Estimating multiple breaks in nonstationary autoregressive models

Chong (1995) and Bai (1997) proposed a sample-splitting method to estimate a multiple-break model. However, their studies focused on stationary time series models, in which the identification of the first break depends on the magnitude and the duration of the break, and a testing procedure is needed to assist the estimation of the remaining breaks in subsamples split by the break points found earlier. In this paper, we focus on nonstationary multiple-break autoregressive models. Unlike the stationary case, we show that the duration of a break does not affect whether it will be identified first. Rather, it depends on the stochastic order of magnitude of signal strength of the break under the case of constant break magnitude and also the square of the magnitude of the break under the case of shrinking break magnitude. Since the subsamples usually have different stochastic orders in nonstationary autoregressive models with breaks, one can therefore determine which break will be identified first. We apply this finding to the models proposed in Phillips and Yu (2011) and Phillips et al. (2011, 2015a, 2015b). We propose an estimation procedure as well as the asymptotic theory for the model. Some extensions to more general models are provided, and the hypothesis test with the null hypothesis being the unit root model is examined. Results of numerical simulations and an empirical study are given to illustrate the finite-sample performance.