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st: ML estimation of equation system with random regressors


From   "David GIVENS" <givens@econ.bsos.umd.edu>
To   <statalist@hsphsun2.harvard.edu>
Subject   st: ML estimation of equation system with random regressors
Date   Thu, 22 Mar 2007 12:19:34 -0400

This is my first posting to Statalist. Please excuse errors of
inexperience! I am looking for broad advice on an estimating strategy.
SETUP: 
I am trying to estimate the following system of equations model by
maximum likelihood (ML):
   y_jk = a_k + b_k*(g_j + e_jk)       j = 1, ..., J; k = 1,...,K 
        where,
  g_j ~ N(0,1), iid
  e_jk ~ N(0, s_k), iid
  E(g_j * e_jk) = 0
  E(e_jk*e_j'k')=0, if j not equal j' OR k not equal k'
  W (K x K) = VC(b_k*(g_j + e_jk)) = bb' + diag{b_k^2*s_k^2}
    - K equations, 3K parameters - an alpha, beta and sigma for each
equation 
    - J observations per equation 
    - Observations within the k-th equation are independent
    - j-th observations across equations are correlated (they have the
same random component, g_j).
    - the model is identified for K >= 3
    - I have the likelihood function written down - it meets STATA's
"lf" restriction
    - no non-stochastic regressors (aside from column of ones)!
    - use seemingly unrelated regression (SUR) model?
    
    QUESTION: 
    Is SUR the paradigm to use for this model, given that I want to go
ML? 
           -If so, is there a way to explicitly define the elements of
STATA's "sigma" matrix in the likelihood equation so it equals W
(defined above)? 
           -If not, is there another, more general technique of ML
estimation for systems of equations I can use? 
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