We propose robust methods for inference about the effect of a treatment variable on scalar outcome in presence very many regressors model with possibly non-Gaussian and heteroscedastic disturbances. allow number to be larger than sample size. To make informative feasible, we require approximately sparse; that is, confounding factors can controlled up small approximation error by including relatively variables whose identities are unknown. The latter condition makes it possible estimate...

Journal Article Square-root lasso: pivotal recovery of sparse signals via conic programming Get access A. Belloni, Belloni Duke University, Fuqua School Business, 100 Street, Durham, North Carolina 27708, U.S.A.abn5@duke.edu Search for other works by this author on: Oxford Academic Google Scholar V. Chernozhukov, Chernozhukov Massachusetts Institute Technology, Department Economics, 52 Memorial Drive, Cambridge, 02142, U.S.A.vchern@mit.edu L. Wang Mathematics, 77 Avenue, 02139,...

Data with a large number of variables relative to the sample size—“high-dimensional data”—are readily available and increasingly common in empirical economics. Highdimensional data arise through combination two phenomena. First, may be inherently high dimensional that many different characteristics per observation are available. For example, US Census collects information on hundreds individual scanner datasets record transaction-level for households across wide range products. Second, even...

We consider median regression and, more generally, a possibly infinite collection of quantile regressions in high-dimensional sparse models. In these models, the number regressors p is very large, larger than sample size n, but only at most s have nonzero impact on each conditional response variable, where grows slowly n. Since ordinary not consistent this case, we ℓ1-penalized (ℓ1-QR), which penalizes ℓ1-norm coefficients, as well post-penalized QR estimator (post-ℓ1-QR), applies to model...

In this article we study post-model selection estimators that apply ordinary least squares (OLS) to the model selected by first-step penalized estimators, typically Lasso. It is well known Lasso can estimate nonparametric regression function at nearly oracle rate, and thus hard improve upon. We show OLS post-Lasso estimator performs as in terms of rate convergence, has advantage a smaller bias. Remarkably, performance occurs even if Lasso-based “fails” sense missing some components “true”...

In this paper, we provide efficient estimators and honest confidence bands for a variety of treatment effects including local average (LATE) quantile (LQTE) in data-rich environments.We can handle very many control variables, endogenous receipt treatment, heterogeneous effects, function-valued outcomes.Our framework covers the special case exogenous either conditional on controls or unconditionally as randomized trials.In latter case, our approach produces (functional) (ATE) (QTE).To make...

Journal Article Uniform post-selection inference for least absolute deviation regression and other Z-estimation problems Get access A. Belloni, Belloni Fuqua School of Business, Duke University, 100 Drive, Durham, North Carolina 27708, U.S.A., abn5@duke.edu Search works by this author on: Oxford Academic Google Scholar V. Chernozhukov, Chernozhukov Department Economics, Massachusetts Institute Technology, 50 Memorial Cambridge, 02142, vchern@mit.edu K. Kato Graduate University Tokyo, 7-3-1...

Alexandre Bellonia, Victor Chernozhukovb & Ying Weica The Fuqua School of Business, Duke University, Durham, NC 27708 ()b Dept. Economics, Massachusetts Institute Technology, Cambridge, MA 02139 ()c Department Biostatistics, Columbia New York, NY 10032 ()

We consider estimation and inference in panel data models with additive unobserved individual specific heterogeneity a high-dimensional setting. The setting allows the number of time-varying regressors to be larger than sample size. To make informative feasible, we require that overall contribution variables after eliminating can captured by relatively small available whose identities are unknown. This restriction problem proceed as variable selection problem. Importantly, treat fixed...

We propose a self-tuning $\sqrt{\mathrm {Lasso}} $ method that simultaneously resolves three important practical problems in high-dimensional regression analysis, namely it handles the unknown scale, heteroscedasticity and (drastic) non-Gaussianity of noise. In addition, our analysis allows for badly behaved designs, example, perfectly collinear regressors, generates sharp bounds even extreme cases, such as infinite variance case noiseless case, contrast to Lasso. establish various...

We take advantage of recent advances in optimization methods and computer hardware to identify globally optimal solutions product line design problems that are too large for complete enumeration. then use this guarantee global optimality benchmark the performance more practical heuristic methods. two sources data: (1) a conjoint study previously conducted real problem, (2) simulated various sizes. For both data sources, several consistently find or near-optimal solutions, including...

We propose a pivotal method for estimating high-dimensional sparse linear regression models, where the overall number of regressors p is large, possibly much larger than n, but only s are significant. The modification lasso, called square-root lasso. in that it neither relies on knowledge standard deviation σ or nor does need to pre-estimate σ. Moreover, not rely normality sub-Gaussianity noise. It achieves near-oracle performance, attaining convergence rate σ{(s/n) log p}1/2 prediction...

We develop results for the use of LASSO and Post-LASSO methods to form first-stage predictions estimate optimal instruments in linear instrumental variables (IV) models with many instruments, p, that apply even when p is much larger than sample size, n. rigorously asymptotic distribution inference theory resulting IV estimators provide conditions under which these are asymptotically oracle-efficient. In simulation experiments, LASSO-based estimator a data-driven penalty performs well...

In this paper, we develop procedures to construct simultaneous confidence bands for ${\tilde{p}}$ potentially infinite-dimensional parameters after model selection general moment condition models where is much larger than the sample size of available data, $n$. This allows us cover settings with functional response data each a function. The procedure based on construction score functions that satisfy Neyman orthogonality approximately. proposed rely uniform central limit theorems...

Summary We consider the linear regression model with observation error in design. In this setting, we allow number of covariates to be much larger than sample size. Several new estimation methods have been recently introduced for model. Indeed, standard lasso estimator or Dantzig selector turns out become unreliable when only noisy regressors are available, which is quite common practice. work, propose and analyse a errors-in-variables Under suitable sparsity assumptions, show that attains...

We consider median regression and, more generally, quantile in high-dimensional sparse models. In these models the overall number of regressors p is very large, possibly larger than sample size n, but only s have non-zero impact on conditional response variable, where grows slower n. Since this case ordinary not consistent, we penalized by 1-norm coefficients (L1-QR). First, show that L1-QR up to a logarithmic factor, at oracle rate which achievable when minimal true model known. The affects...

In this note, we propose the use of sparse methods (e.g. LASSO, Post-LASSO, p and Post-p LASSO) to form first-stage predictions estimate optimal instruments in linear instrumental variables (IV) models with many canonical Gaussian case. The apply even when number is much larger than sample size. We derive asymptotic distributions for resulting IV estimators provide conditions under which these sparsity-based are asymptotically oracle-efficient. simulation experiments, a estimator data-driven...

Quantile regression (QR) is a principal method for analyzing the impact of covariates on outcomes.The described by conditional quantile function and its functionals.In this paper we develop nonparametric QR-series framework, covering many regressors as special case, performing inference entire linear approximate combination series terms with quantile-specific coefficients estimate function-valued from data.We large sample theory coefficient process, namely obtain uniform strong...