Testing identifying assumptions in Tobit models

We develop testable implications for the identifying assumptions of Tobit and IV-Tobit models: linear index, (joint) normality of errors, treatment (instrument) exogeneity, and relevance. The new testable equalities can detect all possible observable violations of the identifying conditions. The proposed test procedure for the model's validity uses existing inference methods for intersection bounds. Simulations suggest adequate test size and power in detecting exogeneity and error structure violations. We review and propose alternatives to partially identify the parameters of interest under less restrictive assumptions. We revisit a study of married women's labor supply in Lee (1995) to demonstrate the test's practical implementation. We qualitatively replicate their original findings, but our validity test rejects the IV-Tobit model. Estimating our proposed robust lower bound, we find that an additional $1,000 in other household income cannot reduce female labor supply by more than 4.2 hours annually, but cannot rule out that the effect is zero.