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Estimating multiple breaks in mean sequentially with fractionally integrated errors. (English) Zbl 1480.60057

Summary: This paper studies the estimation issue for multiple breaks in mean when the model errors are fractionally integrated processes of order \(d\) with \(-0.5<d<0.5\). For this problem with \(0<d<0.5\), M. Lavielle and E. Moulines [J. Time Ser. Anal. 21, No. 1, 33–59 (2000; Zbl 0974.62070)] estimated the break fractions simultaneously by least squares method. The computational complexity of this method is of order \(O(T^2)\) even employing the dynamic programming algorithm [J. Bai and P. Perron, “Computation and analysis of multiple structural change models”, J. Appl. Econ. 18, No. 1, 1–22 (2003; doi:10.1002/jae.659)], where \(T\) is the sample size. It is well-known that the computational complexity of the sequential method is of order \(O(T)\) [J. Bai, “Estimating multiple breaks one at a time”, Econ. Theory 13, No. 3, 315–352 (1997; doi:10.1017/S0266466600005831)], which is obviously more efficient than the simultaneous method in terms of the computational cost. Therefore, in this paper, we revisit the issue of estimating multiple breaks in mean with fractionally integrated errors, and examine the sequential method for the above issue. It is found that: (1) when the break magnitudes are fixed, the convergence rates of the estimators for the break fractions are all \(1 / T\), which is invariant to the fractionally differencing parameter \(d\) over the entire range \(d\in (-0.5, 0.5)\); (2) when the break magnitudes shrink to zero as \(T\rightarrow \infty \), both the convergence rates and the asymptotic distributions of the estimators for the break fractions rely on the fractionally differencing parameter \(d\) for all \(0\leq d<0.5\). The convergence rates of the estimators for the break fractions in the case of \(-0.5<d<0\) in this paper are not optimal due to the absence of optimal Hájek-Rényi inequalities. Our theoretical results extend and improve the results established in [Bai, loc. cit.] and [C.-M. Kuan and C.-C. Hsu, J. Time Ser. Anal. 19, No. 6, 693–708 (1998; Zbl 0921.62112)], respectively. Monte Carlo simulations are conducted to examine the finite-sample performance for the estimators. Our theoretical findings are supported by the Monte Carlo simulations.

MSC:

60F05 Central limit and other weak theorems
62F12 Asymptotic properties of parametric estimators
90C39 Dynamic programming
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
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