50 K. Li and N.R. Prabhala
YNE,iYE,i, we observe{NE,YNE,i}. The full model is
C=E≡YE,i>YNE,i, (16)
C=NE≡YE,iYNE,i, (17)
YE,i=XiβE+E,i, (18)
YNE,i=XiβNE+NE,i, (19)where the’s are (as usual) assumed to be bivariate normal. The Roy model is no more
demanding of the data than standard selection models. Two-step estimation is again
fairly straightforward (Maddala, 1983, Chapter 9.1).
The Roy selection mechanism is rather tightly specified on two dimensions. One,
the model exogenously imposes the restriction that firms selectingEwould experience
worse outcomes had they chosenNEand vice versa. This is often plausible. However,
it is unclear whether this should be a hypothesis that one wants to test or a restriction
that one imposes on the data. Two, the outcome differential is theonlydriver of the
self-selection decision in the Roy setup. Additional flexibility can be introduced by
loosening the model of self-selection. This extra flexibility is allowed in models to be
described next, but it comes at the price of requiring additional exclusion restrictions
for model identification.
3.2.2. Structural self-selection models
In the standard Heckman and switching regression models, the explanatory variables in
the selection equation are exogenous. At the other end of the spectrum is the Roy model
of Section3.2.1, in which self-selection is driven solely by the endogenous variable. The
interim case is one where selection is driven by both exogenous and outcome variables.
This specification is
C=E≡Ziγ+δ(YE,i−YNE,i)+ηi> 0 , (20)
C=NE≡Ziγ+δ(YE,i−YNE,i)+ηi 0 , (21)
YE,i=XiβE+E,i, (22)
YNE,i=XiβNE+NE,i. (23)The structural model generalizes the switching regression model of Section3.1,byin-
corporating the extra explanatory variableYE,i−YNE,i, the net outcome gain from
choosingEoverNE, in the selection decision, and generalizes the Roy model by per-
mitting exogenous variablesZito enter the selection equation. Estimation of the system
(20)–(23)follows the route one typically treads in simultaneous equations systems
estimation—reduced form estimation followed by a step in which we replace the de-
pendent variables appearing in the RHS by their fitted projections. A trivariate normal
assumption is standard (Maddala, 1983, pp. 223–239). While structural self-selection
models have been around for a while in the labor economics literature, particularly