This paper investigates four topics. (1) It examines the different roles played by the propensity score (the probability of selection into treatment) in matching, instrumental variable, and control function methods. (2) It contrasts the roles of exclusion restrictions in matching and selection models. (3) It characterizes the sensitivity of matching to the choice of conditioning variables and demonstrates the greater robustness of control function methods to misspecification of the conditioning variables. (4) It demonstrates the problem of choosing the conditioning variables in matching and the failure of conventional model selection criteria when candidate conditioning variables are not exogenous in a sense defined in this paper.

We study the nonparametric identification of gross output production functions under the environment of the commonly employed proxy variable methods. We show that applying these methods to gross output requires additional sources of variation in the demand for flexible inputs (e.g., prices). Using a transformation of the firm’s first-order condition, we develop a new nonparametric identification strategy for gross output that can be employed even when additional sources of variation are not available. Monte Carlo evidence and estimates from Colombian and Chilean plant-level data show that our strategy performs well and is robust to deviations from the baseline setting.

This paper examines how estimates of the deterrent effect of capital punishment depend on alternate choices of assumptions concerning the homicide process. Specific models of the homicide process represent bundles of these assumptions, which involve the unobserved heterogeneity, the relevant penalty probabilities for homicide choices, possible cross-polity parameter variation, and exchangeability between polity-time pairs that do and do not experience positive numbers of murders. We demonstrate how various assumptions have driven the conflicting findings from studies on capital punishment, and isolate a particular set of assumptions that are required to find a positive deterrent effect.