st: recursive bivariate probit and insignificant rho

[David Quinn <dxquinnx@gmail.com>]

Hello,

I am running the following recursive bivariate probit:

Y1 = Y2 + X1 + X3 (outcome equation)
Y2 = X1 + X2 (selection equation)

When I run the biprobit, I get an insignificant rho, which seemingly
indicates little to no correlation between unobserved factors
affecting decision Y1 and decision Y2.  In other words, this would
tell me that I need not correct for sample selection, and that the
univariate probits are independent and can be estimated separately
(see Greene 2003, 712).

However, the results are slightly different in the univariate and
bivariate probits, and specifically regarding two variables of
interest: X1 and Y2.  When I run the independent univariate probits:
A.) X1 has a statistically significant effect on Y2, B.) X1 has a
statistically insignificant effect on Y1, and C.) Y2 has a
statistically insignificant effect on Y1.  But when I run the
biprobit, relationships B and C become statistically significant (the
signs/directionality do not change, but the relationships become
statistically significant).  And as I said, this is despite the fact
that rho is insignificant in the biprobit.  The biprobit model's LR
chi-square value is large and statistically significant, indicating
that the model is fit well.

So, my question is what to do for a case like this?  When rho is
insignificant in a biprobit, are the estimates produced by biprobit
more accurate or more biased relative to the estimates produced by the
independent univariate probits?  It seems that most studies I have
found just stick with the independent univariate probits when the
biprobit demonstrates no selection effects.  But in my case, the
results change slightly between the biprobit and univariate probit for
two variables of interest (the other effects remain the same).

Thanks in advance for any insight!
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