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?



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