Watch a regression invent predictability

Below, the true predictive slope is exactly zero. There is no predictability at all. Yet when we simulate this world thousands of times and run the standard regression on each history, the estimated slopes pile up to the right of zero. Drag the sliders and watch the pile move.

Innovation correlation fixed at −0.95 and σ_uv/σ_vv at the value estimated from CRSP data, 1927–1996. 1,500 simulated histories per update.

What you should notice

Push ρ toward 1. The pile slides right. A more persistent predictor produces a larger bias, because the regression's estimate of persistence is itself biased downward by roughly (1+3ρ)/T, and that error transmits into the slope with its sign flipped.

Push T up. The pile slides back toward zero. The bias is a finite-sample problem: it decays like 1/T and vanishes in the limit. OLS is not wrong here, it is biased — a different and more insidious thing.

Notice the shape. The distribution is not symmetric. It has a long right tail, which is why a p-value computed from a normal approximation understates how easily a zero-predictability world produces a large positive slope.

Why negative correlation? The dividend yield has price in its denominator. A shock that lifts the price lifts this month's return and lowers the yield — one shock, two opposite effects. That single fact drives everything above.