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You see $(Y,X)$, you want a relation ship between $X$ and $Y$. You will assume Linear regression I.e you assume it exists $\beta$ such that $Y=X\beta + \epsilon$ and you want to find $\beta$. Solution: $\hat{\beta}=(X'X)^{-1}X'Y$ and $\epsilon = Y-\hat{Y}=Y-X\hat{\beta}=(I-X(X'X)^{-1}X')Y$ So if you apply to your case : $X\to B$ $\beta \to f$ $Y\to R$ ...


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Statistically speaking you should not include the factor that aren't significant. Economically speaking you should take all the factors because intuitively they explain the returns of the assets, and if you don't do it will incur in specification bias by omit the factors and will cause the the estimates aren't efficient, unbiased and consistent



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