On Saturday, November 1, 2003, at 02:33 AM, David wrote:
I agree. If a term involves parameters to be estimated, then it should
not be dropped, but retained. Sometimes statistics books present
artificial circumstances to make a point. For example, introductory
texts often present a t test for a single mean based on the assumption
that the standard deviation of the variable in the population is
known. Often, if one knows the population standard deviation, one
would also have the original data and could compute the population
mean directly, rather than having to estimate it from a simple random
sample. Incidentally, I just checked my copy of the 4th edition of
Greene's ECONOMETRIC ANALYSIS, and on page 247 he gives the
log-likelihood for a regression equation that includes the term in
sigma squared. - David Greenberg, Sociology Department, NYU
The issue here is very well presented in Greene 5th ed. On p.493 he
presents the LLF for the standard OLS regression problem, and then goes
on to show that one can work with the concentrated LLF on p.495. The
latter object is not a function of sigma^2. He refers to an Appendix,
E.3, in which he lays out the formal definition of the concentrated
LLF: partition the parameter vector into theta_1 and theta_2 such that
the solution theta_2(hat) can be written as an explicit function of
theta_1(hat). Then F(theta_1,theta_2) = F(theta_1, t(theta_1))=
F*(theta_1) where theta_2 = t(theta_1). This is naturally the case with
OLS, since one can solve the normal equations for the beta-hats,
generate residuals, and compute sigma^2(hat) as their sum of squares,
appropriately normalized. So there is surely no conflict between
Greene's description of OLS as a MLE and that of Stata's book on MLE.
Kit
*
* For searches and help try:
* http://www.stata.com/support/faqs/res/findit.html
* http://www.stata.com/support/statalist/faq
* http://www.ats.ucla.edu/stat/stata/
