The Noise in Observed Factors: Spectral Corrections and Amortized Bayesian Inference in Large Return Panels

Summary

A factor is a measurement

Empirical asset pricing leans on a handful of published factor series: the market's excess return, the size and value spreads, momentum, quality. People regress on them and feed them to risk models as though they were the common forces behind returns. Yet the same literature says those forces are latent. They are identified only up to rotation, and in practice they are estimated by principal components. A published factor is therefore a measurement. Some of it replicates the latent structure, and the rest is noise left by construction choices such as the sorting variable, the rebalancing calendar and the universe.

How big is that noise? Few people ask, even though every errors-in-variables correction needs the answer. A noisy factor pulls betas toward zero, and two-pass estimates of risk premia inherit the bias. No dataset reports the noise variance. We estimate it from a single return panel, one factor at a time, with a calibrated credible interval around each estimate.

Reading the noise off a panel

Our identification is spectral. From a panel of excess returns we take the top eigenvectors of its Gram matrix, which gives the panel's own estimate of the latent factor space, built without ever looking at the published factor. Then we project the factor onto that space. If the fit falls short of one, only two things can be responsible: the factor's noise, or sampling error in the eigenvectors. Random matrix theory describes the second. Each true spike lines up with its sample eigenvector at a predictable squared overlap, and that overlap drops to zero at the detection threshold (Figure 1). Take away the panel's share of the shortfall. What is left belongs to the factor.

Geometry matters here. An observed factor usually loads on several sample eigenvectors at once, because of the rotation problem, so comparing it with any single eigenvector counts rotation as noise. In simulation that shortcut has median absolute errors of 2.8×10² and 8.3×10² for factors aligned with the second and third latent factors (Figure 2). Projecting on the whole space, with a correction for each spike, cuts those to 0.012 and 0.018.

Real equity residuals are not white noise either. Their variances differ widely across names and move together within industries. We replace the white-noise law with the D-transform of the panel's own leftover spectrum, which needs no model of the residual covariance. On individual stocks the white-noise version is wrong by a factor of six, and under industry-block structure the colored version cuts the 90th-percentile error from 0.171 to 0.102.

For uncertainty we train a mixture-density network on simulated panels. After one ninety-second training run, an interval for any new panel costs a single forward pass. On 1,770 held-out panels the nominal 90% interval covers 0.910 of cases (Figure 6). Its median is also the steadiest point estimate we have: an RMSE of 0.28 over a prior that includes near-threshold and heavily colored panels, where the plug-in chain reaches 1777.

Five factors against three universes

We test Mkt-RF, SMB, HML, momentum and AQR's quality factor against three daily panels from 1990-2026. They are 100 portfolios sorted on size and book-to-market, 49 industry portfolios, and 421 individual large-cap stocks.

Figure 8 maps the results. For the market factor the measurement is clean in every universe, with R̂² of 0.96 or 0.97 and noise between zero and four percent of its systematic variance. Size behaves very differently: SMB carries noise of 0.007 on the size and book panel, while its R̂² drops to 0.22 on industries and 0.23 on stocks, too low to count as spanned. Value is sharp on its own sort, with noise of 0.2%, and about a third of its variance is noise at the stock level. Momentum is spanned by none of the three panels.

Could the pipeline be inventing these numbers? We added white noise of known size to the real factor series and asked for it back. Injected noise of 0.10 returns as 0.1027 on the size and book panel and 0.0993 on industries (Figure 9), and Figure 10 sets first-half against second-half estimates for the factors that are spanned.

Why the panel matters

What surprised us most was how much the choice of universe matters. Only the market factor measures the same thing everywhere; size and value look good on the sorts that built them and poor or unspanned elsewhere. So a factor's quality belongs to the pair, factor and panel together. Researchers can measure it from public data, with calibrated intervals, and the whole pipeline runs in minutes on a single laptop CPU.

Who this is for: Investors and journalists who want to know what a factor label measures, and researchers and quants who correct betas or risk premia for errors in variables.