ivpoisson control function (with multiplicative error)

[John Antonakis]

Hi:

Suppose:

clear
webuse website
ivpoisson cfunction visits ad female (time = phone frfam)

How could I manually reproduce the point estimates from the ivpoisson estimator after having obtained the residuals from the first stage? That is,

reg time phone frfam ad female
predict residuals, residuals

Thanks.
John.

[Fei Wang]

-ivpoisson cfunction- uses one-step -gmm- to estimate coefficients. Below is a replica of -ivpoisson cfunction-.

webuse website, clear
ivpoisson cfunction visits ad female (time = phone frfam), winit(identity)

Step 1
Iteration 0:   GMM criterion Q(b) =  .03944785  
Iteration 1:   GMM criterion Q(b) =   .0002822  
Iteration 2:   GMM criterion Q(b) =  2.421e-08  
Iteration 3:   GMM criterion Q(b) =  1.757e-17  

note: model is exactly identified

Exponential mean model with endogenous regressors

Number of parameters =  10                         Number of obs  =        500
Number of moments    =  10
Initial weight matrix: Identity
GMM weight matrix:     Robust

------------------------------------------------------------------------------
             |               Robust
      visits |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
visits       |
          ad |    .142332   .0127635    11.15   0.000     .1173161     .167348
      female |   .0014383   .0436521     0.03   0.974    -.0841183    .0869949
        time |   .0683575   .0182084     3.75   0.000     .0326697    .1040452
       _cons |   .9422102   .0541605    17.40   0.000     .8360576    1.048363
-------------+----------------------------------------------------------------
time         |
          ad |   .1655074   .0546634     3.03   0.002     .0583691    .2726457
      female |   -.052156   .2011019    -0.26   0.795    -.4463084    .3419964
       phone |   .2908914   .0748883     3.88   0.000     .1441131    .4376696
       frfam |   .2756408   .0651957     4.23   0.000     .1478597     .403422
       _cons |   .9347438   .2602415     3.59   0.000     .4246798    1.444808
-------------+----------------------------------------------------------------
     /c_time |   .0796761   .0215887     3.69   0.000     .0373631    .1219891
------------------------------------------------------------------------------
Instrumented:  time
Instruments:   ad female phone frfam

reg time ad female phone frfam

      Source |       SS           df       MS      Number of obs   =       500
-------------+----------------------------------   F(4, 495)       =     49.66
       Model |  1007.50594         4  251.876486   Prob > F        =    0.0000
    Residual |  2510.78671       495  5.07229639   R-squared       =    0.2864
-------------+----------------------------------   Adj R-squared   =    0.2806
       Total |  3518.29265       499  7.05068668   Root MSE        =    2.2522

------------------------------------------------------------------------------
        time |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
          ad |   .1655074   .0599471     2.76   0.006     .0477252    .2832896
      female |   -.052156   .2015858    -0.26   0.796    -.4482253    .3439133
       phone |   .2908914   .0622642     4.67   0.000     .1685567     .413226
       frfam |   .2756408   .0602579     4.57   0.000     .1572481    .3940335
       _cons |   .9347438   .2216227     4.22   0.000     .4993065    1.370181
------------------------------------------------------------------------------

predict v, r
gmm (eq1: visits / exp({visits: ad female time _cons}+{c_time}*(time-{time: ad female phone frfam _cons})) - 1) ///
    (eq2: time - {time:ad}*ad - {time:female}*female - {time:phone}*phone - {time:frfam}*frfam - {time:_cons}), ///
    instruments(eq1: ad female time v) ///
    instruments(eq2: ad female phone frfam) onestep winit(identity)

Step 1
Iteration 0:   GMM criterion Q(b) =  1161.5345  (not concave)
Iteration 1:   GMM criterion Q(b) =  119.23133  
Iteration 2:   GMM criterion Q(b) =  4.0049842  
Iteration 3:   GMM criterion Q(b) =  .06717307  
Iteration 4:   GMM criterion Q(b) =  .00018589  
Iteration 5:   GMM criterion Q(b) =  3.015e-09  
Iteration 6:   GMM criterion Q(b) =  1.107e-18  

note: model is exactly identified

GMM estimation 

Number of parameters =  10
Number of moments    =  10
Initial weight matrix: Identity                   Number of obs   =        500

------------------------------------------------------------------------------
             |               Robust
             |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
visits       |
          ad |    .142332   .0127635    11.15   0.000     .1173161     .167348
      female |   .0014383   .0436521     0.03   0.974    -.0841183    .0869949
        time |   .0683575   .0182084     3.75   0.000     .0326697    .1040452
       _cons |   .9422102   .0541605    17.40   0.000     .8360576    1.048363
-------------+----------------------------------------------------------------
c_time       |
       _cons |   .0796761   .0215887     3.69   0.000     .0373631    .1219891
-------------+----------------------------------------------------------------
time         |
          ad |   .1655074   .0546634     3.03   0.002     .0583691    .2726457
      female |   -.052156   .2011019    -0.26   0.795    -.4463084    .3419964
       phone |   .2908914   .0748883     3.88   0.000     .1441131    .4376696
       frfam |   .2756408   .0651957     4.23   0.000     .1478597     .403422
       _cons |   .9347438   .2602415     3.59   0.000     .4246798    1.444808
------------------------------------------------------------------------------
Instruments for equation eq1: ad female time v _cons
Instruments for equation eq2: ad female phone frfam _cons

[John Antonakis]

Perfect! Thank you much.

Out of curiosity, how would we run the above with additive errors?

[Fei Wang]

-ivpoisson cfunction- doesn't support additive errors. Actually, additive errors do not make much sense to a poisson-distributed outcome. See discussions at https://www.statalist.org/forums/for...ive-iv-poisson.

[John Antonakis]

Right. I have seen that post before. In practice have you seen much of a difference in estimates when using multiplicative or additive errors with gmm? Any studies on this--simulations in particular?

[Fei Wang]

I'm unfamiliar with such literature. But if we apply control function to additive errors anyway, the -gmm- code would be

reg time ad female phone frfam
predict v, r
gmm (eq1: visits - exp({visits: ad female time _cons}+{c_time}*(time-{time: ad female phone frfam _cons}))) ///
    (eq2: time - {time:ad}*ad - {time:female}*female - {time:phone}*phone - {time:frfam}*frfam - {time:_cons}), ///
    instruments(eq1: ad female time v) ///
    instruments(eq2: ad female phone frfam) onestep winit(identity)

Whose results may substantially differ from those with multiplicative errors, but may be quite similar to an alternative approach below.

reg time ad female phone frfam
predict v, r
poisson visits ad female time v, vce(robust)

[John Antonakis]

Right. That is the approach I used initially too with a basic augmented regression:

reg time ad female phone frfam
predict v,r
poisson visits ad female time v, vce(robust)

As discussed here in section 17.5.1: Cameron, A. C., & Trivedi, P. K. (2005). Microeconometrics: Methods and applications. New York: Cambridge University Press.

Thank you for your help!

[John Antonakis]

Oops--just referred to this post. The reference is wrong. It is section 17.5.1 in Cameron, A. C., & Trivedi, P. K. 2009. Microeconometrics Using Stata. College Station, Tex.: Stata Press.