Empirical Asset Pricing via Machine Learning

Empirical Asset Pricing via Machine Learning Shihao Gu Bryan Kelly Dacheng Xiu

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Empirical Asset Pricing via Machine Learning 2020 Review of Financial Studies / arXiv 10.1093/rfs/hhaa009 gu2020-ml-asset-pricing gu2020-ml-asset-pricing.txt, the project's staged copy 65d08196367dad03a2ed4c6d50467b0f395b6172 arxiv-open quantitative-finance

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0 spans located by searching the stored text for the quote's exact words, allowing any whitespace between them Measurement of equity risk premia, treated as a prediction problem, across the machine learning method repertoire. Quotation withheld because this record is not cleared for redistribution. How do machine learning methods compare for the canonical problem of empirical asset pricing, measuring asset risk premia? Quotation withheld because this record is not cleared for redistribution. Provides benchmarks for the predictive accuracy of machine learning methods in measuring risk premia, robust across specifications. Quotation withheld because this record is not cleared for redistribution. method Quotation withheld because this record is not cleared for redistribution. trees and neural networks improve return prediction over linear and generalized linear models, traced to their allowance of interactions among predictors Quotation withheld because this record is not cleared for redistribution. comparative comparative Quotation withheld because this record is not cleared for redistribution. mechanisms the sample is cut into three disjoint periods that keep the data's time ordering: the model is estimated on the training period alone, while the hyperparameters are chosen by scoring forecasts on the second period and re-estimating from the training data at every step Quotation withheld because this record is not cleared for redistribution. mechanisms the third period is used for neither estimation nor tuning, which is what makes it the only one of the three that is truly out of sample - the validation fits are not, because they are an input to the estimation Quotation withheld because this record is not cleared for redistribution. mechanisms the out-of-sample R-squared is computed against a forecast of zero rather than against the historical mean return, because the historical mean of an individual stock is so noisy that benchmarking against it would lower the bar - and the paper states the size of the choice: every method's monthly R-squared rises by roughly three percentage points under the conventional denominator Quotation withheld because this record is not cleared for redistribution. mechanisms the Diebold-Mariano comparison is applied to the cross-sectional AVERAGE of each month's prediction errors rather than to individual stock errors, because the test's weak-dependence condition is unlikely to hold across a cross-section of stocks with potentially strong dependence Quotation withheld because this record is not cleared for redistribution. mechanisms the network architectures are fixed in advance instead of searched over, because searching uncountably many is called unrealistic and unnecessary - and what the authors say they hope to obtain is a reasonable LOWER BOUND on machine-learning performance, not the best network Quotation withheld because this record is not cleared for redistribution. mechanisms the squared-error objective is replaced by a hybrid of squared loss for relatively small errors and absolute loss for relatively large ones, with the crossover controlled by a tuning parameter optimised adaptively from the data about this quote: 2026-08-16: THE EXTRACTION HAS STRIPPED THE MATH SYMBOLS INSIDE THIS QUOTE. It reads "The Huber loss, H(?), is a hybrid ... controlled by a tuning parameter, , that can be optimized adaptively" - the argument of H and the NAME OF THE TUNING PARAMETER are both absent from the staged file, leaving a bare comma where the symbol was. The quote is an exact substring and the file is incomplete here. The tag deliberately describes the parameter by its role rather than naming it, because naming it would mean supplying a character the staged text does not contain. Quotation withheld because this record is not cleared for redistribution. mechanisms trees in the forest are de-correlated by offering each branch only a randomly drawn subset of the predictors, so that a single dominant predictor cannot force the same early split in every bootstrap sample and collapse the variance reduction that averaging is supposed to provide Quotation withheld because this record is not cleared for redistribution. mechanisms the network is estimated from several different random initialisations and the forecasts are averaged, because the optimiser is stochastic and a single seed's forecast carries variance that has nothing to do with the data Quotation withheld because this record is not cleared for redistribution. mechanisms the interpretation of which methods win is checked against two simulated data sets whose generating processes are known - one linear and additive, one nonlinear with pairwise interactions - so that the ranking can be shown to track the property it is being credited to Quotation withheld because this record is not cleared for redistribution. The headline R2 uses a denominator without demeaning; the paper states the common alternative convention raises every method's R2 by roughly three percentage points. what this limits: comparing this paper's R2 against an R2 computed the usual way, against historical mean returns: the paper says that convention raises all its numbers by roughly three percentage points Quotation withheld because this record is not cleared for redistribution. The paper states its improved predictions are measurements only, and do not identify economic mechanisms or equilibria. what this limits: reading the results as evidence about economic mechanisms: the paper says the measurements do not tell us about mechanisms or equilibria Quotation withheld because this record is not cleared for redistribution. The set of methods compared is explicitly not exhaustive, and support vector machines are excluded with a stated reason. what this limits: reading the comparison as covering the machine learning toolkit: the paper says it is not exhaustive and names an exclusion Quotation withheld because this record is not cleared for redistribution. The paper describes its own interpretive goal as modest. what this limits: reading the variable-importance results as an account of why a predictor works: the stated goal is only to identify influential covariates Quotation withheld because this record is not cleared for redistribution. The conditions underlying the Diebold-Mariano test are stated to be unlikely to hold for the stock-level analysis; the paper modifies the test rather than proceeding. what this limits: the reported model-comparison p-values as standard Diebold-Mariano results: the test was adapted because its conditions were judged unlikely to hold Quotation withheld because this record is not cleared for redistribution. The predictor literature the study draws on is stated to exclude an unknown larger set of predictors that were tested, failed and never reported. what this limits: the 900+ signal set as the space of tested predictors: the paper says an unknown larger set failed and was never reported Quotation withheld because this record is not cleared for redistribution. The finding that shallow networks beat deeper ones is attributed, with hedging, to the data available rather than established as a property of the problem. what this limits: generalising shallow-beats-deep beyond this setting: the paper attributes it, hedged, to the small data and low signal-to-noise ratio here Quotation withheld because this record is not cleared for redistribution. The estimation target is defined in terms of an information set the paper calls true and unobservable. what this limits: treating the target as an observable quantity: the conditioning information set is stated to be unobservable Quotation withheld because this record is not cleared for redistribution. The paper's central term is stated to have no settled definition, and the definition used is stipulated by the authors. what this limits: reading "machine learning" here as a standard term: the paper says the definition is inchoate and stipulates its own three-part one Quotation withheld because this record is not cleared for redistribution. the architecture and vocabulary of the biological nervous system - neurons, synapses, axons, signals passed between layers - carried into empirical asset pricing as the description of a feed-forward network Quotation withheld because this record is not cleared for redistribution. parameter penalisation and deliberately conservative model-selection criteria applied across the whole method comparison eliminates: overfitting and false discovery as the source of apparent predictive power - the flexible functional forms being compared can fit noise, and without the penalty the comparison would reward whichever method fits it best · outcome (stated by the tagger, no quotation): held Quotation withheld because this record is not cleared for redistribution. machine learning machine learning Quotation withheld because this record is not cleared for redistribution. risk premium risk premium Quotation withheld because this record is not cleared for redistribution. expected return expected return Quotation withheld because this record is not cleared for redistribution. neural networks neural networks Quotation withheld because this record is not cleared for redistribution. regression trees regression trees Quotation withheld because this record is not cleared for redistribution. elastic net elastic net Quotation withheld because this record is not cleared for redistribution. support vector machines support vector machines Quotation withheld because this record is not cleared for redistribution. variable importance variable importance Quotation withheld because this record is not cleared for redistribution. out-of-sample R2 out-of-sample R2 Quotation withheld because this record is not cleared for redistribution. predictive performance predictive performance about this quote: This quote ends on 'as' because what it introduces is a displayed formula that the staged extraction does not carry as readable text (the text continues: "Roos 2 = 1 - (i,t)T3 (ri,t+1 - ri,t+1)2 (i,t)T3 "). The quote is verbatim and the formula is in the paper; the quote was not extended, because extending it would put extraction debris inside a quotation. Quotation withheld because this record is not cleared for redistribution. traditional econometric prediction techniques traditional econometric prediction techniques Quotation withheld because this record is not cleared for redistribution. Diebold-Mariano test Diebold-Mariano test Quotation withheld because this record is not cleared for redistribution. out-of-sample R2 - denominator is the sum of squared excess returns, NOT demeaned, so the benchmark forecast is zero Quotation withheld because this record is not cleared for redistribution. out-of-sample R2 - the conventional sense, benchmarked against historical mean returns, which the paper says would raise all its numbers by about three percentage points Quotation withheld because this record is not cleared for redistribution. machine learning - the authors' stipulated three-part sense: high-dimensional prediction models, regularization, and search algorithms Quotation withheld because this record is not cleared for redistribution. risk premium - used interchangeably with expected return, while the paper notes components one may wish to distinguish Quotation withheld because this record is not cleared for redistribution. variable importance - a ranking used to identify influential covariates, explicitly not an account of mechanism Quotation withheld because this record is not cleared for redistribution. qualifies → Green et al. (2013), Harvey et al. (2016), Welch and Goyal (2008) — the predictor-counting literature differs in: WHETHER THE COUNT IS OF PREDICTORS OR OF REPORTED PREDICTORS. Three surveys are cited with their totals - 330 stock-level signals, 316 factors, nearly 20 aggregate-market predictors - and then: 'Quotation withheld.' The denominator of the search is unobservable, so the counts bound the published literature and not the work done Quotation withheld because this record is not cleared for redistribution. builds-on → Welch and Goyal (2008) differs in: WHAT THE BENCHMARK IS. The comparison point is not a rival method's success but a documented FAILURE: 'nearly all of the macroeconomic return predictor variables in the survey of Welch and Goyal (2008) Quotation withheld.' A negative result is doing the work of a baseline, which is only legible if the reader knows the sign convention of this R2 - and the record already carries two senses of it Quotation withheld because this record is not cleared for redistribution. distinguishes-from → principal components regression (PCA) differs in: WHETHER THE DIMENSION REDUCTION KNOWS WHAT IT IS PREDICTING. 'Quotation withheld' Two reductions of the same data differing only in whether the target is in scope when the axes are chosen Quotation withheld because this record is not cleared for redistribution. risk premium —same-as→ expected return risk premium expected return strength of the relation: 1 Quotation withheld because this record is not cleared for redistribution. predictive performance —operationalised-as→ out-of-sample R2 predictive performance out-of-sample R2 strength of the relation: 1 · about this quote: This quote ends on 'as' because what it introduces is a displayed formula that the staged extraction does not carry as readable text (the text continues: "Roos 2 = 1 - (i,t)T3 (ri,t+1 - ri,t+1)2 (i,t)T3 "). The quote is verbatim and the formula is in the paper; the quote was not extended, because extending it would put extraction debris inside a quotation. Quotation withheld because this record is not cleared for redistribution. machine learning —contrasted-with→ traditional econometric prediction techniques machine learning traditional econometric prediction techniques strength of the relation: 1 Quotation withheld because this record is not cleared for redistribution.

The full text of this article is not redistributable, so it is not included here. The evidence sentences are withheld with it; the tag layer and its grounding status remain. The article itself is at https://doi.org/10.1093/rfs/hhaa009.