Poisson / Negative Binomial

Motivating Example: Patent Citations#

You are studying what predicts how many citations a patent receives. Citations are a widely used proxy for patent value — more citations typically suggest a more influential invention. Your dataset has 50,000 patents, and the citation counts look like this distribution: a huge spike at zero (many patents are never cited), a long right tail (a few blockbuster patents get hundreds of citations), and everything in between.

The mean number of citations is 8.2, but the variance is 142.6. That gap is a variance-to-mean ratio of about 17:1.

If you run OLS on this outcome, three problems arise:

1. You predict negative citations for some patents (impossible).
2. Your residuals are wildly heteroskedastic (variance increases with the mean).
3. The normality assumption for inference is badly violated.

Count models are designed for exactly this situation.

The Poisson Model#

The Poisson regression models the conditional mean of a count outcome as:

The exponential function ensures that predicted counts are always positive. Estimation is by maximum likelihood, assuming $Y_i | X_i \sim \text{Poisson}(\mu_i)$.

The key property of the Poisson distribution: the mean equals the variance. That property is:

This restriction is called equidispersion. In practice, real count data very frequently violate this assumption — the variance exceeds the mean, a condition called overdispersion.

The Negative Binomial Model#

The negative binomial relaxes equidispersion by adding a dispersion parameter $\alpha$:

When $\alpha = 0$, this collapses to Poisson. When $\alpha > 0$, the variance exceeds the mean, and the negative binomial accommodates the extra variability. This specification is the Negative Binomial type 2 (NB2) parameterization (Cameron & Trivedi, 1986). An alternative is Negative Binomial type 1 (NB1), where $\text{Var}(Y_i | X_i) = \sigma^2 \mu_i$ — overdispersion proportional to the mean rather than quadratic (Wooldridge, 2010).

An Important Subtlety: Poisson with Robust SEs#

An important result is that even if the Poisson variance assumption is wrong (and it usually is), the Poisson model with robust standard errors still gives consistent estimates of the coefficients. This consistency holds because the Poisson maximum likelihood estimator (MLE) only requires the mean to be correctly specified — you do not need the variance to be correct (Gourieroux et al., 1984).

This result is the basis for Poisson Pseudo-Maximum Likelihood (PPML). You use the Poisson likelihood as a working model but make no assumption about the variance. With robust SEs, inference is valid as long as the conditional mean is correctly specified.

Common Confusions#

Like OLS and logit/probit, count models require exogeneity for causal interpretation:

where $\varepsilon_i = Y_i - \exp(X_i'\beta)$. If your key regressor is endogenous, you need an identification strategy. Common approaches include:

• Poisson with fixed effects — removes time-invariant confounders, just like linear FE.
• Control function approach — the nonlinear equivalent of IV for exponential models.
• PPML with instrumental variables — available in specialized software.

Hausman et al. (1984) first applied Poisson and negative binomial models to patent data, establishing the methodology that many innovation researchers still use. In management, count models are widely used to study innovation and organizational behavior — for example, Ahuja (2000) used negative binomial regression to examine how network structure and structural holes affect firm innovation output, and Greve (2003) modeled R&D expenditures and innovation counts to test behavioral theories of the firm.

Picture the distribution of patent citations. It looks nothing like a bell curve. It is a histogram bunched up at zero, rising to a peak around 2-5 citations, and then trailing off in a long right tail. A few patents have 50, 100, or even 500 citations.

The Poisson model says: the average number of citations depends on covariates through an exponential function, and the spread around that average follows a Poisson distribution. But if the actual spread is much wider than the Poisson predicts (which it very frequently is), you have overdispersion.

Visually, overdispersion means the histogram is wider and has thicker tails than the Poisson would predict. The negative binomial adds a "mixing" distribution that spreads things out more, better matching the observed data.

