Spurious Factors in Linear Asset Pricing Models

The standard way to test whether a proposed risk factor deserves a return premium is a two-pass regression: first estimate each asset's exposure (beta) to the factor from time-series data, then run a single cross-sectional regression of average returns on those betas to see if a higher beta earns a higher return. Bryzgalova's result is an uncomfortable one for this workhorse: a factor that has literally zero true explanatory power, pure noise, weakly correlated with the assets purely by chance, can still produce a statistically significant, large-looking risk premium estimate in the second-pass regression.

The mechanism is that the second-pass regression divides by the cross-sectional spread in estimated betas. If a useless factor happens to produce betas that are all close to each other (weak identification), that small denominator inflates the estimated premium and shrinks its apparent standard error, so a t-statistic can look impressively significant even though the underlying factor explains nothing. This is a version of the "weak instruments" problem familiar from econometrics, imported into asset pricing: standard tests were not built to detect when the factor itself is barely identified by the data.

The practical fix is to test factors with methods robust to weak identification, checking how an estimate behaves as betas are jointly, not separately, estimated, rather than trusting a single cross-sectional t-statistic at face value. This is part of why the "factor zoo" (hundreds of published factors, most from underpowered tests) is treated skeptically: some fraction of published risk premia are statistical artefacts of exactly this mechanism, not genuine compensation for risk.