Why Fundamental Factor Models Use Weighted Least Squares

Summary

The document explains two reasons to weight cross-sectional regressions when estimating factor returns in a fundamental factor model. The technical reason is to address heteroskedasticity: smaller companies’ returns may be more volatile than larger companies’ returns, so weights can approximate inverse variance and give observations a more suitable influence. This is presented as a practical correction, not a guarantee that the variance is modeled perfectly.

The practical reason is that a risk or return model may be intended to serve particular users or markets. Choosing weights can emphasize large companies or specific countries, shaping which observations the fitted model explains better. Square root or logarithm of market capitalization are given as common weight choices, with the resulting fit tending to favor larger companies. The document contrasts fundamental models, where loadings are known and factor returns are estimated, with macroeconomic models, where factors are known; it does not provide a full derivation of the estimation differences.

Key ideas

• Cross-sectional return regressions may use weights to account for different return variances across assets.
• Weights can serve as proxies for inverse variance when residual volatility is heterogeneous.
• Weights can also prioritize the companies or regions the model is intended to represent.
• Square root or logarithm of market capitalization can give larger firms more influence in the fitted model.

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# Why is Weighted Least Squares necessary in fundamental factor model?


# Why is Weighted Least Squares necessary in fundamental factor model?












Why is Weighted Least Squares necessary in fundamental factor model while it is not in a standard Macroeconomic factor model? I understand that $\mathbb{E}[\epsilon^2_{it}]=\sigma_i^2$ varies across observations $i$, but isn't this the same in a macroeconomic factor model?

For reference: in the following model of returns, for a macroeconomic model the factors are known, whereas for a fundamental model the loadings are known and the factors are not.

$R_{it}=\alpha_i + \beta_{i,1} f_{1,t}+ \beta_{i,2}f_{2,t}+ \dots + \beta_{i,k}f_{k,t} + \epsilon_{i,t} \quad \forall i = 1, \dots, N$

## Answer by Alexandre Oliveira (score 3)

https://quant.stackexchange.com/a/47456

There are two main reasons for using weights when estimating factor returns with cross-sectional regressions:

a. The 'technical argument': To fix for heteroskedasticity as cross-sectional returns of small companies are more volatile than large ones, so you assign weights for correcting for this fact, hoping that it will be a good proxy for reciprocal of variance of assets.

b. The 'practical argument': The risk or return model must fit a set of different use cases, so when you assign weights for biasing your model for your needs, like Large companies or specific Countries.

Usually square root or log of market cap are used as weights, what means that the model created will better explain suit those large companies.

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.