Nonparametric estimation of the density of a change-point

Abstract

The paper considers a panel model where the regression coe¢ cients undergo changes at an unknown time point, di§erentfor each series. The timings of changes are assumed to be independent, identically distributed, and drawn from some com-mon distribution, the density of which we aim to estimate nonparametrically. The estimation procedure involves two steps. First, changepoints are estimated indi-vidually for each series using the least-squares method. While these estimators are not consistent, they can be regarded as noisy signals of the true change-points. To address the inherent estimation error, a deconvolution kernel estimator is applied to estimate the density of the change-point. The paper establishes the consistency of this estimator and demonstrates that the rate of convergence of the Mean Inte-grated Squared error (MISE) is faster than that obtained with normal or Laplacian errors. Finally, using a Bayesian approach, we propose an estimator of the poste-rior means of the breakpoints, utilizing nonparametric estimates of the required densities. An application of the proposed methodology to portfolio returns reveals how quickly the markets responded to the Covid shock.