τ-estimators of regression models with structural change of unknown location

Abstract This paper concerns with robust estimation of linear regression models with structural change of unknown location under possibly contaminated distributions both for the regressors and for the perturbance term. Existing estimators will be inefficient in this context. Furthermore, they may protect against outlying Yt, but cannot cope with leverage points, namely outliers in the factor space, which could have large influence on the fit. As a result, these estimators could not discriminate between outlier observations and structural break points, misplacing the shift location. This fact can be of special importance in practice. Therefore, it may be advisable to consider robust estimators under possible leverage points in a structural change context. Thus, we propose the τ-estimator, introduced by Yohai and Zamar (J. Am. Stat. Assoc. 83 (1988) 406) in the standard context of no change. This type of estimator is qualitatively robust, with the best possible breakdown-point and highly efficient under normal errors. The asymptotic distribution of the break location estimator is obtained both for fixed magnitude of shift and for shift with magnitude converging to zero as the sample size increases. The analysis is carried out in the framework of general NED dependence conditions for the data. Monte Carlo experiments illustrates the performance of our estimators in finite samples.

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