Regime Dependence of the Size Premium: Identification Versus Ex-Ante Forecastability in the Carhart Four-Factor Model
Abstract
Among the four factors of the Carhart four-factor model, which premium is regime-dependent — and can that dependence be exploited in real time? A factor-attribution methodology isolates each factor’s regime contribution across 18 size-sorted portfolios and three evaluation windows (January 1927–November 2025). As a regime-identification exercise, the size factor (SMB) is uniquely regimedependent: forecast improvement scales with SMB loading (Spearman ρ = 0.948; pooled dependencerobust p = 0.12 at the automatically selected block length, falling to 0.03 at fixed 12-month blocks), and regime parameters show the premium is approximately zero in the calm regime and concentrated in the turbulent regime at 0.5–1.1% per month (calm-stress difference positive in all three windows, one-sided bootstrap p ≤ 0.030, though the windows are nested and two lean heavily on one historical episode), consistent with countercyclical risk compensation; a parametric Monte Carlo test against a single-state GARCH null indicates this difference exceeds what a volatility-clustered process without switching manufactures (p = 0.015–0.090 across windows); high-minus-low (HML) switching is counterproductive. These identification results do not, however, translate into ex-ante forecasting skill. Under a fully ex-ante design — parameters estimated on training data only, combined with one-step-ahead predicted regime probabilities — the regime model’s out-of-sample R2 is 0.07–0.08% depending on the regime-mean estimator, no portfolio-window test is significant even before adjustment (minimum Clark-West p = 0.081), and none survives a 5% false-discovery rate across the full 54-test family (minimum adjusted p = 0.709). Moving-block bootstrap inference preserving serial and cross-sectional dependence renders even the strongest single-window cross-sectional correlation (2010, naive p ≈ 7×10−9) indistinguishable from zero (bootstrap p = 0.40; pooled p = 0.86). A three-stage decomposition attributes the apparent gains of less disciplined designs to parameter and probability-timing look-aheads, with the timing channel dominating. The size premium is regime-dependent in identification but not exploitably forecastable.