*Rev. Date August 18, 2023*

All factor analysis using the Two Sigma Factor lens calculates factor exposures through a process involving two sets of regressions. The first, as described in the factor selection methodology, serves as variable selection. The full factor lens is run through a lasso regression to select a subset of the factors to retain in the analysis. The second step is an ordinary least squares (or OLS) regression (explained in more detail below), which applies the selected factors as independent variables against each return stream. The betas in this final regression are reported as the factor exposure per period.

Ordinary Least Squares Regression:

At the core of modern finance is the relation that realized excess asset returns (net of the risk-free rate),

, should equal the expected excess return, α, plus exposures to common factors,

, and an uncorrelated residual term,

, as shown in the following equation:

The asset-pricing implication of the equation above is that assets with a higher sensitivity (β) to these factors should offer a higher return (assuming these factors offer long-term positive returns), as a compensation for investors for bearing non-diversifiable or systematic risk.

Both α and β are statistically inferred using OLS regressions of the portfolio or investment returns on a constant and a set of pre-chosen observed factors.

For daily data, Venn runs the regressions on rolling 5-day average returns for both factor returns and portfolio/fund returns to minimize impact of market asynchronicity. Newey-West correction is applied to the t-stat to correct for the heteroskedasticity and autocorrelation in the time series data.

*This document highlights certain aspects of this feature. As an overview, it does not discuss all material facts or assumptions. Please see **Important Disclosure and Disclaimer Information**.*