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st: RE: Likelihood Function


From   "Julian Fennema" <j.a.fennema@hw.ac.uk>
To   <statalist@hsphsun2.harvard.edu>
Subject   st: RE: Likelihood Function
Date   Thu, 24 Jul 2003 08:31:34 +0100

Chuntao,

The function -phi- is, as you say, defined for the standard normal. The
relationship between this -phi- and the probability distribution
function for a generalised normal, i.e. where sigma=/=1, is 
(1/sigma)phi((y-x*beta)/sigma).
Note that the (1/sigma) comes from the differentiation of the normalised
cumulative density function Phi((y-x*beta)/sigma) with respect to
x*beta.
Take logs, and you're there.

Julian

-----Original Message-----
From: owner-statalist@hsphsun2.harvard.edu
[mailto:owner-statalist@hsphsun2.harvard.edu] On Behalf Of Lee Chuntao
Sent: 24 July 2003 06:13
To: statalist@hsphsun2.harvard.edu
Subject: st: Likelihood Function

Dear Listers,

     In page 29 of  Maximum Likelihood Estimation with Stata (Gould and
Sribney 1999), the likelihood function for the linear regression model
is
written as:

                   lnL=SUM (ln (phi((y-x*beta)/sigma)) - ln(sigma))
where phi() is the standard normal PDF.

My Question is: How the last term, ln(sigma), comes to the likelihood
function?


Can you knidly give me some ideas?


thanks in advance


Chuntao

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