Hi Alejandra,
The non-significant chi-square statistic shows that your overidentifying constraint cov(e1,e2) = 0 cannot be rejected, but I don't know for sure that it "proves" endogeneity, or that the causal specification is true.
Now, it has indeed been noted that intrumental variable and path-analytic (simultaneous equation) specifications are equivalent, parallel to what you found regarding the bivariate probit (i.e., Knapp & Seakes, 1998):
Angrist, J.D, & Krueger, A.B. (2001). Instrumental variables and the search for identification: from supply and demand to natural experiments. Journal of Economic Perspectives, 15(4), 69–85.
http://www.hec.unil.ch/hec/doctorats/phdmanagement/researchhandbook/instrumental%20variables%202.pdf
I think you are reasonably safe in concluding that the model is "a possible candidate" for the true causal process and consequently that the parameter estimates "seem trustworthy." But in causal modeling, remember that the equivalent models problem and not having direct knowledge of the true causal forces renders your claims tentative. This is why I am cautioning you in the use of the word "prove." Have a look at:
Raykov, T., & Marcoulides, G. A. (2007). Equivalent structural equation models: A challenge and responsibility. Structural Equation Modeling, 14(4), 695-700.
Hershberger, S. L. (2006). The problem of equivalent structural models. In G. R. Hancock & R. O. Mueller (Eds.), Structural equation modeling: A second course (pp. 13-41). Greenwich, CN: Information Age Publishing.
James, L. R., Mulaik, S. A., & Brett, J. M. (1982). Causal analysis: Models, assumptions and data. Beverly Hills, CA: Sage.
Cam
> Date: Sun, 9 Aug 2009 18:07:46 +0200
> Subject: st: [Fwd: Endogeneity in Bivariate Probit]
> From: [email protected]
> To: [email protected]
>
> Thank you Cameron.
> Mi model is:
> y1* = alpha1 + betaY2* + e1
> y2* = alpha2 + betaX + e2
> where y1 (0,1) is innovation in product and y2 (0,1) is innovation in
> process.
> I supossed that both inlfuence each other and put them in a system. X
> represent other factors such as foreign ownership, quality certification
> and also two dummies (for 3) different sizes.
> For two of the selected countries I received low Prob> chi(2) and for the
> third, these results.
>
> Likelihood-ratio test of rho = 0 : chi2(1) = 2,1962
> Prob> chi(2) = 0,1383
>
> I was wondering if these results imply endogenity.
> Best,
> Alejandra
>
>
> 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
>
>
>
>
>
>
> -------------------------- Messaggio originale ---------------------------
> Oggetto: Endogeneity in Bivariate Probit
> Da: [email protected]
> Data: Dom, 9 Agosto 2009 1:51 am
> A: [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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