Ch. 2: Self-Selection Models in Corporate Finance 51
those studying unionism and the returns to education (seeMaddala, 1983, Chapter 8),
applications in finance are of very recent origin.
The structural self-selection model clearly generalizes every type of selection model
considered before. The question is why one should not always use it. Equivalently, what
additional restrictions or demands does it place on the data? Because it is a type of the
switching regression model, it comes with all the baggage and informational require-
ments of the switching regression. As in simultaneous equations systems, instruments
must be specified to identify the model. In the diversification example at the begin-
ning of this section, the identification requirement demands that we have at least one
instrument that determines whether a firm diversifies but does not determine the ex-
post productivity of the diversifying firm. The quality of one’s estimates depends on
the strength of the instrument, and all the caveats and discussion of Section2.3.1apply
here.
- Matching models and self-selection
This section reviews matching models, primarily those based on propensity scores.
Matching models are becoming increasingly common in applied work. They represent
an attractive means of inference because they are simple to implement and yield read-
ily interpretable estimates of “treatment effects.” However, matching models are based
on fundamentally different set of assumptions relative to selection models. Matching
models assume that unobserved private information is irrelevant to outcomes. In con-
trast, unobserved private information is the essence of self-selection models. We discuss
these differences between selection and matching models as well as specific techniques
used to implement matching models.
To clarify the issues, consider the switching regression selection model of Section3.1,
but relabel the choices to be consistent with the matching literature. Accordingly, firms
aretreatedand belong to groupEoruntreatedand belong to groupNE. This assignment
occurs according to the probit model
pr(E|Z)=pr(Zγ+η) > 0 , (24)whereZdenotes explanatory variables,γis a vector of parameters and we drop firm
subscriptifor notational convenience. The probability of being untreated is 1−pr(E|Z).
We write post-selection outcomes asYEfor treated firms andYNEfor untreated firms,
and for convenience, write
YE=XEβE+E, (25)
YNE=XNEβNE+NE, (26)where (again suppressing subscripti)Cdenotes error terms,XCdenotes explanatory
variables,βCdenotes parameter vectors, andC∈{E,NE}. We emphasize that the basic
setup is identical to that of a switching regression of Section3.1.