48 K. Li and N.R. Prabhala
by two regressions. The complete model is as follows:
C=E≡Ziγ+ηi> 0 , (11)
C=NE≡Ziγ+ηi 0 , (12)
YE,i=XE,iβE+E,i, (13)
YNE,i=XNE,iβNE+NE,i, (14)whereC∈{E,NE}. Along with separate outcome regression parameter vectorsβEand
βNE, there are also two covariance coefficients for the impact of private information
on outcomes, the covariance between private informationηandEand that between
ηandNE. Two-step estimation is again straightforward, and is usually implemented
assuming that the errors{ηi,E,i,NE,i}are trivariate normal.^9
Given the apparent flexibility in specifying two outcome regressions(13) and (14)
compared to the one outcome regression in the standard selection model, it is natural to
ask why we do not always use the switching regression specification. There are three
issues involved. First, theory should say whether there is a single population regression
whose LHS and RHS variables are observed conditional on selection, as in the Heckman
model, or whether we have two regimes in the population and the selection mechanism
dictates which of the two we observe. In some applications, the switching regression is
inappropriate: for instance, it is not consistent with the equilibrium modeled inAcharya
(1988). A second issue is that the switching regression model requires us to observe
outcomes of firms’ choices in both regimes. This may not always be feasible because we
only observe outcomes of firms self-selectingEbut have little data on firms that choose
not to self-select. For instance, if we were analyzing stock market responses to merger
announcements as inEckbo, Maksimovic and Williams (1990), implementing switching
models literally requires us to obtain a sample of would-be acquirers that had never
announced to the market and the market reaction on the dates that the markets realize
that there is no merger forthcoming. These data may not always be available (Prabhala,
1997 ).^10 A final consideration is statistical power: imposing restrictions such as equality
of coefficients{β, π}forEandNEfirms (when valid), lead to greater statistical power.
A key advantage of the switching regression framework is that we obtain more useful
estimates of (unobserved) counterfactual outcomes. Specifically, if firmichoosesE,
we observe outcomeYE,i. However, we can ask what the outcome might have been had
(^9) Write equations(13) and (14)in regression form as
YC,i=XC,iβC+πCλC(Ziγ), (15)
whereC∈{E,NE}. The two-step estimator follows: the probit model(11) and (12)gives estimates ofγand
hence the inverse Mills ratioλC(.), which is fed into regression(15)to give parameters{βE,βNE,πE,πNE}.
As before, standard errors in the second step regression require adjustment becauseλC(Zγ)ˆ is a generated
regressor (Maddala, 1983, pp. 226–227).
(^10) Li and McNally (2004)andScruggs (2006)describe how we can use Bayesian methods to update priors
on counterfactuals. More details on their approach are given in Section6.