Ch. 2: Self-Selection Models in Corporate Finance 69
of self-selection in explaining this choice, and in particular, whether firms choose the
alternative that minimizes their underwriting costs.
LetWdenote the decision to use warrants to compensate underwriters andNif not,
subscriptswandndenote the costs if warrants are used or not, respectively,Udenote
underpricing costs andCthe other costs of going public. If firmichooses underwriter
warrant compensation, we observe the pair{Uwi,Cwi}while we observe{Uni,Cni}if it
chooses just straight cash compensation. The key self-selection issue is that we observe
the choice made by firmibut not the costs of the alternative not chosen by firmi.
Without knowing the unchosen counterfactuals, we cannot tell how much a company
saved by choosing to include or exclude warrants to compensate its underwriters.
Dunbar models the decision to use warrants using a probit model
W=ξ(Uni+Cni−Uwi−Cwi)−εi> 0 , (57)N=ξ(Uni+Cni−Uwi−Cwi)−εi 0. (58)The expression in parentheses in equation(57)is the reduction in offering costs if war-
rants are used as compensation instead of straight cash compensation. Each component
of costs is written as a function of observables and unobservables as follows:
Uni=Xniβn+εuni, (59)
Uwi=Xwiβw+εuwi, (60)Cni=Zniγn+εcni, (61)Cwi=Zwiγw+εcwi. (62)Assuming that the errors in equations(59)–(62)are i.i.d. normal but potentially cor-
related with the probit error term, Dunbar’s system is a version of theRoy (1951)
self-selection model.
Dunbar reports that variables such as offering size, underwriter reputation, and a
hot issue dummy explain underpricing in the warrant and cash compensation sam-
ples. The self-selection term is significant in the non-warrant sample but not in the
warrant compensation sample. Most interesting are Dunbar’s estimates of unobserved
counterfactuals. For firms that do not use warrants, underpricing (other costs) would
be 11.6% (19.2%) on average had warrants been used compared to actual underpricing
(other costs) of 12.8% (9.8%). For firms that do use warrants, underpricing (other costs)
would be 36.4% (14.6%) if warrants had not been used, compared to actual costs of
23.3% (23.9%). While warrants are associated with high underpricing in reduced form
cross-sectional regressions, it is incorrect to conclude that warrants result in higher
underpricing. Estimates of the self-selection model indicates that the use of warrants
actuallyreducesunderpricing compared to what it would be without warrants. Firms
appear to use warrants to reduce underpricing costs.