Ch. 2: Self-Selection Models in Corporate Finance 43
using sub-samples of firms that self-select into choiceE. To estimate population para-
meters from self-selected subsamples, we first specify a self-selection mechanism. This
usually takes the form of a probit model in which firmichoosesEif the net benefit
from doing so, a scalarWi, is positive. Writing the selection variableWias a function
of explanatory variablesZi, which are assumed for now to be exogenous,^4 we have the
system
C=E≡Wi=Ziγ+ηi> 0 , (3)
C=NE≡Wi=Ziγ+ηi 0 , (4)
Yi=Xiβ+i, (5)whereZidenotes publicly known information influencing a firm’s choice,γis a vec-
tor of probit coefficients, andηiis orthogonal to public variablesZi. In the standard
model,Yi is observed only when a firm picks one ofEorNE(but not both), so
equation(5)would require the appropriate conditioning. Assuming thatηiandiare
bivariate normal, the likelihood function and the maximum likelihood estimators for
equations(3)–(5)follow, although a simpler two-step procedure (Heckman, 1979, and
Greene, 1981) is commonly used for estimation. Virtually all applied work is based on
the bivariate normal structure discussed above.
2.2. Self-selection and private information
In the above setup, self-selection is a nuisance problem. We model it because not do-
ing so leads to inconsistent estimates of parametersβin regression(1). Self-selection
is, by itself, of little interest. However, this situation is frequently reversed in corpo-
rate finance, because tests for self-selection can be viewed as tests of private infor-
mation theories. We develop this point in the context of theHeckman (1979)model
outlined above, but we emphasize that this private information interpretation is more
general.
We proceed as follows. Following a well-established tradition in econometrics, Sec-
tion2.2.1presents selection as an omitted variable problem. Section2.2.2interprets
the omitted variable as a proxy for unobserved private information. Thus, including
the omitted self-selection variable controls for and tests for the significance of private
information in explaining ex-post outcomes of corporate finance choices.
2.2.1. Selection: An omitted variable problem
Suppose that firmiself-selects choiceE. For firmi, we can take expectations of equa-
tion(5)and write
(^4) Thus, we preclude for now the possibility thatZincludes the outcome variableY. This restriction can be
relaxed at a cost, as we show in later sections.