Skip to main content
×
×
Home

INTEGRATED SCORE ESTIMATION

  • Sung Jae Jun (a1), Joris Pinkse (a1) and Yuanyuan Wan (a2)
Abstract

We study the properties of the integrated score estimator (ISE), which is the Laplace version of Manski’s maximum score estimator (MMSE). The ISE belongs to a class of estimators whose basic asymptotic properties were studied in Jun, Pinkse, and Wan (2015, Journal of Econometrics 187(1), 201–216). Here, we establish that the MMSE, or more precisely $$\root 3 \of n |\hat \theta _M - \theta _0 |$$ , (locally first order) stochastically dominates the ISE under the conditions necessary for the MMSE to attain its $\root 3 \of n $ convergence rate and that the ISE has the same convergence rate as Horowitz’s smoothed maximum score estimator (SMSE) under somewhat weaker conditions. An implication of the stochastic dominance result is that the confidence intervals of the MMSE are for any given coverage rate wider than those of the ISE, provided that the input parameter α n is not chosen too large. Further, we introduce an inference procedure that is not only rate adaptive as established in Jun et al. (2015), but also uniform in the choice of α n . We propose three different first order bias elimination procedures and we discuss the choice of input parameters. We develop a computational algorithm for the ISE based on the Gibbs sampler and we examine implementational issues in detail. We argue in favor of normalizing the norm of the parameter vector as opposed to fixing one of the coefficients. Finally, we evaluate the computational efficiency of the ISE and the performance of the ISE and the proposed inference procedure in an extensive Monte Carlo study.

Copyright
Corresponding author
*Address correspondence to Sung Jae Jun, Department of Economics, The Pennsylvania State University, 303 Kern Graduate Building, University Park, PA 16802, USA; e-mail: sjun@psu.edu.
Footnotes
Hide All

This paper is based on research supported by NSF grant SES–0922127. We thank the Human Capital Foundation (www.hcfoundation.ru), especially Andrey P. Vavilov, for their support of CAPCP (http://capcp.psu.edu) at the Pennsylvania State University. We thank Don Andrews, Miguel Delgado, Jeremy Fox, Bo Honoré, Joel Horowitz, Roger Koenker, Arthur Lewbel, Runze Li, Oliver Linton, Peter Robinson, Neil Wallace, Haiqing Xu, Vicky Zinde–Walsh, six anonymous referees, and numerous departmental seminar and conference participants for helpful suggestions.

Footnotes
References
Hide All
Abrevaya, J. & Huang, J. (2005) On the bootstrap of the maximum score estimator. Econometrica 73(4), 11751204.
Chamberlain, G. (1986) Asymptotic efficiency in semi-parametric models with censoring. Journal of Econometrics 32(2), 189218.
Chen, S. & Zhang, H. (2014) Binary Quantile Regression with Local Polynomial Smoothing. Discussion paper, HKUST.
Chernozhukov, V. & Hong, H. (2003) An mcmc approach to classical estimation. Journal of Econometrics 115(2), 293346.
Cowles, M.K. & Carlin, B.P. (1996) Markov chain monte carlo convergence diagnostics: a comparative review. Journal of the American Statistical Association 91(434), 883904.
Davidson, J. (1994) Stochastic Limit Theory: An Introduction for Econometricians. Oxford University Press.
Florios, K. & Skouras, S. (2008) Exact computation of max weighted score estimators. Journal of Econometrics 146(1), 8691.
Hong, H., Mahajan, A., & Nekipelov, D. (2010) Extremum Estimation and Numerical Derivatives. Discussion paper, The University of California at Berkeley.
Horowitz, J. (1992) A smoothed maximum score estimator for the binary response model. Econometrica 60(3), 505–31.
Ichimura, H. (1993) Semiparametric least squares (SLS) and weighted SLS estimation of single-index models. Journal of Econometrics 58(1), 71120.
Jun, S.J., Pinkse, J., & Wan, Y. (2009) Cube Root N and Faster Convergence, Laplace Estimators, and Uniform Inference. Discussion paper, Pennsylvania State University.
Jun, S.J., Pinkse, J., & Wan, Y. (2015) Classical laplace estimation for-consistent estimators: Improved convergence rates and rate-adaptive inference. Journal of Econometrics 187(1), 201216.
Khan, S. & Tamer, E. (2010) Irregular identification, support conditions, and inverse weight estimation. Econometrica 78(6), 20212042.
Kim, J. & Pollard, D. (1990) Cube root asymptotics. Annals of Statistics 18(1), 191219.
Klein, R. & Spady, R. (1993) An efficient semiparametric estimator for binary response models. Econometrica 61, 387421.
Kosorok, M. (2008) Introduction to Empirical Processes and Semiparametric Inference. Springer Verlag.
Kotlyarova, Y. & Zinde-Walsh, V. (2006) Non– and semi–parametric estimation in models with unknown smoothness. Economics Letters 93(3), 379386.
Lee, S.M.S. & Pun, M. (2006) On m out of n bootstrapping for nonstandard M-estimation with nuisance parameters. Journal of the American Statistical Association 101(475), 11851197.
Lewbel, A. (1998) Semiparametric latent variable model estimation with endogenous or mismeasured regressors. Econometrica 66(1), 105121.
Lewbel, A. (2000) Semiparametric qualitative response model estimation with unknown heteroscedasticity or instrumental variables. Journal of Econometrics 97(1), 145177.
Manski, C. (1975) Maximum score estimation of the stochastic utility model of choice. Journal of Econometrics 3(3), 205228.
Manski, C. (1985) Semiparametric analysis of discrete response. Asymptotic properties of the maximum score estimator. Journal of Econometrics 27(3), 313333.
Manski, C. (1987) Semiparametric analysis of random effects linear models from binary panel data. Econometrica 55, 357362.
Müller, M.E. (1959) A note on a method for generating points uniformly on n-dimensional spheres. Communications of the ACM 2(4), 1920.
Patra, R.K., Seijo, E., & Sen, B. (2015) A consistent bootstrap procedure for the maximum score estimator. arXiv:1105.1976v5.
Phillips, P.C.B. (1984) The exact distribution of LIML: I. International Economic Review 25(1), 249261.
Phillips, P.C.B. (1989) Partially identified econometric models. Econometric Theory 5(02), 181240.
Pinkse, C. (1993) On the computation of semiparametric estimates in limited dependent variable models. Journal of Econometrics 58, 185205.
Pollard, D. (1993) The Asymptotics of a Binary Choice Model. Discussion paper, Yale University.
Powell, J., Stock, J., & Stoker, T. (1989) Semiparametric estimation of index coefficients. Econometrica 57(6), 14031430.
Stroud, A.H. (1971) Approximate Calculation of Multiple Integrals. Prentice-Hall.
van der Vaart, A. & Wellner, J. (1996) Weak Convergence and Empirical Processes. Springer.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Econometric Theory
  • ISSN: 0266-4666
  • EISSN: 1469-4360
  • URL: /core/journals/econometric-theory
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed