Identification and decompositions in probit and logit models
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Identification and decompositions in probit and logit models Chung Choe1 · SeEun Jung2 · Ronald L. Oaxaca3,4,5,6,7 Received: 15 May 2017 / Accepted: 18 April 2019 © Springer-Verlag GmbH Germany, part of Springer Nature 2019
Abstract Probit and logit models typically require a normalization on the error variance for model identification. This paper shows that in the context of decompositions of group sample mean proportions, error variance normalizations preclude estimation of the effects of group differences in the latent variable model parameters. This problem applies equally to decompositions of group differences in the underlying latent outcome variable. An empirical example is provided for a probit model in which the error variances are identified if an underlying random utility/latent variable theoretical model contains a variable whose coefficient is equal to 1. In the resulting probit model, for example, the coefficient of this variable is the reciprocal of the error term standard deviation. From this information, one can back out estimates of all of the coefficients in the underlying random utility/latent variable model and thereby allow the effects of group differences in the latent variable model parameters to be estimated. Keywords Decompositions · Probit · Logit · Identification JEL Classification C35 · J16 · D81 · J71
1 Introduction The objective of decomposition methodology is to identify and estimate the separate contributions of differences in characteristics (explained) and differences in parameters (unexplained) when accounting for mean differences in outcome variables for two population groups. Standard decomposition approaches appear in the context of a
We thank Michael Lechner and two anonymous referees for comments that greatly improved the paper. This work is supported by the research fund of Hanyang University (HY-2016-00000002971).
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SeEun Jung [email protected]
Extended author information available on the last page of the article
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linear model. For nonlinear models, some modification of the standard decomposition methodology is required. Our contribution should be viewed as a specific example in the broader literature on identification in detailed wage decompositions (Oaxaca and Ransom 1999). The central objective of this paper is to draw attention to an important issue associated with identifying explained and unexplained decomposition components arising from the estimated parameters in probit/logit models. The issue here is the dependence of this decomposition identification on identification of the variance of the error term in the latent variable model. While the normalization of the error term variance in probit/logit models is well known and is innocuous for many applications, awareness of the implications of the routine normalization for decomposition analysis is not very apparent. Below we describe how identification arises when a theoretical random utility/latent variable model with a normally distributed random error term happens to restrict a coefficient
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