IDEAS home Printed from https://ideas.repec.org/a/eee/econom/v144y2008i1p62-80.html
   My bibliography  Save this article

Likelihood approximation by numerical integration on sparse grids

Author

Listed:
  • Heiss, Florian
  • Winschel, Viktor

Abstract

The calculation of likelihood functions of many econometric models requires the evaluation of integrals without analytical solutions. Approaches for extending Gaussian quadrature to multiple dimensions discussed in the literature are either very specific or suffer from exponentially rising computational costs in the number of dimensions. We propose an extension that is very general and easily implemented, and does not suffer from the curse of dimensionality. Monte Carlo experiments for the mixed logit model indicate the superior performance of the proposed method over simulation techniques.

Suggested Citation

  • Heiss, Florian & Winschel, Viktor, 2008. "Likelihood approximation by numerical integration on sparse grids," Journal of Econometrics, Elsevier, vol. 144(1), pages 62-80, May.
    • Handle: RePEc:eee:econom:v:144:y:2008:i:1:p:62-80
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0304-4076(07)00255-2
    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. Kenneth L. Judd, 1998. "Numerical Methods in Economics," MIT Press Books, The MIT Press, edition 1, volume 1, number 0262100711, December.
    2. Lee, Lung-fei, 2000. "A numerically stable quadrature procedure for the one-factor random-component discrete choice model," Journal of Econometrics, Elsevier, vol. 95(1), pages 117-129, March.
    3. Chiou, Lesley & Walker, Joan L., 2007. "Masking identification of discrete choice models under simulation methods," Journal of Econometrics, Elsevier, vol. 141(2), pages 683-703, December.
    4. 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.
    5. Brownstone, David & Train, Kenneth, 1998. "Forecasting new product penetration with flexible substitution patterns," Journal of Econometrics, Elsevier, vol. 89(1-2), pages 109-129, November.
    6. Krueger, Dirk & Kubler, Felix, 2004. "Computing equilibrium in OLG models with stochastic production," Journal of Economic Dynamics and Control, Elsevier, vol. 28(7), pages 1411-1436, April.
    7. 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.
    8. Rabe-Hesketh, Sophia & Skrondal, Anders & Pickles, Andrew, 2005. "Maximum likelihood estimation of limited and discrete dependent variable models with nested random effects," Journal of Econometrics, Elsevier, vol. 128(2), pages 301-323, October.
    9. Sandor, Zsolt & Andras, P.Peter, 2004. "Alternative sampling methods for estimating multivariate normal probabilities," Journal of Econometrics, Elsevier, vol. 120(2), pages 207-234, June.
    10. Geweke, John, 1996. "Monte carlo simulation and numerical integration," Handbook of Computational Economics, in: H. M. Amman & D. A. Kendrick & J. Rust (ed.), Handbook of Computational Economics, edition 1, volume 1, chapter 15, pages 731-800, Elsevier.
    11. Viktor Winschel, 2005. "Solving, Estimating and Selecting Nonlinear Dynamic Economic Models without the Curse of Dimensionality," GE, Growth, Math methods 0507014, University Library of Munich, Germany.
    12. repec:pit:wpaper:321 is not listed on IDEAS
    13. Daniel McFadden & Kenneth Train, 2000. "Mixed MNL models for discrete response," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 15(5), pages 447-470.
    14. Train,Kenneth E., 2009. "Discrete Choice Methods with Simulation," Cambridge Books, Cambridge University Press, number 9780521747387, December.
    15. Butler, J S & Moffitt, Robert, 1982. "A Computationally Efficient Quadrature Procedure for the One-Factor Multinomial Probit Model," Econometrica, Econometric Society, vol. 50(3), pages 761-764, May.
    16. Bhat, Chandra R., 2001. "Quasi-random maximum simulated likelihood estimation of the mixed multinomial logit model," Transportation Research Part B: Methodological, Elsevier, vol. 35(7), pages 677-693, August.
    17. Lee, Lung-Fei, 1997. "Simulated maximum likelihood estimation of dynamic discrete choice statistical models some Monte Carlo results," Journal of Econometrics, Elsevier, vol. 82(1), pages 1-35.
    18. 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.
    19. Geweke, John F. & Keane, Michael P. & Runkle, David E., 1997. "Statistical inference in the multinomial multiperiod probit model," Journal of Econometrics, Elsevier, vol. 80(1), pages 125-165, September.
    20. Naylor, J. C. & Smith, A. F. M., 1988. "Econometric illustrations of novel numerical integration strategies for Bayesian inference," Journal of Econometrics, Elsevier, vol. 38(1-2), pages 103-125.
    21. Hess, Stephane & Train, Kenneth E. & Polak, John W., 2006. "On the use of a Modified Latin Hypercube Sampling (MLHS) method in the estimation of a Mixed Logit Model for vehicle choice," Transportation Research Part B: Methodological, Elsevier, vol. 40(2), pages 147-163, February.
