Re: st: nonlinear regression with restrictions

[Ed Blackburne <blackburne@shsu.edu>]

Michael,

I have attached two pieces of code that I believe solves your problem. 
Please review and see if it works for you. Also, I would appreciate any 
feedback to this thread from the Stata power users who know of a better 
(i.e. more efficient) method of solving this problem. I especially 
prefer not to use globals, but I can't see another way around the problem.



using the infamous auto data, the code will estimate a model of

price=a1+a2*trunk+a3*weight+a4*(1-dk)*((b^2 - 2 * b * displacement + 
displacement^2)
where dk is defined as by you below.

You can, of course, use any variable as the threshold variable.


After copying the files to your working directory, the output is

. michael
(1978 Automobile Data)

initial:       log likelihood = -681.03832
alternative:   log likelihood = -681.03317
rescale:       log likelihood = -675.90404
Iteration 0:   log likelihood = -675.90404 
Iteration 1:   log likelihood = -675.37653 
Iteration 2:   log likelihood = -675.34488 
Iteration 3:   log likelihood = -675.34441 
Iteration 4:   log likelihood = -675.34441 
------------------------------------------------------------------------------
Threshold nonlinear estimation
Threshold value of displacement = 224.98074 with standard error = 26.895476
------------------------------------------------------------------------------

     Source |       SS       df       MS              Number of obs 
=      74
-------------+------------------------------           F(  3,    70) =   
17.13
      Model |   268847078     3  89615692.7           Prob > F      =  
0.0000
   Residual |   366218318    70  5231690.26           R-squared     =  
0.4233
-------------+------------------------------           Adj R-squared =  
0.3986
      Total |   635065396    73  8699525.97           Root MSE      =  
2287.3

------------------------------------------------------------------------------
      price |      Coef.   Std. Err.      t    P>|t|     [95% Conf. 
Interval]
-------------+----------------------------------------------------------------
      trunk |  -65.54658   84.54591    -0.78   0.441    -234.1681    
103.0749
     weight |   1.207276   .5366726     2.25   0.028     .1369165    
2.277636
       term |   .1762172   .0445175     3.96   0.000     .0874298    
.2650046
      _cons |   2875.371   1294.126     2.22   0.030     294.3173    
5456.424
------------------------------------------------------------------------------


The first piece of code is a simple do file that defines the problem, etc

=====================================================
capture program drop michael
program define michael
   webuse auto, clear
   tempname b V
   global dep_var price
   global indep_var trunk weight
   global thresh displacement
  
   generate double term=0
   global term term
  
   ml model d0 michael_ml /gamma, max
   matrix b=e(b)
   matrix V=e(V)
   scalar `b'=b[1,1]
   scalar `V'=V[1,1]
   scalar `V'=sqrt(`V')
  
   di _n in smcl in gr "{hline 78}"
   di in gr "Threshold nonlinear estimation"
   di in gr "Threshold value of $thresh = " in ye `b' in gr " with 
standard error = " in ye `V'
   di in smcl in gr "{hline 78}"

   regress $dep_var $indep_var $term
  
end

===============================================================

The michael_ml procedure is an ado file that uses the undocumented 
ml_hold option of Stata's ml command

===============================================================
capture program drop michael_ml
program define michael_ml
   args todo b lnf
   tempname beta
   marksample touse
   mleval `beta' = `b', scalar
   quie replace term=0
   quie replace term=(`beta'^2-2*`beta'* $thresh + $thresh ^2) if 
$thresh > `beta'
   ml hold
       quie regress $dep_var $indep_var term
   ml unhold
   scalar `lnf'=e(ll)
end

================================================================





Michael Stobernack wrote:

> Dear all,
>
> I would like to run the following nonlinear regression:
>
> Y = a1 + a2 * X +a3 * (1 - dk) * (b**2 - 2 * b * x + x**2) + u
>
> with dk = 1 if x <= b  and dk = 0 if x > b
>
> That is, there is a variable (dk) which depends on a parameter (b) 
> which itself
> is still to be estimated.
>
> The background is: I would like to get a linear regression between x=0 
> and x=b,
> but a nonlinear regression for x>b. And I don't know the value of b in 
> advance.
> The regression should estimate the value of b.
>
> I would be very glad about any hints how to do this in Stata.
>
> Best regards,
>
> Michael
>
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