52 K. Li and N.R. Prabhala
4.1. Treatment effects
Matching models focus on estimatingtreatment effects. A treatment effect, loosely
speaking, is the value added or the difference in outcome when a firm undergoes treat-
mentErelative to not undergoing treatment, i.e., choosingNE. Selection models such as
the switching regression specification (equations(11)–(14)) estimate treatment effects.
Their approach is indirect. In selection models, we estimate a vector of parameters and
covariances in the selection equations and use these parameters to estimate treatment
effects. In contrast, matching models go directly to treatment effect estimation, setting
aside the step of estimating parameters of regression structures specified in selection
models.
The key question in the matching literature is whether treatment effects are signifi-
cant. In the system of equations(24)–(26), this question can be posed statistically in a
number of ways.
- At the level of an individual firmi, the effectiveness of a treatment can be judged by
asking whetherE(YE,i−YNE,i)=0. - For the group of treated firms, the effectiveness of the treatment for treated firms is
assessed by testing whether thetreatment effect on treated(TT), equals zero, i.e.,
whetherE[(YE−YNE)|C=E]=0. - For the population as a whole whether treated or not, we test the significance of the
average treatment effect(ATE) by examining whetherE(YE−YNE)=0.
The main issue in calculating any of the treatment effects discussed above, whether by
selection or matching models, is the fact that unchosen counterfactuals are not observed.
If a firmichoosesE, we observe outcome of its choiceYE,i. However, because firm
ichoseE, we do not explicitly observe the outcomeYNE,ithat would occur had the
firm instead made the counterfactual choiceNE. Thus, the differenceYE,i−YNE,iis
never directly observed for any particular firmi, so its expectation—whether at the
firm level, or across treated firms, or across treated and untreated firms—cannot be
calculated directly. Treatment effects can, however, be obtained via selection models
or by matching models, using different identifying assumptions. We discuss selection
methods first and then turn to matching methods.
4.2. Treatment effects from selection models
Self-selection models obtain treatment effects by first estimating parameters of the sys-
tem of equations(24)–(26). Given the parameter estimates, it is straightforward to
estimate treatment effects described in Section4.1, as illustrated, e.g., in Section3.1
for the switching regression model. The key identifying assumption in selection mod-
els is the specification of the variables entering selection and outcome equations, i.e.,
variablesXandZin equations(24)–(26).
Two points deserve emphasis. The first is that the entire range of selection models dis-
cussed in Section2 through Section3.2can be used to estimate treatment effects. This
point deserves special mention because in received corporate finance applications, the