Factor-Models

Help understanding factor modeling, solving for residuals

  • June 2, 2016

I am trying to understand and implement a factor model, and I think I might be having some issues. I am trying to solve for the residuals in the equation:

$$ R_{i} = \sum_{A=1}^{K}\beta_{iA} f_{A} + \epsilon_{i} $$ where R is an N x 1 (i = 1, …, N) matrix, and there are K latent factors. I know the values inside the B (factor loadings) matrix and R (returns) matrices beforehand.

My understanding is that this equation gives the values the residuals matrix:

$$ \epsilon = (I_{N} - H)R $$ $$ H = \beta(\beta’\beta)^{-1}\beta' $$ Is this correct way to solve for the residuals?

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 $

$ \epsilon \to \epsilon $

引用自:https://quant.stackexchange.com/questions/26389