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Another Look at the Linear Probability Model and Nonlinear Index Models

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  • Kaicheng Chen
  • Robert S. Martin
  • Jeffrey M. Wooldridge

Abstract

We reassess the use of linear models to approximate response probabilities of binary outcomes, focusing on average partial effects (APE). We confirm that linear projection parameters coincide with APEs in certain scenarios. Through simulations, we identify other cases where OLS does or does not approximate APEs and find that having large fraction of fitted values in [0, 1] is neither necessary nor sufficient. We also show nonlinear least squares estimation of the ramp model is consistent and asymptotically normal and is equivalent to using OLS on an iteratively trimmed sample to reduce bias. Our findings offer practical guidance for empirical research.

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  • Kaicheng Chen & Robert S. Martin & Jeffrey M. Wooldridge, 2023. "Another Look at the Linear Probability Model and Nonlinear Index Models," Papers 2308.15338, arXiv.org, revised Oct 2023.
    • Handle: RePEc:arx:papers:2308.15338
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    1. Joshua D. Angrist & Jörn-Steffen Pischke, 2009. "Mostly Harmless Econometrics: An Empiricist's Companion," Economics Books, Princeton University Press, edition 1, number 8769.
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    3. Horrace, William C. & Oaxaca, Ronald L., 2006. "Results on the bias and inconsistency of ordinary least squares for the linear probability model," Economics Letters, Elsevier, vol. 90(3), pages 321-327, March.
    4. Hunter, William C & Walker, Mary Beth, 1996. "The Cultural Affinity Hypothesis and Mortgage Lending Decisions," The Journal of Real Estate Finance and Economics, Springer, vol. 13(1), pages 57-70, July.
    5. Jeffrey M Wooldridge, 2010. "Econometric Analysis of Cross Section and Panel Data," MIT Press Books, The MIT Press, edition 2, volume 1, number 0262232588, April.
    6. Ruud, Paul A, 1983. "Sufficient Conditions for the Consistency of Maximum Likelihood Estimation Despite Misspecifications of Distribution in Multinomial Discrete Choice Models," Econometrica, Econometric Society, vol. 51(1), pages 225-228, January.
    7. Joel L. Horowitz & N. E. Savin, 2001. "Binary Response Models: Logits, Probits and Semiparametrics," Journal of Economic Perspectives, American Economic Association, vol. 15(4), pages 43-56, Fall.
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    1. Roychowdhury, Punarjit, 2024. "Social Identity and Depression Among the Elderly: Evidence from India," GLO Discussion Paper Series 1466, Global Labor Organization (GLO).

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