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Journal of Econometric Methods

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A Simple GMM Estimator for the Semiparametric Mixed Proportional Hazard Model

Govert E. Bijwaard
  • Corresponding author
  • Netherlands Interdisciplinary Demographic Institute (NIDI), PO Box 11650, NL-2502, AR, The Hague, The Netherlands
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/ Geert Ridder / Tiemen Woutersen
Published Online: 2013-05-28 | DOI: https://doi.org/10.1515/jem-2012-0005


Ridder and Woutersen (Ridder, G., and T. Woutersen. 2003. “The Singularity of the Efficiency Bound of the Mixed Proportional Hazard Model.” Econometrica 71: 1579–1589) have shown that under a weak condition on the baseline hazard, there exist root-N consistent estimators of the parameters in a semiparametric Mixed Proportional Hazard model with a parametric baseline hazard and unspecified distribution of the unobserved heterogeneity. We extend the linear rank estimator (LRE) of Tsiatis (Tsiatis, A. A. 1990. “Estimating Regression Parameters using Linear Rank Tests for Censored Data.” Annals of Statistics 18: 354–372) and Robins and Tsiatis (Robins, J. M., and A. A. Tsiatis. 1992. “Semiparametric Estimation of an Accelerated Failure Time Model with Time-Dependent Covariates.” Biometrika 79: 311–319) to this class of models. The optimal LRE is a two-step estimator. We propose a simple one-step estimator that is close to optimal if there is no unobserved heterogeneity. The efficiency gain associated with the optimal LRE increases with the degree of unobserved heterogeneity.

This article offers supplementary material which is provided at the end of the article.

Keywords: counting process; linear rank estimation; mixed proportional hazard; JEL Classification: C41; C14


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About the article

Corresponding author: Govert E. Bijwaard, Netherlands Interdisciplinary Demographic Institute (NIDI), PO Box 11650, NL-2502, AR, The Hague, The Netherlands

Published Online: 2013-05-28

Published in Print: 2013-07-01

Horowitz (2001, theorem 2.2) averages gn (Xi); the STATA program on our website is sufficiently fast to apply the bootstrap to most survey datasets.

The Brent’s method combines the bisection method, the secant method and inverse quadratic interpolation. The idea is to use the secant method or inverse quadratic interpolation if possible, because they converge faster, but to fall back to the more robust bisection method if necessary. The secant method can be thought of as a finite difference approximation of the Newton-Raphson method. The Powell method extends the Brent method by searching in a specific direction, rather than changing one parameter at the time.

See http://publ.nidi.nl/output/other/LRE.zip for the program and http://publ.nidi.nl/output/other/LRE_help.pdf for the help file.

In the MLE for models with duration dependence, we do not need the standard identification restriction that the unobserved heterogeneity term has mean one because the baseline hazard is normalized to be equal to 1 in the first interval.

The Gâteaux derivative is a directional derivative; let

and η>0 then df(x, a)=limη0[{f(x+)–f(x)}/η].

Our calculations were done in Gauss 6.0 on 3 parallel computers: a Pentium 2.1 PC, a Pentium 2.8 PC and a Pentium 2.0 laptop. The calculations took about 9 weeks of CPU time.

The LRE with a duration dependence on 10 intervals for a sample size of 500 did not converge in seven of the experiments. The average is therefore base on 93 experiments instead of 100.

The results for the parameters of the piecewise constant duration dependence, α2 and α3, are given in Tables A3 and A4 in Appendix A.

The Doob-Meyer decomposition theorem is a theorem in stochastic calculus stating the conditions under which a submartingale may be decomposed in a unique way as the sum of a martingale and a continuous increasing process, see Meyer (1963) and Protter (2005).

Citation Information: Journal of Econometric Methods, Volume 2, Issue 1, Pages 1–23, ISSN (Online) 2156-6674, ISSN (Print) 2194-6345, DOI: https://doi.org/10.1515/jem-2012-0005.

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