Non linear correlated random effects models with endogeneity and unbalanced panels

Abstract

We present simple procedures for estimating non linear panel data models in the presence of unobserved heterogeneity and possible endogeneity with respect to time-varying unobservables. We combine a correlated random effects approach with a control function approach while accounting for missing time periods for some units. We examine the performance of the approach in comparisons with standard estimators using Monte Carlo simulation. We apply the methods to estimate the effects of school spending on student pass rates on a standardized math exam. We find that a 10% increase in spending leads to an approximately 2 percentage point increase in math pass rates.

Keywords:

• Control function
• correlated random effects
• quasi-maximum likelihood estimation
• sufficient statistic

Disclosure statement

Michael Bates, Leslie Papke, and Jeffrey Wooldridge declare no potential conflicts of interest with respect to the research, authorship, and/or publication of this article. The authors disclose receipt of the following financial support for the research, authorship, and/or publication of this article. The opinions expressed are those of the authors and do not represent the views of the Institute or the U.S. Department of Education. The data used for this project lies outside the scope of IRB approval, and no IRB approval was sought.

Notes

1. While this assumption can be violated, all standard estimation methods – even in linear models – rule out the possibility that selection is correlated with shocks. As with the usual fixed effects and fixed effects IV estimators of linear models, we allow selection to be correlated with unobserved heterogeneity. Allowing selection to be correlated with the shocks is considerably more difficult. Wooldridge (1995) and Semykina and Wooldridge (2010) show how to allow this in the context of linear models. We leave to future research the possibility of extending selection methods to fractional responses.

2. The resulting estimator is now often called “fractional” probit to emphasize that the outcome variable, y_it1, can take on any value in the unit interval. That we are using the Bernoulli log-likelihood function when y_it1 is not a binary variable is what makes our procedure a pooled quasi-maximum likelihood estimator. It is true that we have made functional form assumptions and normal distributional assumptions on unobservables to derive the conditional expectation in (12(12) $$\begin{array}{l}\begin{array}{ll}E(y_{\mathrm{it}1}|y_{\mathrm{it}2},& {\mathbf{z}}_i,v_{\mathrm{it}2},s_{\mathrm{it}}=1)=\\ & \mathrm{\Phi }({\beta }_{1e}{y}_{\mathrm{it}2}+{\mathbf{z}}_{\mathrm{it}1}{\delta }_1+\sum _{r=1}^T{\psi }_{r1e}1[T_i=r]+\sum _{r=1}^T\left(1[T_i=r]\cdot {\overline{\mathit{z}}}_i\right)\mathit{\xi }{}_{r1e}+{\eta }_{1e}{v}_{\mathrm{it}2}),\end{array}\end{array}$$(12) ). Nevertheless, it is important to understand that only (12(12) $$\begin{array}{l}\begin{array}{ll}E(y_{\mathrm{it}1}|y_{\mathrm{it}2},& {\mathbf{z}}_i,v_{\mathrm{it}2},s_{\mathrm{it}}=1)=\\ & \mathrm{\Phi }({\beta }_{1e}{y}_{\mathrm{it}2}+{\mathbf{z}}_{\mathrm{it}1}{\delta }_1+\sum _{r=1}^T{\psi }_{r1e}1[T_i=r]+\sum _{r=1}^T\left(1[T_i=r]\cdot {\overline{\mathit{z}}}_i\right)\mathit{\xi }{}_{r1e}+{\eta }_{1e}{v}_{\mathrm{it}2}),\end{array}\end{array}$$(12) ) is assumed to hold in our analysis; no other feature of the distribution of y_it1 given y_it2, 𝐳_i, v_it2) and s_it1 = 1 is being assumed to be correctly specified. Moreover, we are not restricting the serial dependence across time in other y_it1 or any of the explanatory variables.

3. We reran the simulation generating using data-generating processes from two non normal distributions to investigate the performance of the estimator when the distribution is misspecified. The description and results appear in the online appendix and reveal that the estimators perform well in both cases.

4. Papke (2005), Chaudhary (2009), and Roy (2011) provide fuller discussion of this school finance reform.

5. As discussed above, the coefficients remain scaled by the variance of the unobserved heterogeneity.

6. Note that due to collinearity, interactions between the indicator for 5 time-observations and time averages for indicators for years 1995, 1996, 1997, and 1998 are omitted as is the interaction between the indicator for four time-observations and the time average of the indicator for year 1997 and 1998.

7. Note that due to the unbalancedness of the data, we also include time averages of the year indicators – ${\overline{y96}}_i$, ${\overline{y97}}_i$, and ${\overline{y98}}_i$.

8. We provide additional robustness checks in the appendix where we include interactions between time averages and time-observation indicators in the linear estimation (FEIVU), and standard differencing in the first-stage estimation of our non linear approaches (CRE FECF, CREU FECF, CREU1 FECF).

9. Note that due to collinearity, interactions between the indicator for five time-observations and time averages for indicators for years 1996, 1997, and 1998 are omitted, as are the interactions between the indicator for four time-observations and the time averages of the indicators for years 1996 and 1997.

Additional information

Funding

The research reported here was supported by the Institute of Education Sciences, U.S. Department of Education, through R305B090011 to Michigan State University.