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st: loglikelihood


From   Christopher F Baum <baum@bc.edu>
To   statalist@hsphsun2.harvard.edu
Subject   st: loglikelihood
Date   Sat, 1 Nov 2003 12:49:34 -0500

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

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