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The basic model is as follows:

[tex] y_{1i}^{*}=\alpha z_i+\eta_{1i} [/tex]

[tex] y_{2i}^{*}=\beta x_i+\gamma y_{1i}+\eta_{2i} [/tex]

Where [tex]y_{1i}^{*}[/tex] and [tex] y_{2i}^{*} [/tex] are latent variables. Let [tex]1(m)[/tex] be an indicator function which equals one if [tex] m>0 [/tex] and zero otherwise. Let [tex]y_{1i}=1(y_{1i}^{*})[/tex] and [tex]y_{2i}=1(y_{2i}^{*})[/tex].

Note that it is [tex] y_{1i}[/tex] and not [tex] y_{1i}^{*} [/tex] which enters the equation for [tex] y_{2i}[/tex]. It turns out that one can estimate this model using a bivariate probit and ignoring the simultaneity (Greene 1998). I.e. it can be estimated using maximum likelihood by using [tex] y_{1i}[/tex] (not [tex] \hat y_{1i}^{*}[/tex], not even [tex] 1(\hat y_{1i}^{*})[/tex]) as an independent variable in the equation for [tex] y_{2i}^{*}[/tex].

A bivariate probit assumes that [tex] \eta_{\cdot i}[/tex] follow a bivariate normal distribution with mean zero, variance one and covariance [tex]\rho[/tex]. It is estimated by full information maximum likelihood. (eg. biprobit in STATA)

Please note that this is not a general finding. It just turns out that in this particular model one can safely ignore the simultaneity and straightforwardly estimate the bivariate model.

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!!References

”Gender Economics Courses in Liberal Arts Colleges: Further Results” William Greene. Journal of Economic Education, fall 1998.

”The effects of Catholic Secondary Schooling on Educational Achievement” Derek Neal, Journal of Labor Economics 1997.

Greene, W. Econometric Analysis 2001

Wooldridge, J.M. Econometric analysis of cross section and panel data MIT Press, 2002