The Machine Learning Horse Race: Gu, Kelly and Xiu Test Everything on Stock Returns

For about fifty years, the standard way to study what drives stock returns was to run a linear regression. You line up some company characteristics on one side, returns on the other, and fit a straight line. The Fama-French model is exactly this. So is almost every factor paper ever published. The method is transparent, it is easy to defend, and it has one enormous blind spot: it assumes the world is a straight line.

In 2020, Shihao Gu, Bryan Kelly and Dacheng Xiu published a paper that took that assumption out behind the shed. They ran what is essentially a giant, disciplined horse race: nearly every serious machine learning method, applied to nearly every US stock, over nearly sixty years, all predicting the same thing. The result reshaped how the profession thinks about predicting returns.

The problem: linear models may be leaving money on the table

Suppose the true relationship between "a company's characteristics" and "its future return" is not a straight line. Maybe cheapness only pays off when a company is also profitable. Maybe momentum works, but only for small stocks, and only when volatility is low. These are interactions: the effect of one variable depends on the value of another.

A standard linear regression cannot see any of that unless you tell it exactly where to look, by hand, in advance. And with dozens of candidate predictors, the number of possible interactions explodes into the thousands. No human is going to guess the right ones.

There is a second problem, and it is nastier. Return prediction is a low signal, high noise exercise. The vast bulk of what a stock does next month is unpredictable. If you throw a flexible model at noisy data with hundreds of variables, it will happily memorize the noise and produce a beautiful in-sample fit that is worth exactly nothing out of sample. So flexible models are tempting and dangerous at the same time.

Gu, Kelly and Xiu set out to answer: can machine learning actually help here, or does the noise eat it alive?

The key idea via analogy: a fair, brutal bake-off

Think of it as a cooking competition with very strict rules. Every contestant gets the same ingredients, the same kitchen, and the same judges. The only thing that varies is the technique.

The ingredients were the same for all: a large set of firm characteristics (things like size, valuation ratios, past returns, profitability, trading volume) plus broad economic indicators, plus interactions between them. The dish was the same for all: predict the stock's return over the next month. And the judging was ruthless: performance was measured only on data the model had never seen.

The contestants were the main families of prediction methods:

• Plain linear regression, the incumbent champion.
• Penalized linear models (lasso, ridge, elastic net), which are linear regressions with a built-in brake that forces them to keep only the predictors that really earn their keep.
• Dimension reduction methods (principal components and partial least squares), which first squash many correlated predictors into a handful of summary indices, then regress on those.
• Regression trees and random forests and boosted trees, which carve the data into regions with a sequence of yes/no splits and can therefore capture interactions automatically.
• Neural networks, which stack layers of simple functions to build up flexible, non-linear shapes.

Everything was tuned honestly, using only past data to choose settings, then tested forward. That last detail is what makes the paper trustworthy: it respects the arrow of time, which a shocking amount of finance research does not.

The verdict: non-linearity is where the money is

Two results stand out.

First, the flexible models won. Trees and neural networks produced meaningfully better out-of-sample return predictions than linear regression, and the improvement was not a rounding error. When the authors translated those predictions into portfolios (buy the stocks the model likes, sell the ones it does not), the machine learning portfolios delivered substantially better risk-adjusted performance than the linear ones.

Second, and more interesting, they diagnosed why. The gain did not come from the machine learning models discovering exotic new predictors that humans had never heard of. It came from how the models combined the predictors they already had. The winning methods were the ones that could represent interactions and non-linear shapes. When the authors stripped that ability away, the advantage largely vanished.

That is a genuinely important finding, and it is worth saying in plain English: the standard predictors of stock returns are mostly the right ones, but the standard way of combining them (adding them up in a straight line) was throwing away real information.

A related result deserves a mention. The most powerful models were not the biggest ones. Very deep neural networks did not beat modest ones. In a low signal environment, extra flexibility mostly buys you extra ways to fit noise. The paper is, quietly, an argument for restraint.

Why it mattered

• It made machine learning respectable in academic finance. Before this paper, "we used a neural network" was often taken as a confession of data mining. Gu, Kelly and Xiu did the work carefully enough, and reported enough honest out-of-sample results, that the top journals could no longer wave it away. A large research literature followed directly.
• It reframed the question. The debate shifted from "which characteristic predicts returns?" to "how do characteristics combine to predict returns?" That is a different, and probably more productive, question.
• It set a methodological template. The paper's approach, expanding rolling windows, honest tuning on past data only, out-of-sample evaluation, ensembles of models with different random seeds, became a standard that later papers were expected to meet.
• It gave practitioners cover, and a checklist. Quant funds had been using these tools quietly for years. This paper made it possible to talk about them, and, more usefully, laid out how to do it without fooling yourself.

The honest limitations

• The predictive power is real but small. The out-of-sample accuracy in this literature is measured in fractions of a percent of explained variance. That sounds pathetic, and in an absolute sense it is. It is enough to build a profitable portfolio only because you can spread tiny edges across thousands of stocks. It is not a crystal ball, and anyone who reads this paper as promising one has misread it.
• Trading costs are the elephant in the room. The headline portfolio results in this kind of study are typically before transaction costs, and the machine learning portfolios tend to trade actively and to lean on smaller, less liquid stocks, exactly where costs bite hardest. Later research has shown that realistic costs eat a serious chunk of the advantage.
• It is a black box. A neural network that predicts returns cannot easily tell you why. The paper does careful work on variable importance to open the box a little, but you fundamentally cannot get a clean economic story out of it. If you cannot explain your edge, it is much harder to know when it has stopped working.
• The results are one long backtest. Everything is a simulation on historical US stock data. It is a good simulation, run with unusual discipline, but the same warnings that apply to every backtest apply here.
• Success invites its own destruction. If everyone runs the same models on the same standard data, the edge those models find should get arbitraged away. The paper's own logic implies its results will decay.

The one-line takeaway

Gu, Kelly and Xiu ran the definitive horse race between machine learning methods and old-fashioned linear regression for predicting stock returns, and found that the flexible models (trees and neural networks) genuinely win, not by finding new predictors but by capturing the non-linear interactions between the predictors we already knew about.