Wavelet analysis of change-points in a non-parametric regression with heteroscedastic variance

In this paper we develop wavelet methods for detecting and estimating jumps and cusps in the mean function of a non-parametric regression model. An important characteristic of the model considered here is that it allows for conditional heteroscedastic variance, a feature frequently encountered with economic and financial data. Wavelet analysis of change-points in this model has been considered in a limited way in a recent study by Chen et al. (2008) with a focus on jumps only. One problem with the aforementioned paper is that the test statistic developed there has an extreme value null limit distribution. The results of other studies have shown that the rate of convergence to the extreme value distribution is usually very slow, and critical values derived from this distribution tend to be much larger than the true ones. Here, we develop a new test and show that the test statistic has a convenient null limit N(0,1) distribution. This feature gives the proposed approach an appealing advantage over the existing approach. Another attractive feature of our results is that the asymptotic theory developed here holds for both jumps and cusps. Implementation of the proposed method for multiple jumps and cusps is also examined. The results from a simulation study show that the new test has excellent power and the estimators developed also yield very accurate estimates of the positions of the discontinuities.