Thank you Dr Denham. Thanks for your suggestions. My question is specific on how to derive the MLE in the case of Probit regression with binary and continuous endogenous variables. I read the SAS link you provide and Econometric Analysis of Cross Section and Panel Data (Wooldridge 2011). I derive the MLE myself,It's long and tedious and I am not sure it's correct or not. suppose I want to estimate the list of equations, 1[.] is the indicator function, I ignore intercept and coefficients for simplicity. y1=1[x+y2+y3+y4+e1>0] (1) y2=1[z2+e2>0] (2) y3=z3+e3 (3) y4=z4+e4 (4) a couple assumptions, e1 e2 are standard normal, a variance/covariance structure of e1-e4 is assumed to imply the assumption of endogeneity. The goal here is to show the joint MLE function condition on exogenous and instrumental variables. Specifically, f(y1, y2, y3, y4|x, z) = f(y1|y2,y3,y4,x,z)*f(y2,y3,y4|x,z). The second term on the right hand side, f(y2,y3,y4|x,z) = f(y2|y3,y4,x,z)*f(y3,y4|x,z) It's straightforward to derive with the properties of joint and conditional distribution of normal variables. (Wooldridge 2011) The first term f(y1|y2,y3,y4,x,z) is somehow tricky. it requires to derive 4 combinations of y1 and y2 separately, 1. f(y1=1|y2=1,y3,y4,x,z) 2. f(y1=1|y2=0,y3,y4,x,z) 3. f(y1=0|y2=1,y3,y4,x,z) 4. f(y1=0|y2=0,y3,y4,x,z) Take #1 for example, p(y1=1│y2=1,y3,y4,x,z) = E[p(e1>-x-y2-y3-y4|e2,e3,e4,x,z)|y2=1,y3,y4,x,z] p(e1>-x-y2-y3-y4|e2,e3,e4,x,z) is a function of random variable e2,e3, and e4 let g(e2,e3,e4) = p(e1>-x-y2-y3-y4|e2,e3,e4,x,z), then p(y1=1│y2=1,y3,y4,x,z) = Integal[g(e2,e3,e4)*f(e2,e3,e4|y2=1,y3,y4,x,z)] d(e2)*d(e3)*(de4) The Integal[.] is operating on the high dimensional space of e2, e3, and e4, however on e3 and e4 the Integal a single point of value and on e2 is a range to allow y2=1. Do you think this is on the right direction? Thanks for your help anyway.
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