Estimation of Matrix Exponential Unbalanced Panel Data Models with Fixed Effects: An Application to US Outward FDI Stock

In this article, we consider a matrix exponential unbalanced panel data model that allows for (i) spillover effects using matrix exponential terms, (ii) unobserved heterogeneity across entities and time, and (iii) potential heteroscedasticity in the error terms across entities and time. We adopt a likelihood based direct estimation approach in which we jointly estimate the common parameters and fixed effects. To ensure that our estimator has the standard large sample properties, we show how the score functions should be suitably adjusted under both homoscedasticity and heteroscedasticity. We define our suggested estimator as the root of the adjusted score functions, and therefore our approach can be called the M-estimation approach. For inference, we suggest an analytical bias correction approach involving the sample counterpart and plug-in methods to consistently estimate the variance-covariance matrix of the suggested M-estimator. Through an extensive Monte Carlo study, we show that the suggested M-estimator has good finite sample properties. In an empirical application, we use our model to investigate the third country effects on the U.S. outward foreign direct investment (FDI) stock at the industry level.