# Working Papers

## Most powerful quadratic tests for high dimensional free alternatives.

**Yi He**, Sombut Jaidee and Jiti Gao

We propose a powerful quadratic test for the overall significance of many weak exogenous variables in a dense autoregressive model. By shrinking the classical weighting matrix on the sample moments to be identity, the test is asymptotically correct in high dimensions even when the number of coefficients is larger than the sample size. Our theory allows a non-parametric error distribution and estimation of the autoregressive coefficients. Using random matrix theory, we show that the test has the optimal asymptotic testing power among a large class of competitors against local dense alternatives whose direction is free in the eigenbasis of the sample covariance matrix among regressors. The asymptotic results are adaptive to the predictors’ cross-sectional and temporal dependence structure, and do not require a limiting spectral law of their sample covariance matrix. The method extends beyond autoregressive models, and allows more general nuisance parameters. Monte Carlo studies suggest a good power performance of our proposed test against high dimensional dense alternative for various data generating processes. We apply our tests to detect the overall significance of over one hundred exogenous variables in the latest FRED-MD database for predicting the monthly growth in the US industrial production index.

Manuscript available soon

## Extreme Value Statistics for High Dimensional Data

John H.J. Einmahl and **Yi He**

We propose a novel statistical formulation of the empirical power laws widely observed in high-dimensional data sets. Our approach extends classical extreme value theory to specifying the behavior of the empirical distribution of data with a complex dependence structure and possibly different marginal distributions, for a diverging number of dimensions. The main assumption is that in the intermediate tail the empirical distribution function approaches some heavy tailed distribution function that is in the max-domain of attraction. In this setup the Hill estimator consistently estimates the extreme value index and extreme quantiles are consistently estimated, on a log-scale. We discuss several model examples that satisfy our conditions. We also consider applications to finance.

Manuscript available soon