Threshold spatial autoregressive model

In this paper, we consider the estimation and inferential issues of the threshold spatial autoregressive (TSAR) model, which is a hybrid of the threshold and spatial autoregressive models. We use the quasi maximum likelihood (QML) method to estimate the model. In addition, we prove the tightness and the Hájek–Rényi type inequality for a quadratic form and establish a full inferential theory of the QML estimator under the setup that threshold effect shrinks to zero as the sample size increases. We conduct hypothesis testing on the presence of the threshold effect, using three super-type statistics. Their asymptotic behaviors are studied under the Pitman local alternatives. A bootstrap procedure is applied to obtain the asymptotically correct critical value. We also consider hypothesis testing on the threshold value set equal to a prespecified one. We run Monte Carlo simulations to investigate the finite sample performance of the QML estimators and find that the estimators perform well. In an empirical application, we apply the proposed TSAR model to study the relationship between financial development and economic growth, and we find firm evidence to support the TSAR model.