12  B5. Count Data Models

12.1 About

Topics covered:

  • Poisson regression: MLE and QMLE (consistency under correct mean specification)
  • Marginal effects as semi-elasticities
  • Equidispersion assumption and overdispersion: robust standard errors
  • Negative Binomial model
  • Zero-inflated models (ZIP, ZINB)

12.2 Lecture Notes

 

12.3 Overview

Count data models are appropriate when \(y_i \in \{0,1,2,\ldots\}\) — a non-negative integer. Examples include number of doctor visits, patent applications, arrests, children ever born.

OLS applied to count outcomes produces consistent estimates under mild conditions, but is inefficient and can predict negative counts. The natural baseline is Poisson regression, which respects the non-negativity constraint by modeling the conditional mean exponentially.

12.4 The Poisson Regression Model

Assume \(y_i \mid x_i \sim \text{Poisson}(\lambda_i)\) where: \[\lambda_i = E[y_i \mid x_i] = \exp(x_i'\beta)\]

The Poisson PMF is: \[P(y_i = k \mid x_i) = \frac{e^{-\lambda_i}\lambda_i^k}{k!}, \quad k=0,1,2,\ldots\]

The exponential mean function ensures \(\lambda_i > 0\) for all \(x_i\).

12.5 Maximum Likelihood Estimation

The log-likelihood for \(n\) independent observations is:

\[\ell(\beta) = \sum_{i=1}^n \left[ y_i x_i'\beta - e^{x_i'\beta} - \ln(y_i!) \right]\]

The score equations are: \[\frac{\partial \ell}{\partial \beta} = \sum_{i=1}^n (y_i - \lambda_i)\,x_i = 0\]

These are solved numerically. The MLE \(\hat\beta\) is consistent and asymptotically normal even if the true conditional distribution is not Poisson, provided the conditional mean is correctly specified as \(\exp(x_i'\beta)\). This is the Poisson QMLE (quasi-MLE) or PMLE result.

12.6 Marginal Effects

The marginal effect of \(x_j\) on the conditional mean is:

\[\frac{\partial E[y\mid x]}{\partial x_j} = \beta_j \exp(x'\beta) = \beta_j\,\lambda\]

So \(\beta_j\) is the semi-elasticity: a one-unit increase in \(x_j\) raises \(E[y\mid x]\) by \(100\times\beta_j\) percent (approximately).

12.7 Equidispersion and Overdispersion

A key feature of the Poisson distribution is equidispersion: \[E[y_i \mid x_i] = \text{Var}(y_i \mid x_i) = \lambda_i\]

In practice, count data often exhibit overdispersion: \(\text{Var}(y_i \mid x_i) > E[y_i \mid x_i]\). This can arise from:

  • Unobserved heterogeneity (omitted variables)
  • Excess zeros
  • Clustering

Consequences of overdispersion

  • Poisson MLE is still consistent for \(\beta\) (QMLE property).
  • Standard errors based on the Poisson information matrix are too small → invalid inference.
  • Solution: Use robust (sandwich) standard errors: \[\widehat{\text{Var}}(\hat\beta) = \left(\sum x_i x_i' \lambda_i\right)^{-1} \left(\sum (y_i-\lambda_i)^2 x_i x_i'\right) \left(\sum x_i x_i' \lambda_i\right)^{-1}\]

12.8 Negative Binomial Model

A fully parametric alternative that allows for overdispersion is the Negative Binomial model. It introduces a gamma-distributed random effect \(v_i\) into the Poisson mean:

\[y_i \mid x_i, v_i \sim \text{Poisson}(v_i\lambda_i), \quad v_i \sim \text{Gamma}\left(\frac{1}{\alpha},\alpha\right)\]

Integrating out \(v_i\) gives a Negative Binomial marginal distribution with: \[E[y_i\mid x_i] = \lambda_i, \quad \text{Var}(y_i\mid x_i) = \lambda_i(1+\alpha\lambda_i)\]

The parameter \(\alpha > 0\) captures overdispersion. A likelihood ratio test of \(H_0: \alpha=0\) vs. \(H_1: \alpha>0\) tests whether Poisson is adequate.

12.9 Zero-Inflated Models

When the data have excess zeros relative to what Poisson or Negative Binomial predicts, zero-inflated models (ZIP, ZINB) may be appropriate. These model zeros as arising from two processes: a structural zero (e.g., non-participation) and a Poisson/NegBin count process.

12.10 References

Cameron y Trivedi (2005), chapter 20. Davidson y MacKinnon (2004).