Expectile and Quantile Kink Regressions with Unknown Threshold

In this study, we propose two non-linear models for explaining the relationship between the response and the predictor variables beyond the conditional mean. We extend the kink approach to quantile and expcetile regressions thus the models provide a more complete picture of the conditional distribution of the response variable in the non-linear context. The proposed models allow us to identify and explore the reputation effect and its heterogeneity in data. The simulation and application studies are also proposed to examine the performance of our models. We find that neither of the approaches is uniformly superior nor both of them have their advantages over each other and it is not clear which model provides the best fit results. However, the application of our models on a service output data shows that expectile kink regression is more conservative than the quantile kink regression.