Al Feiveson <[email protected]> wrote that the estimate of the
covariance matrix of the coefficient errors in a random coefficient model
produced by -xtrchh- on a simulated sample was not very close to its true
value. He also noted that the estimate of the covariance matrix from the
sample produced by MLWIN was closer to its true value.
Since I do not know what MLWIN is doing, I cannot address those results
specifically. However, I can explain why it is possible to obtain more
efficient estimates of this covariance matrix than the ones provided in
-xtrchh-.
As noted on page 188 of the [XT] manual in Stata 8 (page 432 of [Su-Z] in
Stata 7), the natural estimator of the covariance matrix,
1 1
hat(Sigma) = ------- A - --- B
m-1 m
is not guaranteed to be nonnegative definite. (See the manual for the
details of A and B.) For this reason, Swamy (1971) suggested using a
consistent estimator that drops the second term. This estimator, which is
used by -xtrchh-, is clearly less precise other estimators but it is
consistent.
While I suspect that the approximation mentioned above is the cause of the
lack of precision, it should be noted that -xtrchh- provides a Generalized
Least Squares (GLS) estimator of the parameters of the random coefficients
model. There are other estimators, e.g. Maximum Likelihood, that would
produce more efficient estimates, conditional on correct specification.
--David
[email protected]
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