Estimation, inference and model selection for jump regression models

TL;DR

This paper studies estimation and model selection in jump regression models, revealing the asymptotic properties of estimators and proposing new criteria for selecting jump points, with Bayesian methods showing advantages.

Contribution

It provides theoretical analysis of estimator rates and introduces jump information criteria (AJIC and BJIC) for model selection in jump regression.

Findings

Jump point parameters are estimated at n-rate precision.

Level parameters are estimated at √n-rate precision.

Bayesian solutions outperform least squares estimators.

Abstract

We consider regression models with data of the type $y_i=m(x_i)+\varepsilon_i$, where the $m(x)$ curve is taken locally constant, with unknown levels and jump points. We investigate the large-sample properties of the minimum least squares estimators, finding in particular that jump point parameters and level parameters are estimated with respectively $n$-rate precision and $\sqrt{n}$-rate precision, where $n$ is sample size. Bayes solutions are investigated as well and found to be superior. We then construct jump information criteria, respectively AJIC and BJIC, for selecting the right number of jump points from data. This is done by following the line of arguments that lead to the Akaike and Bayesian information criteria AIC and BIC, but which here lead to different formulae due to the different type of large-sample approximations involved.