Poisson regression with endogenous variables

ivpoisson fits a Poisson regression model (a.k.a. exponential conditional mean model) in which one or more of the regressors are endogenous. Poisson is frequently used to model count outcomes or to model nonnegative outcome variables.

Suppose we are modeling the number of automobile accidents involving young male drivers.

We will assume the number of accidents comes from a Poisson distribution with mean

\(exp(b_0 + b_1\) horsepower \(+\;b_2\;x1 + b_3\;x2)\)

In this artificial example, we will assume horsepower, in addition to having a direct effect, also reflects an underlying tendency for risky behavior. We will use x3 and x4 as measures of the tendency, though x3 and x4 might have nothing whatsoever to do with cars. We will use the full set of variables x1 through x4 as instruments for horsepower.

We will estimate our additive model using the efficient two-step GMM. We type

. ivpoisson gmm accidents x1 x2 (horsepower = x3 x4)

Step 1
Iteration 0:   GMM criterion Q(b) =  .00004111  
Iteration 1:   GMM criterion Q(b) =  1.045e-06  
Iteration 2:   GMM criterion Q(b) =  1.036e-06  

Step 2
Iteration 0:   GMM criterion Q(b) =   .0002911  
Iteration 1:   GMM criterion Q(b) =  .00024685  
Iteration 2:   GMM criterion Q(b) =  .00024685

Exponential mean model with endogenous regressors

Number of parameters =   4                         Number of obs  =      1,000
Number of moments    =   5
Initial weight matrix: Unadjusted
GMM weight matrix:     Robust




             
 
               Robust

         accidents 
 
 Coefficient  std. err.      z    P>|z|     [95% conf. interval]

 
 
 

        horsepower 
 
   .0077525   .0010175     7.62   0.000     .0057582    .0097467

                x1 
 
   .1952001   .0068223    28.61   0.000     .1818286    .2085716

                x2 
 
   .1374668   .0064702    21.25   0.000     .1247854    .1501483

             _cons 
 
  -1.861607   .0108662  -171.32   0.000    -1.882904   -1.840309



Instrumented: horsepower
 Instruments: x1 x2 x3 x4

To understand the impact of pure horsepower (holding underlying risky behavior constant) on our young males, we will use Stata's margins to estimate the expected number of accidents using observed horsepower and the expected number of accidents from giving each car 50 more units of horsepower:

. margins, at((asobserved)) at(horsepower=generate(horsepower +50))

Predictive margins                                       Number of obs = 1,000
Model VCE: Robust

Expression: Predicted number of events, predict()
1._at: (asobserved)                 
2._at: horsepower = horsepower +50




       
 
            Delta-method
 

       
 
     Margin   std. err.      z    P>|z|     [95% conf. interval]
 

 
 
 
 

         _at 
 
 
 

          1
 
   .2582595   .0020218   127.74   0.000     .2542969    .2622222
 

          2
 
   .3805387   .0196401    19.38   0.000     .3420448    .4190326
 

We find that the expected number of accidents using observed horsepower is 0.26 and that it increases to 0.38 if each car produces 50 more horsepower.

We can compute the effect of the 50-horsepower increases by contrasting these two estimates:

. margins, at((asobserved)) at(horsepower=generate(horsepower +50))
> contrast(at(r._at))

Contrasts of predictive margins                          Number of obs = 1,000
Model VCE: Robust

Expression: Predicted number of events, predict()
1._at: (asobserved)                 
2._at: horsepower = horsepower +50


 

 
        
 
         df        chi2     P>chi2
 

 
         _at 
 
          1       39.73     0.0000
 




 

 
        
 
            Delta-method
 
        
 
   Contrast   std. err.     [95% conf. interval]
 

 
         _at 
 
 
 
   (2 vs 1) 
 
   .1222792   .0194003      .0842553    .1603031
 

We find that increasing horsepower by 50 increases the expected number of accidents per young man by 0.12 on average. (This Poisson model is nonlinear, so the amount of increase varies across young men.)

The above results would be of interest to insurance companies that want to judge the effect of increasing the horsepower of modern cars.