Ch. 2: Self-Selection Models in Corporate Finance 45
underlying a firm’s choice and testing its significance is a test of whether private infor-
mation possessed by a firm explains ex-post outcomes. In fact, a two-step procedure
most commonly used to estimate selection models follows this logic.^6
Our main purpose of incorporating the above discussion of the Heckman model is
to highlight the dual nature of self-selection “corrections”. One can think of them as
a way of accounting for a statistical problem. There is nothing wrong with this view.
Alternatively, one can interpret self-selection models as a way of testing private in-
formation hypotheses, which is perhaps an economically more useful perspective of
selection models in corporate finance. Selection models are clearly useful if private in-
formation is one’s primary focus, but even if not, the models are useful as means of
controlling for potential private information effects.
2.3. Specification issues
Implementing selection models in practice poses two key specification issues: the need
for exclusion restrictions and the assumption that error terms are bivariate normal. While
seemingly innocuous, these issues, particularly the exclusion question, are often impor-
tant in empirical applications, and deserve some comment.
2.3.1. Exclusion restrictions
In estimating equations(3)–(5), researchers must specify two sets of variables: those de-
termining selection (Z) and those determining the outcomes (X). An issue that comes up
frequently is whether the two sets of variables can be identical. This knotty issue often
crops up in practice. For instance, consider the self-selection eventEin equations(3)
and (4)as the decision to acquire a target and suppose that the outcome variable in
equation(5)is post-diversification productivity. Variables such as firm size or the relat-
edness of the acquirer and the target could explain the acquisition decision. The same
variables could also plausibly explain the ex-post productivity gains from the acquisi-
tion. Thus, these variables could be part of bothZandXin equations(3)–(5). Similar
arguments can be made for several other explanatory variables: they drive firms’ deci-
sion to self-select into diversification and the productivity gains after diversification. Do
we need exclusion restrictions so that there is at least one variable driving selection, an
instrument inZthat is not part ofX?
Strictly speaking, exclusion restrictions are not necessary in the Heckman selection
model because the model is identified by non-linearity. The selection-adjusted outcome
regression(10)regressesYonXandλC(Z′γ).IfλC(.)were a linear function ofZ,
we would clearly need some variables inZthat are not part ofXor the regressors
(^6) Step 1 estimates the probit model(3) and (4)to yield estimates ofγ,sayγˆ, and hence the private infor-
mation functionλC(Ziγ)ˆ. In step 2, we substitute the estimated private information in lieu of its true value in
equation(10)and estimate it by OLS. Standard errors must be corrected for the fact thatγis estimated in the
second step, along the lines ofHeckman (1979), Greene (1981),andMurphy and Topel (1985).