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Structural Threshold Quantile Regression

In this paper, a simultaneous equation model with an endogenous variable and an exogenous threshold variable is analysed and estimated thereby extending Caner and Hansen (2004) model to quantile regression. In our framework, we allow both the reduced-form and the structural equation to exhibit regime-change behavior. An two-step estimation procedure for the model parameters is proposed assuming an unknown threshold value while existing estimation procedures are extended to deal with the case of a known threshold value. We develop an asymptotic frame-work for the parameter estimators and the estimated threshold value assuming the size of the regime-change shrinks as the sample size increases and we derive the limiting distribution of the last. A likelihood-ratio-type statistic is employed to test hypotheses of interest concerning the estimated threshold and its limiting distribution is derived. We also form confidence regions robust for heteroskedasticity for the estimated threshold by inverting the likelihood-ratio-type statistic as in Hansen (2000) and simulate its coverage probability. Extensive simulation assesses the accuracy of our estimation procedure in detecting the unknown threshold value under different error distributions and size of the threshold.