macrocephalopod

1. How do you get the exposure matrix Xt? There are different ways to estimate it, depending on the factor. Simplest is factors like industry or country exposure where the entries can be 0/1 depending on whether the stock is in that industry/country or not.

Some exposures can be estimated by linear regression on historical data, if you already have a time series which approximates the factor returns. E.g. exposure to the market factor (beta) is estimated this way, by regressing each stock against the S&P 500 (or some other index)

This also works for "macro" factors e.g. you can estimate exposures for each stock to commodity prices, exchange rates, interest rates, GDP or inflation surprises etc by regressing stock returns against the relevant historical time series.

Finally you can have exposures which are heuristically derived from other observable data about the stock, e.g. accounting data, analyst reports, past price movements etc. In this case you find some metric which measures the factor you care about (e.g. price to earnings) and

transform it so that it has a nice distribution in the cross-section - common approaches are z-scoring (subtract mean and divide by standard deviation) or ranking (the stock with the lowest metric gets exposure -1, highest gets +1 and others are linearly spaced between -1 and +1)

You want all the entries in the exposure matrix to have a similar scale (generally -3 <= X(i,j) <= +3 for all entries) as this makes it much easier to compare the factors with each other.

Implementation note -- with linearly dependent factors (e.g. each stock is in exactly one industry so sum of industry exposures equals market exposure) you can't use the normal equations below. You need a constraint on the factor returns.https://twitter.com/macrocephalopod/status/1356735696468377600?s=20 …

(Normally you would require the sum of all industry factor returns to be zero, sum of all country factor returns to be zero etc)