The fundamental family inverts the estimation. Instead of estimating exposures from a time series and treating factor values as known, it treats the exposures as known, straight from company data, and estimates the factor returns from a cross-section.
How does a fundamental factor model work?
An exposure here is a standardised attribute. Take every company's book-to-price ratio in a given month, subtract the cross-sectional mean and divide by the cross-sectional standard deviation, and you have a z-score: a name at +1.4 is 1.4 standard deviations cheaper than the average name that month. The construction of these scores, including the point-in-time discipline that keeps a filing date from leaking future information into a past exposure, belongs to the signal-construction work in the quant branch and is not rebuilt here.
With exposures known, each period's factor returns come from one regression across names:
r_i = f_MKT + z_i,VAL f_VAL + z_i,QUAL f_QUAL + ... + u_i
The market factor is the column of ones, so every asset has unit exposure to it and f_MKT is the return of a portfolio with average exposure to every style. The style columns are the z-scores. The regression is run afresh every period, and the estimated f values become a factor-return time series that everything downstream consumes.
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What do the regression residuals tell you?
Two orthogonality conditions come free with ordinary least squares and both are worth checking every time, because they catch data errors that nothing else catches. The residuals sum to zero across names. And the residuals are uncorrelated with every exposure column, so Σ z_i u_i = 0 for each style. On the five-name Indian panel in Worked Example 3 both hold to fifteen decimal places, and the value factor returns 1.3430% for the month with an R-squared of 0.7252.
The interpretation of u_i is the whole reason to build the thing. It is the part of a company's return that its measured characteristics do not explain: the earnings surprise, the order win, the promoter pledge, the fraud. That is the part a stock-picker claims to be paid for, and now it has a number.
What are the strengths and limits of fundamental factor models?
Fundamental models estimate cleanly because a cross-section of five hundred names each month is far more data than sixty months of macro history. They also carry a specific fragility. The exposures are only as good as the accounting behind them, so a book value distorted by acquisitions or by lease capitalisation produces a value score that is measuring the accounting rather than the cheapness. Anyone building one should be able to say which line of which statement each z-score came from.