Estimating a gradual parameter change in an AR(1)-process

We discuss the estimation of a change-point t(0) at which the parameter of a (non-stationary) AR(1)-process possibly changes in a gradual way. Making use of the observations X-1,..., X-n, we shall study the least squares estimator (t(0)) over cap for t(0), which is obtained by minimizing the sum of squares of residuals with respect to the given parameters. As a first result it can be shown that, under certain regularity and moment assumptions, (t(0)) over cap /n is a consistent estimator for t(0), where t(0) = left perpendicularn tau(0)right perpendicular, with 0 (P) tau(0) (n ->infinity). Based on the rates obtained in the proof of the consistency result, a first, but rough, convergence rate statement can immediately be given. Under somewhat stronger assumptions, a precise rate can be derived via the asymptotic normality of our estimator. Some results from a small simulation study are included to give an idea of the finite sample behaviour of the proposed estimator

Robust monitoring of CAPM portfolio betas II

In this work, we extend our study in Chochola et al. [7] and propose some robust sequential procedure for the detection of structural breaks in a Functional Capital Asset Pricing Model (FCAPM). The procedure is again based on M-estimates and partial weighted sums of M-residuals and robustifies the approach of Aue et al. [3], in which ordinary least squares (OLS) estimates have been used. Similar to Aue et al. [3], and in contrast to Chochola et al. [7], high-frequency data can now also be taken into account. The main results prove some null asymptotics for the suggested test as well as its consistency under local alternatives. In addition to the theoretical results, some conclusions from a small simulation study together with an application to a real data set are presented in order to illustrate the finite sample performance of our monitoring procedure. (C) 2014 Elsevier Inc. All rights reserved