Can You Predict Which Way a Stock Will Move Tomorrow?

A simple nonparametric sign-prediction rule earns statistically significant directional accuracy above a drift-adjusted random walk benchmark in both small-cap and large-cap stocks.

Hello and welcome back to another paid post!

Today we will take a look at a simple nonparametric sign-prediction rule which earns statistically significant directional accuracy above a drift-adjusted random walk benchmark in both small-cap and large-cap stocks.

Let’s dive right in.

---

Introduction

Every retail trader has asked the same question at some point: if a stock just had a big day up, is tomorrow more likely to be up too, or is a reversal coming? The answer turns out to depend enormously on which stock you are asking about, what the broader drift of that stock looks like, and how you define a “big” day. Getting any one of those three things wrong produces results that look like forecasting ability when they are really just artifacts of the data construction.

Andrei Semenov of York University sets a nonparametric statistic called the Probability Difference (PD), designed specifically to measure the genuine sign predictability of daily equity returns after stripping out the mechanical component that comes from a positive expected return. The logic is elegant: if a stock earns 5 percent per year on average, then same-direction sequences will be slightly more common than reversals even if daily returns are purely random. Semenov’s PD statistic accounts for this, then asks whether the residual predictability is statistically significant.

There are 3 main claims. First, small-cap stocks show meaningful directional predictability, with average out-of-sample accuracy around 53 percent above the drift-adjusted benchmark. Second, large-cap stocks show essentially no edge. Third, predictability after positive extreme return days substantially exceeds predictability after negative extreme return days for small-caps, consistent with a behavioral story about the disposition effect creating predictable selling pressure after gains.

---

The Methodology: What the PD Statistic Actually Measures

Standard momentum and reversal research measures whether returns are correlated across time. The PD statistic takes a different approach: it only cares about the sign of the return, not its magnitude. Define Dt as plus one if today’s return is positive and minus one if it is negative. The PD statistic for a given estimation window of W trading days is: