Can You Predict Which Way a Stock Will Move Tomorrow?

The Probability Difference statistic is an interesting forecasting idea, but this implementation does not establish a small-cap or large-cap predictive edge.

Reviewed and updated July 28, 2026

Hello and welcome back to another paid post!

Today I will take a look at a simple nonparametric sign-prediction rule and what it would take to test it reliably against a drift-adjusted random walk benchmark.

Let’s dive right in.

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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.

The analysis is built around a nonparametric statistic called the Probability Difference (PD), designed to measure the 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. The PD statistic accounts for this, then asks whether the residual predictability is statistically significant.

There are 3 questions at the center of the analysis. First, do smaller stocks show meaningful out-of-sample directional predictability? Second, do larger stocks show the same pattern? Third, does predictability differ after positive and negative extreme-return days? Those questions are worth asking, but the implementation has to identify company size correctly and control its statistical search before the answers can be trusted.

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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: