# 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.

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

• 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? Commented Jan 16, 2015 at 9:38
• @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).
– rbm
Commented Jan 16, 2015 at 9:42
• @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.
– rbm
Commented Jan 16, 2015 at 9:52
• @rbm That's not possible (assuming that the errors are heteroschedastic) that WLS is not necessary. Could you provide the reference for book/papers? Commented Mar 28, 2015 at 20:00