Ch. 2: Self-Selection Models in Corporate Finance 49
firmichosenNE, the unobserved counterfactual, and what the gain is from firmi’s
having made choiceErather thanNE. The switching regression framework provides
an estimate. The net benefit from choosingEis the outcome of choosingEless the
counterfactual had it chosenNE, i.e.,YE,i−YNE,i=YE,i−XiβNE−πNEλNE(Ziγ).
Theexpectedgain for firmiisXi(βE−βNE)+(πEλE(.)−πNEλNE(.)).^11 We return
to the counterfactuals issue when we deal with treatment effects and propensity scores.
We make this point at this stage only to emphasize that selection models do estimate
treatment effects. This fact is often not apparent when reading empirical applications,
especially those employing matching methods.
3.2. Simultaneity in self-selection models
The models considered thus far presume that the variablesZexplaining the self-
selection decision (equations(3) and (4)or equations(11) and (12)) are exogenous.
In particular, the bite of this assumption is to preclude the possibility that the deci-
sion to self-select choiceCdoes not directly depend on the anticipated outcome from
choosingC. This assumption is sometimes too strong in corporate finance applications.
For instance, suppose we are interested in studying the diversification decision and that
the outcome variable to be studied is firm productivity. The preceding models would
assume that post-merger productivity doesnotinfluence the decision to diversify. If
firms’ decisions to diversify depend on their anticipated productivity changes, as theory
might suggest (Maksimovic and Phillips, 2002), the assumption thatZis exogenous is
incorrect.
The dependence of the decision to self-select on outcomes and the dependence of
outcomes on the self-selection decision is essentially a problem of simultaneity. Struc-
tural selection models can account for simultaneity. We review two modeling choices.
TheRoy (1951)model places few demands on the data but it places tighter restrictions
on the mechanism by which self-selection occurs. More elaborate models are less strin-
gent on the self-selection mechanism, but they demand more of the data, specifically
instruments, exactly as in conventional simultaneous equations models.
3.2.1. The Roy model
The Roy model hard-wires the dependence of self-selection on post-selection outcomes.
Firms self-selectEorNEdepending on which of the two alternatives yields the higher
outcome. Thus,{E, YE}is observed for firmiifYE,i>YNE,i. If, on the other hand,
(^11) This expression stands in contrast to the basic Heckman setup. There, in equation(9),βE=βNEand
πE=πNE, so the expected difference isπ(λE(.)−λNE(.)). There, the sign of the expected difference is
fixed: itmustequal to the sign ofπbecause(λE(.)−λNE(.)) >0. Additionally, the expected difference in
the setup of Section2 does not vary withβor variablesXthat are not part ofZ: here, it does. In short, the
counterfactual choices that could be made but were not are less constrained in the switching regression setup.