High Dimensional Regression Analysis

The focus is confidence interval construction in high-dimensional sparse linear models. Motivated by applications, the interest is not just on the high-dimensional regression vector, but also on a variety of functionals of the regression vectors, including (1) linear functionals and quadratic functionals of the regression vector; (2) the inner product of two regression vectors; and (3) the estimation accuracy of a given estimator.

The following papers tackle the inference problems from two perspectives, (i) how to construct confidence intervals; (ii) what is the necessary sample size to construct adaptive confidence intervals achieving optimal length. From both perspectives, the inference problems in high dimensions exibit significant differeces from those in low dimensions due to the non-convexity structure of sparse linear regression.

High Dimensional Causal Inference / Econometrics

Instrumental variable is an important topic in many fileds. However, the traditional low-dimensional instrumental variable analysis suffers from curse of dimensionality and requires strong assumptions on instrumental variables. The following papers study how to construct confidence intervals for treatment effects and test endogeneity in the presence of invalid instrumental variables and high-dimensional covariates/instrumantal variables. One essenstial novel technique, Two Stage Hard Thresholding, is developed to select valid instruments among a set of candidate instrumenal variables.

Causal Inference / Econometrics

One key concern of inference for treatment effects and mediation effects is endogeneity, where the treatment or mediator is correlated with unmeasured confounders. To address the endogeneity problem, instrumental variable approach is broadly applied to estimate treatments and mediation effects consistently. The following papers tackle the inference problem for treatment or mediation effects in the non-linear model, including non-linear additive model, logistic model, possion model and zero-inflated count model. The proposed methods have been demonstrated in economics and health studies.

Collaborative Project

  • Lowder, E. M., Desmarais, S. L., Guo, Z., Coffey, T. and Van Dorn, R. A. (2016).
    Criminal justice outcomes and behavioral health utilization following receipt of SSI/SSDI benefits.

Fractal Analysis