st: Output of ML for Poisson vs. Stata Poisson Command

[Gordon Burtch <gburtch@gmail.com>]

Hello,

I have hit a small wall trying to code the poisson estimator from scratch, using the ML command. I am estimating an intercept only model (i.e., ml model lf mypoisson (y=)). The intercept estimate that my custom code produces does not match the estimate produced by the stock STATA poisson command (i.e., poisson y). The issue here appears to be with my definition of the likelihood function. I am hoping someone can help me understand why. 

Here is my code... as I understand it, the likelihood function should just be the log of the poisson PDF (this is the same approach I took to code the Probit and Logit estimators, which worked correctly):

program mypoisson
	version 10.0
	args lnf lambda
	quietly replace `lnf' = $ML_y1 * ln(`lambda') - `lambda' - lnfactorial($ML_y1)
end

Upon discovering that this code produced an incorrect coefficient, I began scouring the Internet for a clue of what I might be doing wrong. I discovered that the correct estimate is obtained if I use exp(`lambda'), in place of `lambda'. I figured this out by looking at this code posted by Colin Cameron from UC Davis (http://cameron.econ.ucdavis.edu/stata/stmle.do). The output from each command is included at the end of this e-mail for reference. All commands are executed using the same count dataset. 

program mypoisson
	version 10.0
	args lnf xB
	local lambda = exp(`xB')
	quietly replace `lnf' = $ML_y1 * ln(`lambda') - `lambda' - lnfactorial($ML_y1)
end

So, why would one want to take the exponential of the lambda here, given that this does not fit with the derived likelihood function? Or have I missed something in my derivation? It had been suggested to me that taking the exponential might simply have to do with imposing a positive value constraint on lambda, however, since the coefficient output is actually changed by doing this, I am guessing this is not the reason.

Thanks in advance for any assistance!

Gord Burtch
Ph.D. Candidate, MIS
Fox School of Business
Temple University 
Speakman Hall 201F
Philadelphia, PA, 19122
Mob 215 688 3852

========Stata Output Follows

. ml model lf mypoisson (y=)

. ml max

initial:       log likelihood =     -<inf>  (could not be evaluated)
feasible:      log likelihood = -247.18569
rescale:       log likelihood = -169.83341
Iteration 0:   log likelihood = -169.83341  
Iteration 1:   log likelihood = -166.57517  
Iteration 2:   log likelihood = -166.37983  
Iteration 3:   log likelihood = -166.37937  
Iteration 4:   log likelihood = -166.37937  

                                                  Number of obs   =        100
                                                  Wald chi2(0)    =          .
Log likelihood = -166.37937                       Prob > chi2     =          .

------------------------------------------------------------------------------
           y |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
       _cons |       1.64   .1280625    12.81   0.000     1.389002    1.890998
------------------------------------------------------------------------------

. poisson y

Iteration 0:   log likelihood = -166.37937  
Iteration 1:   log likelihood = -166.37937  

Poisson regression                                Number of obs   =        100
                                                  LR chi2(0)      =      -0.00
                                                  Prob > chi2     =          .
Log likelihood = -166.37937                       Pseudo R2       =    -0.0000

------------------------------------------------------------------------------
           y |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
       _cons |   .4946962   .0780869     6.34   0.000     .3416488    .6477437
------------------------------------------------------------------------------


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