Variance Estimation Using Refitted Cross-Validation in Ultrahigh Dimensional Regression

Fan, Jianqing; Guo, Shaojun; Hao, Ning

doi:10.1111/j.1467-9868.2011.01005.x

Cited by 233 publications

(234 citation statements)

References 27 publications

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“…This decrease in the rate of convergence can be substantial for large p, for example if log p ∝ n γ for some positive γ < 1/2. This condition can be relaxed using the sample-splitting method of Fan, Guo, and Hao (2011), which is done in a supplementary appendix. Condition (v) is simply a set of sufficient conditions for consistent estimation of the variance of the double selection estimator.…”

Section: Condition Aste (P)mentioning

confidence: 99%

Inference on treatment effects after selection amongst high-dimensional controls

Belloni¹,

Chernozhukov²,

Hansen³

2013

143

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Abstract. We propose robust methods for inference on the effect of a treatment variable on a scalar outcome in the presence of very many controls. Our setting is a partially linear model with possibly non-Gaussian and heteroscedastic disturbances where the number of controls may be much larger than the sample size. To make informative inference feasible, we require the model to be approximately sparse; that is, we require that the effect of confounding factors can be controlled for up to a small approximation error by conditioning on a relatively small number of controls whose identities are unknown. The latter condition makes it possible to estimate the treatment effect by selecting approximately the right set of controls. We develop a novel estimation and uniformly valid inference method for the treatment effect in this setting, called the "post-double-selection" method.Our results apply to Lasso-type methods used for covariate selection as well as to any other model selection method that is able to find a sparse model with good approximation properties.The main attractive feature of our method is that it allows for imperfect selection of the controls and provides confidence intervals that are valid uniformly across a large class of models. In contrast, standard post-model selection estimators fail to provide uniform inference even in simple cases with a small, fixed number of controls. Thus our method resolves the problem of uniform inference after model selection for a large, interesting class of models. We also present a simple generalization of our method to a fully heterogeneous model with a binary treatment variable. We illustrate the use of the developed methods with numerical simulations and an application that considers the effect of abortion on crime rates.Key Words: treatment effects, partially linear model, high-dimensional-sparse regression, inference under imperfect model selection, uniformly valid inference after model selection, average treatment effects, average treatment effects for the treated

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Section: Condition Aste (P)mentioning

confidence: 99%

Inference on treatment effects after selection amongst high-dimensional controls

Belloni¹,

Chernozhukov²,

Hansen³

2013

143

View full text Add to dashboard Cite

show abstract

“…In particular, Table 1 refers to the case when lags of the predictand are not considered for forecasting using DI and the selection of the PCs is based on the p-values computed on the marginal t * i statistics; Table 2 refers to the case when p lags are considered and the selection is based on the marginal t * i statistics. Tables 3 and 4 deal with the selection based on the t i statistics with σ estimated according to the RCV method by Fan et al (2012): in 3 no lags of the predictands were considered, whereas in 4 they were.…”

Section: Resultsmentioning

confidence: 99%

“…We consider two implementations of the variable selection procedure, the first based on the marginal t * i statistics and the second based on the t i statistics computed using the Fan et al (2012) estimator of the regression error variance.…”

Section: Methodsmentioning

confidence: 99%

“…This can be achieved by weighting the p-values, p i , according to the index i of the factor. A procedure that allows for p-value weighting and achieves control over the FDR is due to Genovese et al (2006), and it works as follows: Fan et al (2012) propose a refitted cross-validatory (RCV) estimator of the error variance which is consistent when the number of factors grows at a faster rate than the number of observations. The procedure has two stages: the sample is split into two independent subsamples.…”

Section: The Selection Of the Common Factors As A Multiple Testing Prmentioning

confidence: 99%

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On the Selection of Common Factors for Macroeconomic Forecasting

Giovannelli

Proietti

2015

SSRN Journal

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We address the problem of selecting the common factors that are relevant for forecasting macroeconomic variables. In economic forecasting using diffusion indexes the factors are ordered, according to their importance, in terms of relative variability, and are the same for each variable to predict, i.e. the process of selecting the factors is not supervised by the predictand. We propose a simple and operational supervised method, based on selecting the factors on the basis of their significance in the regression of the predictand on the predictors. Given a potentially large number of predictors, we consider linear transformations obtained by principal components analysis. The orthogonality of the components implies that the standard t-statistics for the inclusion of a particular component are independent, and thus applying a selection procedure that takes into account the multiplicity of the hypotheses tests is both correct and computationally feasible. We focus on three main multiple testing procedures: Holm's sequential method, controlling the family wise error rate, the Benjamini-Hochberg method, controlling the false discovery rate, and a procedure for incorporating prior information on the ordering of the components, based on weighting the p-values according to the eigenvalues associated to the components. We compare the empirical performances of these methods with the classical diffusion index (DI) approach proposed by Stock and Watson, conducting a pseudo-real time forecasting exercise, assessing the predictions of 8 macroeconomic variables using factors extracted from an U.S. dataset consisting of 121 quarterly time series. The overall conclusion is that nature is tricky, but essentially benign: the information that is relevant for prediction is effectively condensed by the first few factors. However, variable selection, leading to exclude some of the low order principal components, can lead to a sizable improvement in forecasting in specific cases. Only in one instance, real personal income, we were able to detect a significant contribution from high order components.

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“…This rate condition requires the conditional expectations to be sufficiently smooth so that a relatively small number of series terms can be used to approximate them well. As in the case of the IV estimator, this condition can be replaced with the weaker condition that s log(p ∨ n) = o(n) by employing a sample splitting method of Fan, Guo, and Hao (2011). This is done in a companion paper, which also deals with a more general setup, covering non-Gaussian, heteroscedastic disturbances (Belloni, Chernozhukov, and Hansen 2011).…”

Section: Inference On Treatment and Structural Effects Conditional Onmentioning

confidence: 99%

Inference for high-dimensional sparse econometric models

Belloni¹,

Chernozhukov²,

Hansen³

2011

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Abstract. This article is about estimation and inference methods for high dimensional sparse (HDS) regression models in econometrics. High dimensional sparse models arise in situations where many regressors (or series terms) are available and the regression function is wellapproximated by a parsimonious, yet unknown set of regressors. The latter condition makes it possible to estimate the entire regression function effectively by searching for approximately the right set of regressors. We discuss methods for identifying this set of regressors and estimating their coefficients based on ℓ1-penalization and describe key theoretical results. In order to capture realistic practical situations, we expressly allow for imperfect selection of regressors and study the impact of this imperfect selection on estimation and inference results. We focus the main part of the article on the use of HDS models and methods in the instrumental variables model and the partially linear model. We present a set of novel inference results for these models and illustrate their use with applications to returns to schooling and growth regression.

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Variance Estimation Using Refitted Cross-Validation in Ultrahigh Dimensional Regression

Cited by 233 publications

References 27 publications

Inference on treatment effects after selection amongst high-dimensional controls

Inference on treatment effects after selection amongst high-dimensional controls

On the Selection of Common Factors for Macroeconomic Forecasting

Inference for high-dimensional sparse econometric models

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