Using PCA Loadings to Estimate Intraday Factor-Neutral Returns

Summary

The document proposes estimating common return factors and asset loadings with a rolling singular value decomposition of volatility-normalized close-to-close returns. It then applies the estimated loadings to other return intervals, such as overnight or open-to-close returns, estimating interval-specific factor returns by projecting those returns onto the loading space. Subtracting the fitted common component gives residual returns that could be used to evaluate predictors after removing shared movements.

The approach depends on the chosen number of components, normalization, rolling window, and stability of the loadings across return horizons. The projection requires the loading matrix to have an invertible Gram matrix, or an appropriate generalized inverse if it does not. Residuals are orthogonal to the fitted loading space under the projection, but that alone does not establish neutrality to economically meaningful factors or eliminate look-ahead and estimation effects. The document poses the method as a question and supplies no empirical validation.

Key ideas

• A rolling SVD can represent normalized historical returns with a reduced set of common components.
• Factor loadings estimated from daily returns can be used to project returns measured over other intervals.
• Subtracting the projected component produces residuals orthogonal to the selected loading space under the fit.
• The method depends on component selection, normalization, window choice, and loading stability.
• The proposed procedure is not empirically validated in the document, and residual orthogonality does not guarantee economic factor neutrality.

Tags

# Is this a correct approach to computing residual returns with PCA Factors?


# Is this a correct approach to computing residual returns with PCA Factors?












I was wondering if this approach to computing factor neutral returns with PCA/SVD is sounds.

Say we have $R = FB + \epsilon$ where $R$ is returns of some period, $F$ is factor returns, $B$ is factor loadings and $\epsilon$ is residual returns.

We want the residual returns for open to close, close to open, or even intraday returns. So we estimate factor loadings on daily returns, back out intraday factor returns, then compute the residual as follows:

Estimating Factor Loadings:

- Vol normalize raw close to close returns

- With some rolling window, apply SVD for $R_{t-T, t-1} = U\Sigma V^\top$. Then select the first $n$ values for $R_f = U_n \Sigma_n V_n ^\top = FB + \epsilon$. $F = U_n = U[:n]$ are the Factor returns. $B = \Sigma_n V_n^\top$ are you factor loadings. $\epsilon$ are you residual returns which you want to find.

Residualizing Returns:

Now that you have factor loadings at time $t-1$ that use close to close returns, at time $t$ and forward you can apply them to your open to close and close to open returns.

- $R_{co} = F_{co}B + \epsilon_{co}$ and we have already estimated $B$ and know $R_{co}$. So we can back out $F_{co}$ as follows: $RB^\top = FBB^\top \implies RB^\top (BB^\top )^{-1} = FBB^\top(BB^\top)^{-1} \implies F_{oc} = R_{oc}B^\top (BB^\top )^{-1}$

- Now we can compute the residual returns as: $\epsilon_{co} = R_{co} - F_{co}B$

Does this method seem right? Now your residual returns are factor neutral so you can use them to evaluate predictors to see how they are predictive of returns without common factors?

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.