¶ 2.2.1 Example

The Poisson regression model

$\begin{array}{cccc}& f\left({y}_{i}\mathrm{\mid }{\mathbf{x}}_{\mathbf{i}}\right)=\frac{{e}^{-{\mu }_{i}}{\mu }_{i}^{{y}_{i}}}{{y}_{i}!}& & \text{(2.1)}\end{array}\tag\left\{2.1\right\} f\left(y_i|\mathbf\left\{x_i\right\}\right)=\frac\left\{e^\left\{-\mu_i\right\}\mu_i^\left\{y_i\right\}\right\}\left\{y_i!\right\}$

$\begin{array}{cccc}& {\mu }_{i}=\mathrm{exp}\left({\mathbf{x}}_{\mathbf{i}}^{\mathbf{\prime }}\mathbf{\beta }\right)& & \text{(2.2)}\end{array}\tag\left\{2.2\right\} \mu_i=\exp\left(\mathbf\left\{x_i\text{'}\right\}\pmb\left\{\beta\right\}\right)$

Variables:

• Count data: ${y}_{i}=0,1,2,\dots y_i=0,1,2,\dots$
• Regressors: \mathbf

Equation (2.1) is the pdf of the Poisson distribution with mean ${\mu }_{i}\mu_i$.

The model implies that the conditional mean is given by

$\begin{array}{cccc}& \mathsf{E}\left[{y}_{i}\mathrm{\mid }{\mathbf{x}}_{\mathbf{i}}\right]=\mathsf{V}\left[{y}_{i}\mathrm{\mid }{\mathbf{x}}_{\mathbf{i}}\right]=\mathrm{exp}\left({\mathbf{x}}_{\mathbf{i}}^{\mathbf{\prime }}\mathbf{\beta }\right)& & \text{(2.3)}\end{array}\tag\left\{2.3\right\} \mathsf\left\{E\right\}\left[y_i|\mathbf\left\{x_i\right\}\right]=\mathsf\left\{V\right\}\left[y_i|\mathbf\left\{x_i\right\}\right]=\exp\left(\mathbf\left\{x_i\text{'}\right\}\pmb\left\{\beta\right\}\right)$

The standard estimator for the Poisson regression model is the MLE. The log-likelihood function is

$\begin{array}{cccc}& \mathcal{L}\left(\mathbf{\beta }\right)=\sum _{i=1}^{n}\left\{{y}_{i}{\mathbf{x}}_{\mathbf{i}}^{\mathbf{\prime }}\mathbf{\beta }-\mathrm{exp}\left({\mathbf{x}}_{\mathbf{i}}^{\mathbf{\prime }}\mathbf{\beta }\right)-\mathrm{ln}{y}_{i}!\right\}& & \text{(2.5)}\end{array}\tag\left\{2.5\right\} \mathcal\left\{L\right\}\left(\pmb\left\{\beta\right\}\right)=\sum_\left\{i=1\right\}^n \\left\{ y_i\mathbf\left\{x_i\text{'}\right\}\pmb\left\{\beta\right\}-\exp\left(\mathbf\left\{x_i\text{'}\right\}\pmb\left\{\beta\right\}\right)-\ln y_i!\\right\}$

FOCs

$\begin{array}{cccc}& \sum _{i=1}^{n}\left({y}_{i}-\mathrm{exp}\left({\mathbf{x}}_{\mathbf{i}}^{\mathbf{\prime }}\mathbf{\beta }\right)\right){\mathbf{x}}_{\mathbf{i}}=\mathbf{0}& & \text{(2.6)}\end{array}\tag\left\{2.6\right\} \sum_\left\{i=1\right\}^n \big\left( y_i-\exp\left(\mathbf\left\{x_i\text{'}\right\}\pmb\left\{\beta\right\}\right)\big\right) \mathbf\left\{x_i\right\}=\mathbf\left\{0\right\}$

¶ The Generalized linear models approach

Similar to OLS, inference can be performed under assumptions about just the mean and possibly variance.

¶ The Moment-based models approach

Because Equation 2.3 implies that $\mathsf{E}\left[\left({y}_{i}-\mathrm{exp}\left({\mathbf{x}}_{\mathbf{i}}^{\mathbf{\prime }}\mathbf{\beta }\right)\right){\mathbf{x}}_{\mathbf{i}}\right]=\mathbf{0}\mathsf\left\{E\right\}\big\left[\big\left( y_i-\exp\left(\mathbf\left\{x_i\text{'}\right\}\pmb\left\{\beta\right\}\right)\big\right) \mathbf\left\{x_i\right\}\big\right]=\mathbf\left\{0\right\}$, we can define an estimator that is the solution to the Equation 2.6, the corresponding moment condition in the sample.

• Cameron, A.C. and Trivedi, P.K. (2013) Regression Analysis of Count Data. 2nd edn. Cambridge: Cambridge University Press (Econometric Society Monographs). doi:10.1017/CBO9781139013567.