I think something is still missing in David's argument. Suppose the
log-likelihood function has two additive components, L = A + B, and
suppose further that A is always a function of parameters. Now, if B
is a constant (like sqrt(_pi)), i.e., it does not contain any
parameter to be estimated, then, as David correctly pointed out,
dropping B from the likelihood function does not affect the parameter
estimates. It is because in this case parameters that maximize A also
maximize L. But if B is also a function of parameters, then dropping
B is "wrong" in general.
In Leecht's example, the second term (aka B) is -ln(sigma), and
sigma is a parameter to be estimated (not a known number!). So I
don't see how one can drop this term in the estimation.
This is what I think has happened: In Greene's formula, the density
function is not the "standard" normal density(that is, the
variance is not normalized to 1 for the density function). On the
other hand, in Stata's book, the density function is the standard
normal density. In this scenario, both formulas are correct. Maybe
Leecht can check and confirm this.
Eddy
> Equation 1 is correct. The reason some writers drop the second term
is this. For most purposes, the
> likelihood function or its log are not of interest in themselves.
One might want to compare two of them, or
> one might want to find the values of parameters that will maximize
the likelihood function. If one is
> comparing two likelihood functions by taking the difference,
constant terms will drop out. If one is finding
> the values of parameters that will maximize the likelihood function
or its log, one will take the first
> derivative and set it equal to zero. Constant terms will have first
derivatives that are zero, no matter what
> the value of the constant. In your example, sigma represents the
standard deviation in the population. It is
> simply a number, which is assumed to be known. For purposes of
estimating the coefficients in a regression
> equation, it is irrelevant. It can be disregarded. One might as
well drop it, at the potential cost of
> confusing some students. David Greenbe
> rg, Sociology Department, New York University
>
> ----- Original Message -----
> From: leechtcn <[email protected]>
> Date: Thursday, October 30, 2003 6:42 am
> Subject: st: Log Likelihood for Linear Regression Models
>
>> Dear Listers,
>>
>> I have asked this question before. I am posting it a
>> second time in case you guys have not received it.
>>
>> I am sorry for the all convinence caused!
>>
>> I have a question concerning William Gould and William
>> Sribney's "MAximium Likelihood Estimation" (1st
>> edition):
>>
>>
>> In its 29th page, the author write the the following
>> lines:
>>
>> For instance, most people would write the log
>> likelihood for the linear regression model as:
>>
>> LnL = SUM(Ln(Normden((yi-xi*beta)/sigma)))-ln(sigma)
>> (1)
>>
>> But in most econometrics textbooks, such as William
>> Green, the log likelihood for a linear regression is
>> only:
>>
>> LnL = SUM(Ln(Normden((yi-xi*beta)/sigma)))
>> (2)
>>
>>
>> that is, the last item is dropped
>>
>> I have also tried to use (2) in stata, it will give
>> "no concave" error message. In my Monte Carlo
>> experiments, (1) always gives reasonable results.
>>
>> Can somebody tell me why there is a difference between
>> stata's log likelihood and those of the other
>> textbooks?
>>
>> thanks a lot
>>
>> Leecht
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