How weak are weak factors? Uniform inference for signal strength in signal plus noise models

TL;DR

This paper develops a unified method for constructing confidence intervals for signal strength in various signal-plus-noise models, valid across all regimes including weak and critical signals, using a universal transitional distribution.

Contribution

It introduces a universal transitional distribution for uniform inference on signal strength, overcoming limitations of Gaussian approximations in critical regimes.

Findings

Traditional Gaussian approximations fail in critical regimes

The universal transitional distribution enables valid inference across all signal strengths

Applications demonstrated in macroeconomics and finance

Abstract

The paper analyzes four classical signal-plus-noise models: the factor model, spiked sample covariance matrices, the sum of a Wigner matrix and a low-rank perturbation, and canonical correlation analysis with low-rank dependencies. The objective is to construct confidence intervals for the signal strength that are uniformly valid across all regimes - strong, weak, and critical signals. We demonstrate that traditional Gaussian approximations fail in the critical regime. Instead, we introduce a universal transitional distribution that enables valid inference across the entire spectrum of signal strengths. The approach is illustrated through applications in macroeconomics and finance.