An M-estimation and inference approach for matrix exponential unbalanced panel data models

We investigate likelihood-based estimations of unbalanced panel data models with entity and time fixed effects that allow for cross-sectional dependence specified through matrix exponential terms. We consider a hybrid approach to deal with the fixed effects that can create the incidental parameters problem. We first eliminate the time fixed effects from the model using an orthogonal transformation. We then concentrate out the entity fixed effects from the quasi log-likelihood function of the transformed model. We introduce an M-estimator that utilises analytically adjusted score functions of the concentrated quasi log-likelihood function. We show that the suggested M-estimator is consistent and has an asymptotic normal distribution irrespective of whether the number of time periods is large or small. For consistent estimation of the variance-covariance matrix of the M-estimator, we propose an analytical bias correction approach involving the sample counterpart and plug-in methods. Through an extensive Monte Carlo study, we show that the suggested M-estimator has good finite sample properties. Finally, we use our model specification to study spatial correlation in crime rates and total factor productivity.