Irregular N2SLS and LASSO estimation of the matrix exponential spatial specification model

In this paper, we consider estimation of the matrix exponential spatial specification model with the Durbin and endogenous regressors. We find that the nonlinear two-stage least squares (N2SLS) estimator is in general consistent and asymptotically normal. However, when the Durbin and endogenous regressors are irrelevant, the gradient vector of the N2SLS criterion function has a singular covariance matrix with probability approaching one (w.p.a.1.). Some components of the N2SLS estimator have slower rates of convergence and their asymptotic distributions are nonstandard. The distance difference and gradient test statistics, which have irregular asymptotic distributions, are derived to test for the irrelevance of the Durbin and endogenous regressors. As an alternative estimation and model selection approach, we propose the adaptive group LASSO, which penalizes the coefficients of the Durbin and endogenous explanatory variables. We show that the estimator has the oracle properties, so the true model can be selected w.p.a.1. and the estimator always has the n-rate of convergence and asymptotic normal distribution. We propose to select the tuning parameter for the adaptive group LASSO by minimizing an information criterion.