A Student t(5) model reduced joint forecast loss at equal covariance for three US equity factors in a 2005–2025 annual archive test

We tested joint monthly probability forecasts for US market excess, size and value returns in 240 months from October 2005 to September 2025. Each of 20 origins used its official annual archive through July and a 120-month fit. We assumed release by September 30 and froze forecasts for the next 12 months. At the same mean and covariance with sample correlations cut in half, a fixed Student t model with five degrees of freedom reduced mean negative log loss by 0.1352 nats per return vector against normal. The descriptive 95% paired annual-block interval was 0.0144 to 0.2663. The gain stayed positive under planned block-length and data-version checks. The October 2019–September 2020 block supplied 42.4% of the total gain; its removal left a positive point gain of 0.0819 nats. Calibration errors remained: nominal 50% and 90% regions covered 42.08% and 85.42% of outcomes. This historical result does not establish return direction skill or trading profit. No trading strategy was tested. The package preserves all eight fixed model cases, dated forecasts, code, source hashes, scores, uncertainty draws, independent checks, exact replay records, and 96 unscored future forecasts. Raw provider returns are excluded.

equity factors · density forecasts · Student t · point-in-time data · annual archives · chronological test · log score · covariance · negative findings