[Paper Review] High-Dimensional Metrics in R

Victor Chernozhukov, C. Hansen+1 more|arXiv (Cornell University)|Mar 5, 2016

Statistical Methods and Inference6 references3 citations

TL;DR

This paper introduces the R package hdm, which provides theoretically grounded, data-driven methods for inference in high-dimensional approximately sparse models. It enables efficient estimation and uniformly valid confidence intervals for low-dimensional parameters—such as treatment effects and regression coefficients—using rigorous Lasso (rlasso) with heteroscedastic and non-Gaussian errors, outperforming traditional cross-validation in theory and efficiency.

ABSTRACT

The package High-dimensional Metrics (\Rpackage{hdm}) is an evolving collection of statistical methods for estimation and quantification of uncertainty in high-dimensional approximately sparse models. It focuses on providing confidence intervals and significance testing for (possibly many) low-dimensional subcomponents of the high-dimensional parameter vector. Efficient estimators and uniformly valid confidence intervals for regression coefficients on target variables (e.g., treatment or policy variable) in a high-dimensional approximately sparse regression model, for average treatment effect (ATE) and average treatment effect for the treated (ATET), as well for extensions of these parameters to the endogenous setting are provided. Theory grounded, data-driven methods for selecting the penalization parameter in Lasso regressions under heteroscedastic and non-Gaussian errors are implemented. Moreover, joint/ simultaneous confidence intervals for regression coefficients of a high-dimensional sparse regression are implemented, including a joint significance test for Lasso regression. Data sets which have been used in the literature and might be useful for classroom demonstration and for testing new estimators are included. \R and the package \Rpackage{hdm} are open-source software projects and can be freely downloaded from CRAN: exttt{http://cran.r-project.org}.

Motivation & Objective

• Address the challenge of statistical inference in high-dimensional models where the number of parameters exceeds sample size.
• Develop methods for valid inference on low-dimensional components (e.g., treatment effects, regression coefficients) in high-dimensional approximately sparse models.
• Provide a theoretically justified, data-driven method for selecting the Lasso penalty parameter λ, avoiding reliance on cross-validation.
• Enable joint and simultaneous confidence regions for multiple regression coefficients in high-dimensional settings.
• Facilitate causal inference in econometric and data science applications through uniformly valid confidence intervals and significance tests.

Proposed method

• Implement a 'rigorous Lasso' (rlasso) estimator that handles non-Gaussian and heteroscedastic errors, improving robustness in high-dimensional regression.
• Introduce a data-driven, theory-grounded method for selecting the penalty parameter λ in Lasso regression, reducing computational burden compared to cross-validation.
• Apply the orthogonality principle via partialling out to construct confidence intervals for target coefficients in high-dimensional models.
• Utilize the framework from Belloni, Chernozhukov, and Kato (2014) to construct uniformly valid joint confidence regions for high-dimensional regression coefficients.
• Extend inference to average treatment effects (ATE), average treatment effects for the treated (ATET), and endogenous treatment settings using double/debiased machine learning principles.
• Integrate empirical applications and real-world datasets (e.g., CPS, BLP, AJR, EminentDomain) to demonstrate methodological utility in econometrics and beyond.

Experimental results

Research questions

• RQ1 How can we construct uniformly valid confidence intervals for a single regression coefficient in a high-dimensional approximately sparse model?
• RQ2 What is a theoretically grounded, data-driven method for selecting the Lasso penalty parameter λ when errors are heteroscedastic or non-Gaussian?
• RQ3 How can we perform joint inference on multiple regression coefficients in high-dimensional models with valid simultaneous coverage?
• RQ4 Can we estimate and test average treatment effects (ATE and ATET) in high-dimensional models with many confounding variables?
• RQ5 How can instrumental variable estimation be extended to high-dimensional settings with endogenous treatments and many controls?

Key findings

• The rlasso estimator in hdm provides consistent and efficient estimation of high-dimensional regression coefficients under heteroscedastic and non-Gaussian error distributions.
• The data-driven penalty selection method for rlasso reduces computational cost compared to cross-validation while maintaining theoretical validity.
• Uniformly valid confidence intervals for target coefficients are constructed using the orthogonality principle, ensuring correct coverage rates even in high-dimensional settings.
• Joint confidence regions for multiple regression coefficients are implemented and shown to be valid under high-dimensional Z-estimation theory.
• The package enables reliable inference on ATE and ATET in high-dimensional models with many confounders, as demonstrated in applications on 401(k) participation and economic development.
• The inclusion of real datasets (e.g., CPS, BLP, AJR, EminentDomain) supports reproducible research and benchmarking of new estimators in econometric applications.