17.4 Count data
For a count \(y_i \in \{0, 1, 2, \dots \}\), the Poisson regression model is \[ P(y_i = m \mid x_i) = \frac {e^{-\mu _i}\mu _i^m}{m!}, \qquad \mu _i = E(y_i \mid x_i) = \exp (x_i'\beta ) . \] The exponential mean keeps \(\mu _i\) positive, and \(\beta _j\) is the proportional change in the expected count per unit change in \(x_j\); for a dummy, \(e^{\beta _j}\) is the ratio of expected counts.
Theorem 17.4.1. The Poisson log-likelihood \(\ell (\beta ) = \sum _i[y_ix_i'\beta - \exp (x_i'\beta ) - \log y_i!]\) has score \(\sum _i(y_i - \mu _i)x_i\) and Hessian \(-\sum _i\mu _ix_ix_i'\), so it is strictly concave. The maximiser \(\hat \beta \) is consistent whenever the conditional mean \(E(y_i \mid x_i) = \exp (x_i'\beta )\) is correctly specified, whether or not \(y_i\) has a Poisson distribution, with asymptotic variance \[ \Bigl (\sum \mu _ix_ix_i'\Bigr )^{-1}\Bigl (\sum \Var (y_i \mid x_i)x_ix_i'\Bigr )\Bigl (\sum \mu _ix_ix_i'\Bigr )^{-1} . \]
Proof. The score and Hessian follow by differentiating, and the Hessian is negative definite as in Theorem 17.1.2. The score equations \(\frac 1n\sum x_i(y_i - \exp (x_i'\beta )) = 0\) are the sample analogue of the moment condition \(E[x_i(y_i - \exp (x_i'\beta _0))] = 0\), which holds whenever the conditional mean is correct, by the law of iterated expectations. So \(\hat \beta \) is a method of moments estimator, consistent under that condition alone, and Theorem 15.1.3 gives its variance with \(G = -E(\mu xx')\) and \(S = E\bigl (\Var (y \mid x)xx'\bigr )\). □
This pseudo-maximum likelihood property (Gourieroux, Monfort and Trognon, 1984) makes Poisson regression far more useful than the Poisson distribution itself. Real counts are usually overdispersed: their variance exceeds their mean, while the Poisson distribution forces them to be equal. The Poisson estimates remain consistent, but the usual standard errors, \((\sum \hat \mu _ix_ix_i')^{-1}\), are too small, and the sandwich form of the theorem must be used. Alternatively, the negative binomial model, with \(\Var (y \mid x) = \mu + \mu ^2/\theta \), models the overdispersion explicitly; it is more efficient if correct but, unlike Poisson, inconsistent if its variance assumption is wrong.
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