Transforming Factors and Loadings in a Reduced Cross-Sectional Model

Summary

The document describes a cross-sectional factor model in which a vector of security returns is explained by factor exposures multiplied by factor values, plus a Gaussian residual vector. It considers compressing the original factor set through a linear transformation, producing a lower-dimensional set of transformed factors. The question is whether the corresponding asset loadings can be obtained by applying a pseudo-inverse to the transformation and original loadings, or whether the reduced factors should instead be used in a fresh regression.

This is a model-reduction question rather than a worked solution: no transformation, dataset, estimation result, or comparison of the alternatives is supplied. The formulation prompts attention to matrix dimensions and whether the transformed factors span the return variation captured by the original model. A pseudo-inverse gives a least-squares mapping under particular rank and projection conditions; re-estimation can instead fit exposures directly to returns under the reduced specification. The document does not establish which approach is preferable in a given application.

Key ideas

• The original model expresses cross-sectional asset returns through factor loadings and factor values plus residuals.
• A linear map can reduce the dimension of the factor vector.
• Using a pseudo-inverse to transform loadings depends on matrix dimensions, rank, and projection properties.
• A regression on the reduced factors is an alternative, but the document provides no comparison or result.

Tags

# Aggregation of (cross-sectional) Factor model


# Aggregation of (cross-sectional) Factor model












Suppose I have a large factor model for security returns, i.e. I have a vector $\mathbf{Y}(t) \in \mathbb{R}^{P}$, with factor loadings $\mathbf{\beta} \in \mathbb{R}^{P \times K}$ over a set of $K$ factors denoted by $\mathbf{X}$. Thus, we can describe the asset returns as

\begin{align} \mathbf{Y}(t) = \mathbf{\beta} \mathbf{X}(t)+ \mathbf{\epsilon}, \end{align}

$\epsilon$ is a residual error term, taken to be $P$-dimensional Gaussian.

I want to build a reduced version of the factor model. So rather than $\mathbf{X}$ being $K$-dimensional, I construct a linear transformation of the factors such that I now have $\mathbf{\tilde{X}} = \mathbf{L}\mathbf{X}$, with $\mathbf{L} \in \mathbb{R}^{N \times K}$.

Would it be enough to take the pseudo-inverse of $\mathbf{L}$ to compute the new factor loadings $\tilde{\mathbf{\beta}} = \mathbf{L}^{-1} \mathbf{\beta}$ ?

or would it be better to define the transformation of the factor loadings and then re-run a regression to find the new $\tilde{\mathbf{X}}$ returns?

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

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