¶ Likelihood-based models

Models in which the joint density of the dependent variables is specified. The likelihood fucntion is

$\begin{array}{cccc}& \mathsf{L}\left(\mathbf{\theta }\right)=\prod _{i=1}^{n}f\left({y}_{i}\mathrm{\mid }{\mathbf{x}}_{\mathbf{i}},\mathbf{\theta }\right)& & \text{(2.9)}\end{array}\tag\left\{2.9\right\} \mathsf\left\{L\right\}\left(\pmb\theta\right)=\prod_\left\{i=1\right\}^\left\{n\right\} f\left(y_i|\mathbf\left\{x_i\right\},\pmb\theta\right)$

The log-likelihood function is

$\begin{array}{cccc}& \mathcal{L}\left(\mathbf{\theta }\right)=\mathrm{ln}\mathsf{L}\left(\mathbf{\theta }\right)=\sum _{i=1}^{n}\mathrm{ln}f\left({y}_{i}\mathrm{\mid }{\mathbf{x}}_{\mathbf{i}},\mathbf{\theta }\right)& & \text{(2.10)}\end{array}\tag\left\{2.10\right\} \mathcal\left\{L\right\}\left(\pmb\theta\right)=\ln\mathsf\left\{L\right\}\left(\pmb\theta\right)=\sum_\left\{i=1\right\}^\left\{n\right\} \ln f\left(y_i|\mathbf\left\{x_i\right\},\pmb\theta\right)$

A weakness of the ML approach is that it assumes correct specification of the complete density. White (1982) obtained the distribution of the MLE if the density function is incorrectly specified. Then the ML estimator is called the quasi-MLE or pseudo-MLE. In general ${\stackrel{^}{\mathbf{\theta }}}_{\mathbf{Q}\mathbf{M}\mathbf{L}}\stackrel{p}{\to }{\mathbf{\theta }}_{\ast }\pmb\left\{\hat\theta_\left\{QML\right\}\right\}\xrightarrow\left\{p\right\}\pmb\left\{\theta_*\right\}$, where ${\mathbf{\theta }}_{\ast }\pmb\left\{\theta_*\right\}$ is the pseudotrue value.

¶ Generalized linear models

If the density is misspecified the MLE is in general inconsistent. Generalized linear models are notable exception.

• For example, for linear regression model, the essential requirement for consistency of MLE is correct specification of the conditional mean: $\mathsf{E}\left[{y}_{i}\mathrm{\mid }{\mathbf{x}}_{\mathbf{i}}\right]={\mathbf{x}}_{\mathbf{i}}^{\mathbf{\prime }}{\mathbf{\beta }}_{\mathbf{0}}\mathsf\left\{E\right\}\left[y_i|\mathbf\left\{x_i\right\}\right]=\mathbf\left\{x_i\text{'}\right\}\pmb\left\{\beta_0\right\}$.

• A similar situation arises for the Poisson regression model. Consistency essentially requires that population analog of Equation (2.6) holds:

$\mathsf{E}\left[\left({y}_{i}-\mathrm{exp}\left({\mathbf{x}}_{\mathbf{i}}^{\mathbf{\prime }}{\mathbf{\beta }}_{\mathbf{0}}\right)\right){\mathbf{x}}_{\mathbf{i}}\right]=\mathbf{0}\mathsf\left\{E\right\}\left[\big\left( y_i-\exp\left(\mathbf\left\{x_i\text{'}\right\}\pmb\left\{\beta_0\right\}\right)\big\right) \mathbf\left\{x_i\right\}\right]=\mathbf\left\{0\right\}$

• This is satisfied if $\mathsf{E}\left[{y}_{i}\mathrm{\mid }{\mathbf{x}}_{\mathbf{i}}\right]=\mathrm{exp}\left({\mathbf{x}}_{\mathbf{i}}^{\mathbf{\prime }}{\mathbf{\beta }}_{\mathbf{0}}\right)\mathsf\left\{E\right\}\left[y_i|\mathbf\left\{x_i\right\}\right]=\exp\left(\mathbf\left\{x_i\text{'}\right\}\pmb\left\{\beta_0\right\}\right)$.

More generally such results hold for ML estimation of modles with specified density that is a member of the linear exponential family, and estimation of the closely related class of generalized linear models.

Although consistency in these models requires only correct specification of the mean, misspecification of the variance leads to invalid statistical inference due to incorrect reported $tt$-statistics and standard errors.

¶ Moment-based models

• 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.