and FORTRAN implementations of 13 simulation algorithms, and I present results on the impact of vectorization on the relative computational performance of the simulation algorithms. I show that the 13 simulators differ greatly with respect to the degree of vectorizability: in some cases activating the CRAY-Y/MP4 vector facility achieves a speed-up factor in excess of 10 times, while in others the gains in speed are negligible. Evaluating the algorithms in terms of lowest simulation root-mean-squared-error for given computation time, I find that (1) GHK, an importance sampling recursive triangularization simulator, remains the best method for simulating probabilities irrespective of vectorization; (2) the crude Monte Carlo simulator CFS offers the greatest benefits from vectorization; and (3) the GSS algorithm, based on "Gibbs resampling," emerges as one of the preferred methods for simulating logarithmic derivatives, especially in the absence of vectorization.
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  • Vassilis Argyrou Hajivassiliou, 1993. "Simulating Normal Rectangle Probabilities and Their Derivatives: The Effects of Vectorization," Working Papers _025, Yale University.
    • Handle: RePEc:wop:yaluwp:_025
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    References listed on IDEAS

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    1. Vassilis A. Hajivassiliou & Daniel McFadden, 1990. "The Method of Simulated Scores for the Estimation of LDV Models with an Application to External Debt Crisis," Cowles Foundation Discussion Papers 967, Cowles Foundation for Research in Economics, Yale University.
    2. Geweke, John, 1989. "Bayesian Inference in Econometric Models Using Monte Carlo Integration," Econometrica, Econometric Society, vol. 57(6), pages 1317-1339, November.
    3. McFadden, Daniel & Ruud, Paul A, 1994. "Estimation by Simulation," The Review of Economics and Statistics, MIT Press, vol. 76(4), pages 591-608, November.
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    6. Keane, Michael P, 1994. "A Computationally Practical Simulation Estimator for Panel Data," Econometrica, Econometric Society, vol. 62(1), pages 95-116, January.
    7. Geweke, John & Keane, Michael P & Runkle, David, 1994. "Alternative Computational Approaches to Inference in the Multinomial Probit Model," The Review of Economics and Statistics, MIT Press, vol. 76(4), pages 609-632, November.
    8. McFadden, Daniel, 1989. "A Method of Simulated Moments for Estimation of Discrete Response Models without Numerical Integration," Econometrica, Econometric Society, vol. 57(5), pages 995-1026, September.
    9. Pakes, Ariel & Pollard, David, 1989. "Simulation and the Asymptotics of Optimization Estimators," Econometrica, Econometric Society, vol. 57(5), pages 1027-1057, September.
    10. Borsch-Supan, Axel & Hajivassiliou, Vassilis A., 1993. "Smooth unbiased multivariate probability simulators for maximum likelihood estimation of limited dependent variable models," Journal of Econometrics, Elsevier, vol. 58(3), pages 347-368, August.
    11. Charles E. Clark, 1961. "The Greatest of a Finite Set of Random Variables," Operations Research, INFORMS, vol. 9(2), pages 145-162, April.
    12. Vassilis A. Hajivassiliou & Daniel L. McFadden, 1998. "The Method of Simulated Scores for the Estimation of LDV Models," Econometrica, Econometric Society, vol. 66(4), pages 863-896, July.
    13. Vassilis A. Hajivassiliou, 1991. "Simulation Estimation Methods for Limited Dependent Variable Models," Cowles Foundation Discussion Papers 1007, Cowles Foundation for Research in Economics, Yale University.
    14. J. E. Dutt, 1976. "Numerical Aspects of Multivariate Normal Probabilities in Econometric Models," NBER Chapters, in: Annals of Economic and Social Measurement, Volume 5, number 4, pages 547-561, National Bureau of Economic Research, Inc.
    15. Stern, Steven, 1992. "A Method for Smoothing Simulated Moments of Discrete Probabilities in Multinomial Probit Models," Econometrica, Econometric Society, vol. 60(4), pages 943-952, July.
    16. McCulloch, Robert & Rossi, Peter E., 1994. "An exact likelihood analysis of the multinomial probit model," Journal of Econometrics, Elsevier, vol. 64(1-2), pages 207-240.
    17. Hendry, David F., 1984. "Monte carlo experimentation in econometrics," Handbook of Econometrics, in: Z. Griliches† & M. D. Intriligator (ed.), Handbook of Econometrics, edition 1, volume 2, chapter 16, pages 937-976, Elsevier.
    18. van Praag, B. M. S. & Hop, J. P., 1987. "Estimation Of Continuous Models On The Basis Of Set-Valued Observations," Econometric Institute Archives 272362, Erasmus University Rotterdam.
    19. R. F. Engle & D. McFadden (ed.), 1986. "Handbook of Econometrics," Handbook of Econometrics, Elsevier, edition 1, volume 4, number 4.
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    1. Hajivassiliou, Vassilis A. & Ruud, Paul A., 1986. "Classical estimation methods for LDV models using simulation," Handbook of Econometrics, in: R. F. Engle & D. McFadden (ed.), Handbook of Econometrics, edition 1, volume 4, chapter 40, pages 2383-2441, Elsevier.
    2. Daniel Ackerberg, 2009. "A new use of importance sampling to reduce computational burden in simulation estimation," Quantitative Marketing and Economics (QME), Springer, vol. 7(4), pages 343-376, December.
    3. Horowitz, Joel & Keane, Michael & Bolduc, Denis & Divakar, Suresh & Geweke, John & Gonul, Fosun & Hajivassiliou, Vassilis & Koppelman, Frank & Matzkin, Rosa & Rossi, Peter & Ruud, Paul, 1994. "Advances in Random Utility Models," MPRA Paper 53026, University Library of Munich, Germany.
    4. Charles Romeo, 2007. "A Gibbs sampler for mixed logit analysis of differentiated product markets using aggregate data," Computational Economics, Springer;Society for Computational Economics, vol. 29(1), pages 33-68, February.
    5. Geweke, John & Keane, Michael P & Runkle, David, 1994. "Alternative Computational Approaches to Inference in the Multinomial Probit Model," The Review of Economics and Statistics, MIT Press, vol. 76(4), pages 609-632, November.
    6. Vijverberg, Wim P. M., 1997. "Monte Carlo evaluation of multivariate normal probabilities," Journal of Econometrics, Elsevier, vol. 76(1-2), pages 281-307.
    7. Hajivassiliou, Vassilis & McFadden, Daniel & Ruud, Paul, 1996. "Simulation of multivariate normal rectangle probabilities and their derivatives theoretical and computational results," Journal of Econometrics, Elsevier, vol. 72(1-2), pages 85-134.

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