Further reading the proc QLIM documentation. I assume SAS is using simulated MLE for my case. y1=1[x+y2+y3+y4+e1>0] (1) y2=1[z2+e2>0] (2) y3=z3+e3 (3) y4=z4+e4 (4) So, the simplified MLE derivation goes as: f(y1,y2,y3,y4|x, z) = f(y1, y2|y3, y4, x, z) * f(y3, y4|x, z) 1. Derive the joint density of (y3,y4) in second term of RHS is trival. 2. Derive first term require separate discussion of 4 combinations. e.g, f(y1=1,y2=1|y3,y4,x,z) = f(e1 > -x-y2-y3-y4, e2 > -z2|y3,y4,x,z) the joint density of e1, e2 condition on e3 and e4 can be derived as bivariate normal by the conditional distribution property of multivariate normal distribution. (e1, e2|e3=y3-z3, e4=y4-z4) ~ MVN(mu, sigma) So the MLE formula has more integrals compare to using WooldRidge's method (2011). I assume SAS is using the simplified version and let computer do solve the multiple integrals in the MLE function. I am seeing the output from QLIM, Seed for Monte Carlo IntegrationNumber of Draws 1923567609 20 Assuming the "Monte Carlo" method is applied for simulated MLE, but I am not sure my above understanding is correct. I hope someone can answer it. Thank you in advance.
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