Endogeneity with an exponential regression function

In statistics as applied to econometrics, exponential regression models constitute a very large and popular class of regression models. Standard econometric concerns such as endogeneity or omitted variables can be accounted for in a more general framework. Wooldridge and Terza provide a methodology to both deal with and test for endogeneity within the exponential regression framework, which the following discussion follows closely. While the example focuses on a Poisson regression model, it is possible to generalize the test to other exponential regression models, although this may come at the cost of additional assumptions (e.g. for binary response or censored data models).

Assume the following exponential regression model, where $a_{i}$ is an unobserved term in the latent variable. We allow for correlation between $a_{i}$ and $x_{i}$ (implying $x_{i}$ is possibly endogenous), but allow for no such correlation between $a_{i}$ and $z_{i}$.

$\operatorname {E} [y_{i}\nu x_{i},z_{i},a_{i}]=\exp(x_{i}b_{0}+z_{i}c_{0}+a_{i})$ (1)

The variables $z_{i}$ serve as instrumental variables for the potentially endogenous $x_{i}$. One can assume a linear relationship between these two variables or alternatively project the endogenous variable $x_{i}$ onto the instruments to get the following reduced form equation:

$x_{i}=z_{i}\Pi +v_{i}$ (2)

The usual rank condition is needed to ensure identification. The endogeneity is then modeled in the following way, where $\rho$ determines the severity of endogeneity and $v_{i}$ is assumed to be independent of $e_{i}$.

$a_{i}=v_{i}\rho +e_{i}$ (3)

Imposing these assumptions, assuming the models are correctly specified, and normalizing $\operatorname {E} [\exp(e_{i})]=1,$, we can rewrite the conditional mean as follows:

$\operatorname {E} [y_{i}vx_{i},z_{i},a_{i}]=\exp(x_{i}b_{0}+z_{i}c_{0}+e_{i}\rho )$ (4)

If $e_{i}$ were known at this point, it would be possible to estimate the relevant parameters by quasi-maximum likelihood estimation. Following the two step procedure strategies, Wooldridge and Terza propose estimating equation [2] by standard OLS methods. The fitted residuals from this regression can then be plugged into the estimating equation [4] and QMLE methods will lead to consistent estimators of the parameters of interest. Significance tests on ${\hat {\rho }}$ can then be used to test for endogeneity within the model.

The methodology proposed here is often used for exponential regression functions. However, the specific assumptions that need to be made can differ across models. Binary response models impose distributional assumptions on y_i and x_i, whereas this model imposed independence between $v_{i}$ and $e_{i}$.

See also

• Binary response model with continuous endogenous explanatory variables
• Endogeneity in multinomial response model

Bibliography

• Wooldridge, J. (1997): Quasi-Likelihood Methods for Count Data, Handbook of Applied Econometrics, Volume 2, ed. M. H. Pesaran and P. Schmidt, Oxford, Blackwell, pp. 352–406
• Terza, J. V. (1998): "Estimating Count Models with Endogenous Switching: Sample Selection and Endogenous Treatment Effects." Journal of Econometrics (84), pp. 129–154
• Wooldridge, J. (2002): "Econometric Analysis of Cross Section and Panel Data", MIT Press, Cambridge, Massachusetts.