- Estimators
- GMM: one-step, two-step, and iterated
- Control function

- Models
- additive
- multiplicative

- Robust SEs to relax distributional assumptions
- Cluster–robust SEs for correlated data

**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)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 | Coef. 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 | |

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 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.

See the manual entry.

To learn more about **margins**, see its manual entry.