Partially time invariant panel data regression

When dealing with panel data, considering the variation over time of the variable of interest allows to get rid of potential individual effects. Even though the outcome variable has a continuous distribution, its variation over time can be equal to zero with a strictly positive probability and thus its distribution is a mixture of a mass at zero and a continuous distribution. We introduce a parametric statistical model based on conditional mixtures, build estimators for the parameters related to the conditional probability of no variation and to the conditional expectation related to the continuous part of the distribution and derive their asymptotic consistency and normality under a specific conditional independence assumption. Consistent confidence intervals are built via an empirical bootstrap approach. In the framework of policy evaluation, we study estimates of treatment effects based on difference-indifferences under the same zero inflation phenomenon and propose estimators of the average treatment effect that are proven to be consistent and asymptotically Gaussian. A small Monte Carlo simulation study assesses the good behavior of the estimators for finite samples and highlights that miss specified models that do not take account of the zero inflation may have a substantial bias. Empirical illustrations based on long time difference for the Mincer wage equation as well as the evaluation of European rural development policies based on the difference-indifferences approach confirm the interest of the proposed statistical modeling, bringing new insights on the size of the bias in commonly used regression models.