  Expectile regression is a useful tool for exploring the relation between the
response and the explanatory variables beyond the conditional mean. This
article develops a continuous threshold expectile regression for modeling data
in which the effect of a covariate on the response variable is linear but
varies below and above an unknown threshold in a continuous way. Based on a
grid search approach, we obtain estimators for the threshold and the regression
coefficients via an asymmetric least squares regression method. We derive the
asymptotic properties for all the estimators and show that the estimator for
the threshold achieves root-n consistency. We also develop a weighted CUSUM
type test statistic for the existence of a threshold in a given expectile, and
derive its asymptotic properties under both the null and the local alternative
models. This test only requires fitting the model under the null hypothesis in
the absence of a threshold, thus it is computationally more efficient than the
likelihood-ratio type tests. Simulation studies show desirable finite sample
performance in both homoscedastic and heteroscedastic cases. The application of
our methods on a Dutch growth data and a baseball pitcher salary data reveals
interesting insights.
