Uniform post selection inference for LAD regression models
Alexandre Belloni,
Victor Chernozhukov and
Kengo Kato
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Alexandre Belloni: Institute for Fiscal Studies
Kengo Kato: Institute for Fiscal Studies
No CWP24/13, CeMMAP working papers from Centre for Microdata Methods and Practice, Institute for Fiscal Studies
Abstract:
We develop uniformly valid confidence regions for a regression coefficient in a high-dimensional sparse LAD (least absolute deviation or median) regression model. The setting is one where the number of regressors p could be large in comparison to the sample size n, but only s « n of them are needed to accurately describe the regression function. Our new methods are based on the instrumental LAD regression estimator that assembles the optimal estimating equation from either post l- penalised LAD regression or l1- penalised LAD regression. The estimating equation is immunised against non-regular estimation of nuisance part of the regression function, in the sense of Neyman. We establish that in a homoscedastic regression model, under certain conditions, the instrumental LAD regression estimator of the regression coefficient is asymptotically root-n normal uniformly with respect to the underlying sparse model. The resulting confidence regions are valid uniformly with respect to the underlying model. The new inference methods outperform the naive, 'oracle based' inference methods, which are known to be not uniformly valid- with coverage property failing to hold uniformly with respect the underlying model- even in the setting with p = 2. We also provide Monte-Carlo experiments which demonstrate that standard post-selection inference breaks down over large parts of the parameter space, and the proposed method does not.
Keywords: median regression; uniformly valid inference; instrumets; Neymanisation; optimality; sparsity; post selection inference (search for similar items in EconPapers)
Date: 2013-06-03
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