## Multiple breaks detection in general causal time series using penalized quasi-likelihood.(English)Zbl 1337.62210

Summary: This paper is devoted to the off-line multiple breaks detection for a general class of models. The observations are supposed to fit a parametric causal process (such as classical models AR($$\infty$$), ARCH($$\infty$$) or TARCH($$\infty$$)) with distinct parameters on multiple periods. The number and dates of breaks, and the different parameters on each period are estimated using a quasi-likelihood contrast penalized by the number of distinct periods. For a convenient choice of the regularization parameter in the penalty term, the consistency of the estimator is proved when the moment order $$r$$ of the process satisfies $$r\geq 2$$. If $$r\geq 4$$, the length of each approximative segment tends to infinity at the same rate as the length of the true segment and the parameters estimators on each segment are asymptotically normal. Compared to the existing literature, we added the fact that a dependence is possible over distinct periods. To be robust to this dependence, the chosen regularization parameter in the penalty term is larger than the ones from BIC approach. We detail our results which notably improve the existing ones for the AR($$\infty$$), ARCH($$\infty$$) and TARCH($$\infty$$) models. For the practical applications (when $$n$$ is not too large) we use a data-driven procedure based on the slope estimation to choose the penalty term. The procedure is implemented using the dynamic programming algorithm. It is an $$O(n^{2})$$ complexity algorithm that we apply on AR(1), AR(2), GARCH(1,1) and TARCH(1) processes and on the FTSE index data.

### MSC:

 62M07 Non-Markovian processes: hypothesis testing 62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH) 62F12 Asymptotic properties of parametric estimators

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### References:

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