st: re: maximum likelihood problem

[Kit Baum <baum@bc.edu>]

Ben wrote

The program I've written in Stata is the following
(which, for obvious reasons, isn't working):

program mymlprog
	version 10.0
	args y x

	quietly replace `y' = ln(1/sqrt(2*_pi*Sigma_hat^2))-
((1/(2*Sigma_hat^2))*(`y'+Alpha_1-Beta_hat*`x')^2 if $ML_y1 <0

	quietly replace `y' =
ln(normal((Alpha_2-Beta_hat*`x')/Sigma_hat)-normal((Alpha_1- 
Beta_hat*`x')/Sigma_hat))
if $ML_y1 == 0

	quietly replace `y' =
ln[1/sqrt(2*_pi*Sigma_hat^2)]-(1/(2*Sigma_hat^2))*(`y'+Alpha_2- 
Beta_hat*`x')^2
if $ML_y1 > 0

end


I think this will do it. Note that there are some fundamental  
conceptual flaws in what Ben has written above. This can be viewed as  
a linear-form (lf) model of a rather strange sort (given that there  
are two parameter vectors that share an element). I think that could  
be handled by applying constraints that the two slope coefficients  
should be the same.

--------mymlprog.ado--------
program mymlprog
	version 10.0
	args lnf mu1 mu2

	quietly replace `lnf' = log(1/sqrt(2*_pi*$Sigma_hat^2))- ///
	(1/(2*$Sigma_hat^2))*($ML_y1+`mu1')^2 if $ML_y1 <0

	quietly replace `lnf' =log(normal((`mu2')/$Sigma_hat)- ///
	normal((`mu1')/$Sigma_hat)) if $ML_y1 == 0

	quietly replace `lnf' =log(1/sqrt(2*_pi*$Sigma_hat^2))- ///
	(1/(2*$Sigma_hat^2))*($ML_y1+`mu2')^2 ///
    if $ML_y1 > 0

end
---------end of program------

to run, you must first set the global value of Sigma_hat to some  
value (as that does not seem to be a parameter to be estimated)

global Sigma_hat 10

I created a dataset with the desired characteristics with

webuse auto
g mpgzap = mpg - 25
ml model lf mymlprog (mpgzap = price) (mpgzap=price)
ml check
ml maximize


                                                  Number of obs    
=         74
                                                  Wald chi2(1)     
=       4.76
Log likelihood = -243.23578                       Prob > chi2      
=     0.0292

------------------------------------------------------------------------ 
------
             |      Coef.   Std. Err.      z    P>|z|     [95% Conf.  
Interval]
------------- 
+----------------------------------------------------------------
eq1          |
       price |   .0008808   .0004039     2.18   0.029     . 
0000891    .0016725
       _cons |  -2.170081    2.78924    -0.78   0.437      
-7.63689    3.296728
------------- 
+----------------------------------------------------------------
eq2          |
       price |   .0050802   .0023531     2.16   0.031     . 
0004683    .0096922
       _cons |  -19.44901   10.88796    -1.79   0.074     
-40.78903    1.891009
------------------------------------------------------------------------ 
------

If we now do

cons 1 [eq1]price = [eq2]price
ml model lf mymlprog (mpgzap = price) (mpgzap=price) , constraints(1)
ml maximize

the two slope parameters will be equated;

                                                  Number of obs    
=         74
                                                  Wald chi2(0)     
=          .
Log likelihood = -246.37235                       Prob > chi2      
=          .

 ( 1)  [eq1]price - [eq2]price = 0
------------------------------------------------------------------------ 
------
             |      Coef.   Std. Err.      z    P>|z|     [95% Conf.  
Interval]
------------- 
+----------------------------------------------------------------
eq1          |
       price |   .0010458   .0004001     2.61   0.009     . 
0002615    .0018301
       _cons |  -3.476855   2.776426    -1.25   0.210     
-8.918549    1.964839
------------- 
+----------------------------------------------------------------
eq2          |
       price |   .0010458   .0004001     2.61   0.009     . 
0002615    .0018301
       _cons |   -.181225     2.8025    -0.06   0.948     
-5.674024    5.311574
------------------------------------------------------------------------ 
------

I think this is estimating the model that Ben has specified. As he  
has an independent
estimate of the model via Excel Solver, it should be easy to check  
that out.

Kit

Kit Baum, Boston College Economics and DIW Berlin
http://ideas.repec.org/e/pba1.html
An Introduction to Modern Econometrics Using Stata:
http://www.stata-press.com/books/imeus.html


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