Detection of the Threshold Interval for Double-Threshold Polynomial Regression Models

Abstract

In this talk, we considered a double-threshold polynomial regression model, where the mean trend of the model is continuous and is equal to a constant within the threshold interval. We denoted this model by M2, which includes models with one or no threshold points, denoted by M1 and M0, respectively, as special cases. We provided an ordered iterate least squares estimation (OiLSE) method when estimating models Mκ, κ = 1, 2, where the consistency of the OiLSE estimate with the oracle convergence rates are established under some mild conditions. We also applied a model-selection procedure for selecting Mκ; κ = 0, 1, 2, for dependent data with heterogeneous variance, where the threshold points in variance are allowed at any locations. When the underlying model exists, we also established the selection consistency under the aforementioned conditions. Finally, we conducted some simulation experiments to demonstrate the finite-sample performance of our asymptotic results.