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

  • Vassilis Argyrou Hajivassiliou

    (Yale University)

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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Paper provided by Yale University in its series Working Papers with number _025.

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Date of creation: Apr 1993
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Handle: RePEc:wop:yaluwp:_025
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  1. McFadden, Daniel & Ruud, Paul A, 1994. "Estimation by Simulation," The Review of Economics and Statistics, MIT Press, vol. 76(4), pages 591-608, November.
  2. Pakes, Ariel & Pollard, David, 1989. "Simulation and the Asymptotics of Optimization Estimators," Econometrica, Econometric Society, vol. 57(5), pages 1027-57, September.
  3. 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.
  4. J. A. Hausman & D. A. Wise, 1976. "A Conditional Profit Model for Qualitative Choice: Discrete Decisions Recognizing Interdependence and Heterogeneous Preferences," Working papers 173, Massachusetts Institute of Technology (MIT), Department of Economics.
  5. Ruud, Paul A., 1991. "Extensions of estimation methods using the EM algorithm," Journal of Econometrics, Elsevier, vol. 49(3), pages 305-341, September.
  6. Vassilis A. Hajivassiliou & Daniel L. McFadden, 1993. "The Method of Simulated Scores for the Estimation of LDV Models," Working Papers _023, Yale University.
  7. Geweke, John, 1989. "Bayesian Inference in Econometric Models Using Monte Carlo Integration," Econometrica, Econometric Society, vol. 57(6), pages 1317-39, November.
  8. John Geweke & Michael Keane & David Runkle, 1994. "Alternative computational approaches to inference in the multinomial probit model," Staff Report 170, Federal Reserve Bank of Minneapolis.
  9. Keane, Michael P, 1994. "A Computationally Practical Simulation Estimator for Panel Data," Econometrica, Econometric Society, vol. 62(1), pages 95-116, January.
  10. 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.
  11. 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.
  12. 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.
  13. Stern, Steven, 1992. "A Method for Smoothing Simulated Moments of Discrete Probabilities in Multinomial Probit Models," Econometrica, Econometric Society, vol. 60(4), pages 943-52, July.
  14. 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.
  15. 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.
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