The paper studies the partial identifying power of structural single equation threshold crossing models for binary responses when explanatory variables may be endogenous. The paper derives the sharp identified set of threshold functions for the case in which explanatory variables are discrete and provides a constructive proof of sharpness. There is special attention to a widely employed semiparametric shape restriction which requires the threshold crossing function to be a monotone function of a linear index involving the observable explanatory variables. It is shown that the restriction brings great computational benefits, allowing direct calculation of the identified set of index coefficients without calculating the nonparametrically specified threshold function. With the restriction in place the methods of the paper can be applied to produce identified sets in a class of binary response models with mis-measured explanatory variables. -- Binary Response ; Endogeneity ; Incomplete models ; Index Restrictions ; Instrumental variables ; Measurement Error Models ; Partial Identification ; Probit Models ; Shape Restrictions ; Threshold Crossing Models

We examine the roles of sample initial conditions and unobserved individual effects in consistent estimation of the dynamic binary response panel data model. Different specifications of the model are estimated using female welfare and labor force participation data from the Survey of Income and Program Participation (SIPP). These include alternative random effects models, in which the conditional distributions of both the unobserved heterogeneity and the initial conditions are specified, and fixed effects conditional logit models that make no assumptions on either distribution. There are several findings. First, the hypotheses that the sample initial conditions are either exogenous or in equilibrium are rejected by the data. Misspecification of the initial conditions results in drastically overstated estimates of the state dependence and understated estimates of the short- and long-run effects of children on labor force participation. The fixed effects conditional legit estimates are similar to the estimates from the random effects model that is flexible with respect to both the initial conditions and the correlation between the unobserved heterogeneity and the covariates. Heterogeneity appears to explain about 50% and 70% of the overall persistence in welfare and labor force participation, respectively. In addition, for female labor force participation, there is evidence that fertility choices are correlated with both unobserved heterogeneity and pre-sample participation histories.