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Semiparametrically efficient inference based on signs and ranks for median-restricted models. (English) Zbl 1351.62134

Summary: Since the pioneering work of R. Koenker and G. S. Bassett jun. [Econometrica 46, 33–50 (1978; Zbl 0373.62038)], median-restricted models have attracted considerable interest. Attention in these models, so far, has focused on least absolute deviation (auto-)regression quantile estimation and the corresponding sign tests. These methods use a pseudolikelihood that is based on a double-exponential reference density and enjoy quite attractive properties of root \(n\) consistency (for estimators) and distribution freeness (for tests). The paper extends these results to general, i.e. not necessarily double-exponential, reference densities. Using residual signs and ranks (not signed ranks) and a general reference density \(f\), we construct estimators that remain root n consistent, irrespective of the true underlying density \(g\) (i.e. also for \(g\neq f\)). However, instead of reaching semiparametric efficiency bounds under double-exponential \(g\), they reach these bounds when g coincides with the chosen reference density \(f\). Moreover, we show that choosing reference densities other than the double-exponential in applications can lead to sizable gains in efficiency. The particular case of median regression is treated in detail; extensions to general quantile regression, heteroscedastic errors and time series models are briefly described. The performance of the method is also assessed by simulation and illustrated on financial data.

MSC:

62J05 Linear regression; mixed models

Citations:

Zbl 0373.62038
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References:

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