1# Requires: fixest, MASS, pscl, AER
2library(fixest)
3library(MASS)
4library(pscl)
5
6# --- Step 1: Poisson with Robust SEs (PPML) ---
7# fepois() from fixest fits Poisson pseudo-maximum likelihood.
8# Coefficients are semi-elasticities: beta = approx. % change in E[Y] per unit X.
9# vcov = ~firm_id clusters standard errors at the firm level.
10# The "|" syntax absorbs tech_class and year fixed effects.
11pois_fit <- fepois(citations ~ rd_spending + firm_age | tech_class + year,
12                 data = df, vcov = ~firm_id)
13# Output: coefficients in log units. Exponentiate for IRRs: exp(beta).
14summary(pois_fit)
15
16# --- Step 2: Negative Binomial ---
17# glm.nb() from MASS adds a dispersion parameter (alpha) to relax
18# the Poisson equidispersion assumption (mean = variance).
19# Use when you want model-based SEs that account for overdispersion.
20nb_fit <- glm.nb(citations ~ rd_spending + firm_age + factor(tech_class), data = df)
21# Check theta (= 1/alpha): small theta = high overdispersion.
22summary(nb_fit)
23
24# --- Step 3: Zero-Inflated Poisson ---
25# zeroinfl() fits a two-part model: (1) logit for structural zeros
26# (e.g., patents that can never be cited), (2) Poisson for counts.
27# The "|" separates the count model from the zero-inflation model.
28zip_fit <- zeroinfl(citations ~ rd_spending + firm_age | small_firm, data = df)
29# Output includes both count and zero-inflation model coefficients.
30summary(zip_fit)
31
32# --- Step 4: Overdispersion Test ---
33# dispersiontest() from AER tests H0: Var(Y) = E[Y] (equidispersion).
34# Rejection (p < 0.05) indicates overdispersion — use robust SEs or NegBin.
35library(AER)
36dispersiontest(glm(citations ~ rd_spending + firm_age, family = poisson, data = df))

Testing for Overdispersion#

1. Deviance test: Compare the deviance (or Pearson chi-squared) to the degrees of freedom. A ratio much greater than 1 suggests overdispersion.
2. Cameron-Trivedi test: A regression-based test (see derivation above).
3. Compare Poisson and NegBin: If the NegBin dispersion parameter $\alpha$ is significantly different from zero, overdispersion is present.

Zero-Inflation#

If your data have more zeros than the Poisson (or NegBin) predicts, you may need a zero-inflated model. Two options:

• Zero-inflated Poisson/NegBin (ZIP/ZINB): A two-part model where one equation determines the probability of being a "structural zero" (e.g., a patent that could never receive citations), and a second equation models the count for the rest.
• Hurdle model: The first part is a binary model for zero vs. positive, and the second part is a truncated count model for positive counts.

The Vuong test can help distinguish between standard Poisson and zero-inflated Poisson, though it has known size issues.

PPML for Gravity Models#

In international trade, the gravity equation models bilateral trade flows as a function of GDP and distance. PPML has become the standard estimator (Silva & Tenreyro, 2006) because it:

1. Handles zeros naturally (many country pairs have zero trade).
2. Is robust to heteroskedasticity in levels.
3. Provides consistent estimates even with non-integer outcomes.

Incidence Rate Ratios (IRRs)#

The exponentiated coefficient $e^{\beta_j}$ is the incidence rate ratio. If $e^{\beta_j} = 1.15$, a one-unit increase in $X_j$ multiplies the expected count by 1.15 — a 15% increase.

Semi-Elasticities#

The raw coefficient $\beta_j$ is a semi-elasticity: a one-unit increase in $X_j$ is associated with an approximate $100 \times \beta_j$% change in the expected count. This approximation is good for small $\beta_j$ (say, below 0.3 in absolute value).

Comparing Poisson and NegBin Coefficients#

Unlike logit (where rescaling makes comparison difficult), Poisson and NegBin coefficients are directly comparable because both target the same conditional mean function. If the coefficients differ substantially, it suggests the conditional mean may be misspecified.

• OLS on log(Y): If all your counts are large (say, above 20) and you have no zeros, taking the log and running OLS is approximately valid. But with zeros or small counts, do not do this transformation.
• Tobit: If your count is in fact a censored continuous variable (e.g., hours worked), Tobit may be more appropriate.
• Hurdle models: If the process that generates zeros is different from the process that generates positive counts (e.g., the decision to patent at all vs. how many patents to file), a hurdle model captures this two-stage structure.
• Quasi-Poisson: In some fields, quasi-Poisson (which adjusts the variance without specifying a full distribution) is used as a middle ground.