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Mediation analysis draws increasing attention in many scientific areas suchas genomics, epidemiology and finance. In this paper, we propose newstatistical inference procedures for high dimensional mediation models, inwhich both the outcome model and the mediator model are linear with highdimensional mediators. Traditional procedures for mediation analysis cannot beused to make statistical inference for high dimensional linear mediation modelsdue to high-dimensionality of the mediators. We propose an estimation procedurefor the indirect effects of the models via a partial penalized least squaresmethod, and further establish its theoretical properties. We further develop apartial penalized Wald test on the indirect effects, and prove that theproposed test has a $\chi^2$ limiting null distribution. We also propose an$F$-type test for direct effects and show that the proposed test asymptoticallyfollows a $\chi^2$-distribution under null hypothesis and a noncentral$\chi^2$-distribution under local alternatives. Monte Carlo simulations areconducted to examine the finite sample performance of the proposed tests andcompare their performance with existing ones. We further apply the newlyproposed statistical inference procedures to study stock reaction to COVID-19pandemic via an empirical analysis of studying the mediation effects offinancial metrics that bridge company's sector and stock return.

Statistical Inference for Linear Mediation Models with High-dimensional Mediators and Application to Studying Stock Reaction to COVID-19 Pandemic