Identification and decompositions in probit and logit models

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.