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How to estimate the parameters of the variable from which we make draws in MSLE. For example, Consider (example from Cameron and Trivedi-12.4.5 Unobserved heterogeneity) $$ y_i=\theta +u_i+\epsilon$$

I am interested in getting value of theta to get that I use MSLE. For using MSLE we make draws from u but how do I approach the problem if I don't know the parameters of u.

Meaning if I know the distribution of u but it has unknown parameter how do I get theta and parameters of u?

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Lets say you think $u_i$ is normally distributed with mean 0 and an unknown standard deviation $\sigma$. Now you have two parameters in your likelihood function and you maximize both. You start with starting values. Now you draw your sample of $u_i$ given that starting value. The next iteration you will get improved guesses of these parameters, and use those for drawing your $u_i$. You continue iterating till convergence.

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    $\begingroup$ Just to be clear: I draw values of u now I just don't know theta. Now I enter the log likelihood loop to get best theta for that particular draw of u. I take an average for all the draws. I get the best possible theta. But the problem is : I assume some sigma for my draws, how do I get to real sigma. Can you be more descriptive in your explanation? $\endgroup$ – econometrics_things Oct 18 '17 at 9:37
  • $\begingroup$ Sigma isn't fixed in this algorithm, it to is updated each iteration. $\endgroup$ – Maarten Buis Oct 18 '17 at 17:26

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