# 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?