Markov models for digraph panel data: Monte Carlo-based derivative estimation

A parametric, continuous-time Markov model for digraph panel data is considered. The parameter is estimated by the method of moments. A convenient method for estimating the variance-covariance matrix of the moment estimator relies on the delta method, requiring the Jacobian matrix—that is, the matrix of partial derivatives—of the estimating function. The Jacobian matrix was estimated hitherto by Monte Carlo methods based on finite differences. Three new Monte Carlo estimators of the Jacobian matrix are proposed, which are related to the likelihood ratio/score function method of derivative estimation and have theoretical and practical advantages compared to the finite differences method. Some light is shed on the practical performance of the methods by applying them in a situation where the true Jacobian matrix is known and in a situation where the true Jacobian matrix is unknown.