Time series quantile regression kink with an unknown threshold

This article studies a time series quantile regression kink model with an unknown threshold over certain quantile levels in the distribution. We propose to estimate the threshold parameter and regression parameters using a two-stage method. We also propose a weighted CUSUM test for threshold effect at both a given quantile level and multiple quantile levels based on the subgradient of the quantile loss function. In addition, we consider a likelihood-ratio-type test for the presence of a common threshold value across different quantile levels. Excellent finite sample performance of the proposed method is demonstrated by simulation studies. We further apply our proposed method to the S & P500 index data to explore the possible nonlinearity and heteroscedasticity of the return autocorrelations in the S & P500 index.