Threshold spatial autoregressive model

This paper considers the estimation and inferential issues of threshold spatial autoregressive model, which is a hybrid of threshold model and spatial autoregressive model. We consider using the quasi maximum likelihood (QML) method to estimate the model. We prove the tightness and the H\'{a}jek-R\'{e}nyi 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 along with an increasing sample size. We consider the hypothesis testing on the presence of threshold effect. Three super-type statistics are proposed to perform this testing. Their asymptotic behaviors are studied under the Pitman local alternatives. A bootstrap procedure is proposed to obtain the asymptotically correct critical value. We also consider the hypothesis testing on the threshold value equal to some prespecified one. We run Monte carlo simulations to investigate the finite sample performance of the QML estimators and find that the QML estimators have good performance.