Common mistake 1 with Poisson / Negative binomial

Reporting Poisson standard errors when the data are over-dispersed. They are far too small, so everything looks significant.

Common mistake 2 with Poisson / Negative binomial

Using a count model when the outcome is a proportion or a rate that is not a count.

Common mistake 3 with Poisson / Negative binomial

Ignoring excess zeros - if far more zeros appear than the model predicts, you may need a zero-inflated model.

Common mistake 4 with Poisson / Negative binomial

Choosing negative binomial over Poisson purely because a test rejected equidispersion. If the point estimates barely move, the practical answer is often Poisson with robust standard errors - which requires only the conditional mean to be right, while negative binomial requires the full distribution.

EconStatsCraft › Poisson / Negative binomial

Poisson / Negative binomial

For outcomes that are counts (0, 1, 2, …). Fits Poisson and negative binomial side by side and tests for over-dispersion, which decides which one you should report.

What it needs

Count outcome - Non-negative whole numbers.
Independent variables

The intuition

A count cannot be negative and is often mostly small numbers with a long right tail. Modelling log E[y|x] rather than E[y|x] guarantees positive predictions and makes coefficients multiplicative - which is usually the natural economic statement anyway ("15% more visits", not "0.4 more visits").

When to use it

The dependent variable is a count. Using OLS on counts gives negative predictions and wrong standard errors.

Where the estimator comes from

Poisson assumes P(y=k\mid x)=e^{-\lambda}\lambda^k/k! with \lambda=\exp(x'\beta), so the log-likelihood is \sum_i\left[y_i x_i'\beta-\exp(x_i'\beta)-\log y_i!\right], globally concave. The negative binomial adds a gamma-distributed multiplicative error to \lambda, producing \operatorname{Var}(y\mid x)=\lambda+\alpha\lambda^2 - a variance that grows faster than the mean, which is what real count data almost always show.

The model

E[y\mid \mathbf{x}]=\exp(\mathbf{x}^{\prime}\boldsymbol{\beta}),\qquad P(y=k)=\frac{e^{-\lambda}\lambda^{k}}{k!}

What it assumes

The outcome is a non-negative whole number.
Poisson assumes variance = mean. This is the assumption that fails nearly always.
Observations independent, exposure equal (or use an offset).
The counts are of comparable things over comparable windows. Doctor visits in a year and in a month are different outcomes, and mixing them without an offset makes the exposure a hidden regressor.

Statistical properties

Poisson QMLE is consistent for β whenever the CONDITIONAL MEAN is correct, even if the distribution is wrong - a genuinely useful robustness result.
But its standard errors are only right if variance = mean. Use robust errors or the negative binomial otherwise.
Over-dispersion inflates t-statistics: everything looks significant when it is not.
The negative binomial nests Poisson at α = 0; the LR test sits on a boundary, so the p-value is halved.
Excess zeros beyond what the model predicts point to a hurdle or zero-inflated specification, which is a different data-generating story.

How to read the output

Quote the incidence-rate ratio exp(β): 1.15 means "15% more events per unit rise in x".
Variance / mean far above 1 in the summary is over-dispersion.
If the over-dispersion test rejects, report the negative binomial, not Poisson.
α in the negative binomial measures over-dispersion; α = 0 collapses it back to Poisson.

A worked reading

Doctor visits for 2,000 people: mean 1.9410, variance 8.2805, so variance/mean = 4.2661 and a third of the sample reports no visits at all. Poisson puts β̂ on a chronic condition at 1.1275, an incidence rate ratio of 3.0880 - roughly three times as many visits, holding age and insurance fixed. The over-dispersion shows up exactly where theory says it will: classical standard errors are between 1.6 and 1.8 times too small across the four coefficients, so the robust column is not optional here. The formal test rejects equidispersion at 9.96 with p < 0.0001, and the negative binomial estimates a dispersion parameter of 0.4638 that is 12.7 standard errors from zero, beating Poisson on a likelihood-ratio test of 571.42. The coefficients themselves barely move - 1.1275 to 1.1398 on chronic - which is the usual pattern and the reason this matters: over-dispersion is a problem for the standard errors, not for the point estimates, so a Poisson fit with robust errors and a negative binomial will agree about the effect and disagree about how sure you should be.

Symptom → diagnosis → what to do

If you see this	It means	Do this
Variance/mean much above 1	Over-dispersion	Negative binomial, or Poisson with robust standard errors
Far more zeros than predicted	Two processes: whether any event occurs, then how many	Zero-inflated or hurdle model
Exposure varies across units	Counts are not comparable	Add log(exposure) as an offset with coefficient fixed at 1
α significant in the negative binomial	Poisson was inadequate	Report the negative binomial
Predictions far above observed maxima	A regressor with an extreme value in exp()	Check for outliers; exp() amplifies them brutally

Common mistakes

Reporting Poisson standard errors when the data are over-dispersed. They are far too small, so everything looks significant.
Using a count model when the outcome is a proportion or a rate that is not a count.
Ignoring excess zeros - if far more zeros appear than the model predicts, you may need a zero-inflated model.
Choosing negative binomial over Poisson purely because a test rejected equidispersion. If the point estimates barely move, the practical answer is often Poisson with robust standard errors - which requires only the conditional mean to be right, while negative binomial requires the full distribution.

Where to read this properly

Cameron & Trivedi (2013), Regression Analysis of Count Data
Wooldridge (2010), Econometric Analysis of Cross Section and Panel Data, 2nd ed., MIT Press, chapter 18
Gourieroux, Monfort & Trognon (1984) - Poisson QMLE consistency

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Poisson / Negative binomial

What it needs

The intuition

When to use it

Where the estimator comes from

The model

What it assumes

Statistical properties

How to read the output

A worked reading

Symptom → diagnosis → what to do

Common mistakes

Where to read this properly

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