Wu's review puts a tradable label on 0.4% monthly forecast R²

A monthly out-of-sample R² of 0.40% needs more evidence before it earns the label tradable. That is the highest, and only, machine-learning performance figure in Wu's review. The result comes from Gu, Kelly and Xiu, measured against a forecast of zero. Wu's conclusion uses it to argue both that the gains are real and that the problem remains. Wu himself describes the gain as limited. Section 7.1 begins by saying that machine learning's total gains in predicting cross-sectional returns are limited. Why, then, does the paper call the figure tradable without a Sharpe or cost number to support it?

Whose evidence supports the survey?

Wu presents a narrative survey, with no dataset or backtest of his own. After an opening diagnosis, the argument proceeds through four steps: testing, construction, discovery and integration. Costs come last.

The diagnosis concerns the factor zoo. Harvey, Liu and Zhu count 316 published predictors; Hou, Xue and Zhang retest 452 anomalies; Chen and Zimmermann catalogue more than 320 reproducible factors. Applying a single protocol to those 452 anomalies, Hou, Xue and Zhang find that 65% fail at |t| ≥ 1.96 and 82% fail at |t| ≥ 2.78. Across 97 predictors, McLean and Pontiff find returns fall 26% out of sample and 58% after publication. Wu sees these findings as the breakdown of the old rule, significant and therefore valid, when the candidate count reaches the hundreds.

Wu proposes adding another yardstick: does the factor forecast returns out of sample? His account attributes the forecasting gains to interactions and nonlinearities missed by linear models, as well as information lost in sorting. Manual sorts can cross two or three characteristics. Trees and neural nets can search for higher-order combinations without instructions on where to find them. Wu states the condition directly. If the true process is linear, the nonlinear gain disappears and overfitting can make it negative.

The survey also discusses value and momentum across 8 markets and asset classes. Its integration section closes with China. There, the CH-3 model excludes the smallest 30% of A-shares because their prices contain shell value, and replaces book-to-market with earnings-to-price.

Wu makes a restrained claim for machine learning: it "has made genuine instrumental advances in prediction-based tasks, but cannot be regarded as a substitute for causal inference or mechanistic explanation." I accept that sentence. My objection concerns the word attached to the 0.40%: tradable.

The zero-forecast benchmark

On monthly US stock returns, Gu, Kelly and Xiu compare 13 models. The leader, NN3, records 0.40%. GBRT records 0.34, random forest 0.33, PLS 0.27, PCR 0.26 and elastic net 0.11. At -3.46%, ordinary least squares using all variables performs worse than a zero forecast for every stock.

Wu defines the measure correctly. Its denominator sums squared un-demeaned excess returns, with a zero forecast as the benchmark. The 0.40% therefore does not measure the share of variance explained. Wu also warns against comparing it with alpha or a Gibbons, Ross and Shanken (GRS) statistic. Both points deserve credit.

A reporting slip matters when comparing the models. Wu says the 0.40% was computed over 1957 to 2016. The original study uses that span for the full sample, including training and validation; its test window is shorter. The 0.06-point difference between NN3 and GBRT thus rests on fewer test months than Wu's stated span suggests.

Getting from R² to a trade

Wu acknowledges the cost problem. The paper suggests that the ranking may still yield portfolios with fairly high out-of-sample Sharpe ratios, without supplying a Sharpe figure. It also says machine-learning strategies "tend to have high turnover and should therefore be evaluated on net returns after transaction costs."

Yet the note beneath the paper's 0.4% schematic says the figure "is enough to form a tradable strategy, but it falls far short of solving the core challenge of asset pricing." The latter part concerns explanation. Costs never enter that sentence. We found no Sharpe ratio, turnover figure or cost estimate in the paper that supports the word tradable. Within the same subsection, only a few lines separate the concession from the claim. The evidence shown supports only the concession.

Before taking a position on a 0.40% R², a practitioner must check at least two things. Were all input features public at formation? What do turnover and the spread in the smallest names cost?

Thirteen models share a test window

The paper's best sentence is Wu's warning: "Out-of-sample validation can prevent overfitting to a fixed sample, but it does not protect against overfitting the validation step itself." Wu also says machine-learning signals face the same post-publication decay that McLean and Pontiff measured at 58% for conventional factors.

The next step in that reasoning is straightforward. Ranking 13 models on a single test window amounts to multiple testing. Compared with 452 anomalies, the exercise is smaller; the mechanism remains the same. For more than 300 candidates, Harvey, Liu and Zhu raise the hurdle to t > 3.0. We did not find a corresponding margin in the paper specifying how large an R² gap should be before preferring one model to another.

Wu's warning supplies the working rule. Select the model and features during validation, then use a final, untouched period once. Retuning after seeing that period makes it another in-sample year.

Our US-equity substitution has limits

Some of Wu's cited evidence concerns bonds: Bianchi, Büchner and Tamoni study bond risk premia, while Kelly, Palhares and Pruitt study corporate bond returns. We have no bond price data. We therefore implement the idea in US equities, without having shown that the bond results transfer to stocks.

We lack the specialised hardware needed for the full method, too. Our run constructs a strategy from the paper's idea using a classical baseline and daily US equity data. We chose daily data; the cited study uses a monthly horizon. Since Wu makes no empirical claim of his own, our run cannot test Wu, and it will not reproduce the 0.40%.

A net-of-cost Sharpe for the top model over its linear rival, measured on a period nobody tuned on, is the number that would change my view of tradable.