    22. Richard, Jean-Francois & Zhang, Wei, 2007. "Efficient high-dimensional importance sampling," Journal of Econometrics, Elsevier, vol. 141(2), pages 1385-1411, December.
    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. Heiss, Florian & Winschel, Viktor, 2006. "Estimation with Numerical Integration on Sparse Grids," Discussion Papers in Economics 916, University of Munich, Department of Economics.
    2. Ziegler Andreas, 2010. "Z-Tests in Multinomial Probit Models under Simulated Maximum Likelihood Estimation: Some Small Sample Properties," Journal of Economics and Statistics (Jahrbuecher fuer Nationaloekonomie und Statistik), De Gruyter, vol. 230(5), pages 630-652, October.
    3. Andreas Ziegler, 2007. "Simulated classical tests in multinomial probit models," Statistical Papers, Springer, vol. 48(4), pages 655-681, October.
    4. Cherchi, Elisabetta & Guevara, Cristian Angelo, 2012. "A Monte Carlo experiment to analyze the curse of dimensionality in estimating random coefficients models with a full variance–covariance matrix," Transportation Research Part B: Methodological, Elsevier, vol. 46(2), pages 321-332.
    5. 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.
    6. Sándor, Z. & András, P., 2003. "Alternate Samplingmethods for Estimating Multivariate Normal Probabilities," Econometric Institute Research Papers EI 2003-05, Erasmus University Rotterdam, Erasmus School of Economics (ESE), Econometric Institute.
    7. Michael P. Keane, 2013. "Panel data discrete choice models of consumer demand," Economics Papers 2013-W08, Economics Group, Nuffield College, University of Oxford.
    8. Kerem Tuzcuoglu, 2019. "Composite Likelihood Estimation of an Autoregressive Panel Probit Model with Random Effects," Staff Working Papers 19-16, Bank of Canada.
    9. Ricardo A. Daziano & Martin Achtnicht, 2014. "Forecasting Adoption of Ultra-Low-Emission Vehicles Using Bayes Estimates of a Multinomial Probit Model and the GHK Simulator," Transportation Science, INFORMS, vol. 48(4), pages 671-683, November.
    10. Joan L. Walker & Moshe Ben-Akiva, 2011. "Advances in Discrete Choice: Mixture Models," Chapters, in: André de Palma & Robin Lindsey & Emile Quinet & Roger Vickerman (ed.), A Handbook of Transport Economics, chapter 8, Edward Elgar Publishing.
    11. Sandor, Zsolt & Andras, P.Peter, 2004. "Alternative sampling methods for estimating multivariate normal probabilities," Journal of Econometrics, Elsevier, vol. 120(2), pages 207-234, June.
    12. Joan L. Walker & Moshe Ben-Akiva & Denis Bolduc, 2007. "Identification of parameters in normal error component logit-mixture (NECLM) models," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 22(6), pages 1095-1125.
    13. Eduardo Fé & Richard Hofler, 2013. "Count data stochastic frontier models, with an application to the patents–R&D relationship," Journal of Productivity Analysis, Springer, vol. 39(3), pages 271-284, June.
    14. Yu, Jie & Goos, Peter & Vandebroek, Martina, 2010. "Comparing different sampling schemes for approximating the integrals involved in the efficient design of stated choice experiments," Transportation Research Part B: Methodological, Elsevier, vol. 44(10), pages 1268-1289, December.
    15. Chiou, Lesley & Walker, Joan L., 2007. "Masking identification of discrete choice models under simulation methods," Journal of Econometrics, Elsevier, vol. 141(2), pages 683-703, December.
    16. Haghani, Milad & Bliemer, Michiel C.J. & Hensher, David A., 2021. "The landscape of econometric discrete choice modelling research," Journal of choice modelling, Elsevier, vol. 40(C).
    17. Laura Mørch Andersen, 2013. "Obtaining reliable Likelihood Ratio tests from simulated likelihood functions," IFRO Working Paper 2013/1, University of Copenhagen, Department of Food and Resource Economics.
    18. Czajkowski, Mikołaj & Budziński, Wiktor, 2019. "Simulation error in maximum likelihood estimation of discrete choice models," Journal of choice modelling, Elsevier, vol. 31(C), pages 73-85.
    19. Kalouptsidis, N. & Psaraki, V., 2010. "Approximations of choice probabilities in mixed logit models," European Journal of Operational Research, Elsevier, vol. 200(2), pages 529-535, January.
    20. Florian Heiss, 2008. "Sequential numerical integration in nonlinear state space models for microeconometric panel data," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 23(3), pages 373-389.

    More about this item

    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:eee:econom:v:144:y:2008:i:1:p:62-80. 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: http://www.elsevier.com/locate/jeconom .

    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/locate/jeconom .

    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.