Constructing Factor Mimicking Portfolios

External question — Quantitative Finance Stack Exchange Author: rudinable Original post: https://quant.stackexchange.com/questions/83891 License: CC BY-SA 4.0 — https://creativecommons.org/licenses/by-sa/4.0/ Adaptation: HTML converted to plain text; contact email addresses removed. I'm working with some factor data from a third party company. Their factor model is estimated on a broad universe. I'm trying to re-estimate the model on a smaller subset (my own universe) to construct factor mimicking portfolios. Essentially, I want to find portfolios from within my universe that track their factor most closely. Let's begin with a factor model: $r_{i,t} = X_{i, t-1}^{'} F_t + \eta_{i, t}$ with a $k \times 1$ vector of factor returns $F_t$ , or in matrix form: $\mathbf{r}_t = \mathbf{X}_{t-1}F_t+\mathbf{\eta}_t$ . Now let's say I know my betas $\mathbf{X}_{t-1}$ , and then I estimate factor returns with a weighted least squares scheme: $\hat{\mathbf{F}}_t = \arg{\min_{\mathbf{F}}} (\mathbf{r}_t - \mathbf{X}_{t-1}\mathbf{F})'\mathbf{W}_t(\mathbf{r}_t - \mathbf{X}_{t-1}\mathbf{F}).$ Then I have the factor mimicking portfolios as $(\mathbf{X}_{t-1}' \mathbf{W}_t \mathbf{X}_{t-1})^{-1} \mathbf{X}_{t-1}'\mathbf{W}_t.$ The problem that I'm having is that this matrix is ill conditioned, because in the factor model there is exact collinearity. For example, including a country factor along with industry factors (as my model does) leaves two independent variables with the value 1. As a result, an additional constraint is imposed so that industry weights sum to 0 instead of 1. I'm having trouble seeing how this gets incorporated into the solution. How can I find the normalization matrix $\mathbf{Z}_{t-1}$ which incorporates this additional constraint and fixes the estimation? I.e. then we have $\mathbf{Z}_{t-1}(\mathbf{X}_{t-1}' \mathbf{Z}_{t-1}'\mathbf{W}_t \mathbf{Z}_{t-1} \mathbf{X}_{t-1})^{-1} \mathbf{X}_{t-1}' \mathbf{Z}_{t-1}'\mathbf{W}_t$ , i'm just not sure what to use for $\mathbf{Z}_{t-1}$ . Thanks.

External answer — Quantitative Finance Stack Exchange Author: Viat Original post: https://quant.stackexchange.com/a/83910 License: CC BY-SA 4.0 — https://creativecommons.org/licenses/by-sa/4.0/ Adaptation: HTML converted to plain text; contact email addresses removed. I think this is what you are looking for.