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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.

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

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  • $\begingroup$ Could you give an example where "the loadings are known" and the factors not? I can imagine what you mean but is this a rigorous statement? $\endgroup$
    – Richi Wa
    Commented Jan 16, 2015 at 9:38
  • $\begingroup$ @Richard An example would be if you let the first elements of each $\beta$ vector be strongly negative and then let them be increasing so that the last values are strongly positive and the values in the middle are relatively close to $0$: that way you would 'force' the factors to represent the slope of the yield curve for instance (in case we see $R_{it}$ as the yield). $\endgroup$
    – rbm
    Commented Jan 16, 2015 at 9:42
  • $\begingroup$ @Richard I agree with you that the definition is rather vague, and it seems as if the $\beta$'s and the factors just swapped roles, but unfortunately my book doesn't give me a more rigorous definition. $\endgroup$
    – rbm
    Commented Jan 16, 2015 at 9:52
  • $\begingroup$ @rbm That's not possible (assuming that the errors are heteroschedastic) that WLS is not necessary. Could you provide the reference for book/papers? $\endgroup$
    – Quantopik
    Commented Mar 28, 2015 at 20:00

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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.

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