The Factor-Lasso and K-Step Bootstrap Approach for Inference in High-Dimensional Economic Applications

Abstract

We consider inference about coefficients on a small number of variables of interest in a linear panel data model with additive unobserved individual and time specific effects and a large number of additional time-varying confounding variables. We suppose that, in addition to unrestricted time and individual specific effects, these confounding variables are generated by a small number of common factors and high-dimensional weakly dependent disturbances. We allow that both the factors and the disturbances are related to the outcome variable and other variables of interest. To make informative inference feasible, we impose that the contribution of the part of the confounding variables not captured by time specific effects, individual specific effects, or the common factors can be captured by a relatively small number of terms whose identities are unknown. Within this framework, we provide a convenient inferential procedure based on factor extraction followed by lasso regression and show that the procedure has good asymptotic properties. We also provide a simple k-step bootstrap procedure that may be used to construct inferential statements about the low-dimensional parameters of interest and prove its asymptotic validity. We provide simulation evidence about the performance of our procedure and illustrate its use in an empirical application.

An exact and robust conformal inference method for counterfactual and synthetic controls

“…Our proposal delivers a way to perform valid inference for policy effects in these models, including the recent new methods, even though we shall be focusing on Bai (2009)'s alternating least squares estimator when verifying our conditions. The results in Hansen and Liao (2016) imply that our high-level conditions hold for their estimator.…”

Learning Latent Factors from Diversified Projections and its Applications to Over-Estimated and Weak Factors

“…Here (α y , α g , β) are low -dimensional coefficient vectors while (γ, θ) are high-dimensional sparse vectors. Fan et al (2018b) and Hansen and Liao (2018) showed that the penalized regression can be successfully applied to (4.3) to select among components in u t , which are cross-sectionally weakly correlated. Their approaches require crucially that the space of factors needs to be strong so that we can consistently estimate the number of factors r = dim(f t ) first.…”

“…Below we first present the factor-augmented algorithm as in Hansen and Liao (2018) for estimating (4.1). For notational simplicity, we focus on the univariate case dim(β) = 1.…”

“…From here, the desired convergence results follow from the standard argument in the lasso literature, we omit details for brevity, and refer to, e.g., Hansen and Liao (2018).…”

“…(ii) The proof of | J | 0 = O P (|J| 0 ) also follows from the standard argument in the lasso literature, we omit details but refer to the proof of Proposition D.1 of Hansen and Liao (2018) and Belloni et al (2014).…”