The Measurement Quality of Asset-Pricing Factors: Spanning, Noise, and the Limits of Joint Correction across Static and Conditional Test-Asset Universes

Summary

A factor is a measurement

Most empirical work skips the premise our working paper starts from. Returns are driven by a small set of latent factors, and a published series is a measurement of that structure, signal in one part and construction noise in the rest. That noise comes from the sorting variables, the rebalancing calendar, the weighting scheme, and the universe the sort was built on. The instrument already existed. A companion paper built it, a spectral estimator that reads a factor's measurement-error variance off the top eigenstructure of a return panel. This paper takes it out of the laboratory and uses it on the zoo.

Each factor photographs its own sort

We grade thirteen daily factors: the Fama-French five, momentum, AQR's quality, betting-against-beta and monthly-updated value, and the four q-factors. Each is tested against the 100 size/book-to-market portfolios, the 25 size/momentum portfolios, the 49 industry portfolios, and 421 individual large-cap stocks, and the grid of results is the measurement-error atlas in Figure 1. Every cell carries a spanning R2 on the top-five latent space and, where the factor qualifies as a measurement, a noise estimate with a calibrated error bar.

One pattern dominates. A characteristic-sorted factor measures its own sort well and everything else poorly. On the 25 size/momentum portfolios, momentum is a 97%-pure, nearly noise-free measurement of a top-five latent direction. On the size/book-to-market portfolios it is barely visible. That momentum is unspanned by size and value test assets was already known; it turns out to be one instance of the rule. Figure 2 plots the pattern for SMB, HML, and MOM with each factor's home universe marked. The market factor is the one series that measures the same thing cleanly everywhere, with a spanning R2 of 0.96 or 0.97 in all four universes. A placebo check rules out a sorting tautology.

Conditional test assets

Strategies that turn over fast leave no trace in a fixed panel's top eigenvectors, so the atlas has a conditional wing, in which the same factors are re-graded against latent spaces estimated by instrumented PCA and by a conditional autoencoder, with return-derived instruments only. Figure 5 puts static and conditional grades side by side on the same monthly panels. Momentum benefits. On individual stocks, conditioning raises its spanning R2 from 0.33 to somewhere between 0.70 and 0.77 depending on the method, and the gain holds out of sample in both time directions. Value moves the other way. Nor can an arbitrary series fit a five-dimensional conditional span. Three placebo series confirm this. Macro candidates such as industrial-production growth and CPI inflation fail the spanning screen in static and conditional spaces alike, as Figure 11 shows.

Correcting the premia

The paper runs the textbook errors-in-variables correction with measured noise in place of assumed noise. Time-series betas are un-attenuated using the estimated noise covariance, and Fama-MacBeth premia are re-estimated. On the 100 size/book-to-market panel, profitability is the only significant non-market premium, and the correction cuts it from 5.3 to 1.7 percent per year per unit of true factor exposure. Two thirds of the naive premium was attenuation bias recycled. A joint moving-block bootstrap that re-estimates all five noises on every replicate puts the corrected figure in a 90% band from 1.4 to 2.0, so the conclusion survives the sampling error of the noise estimates themselves. Figure 6 shows each premium before and after.

With the zoo assembled in one place, a warning appears. Correcting many noisy, correlated factors at once is ill-posed. Subtracting the measured noise drives the implied true-factor covariance matrix non-positive-definite. Figure 7 traces how far the correction can go. Along the ray from the published covariance toward full correction, the nine-factor zoo stays correctable only up to 16 percent of its measured noise, where the smallest eigenvalue crosses zero. Past that point any joint purification becomes a modeling choice, and the paper labels its nearest-PSD repair as one.

Calibration and factor design

Our original amortized posterior intervals were degenerate. A conformal calibration against a simulation bank replaced them with a global 90% multiplier of 3.66 bootstrap standard errors, shown in Figure 3. A second calibration then drops the simulation prior and injects known noise into the real panels, which gives the multipliers Figure 4 reports: they run from 2.4 to 10.5 bootstrap standard errors by universe, and no cell changes classification. Membership matters as well. Rebuilding the stock universe point in time, with membership as it stood in each window, lowers the characteristic factors' spanning R2 by up to 0.11.

Measurement quality can also discipline construction. We grade four versions of value on universes not used to build them: official HML, AQR's Devil variant, and value-weighted and equal-weighted corner rebuilds from the 25 size/book-to-market portfolios. Official HML carries the least noise. Devil carries slightly more, the corner rebuilds the most, and equal weighting is worst of all, so that on the 49 industries the measured noise is 0.86 for official HML, 1.03 for Devil, and 1.05 for the value-weighted rebuild, as Figure 8 shows. So we propose measured noise as a design criterion: which construction is the better measurement is then a question with an estimable and falsifiable answer.

Why does this bear on the replication debates? Mean-zero construction noise leaves a factor's mean unchanged while dividing its t-statistic by the square root of one plus the noise variance, so a factor can replicate across samples and still be a poor measurement of the latent structure it claims to capture.

Who this is for: Researchers and quants who build or test factor models, and journalists covering the factor-zoo replication debate.