- Daily asset returns r_{i,t} for a liquid universe.
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Daily VIX return f_{\text{VIX},t} and any other observed factors f_{\ell,t}.
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Estimate betas (time-series regressions)
r_{i,t} \;=\; \beta_{i,\text{VIX}}\,f_{\text{VIX},t}\;+\;\sum_{\ell\neq\text{VIX}}\beta_{i,\ell}\,f_{\ell,t}\;+\;\varepsilon_{i,t}.
Collect all \beta_{i,\cdot} into the loading matrix B and The phrase means: once you’ve run each time‐series regression
r_{i,t} \;=\;\sum_{\ell}\beta_{i,\ell}\,f_{\ell,t}\;+\;\varepsilon_{i,t},
you collect all \beta_{i,\ell} into one matrix
\[
B = \begin{pmatrix} \beta_{1,1} & \beta_{1,2} & \cdots\\ \beta_{2,1} & \beta_{2,2} & \cdots\\ \vdots & \vdots & \ddots \end{pmatrix},
\]
and you form the vector of residuals for each asset
\varepsilon_{i,t}. Then
\Omega_{\varepsilon} \;=\; \operatorname{Cov}\bigl(\varepsilon_{t}\bigr)
is the N\times N covariance matrix whose (i,j) entry is
\mathbb{E}[\varepsilon_{i,t}\,\varepsilon_{j,t}].