Dear colleagues,
I am dealing with two bivariate probit models, in which y1 and z1 form
the first one, and y2 and z2 form the second (In X1 and X2, some
variables are overlapped, but some are different do identify from each
other. So do the X3 and X4). A striking feature is (1) and (3) are
simultaneous.
y1= a1y2+X1b1+u1 (1)
z1= X2b2+u2 (2)
y2=a2y1+X3b3+v1 (3)
z2=X4b4+v2 (4)
I am primarily interested in a1 and a2, as well as the predict
probabilities of y1 and y2. According to the theory (or my
hypothesis), these two bivariate probit models are correlated and
should be estimated simultaneously. The -mvprobit- is available in
Stata to estimate the multiprobit estimation.
The difficulties with –mvprobit- (multivariate probit) are as following:
1) My main interest, the relationship y1 and y2, are carried by a1 and
a2. But does is make sense to run –mvprobit- with the structural
specification to get a1 and a2? For example, mvprobit (y1=y2 X1)
(z1=X2) (y2= y1 X3 ) (z2=X4)
2) Since structural equations are used, identification in the system
is tricky. I am not sure if the rank condition can be met. I have to
ensure X1 and X3 contain exogenous variables to identify.
(theoretically, u1, u2 ,v1 ,v2 are correlated because u1 correlates to
u2, and correlates to v1 as well. So u2 correlates to v1 and v2. But
this correlation is not strong as that between u1 and u2, as well as
u1 and v1, which can be justified by economics)
3) Since (1) and (3) are correlated and I am using the structural
equations, how can I put the constraint a1*a2!=1 in the system?
(Wooldridge 2003, p506) Wooldridge calls a special attention at this.
Alternatively, we can treat either y1 or y2 as instrumental variable,
to induce exogenous variables .But I have some reasons to reject this
approach:
1) the relationship between y1 and y2, as shown in structural
equations, is not available any more in instrumental variable
approach. But that is my 100% interest.
2)I am not sure if -biprobit- can run the instrumental variable.
Personally, I prefer to -mvprobit-, unless a1 and a2 are available. I
have been annoyed by this for long time. Hope you can give me some
comments. Tons of thanks.
p.s., I am using Stata version 8.2.
--
Xiangping JIA
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