42 K. Li and N.R. Prabhala
methods are often motivated by the fact that they yield easily interpretabletreatment
effects, selection methods also estimate treatment effects with equal ease. Our review
of methodology closes by briefly touching upon fixed effect models in Section5 and
Bayesian approaches to selection in Section6.
- Self-selection: The statistical issue
To set up the self-selection issue, assume that we wish to estimate parametersβof the
regression
Yi=Xiβ+i (1)for a population of firms. In equation(1),Yiis the dependent variable, which is typically
anoutcomesuch as profitability or return. The variables explaining outcomes areXi,
and the error term isi.Ifisatisfies usual classical regression conditions, standard
OLS/GLS procedures consistently estimateβ.
Now consider a sub-sample of firms who self-select choiceE. For this sub-sample,
equation(1)can be written as
Yi|E=Xiβ+i|E. (2)
The difference between equations(2) and (1)is at the heart of the self-selection prob-
lem. Equation(1)is a specification written for the population but equation(2)is written
for a subset of firms, those that self-select choiceE. If self-selecting firms are not ran-
dom subsets of the population, the usual OLS/GLS estimators applied to equation(2),
are no longer consistent estimators ofβ.
Accounting for self-selection consists of two steps. Step 1 specifies a model for self-
selection, using economic theory to model why some firms selectEwhile others do
not. While this specification step is not often discussed extensively in applications, it
is critical because the assumptions involved ultimately dictate what econometric model
should be used in the empirical application. Step 2 ties the random variable(s) driving
self-selection to the outcome variableY.
- The baseline Heckman selection model
2.1. The econometric model
Early corporate finance applications of self-selection are based on the model analyzed
inHeckman (1979). We spend some time developing this model because most other
specifications used in the finance literature can be viewed as extensions of the Heckman
model in various directions.
In the conventional perspective of self-selection, the key issue is that we have a regres-
sion such as equation(1)that is well specified for a population but it must be estimated