I am trying to understand how factor loadings in a general factor model are computed. For simplicity sake, lets assume a simple model:
$$ R = B \times F + \epsilon $$
$$ R = N \times 1 $$ $$ B = N \times K $$ $$ F = K \times 1 $$
where $B$ are the factor loadings and $F$ is the factor return and $N$ is the number of securities.
The way this equation is setup, a factor that is common to all $N$ assets, like GDP, inflation, or market index, can easily be plugged in as a factor return ($F$) into the above equation. But how about financial ratios? Let's say for example the PE ratio. The PE ratio is different for each security, which would in turn give us a different factor loading for each security. But how can we plugin a factor return for PE ratio into $F$; its different for each security.
Or is my understanding of using fundamental ratios in a factor model totally off?
