Hi Alejandra,
So if I understand your model specification correctly, it is along the lines of some mediational model (let's say the underlying response variate y2* is the mediator between some IV X and the ultimate outcome y1*):
y1* = alpha1 + betaX + betaY2* + e1
y2* = alpha2 + betaX + e2
Then you are asking if a test of cov(e1,e2) = 0 under the 'biprobit' is equivalent to a Hausman test of endogeneity. Indeed, the Knapp and Seakes (1998) paper is oft-cited in the bivariate probit context for this purpose, but I am not 100% sure if it can be invoked in this case. (I could be wrong). Mediational models bring in a number of issues about the proper direction of causal flow among the variables in the system. Of course, cov(e1,e2) = 0 is necessary for getting good overall empirical model fit (implying, for example, that no common causes of y1* and y2* have been omitted), and thus unbiased estimates, but I don'[t know if this automatically implies endogeneity in the sense that cov(y2*,e1) = 0 (unless X is being regarded as an instrument here), and I believe that more is involved for convincingly demonstrating the properness of the postulated causal chain. I would suggest having a look at:
James, L.R., Mulaik, S.A., & Brett, J.M. (2006). A tale of two methods. Organizational Research Methods, 9(2), 233-244.
Further, quantifying the degree of mediation in probit (and logit) models is more complicated than in the linear case:
MacKinnon, D.P., Lockwood, C.M., Brown C.H., Wang W., & Hoffman M. (2007). The intermediate endpoint effect in logistic and probit regression. Clinical Trials, 4(5), 499-513.
For more general readings on mediation see:
MacKinnon, D.P. (2008). Introduction to statistical mediation analysis. Mahwah, NJ: Erlbaum.
MacKinnon, D. P., & Fairchild, A. J. (2009). Current directions in mediation analysis. Current Directions in Psychological Science, 18, 16-20.
Hope this is helpful, and that I have understood the situation correctly.
Cam
> Date: Sun, 9 Aug 2009 01:51:20 +0200
> Subject: st: Endogeneity in Bivariate Probit
> From: [email protected]
> To: [email protected]
>
> Dear Statalist,
> I found on the Statist archive that Knapp and Seaks argue that a
> likelihood-ratio test of the correlation coefficient of the residuals
> (rho) can be used as an endogeneity test.
> On the other hand I read that if "the second dependent variable, y2,
> appears on the right-hand side of the first equation, this is a recursive,
> simultaneous-equations model. Surprisingly, the endogenous nature of one
> of the variables on the right-hand side of the first equation can be
> ignored in formulating the log-likelihood" (Greene, 2002, pp.715).
> I run my model for three countries and only in one case I obtained these
> results:
>
> Likelihood-ratio test of rho = 0 : chi2(1) = 2,1962
> Prob> chi(2) = 0,1383
>
> Should I consider that y2 variable have passed the endogeneity test?
> Thanks in advance and regards,
> Alejandra Molina
>
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