We give semiparametric identification and estimation results for econometric
models with a regressor that is endogenous, bound censored, and selected; it
is called a Tobin regressor. First, we show that the true parameter value is set-
identified and characterize the identification sets. Second, we propose novel es-
timation and inference methods for this true value. These estimation and infer-
ence methods are of independent interest and apply to any problem possessing
the sensitivity structure, where the true parameter value is point-identified con-
ditional on some nuisance parameter values that are set-identified. By fixing the
nuisance parameter value in some suitable region, we can proceed with regular
point and interval estimation. Then we take the union over nuisance parameter
values of the point and interval estimates to form the final set estimates and con-
fidence set estimates. The initial point or interval estimates can be frequentist or
Bayesian. The final set estimates are set-consistent for the true parameter value,
and confidence set estimates have frequentist validity in the sense of covering
this value with at least a prespecified probability in large samples. Our procedure
may be viewed as a formalization of the sensitivity analysis in the sense of Leamer
(1985). We apply our identification, estimation, and inference procedures to study
the effects of changes in housing wealth on household consumption. Our set es-
timates fall in plausible ranges, significantly above low ordinary least squares es-
timates and below high instrumental variables estimates that do not account for
the Tobin regressor structure.
Keywords. Partial identification, endogenous censoring, sample selection.
JEL classification. C14, C24, C26.