Set identification and sensitivity analysis with Tobin regressors

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 estimation and inference methods for this true value. These estimation and inference methods are of independent interest and apply to any problem possessing the sensitivity structure, where the true parameter value is point-identified conditional 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 confidence 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 estimates fall in plausible ranges, significantly above low ordinary least squares estimates and below high instrumental variables estimates that do not account for the Tobin regressor structure.