Weighted Least Squares for Factor Mimicking Portfolios with Collinearity
Summary
The document describes a factor model in which factor returns are estimated by weighted least squares from asset returns and known exposures. It derives the corresponding factor mimicking portfolio weights as a matrix involving the exposure matrix and the observation weights. The practical problem is that the matrix can be ill-conditioned when exposures are exactly collinear, such as when a country indicator duplicates the combined industry indicators.
The question asks how to impose a normalization that makes industry weights sum to zero, and proposes using a transformation matrix in the estimator. However, the only response points elsewhere without explaining the construction or supplying a solution. The document therefore identifies the relationship between weighted regression, factor portfolios, and normalization constraints, but it does not provide enough detail to implement the proposed correction. Readers would need an additional derivation specifying a nonredundant parameterization or constrained least-squares method.
Key ideas
- Weighted least squares estimates factor returns from known exposures and asset returns.
- The factor mimicking portfolio matrix follows from the weighted regression estimator.
- Exact collinearity among factor exposures makes the unnormalized matrix ill-conditioned.
- A zero-sum industry normalization is raised, but the supplied response does not explain how to implement it.
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# Constructing Factor Mimicking Portfolios
# Constructing Factor Mimicking Portfolios
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.
## Answer by Viat (score 0)
https://quant.stackexchange.com/a/83910
I think this is what you are looking for.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.