empirical asset pricing via machine learning

Empirical Asset Pricing via Machine Learning

Shihao Gu1 Bryan Kelly2 Dacheng Xiu1

1University of Chicago Booth School of Business

2Yale School of Management

New Methods for the Cross Section of ReturnsSept. 2018, University of Chicago

The HypeThe Intellectually Honest Answer

4

CNBC, July 12, 2017

Bloomberg, July 15, 2017

Bloomberg, Aug 9, 2017

Same Reporter3 weeks later

Economist, May 2017

Economist, Dec 2017

The Hype

The Intellectually Honest Answer

4

CNBC, July 12, 2017

Bloomberg, July 15, 2017

Bloomberg, Aug 9, 2017

Same Reporter3 weeks later

Economist, May 2017

Economist, Dec 2017

Despite this excitement, understanding of role/value

of these methods in asset pricing setting is barely nascent

What We Do

I Comparative analysis of machine learning methods in context of perhaps most widely

studied problem in finance, measuring equity risk premia

Clearest way to understand the relevance of ML for AP is to apply methods and

compare performance in familiar empirical problem

Two Primary Contributions

1. New benchmark of accuracy in measuring risk premia (aggregate and individual asset)

I Unprecedented out-of-sample predictive R2

I Strategies that leverage ML forecasts earn Sharpe ratios >2.0

2. Synthesize empirical AP with field of machine learning

I Show ML well positioned to push frontier of risk premium measurementI We provide comparative overview of ML methods applied to the two canonical problems of

empirical asset pricing: predicting returns in the cross section and time series

Return prediction is economically meaningful.

Fundamental goal of asset pricing is to understand the behavior of risk premia. If expected

returns were perfectly observed, we would still need theories to explain their behavior and

empirical analysis to test those theories

Efficient markets.

Risk premia are notoriously difficult to measure

— market efficiency forces return variation to be dominated by unforecastable news

— our empirical exercise is non-trivial because of the extremely low signal-to-noise ratio

What is Machine Learning?

Definition of “machine learning” is inchoate and often context specific.

Our definition:

i. a diverse collection of high-dimensional models for statistical prediction

+

ii. regularization methods for model selection and mitigation of overfit

+

iii. efficient algorithms for searching among a vast number of potential model specifications

Why Apply Machine Learning to Asset Pricing?

Reason 1: Measuring an asset’s risk premium is fundamentally a problem of prediction

Fama Nobel Lecture: Two pillars of empirical asset pricing research

1. Describe/understand differences in risk premis across assets

2. Describe/understand dynamics of the market equity risk premium

A risk premium is a conditional expectation of a future realized excess return.

ML methods specialize in prediction tasks, thus ideally suited to risk premium measurement.


Reason 2: The collection of candidate conditioning variables for the risk premium is large

I We’ve accumulated a staggering list of return predictors

I They are often close cousins and highly correlated

Traditional prediction methods break down when predictor count approaches the observation

count and/or predictors are highly correlated

I With its emphasis on variable selection and dimension reduction, ML well suited for

such challenging empirical issues


Reason 3: Functional form is unknown and likely complex

I Theoretical literature offers little guidance for winnowing list of conditioning variables and

functional forms

Three aspects of ML suited to problems of ambiguous functional form

1. Suite of dissimilar methods. Casts wide net in model search

2. Nonparametric design to approximate complex nonlinear associations

3. Parameter penalization and conservative model selection criteria complement complexity,

help avoid overfit and false discovery

The (Very Familiar) Empirical Setting

∼100 stock characteristics (usual suspects)

+

∼10 macroeconomic predictors (a la Goyal-Welch)

⇓

Monthly returns on 1) individual stocks and 2) stock portfolios

Which Machine Learning Methods?

I Linear ModelsI OLS(3) includes value, size, momentumI OLS + Elastic Net + Huber’s Loss

I Dimension ReductionI PCA, PLS

I Generalized Linear ModelsI Additive Series Regression

I Regression TreesI Random ForestI Gradient Boosted Regression Trees

I Neural Networks aka “Deep Learning”I up to 5 hidden layersI around 30,000 parameters

Main Empirical Findings

Machine learning holds promise for empirical asset pricing

1. Vast predictor sets viable in linear prediction when penalization used

2. Non-linearities substantially improve predictions

3. Shallow learning outperforms deeper learning

4. Distance between non-linear methods and benchmark widens when predicting portfolios

5. Gains from machine learning forecasts are economically large

6. Most successful predictors: price trends, liquidity, and volatility

Data and Over-arching Model

I Goal: Find best representation of Et(ri,t+1)

I We consider general model

Et(ri,t+1) = g?(zi,t), zi,t = xt ⊗ ci,t ,

I xt vector of macro predictorsI ci,t vector of stock characteristicsI g?(·) functional form approximated by ML

I Our framework nests typical (time-varying) beta-pricing specification

ri,t+1 = Et(ri,t+1) + β′i,t(Ft+1 − Et(Ft+1)) + εi,t+1︸︷︷︸

Noise

, Et(ri,t+1) = β′i,tλt ,

Training, Validation, and TestingWe divide the 60 years of data into

I 18 years of training sampleI 12 years of validation sampleI and remaining 30 years for out-of-sample testing.

Empirical Assessments

I Predictive performance in statistical terms

I R2OOS = 1 −

∑(i,t)∈T3

(ri,t+1−ri,t+1)2∑

(i,t)∈T3(ri,t+1−0)2

I Predictive performance in economic terms

I Sharpe ratio

I Model comparison

I Diebold-Mariano tests

I Variable importance

I Decrease in R2 from exclusion

I Benchmark predictive model (“OLS-3”)

I Linear model using size, value, and momentum

Individual Stock Return Prediction: Monthly

OLS OLS-3 PLS PCR ENet GLM RF GBRT NN1 NN2 NN3 NN4 NN5

All -4.60 0.16 0.18 0.28 0.09 0.19 0.27 0.30 0.35 0.38 0.39 0.37 0.35

Top 1000 -14.21 0.15 -0.10 -0.05 0.10 0.17 0.62 0.53 0.44 0.58 0.72 0.67 0.69

Bottom 1000 -2.13 0.37 0.29 0.36 0.18 0.28 0.29 0.27 0.41 0.45 0.46 0.42 0.40

OLS-3+H

PLSPC

REN

et+H

GLM

+H

RF

GBRT+H

NN

1N

N2

NN

3N

N4

NN

5

R2 oos

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8All

Top

Bottom

Similar for big/small stocks, monthly/annual returns

“Bottom-up” Prediction of Portfolio Returns

Predict portfolio returns by aggregating individual stock return predictions

Given weights wpi,t in portfolio p, and individual return predictions ri,t+1,

rpt+1 =N∑i=1

wpi,t × ri,t+1

Predicting Pre-specified PortfoliosMonthly R2

OLS-3 PLS PCR ENet GLM RF GBRT NN1 NN2 NN3 NN4 NN5

S&P 500 -0.11 -0.86 -2.62 -0.38 0.86 1.39 1.13 0.84 0.96 1.80 1.46 1.60

Big Growth 0.41 0.75 -0.77 -1.55 0.73 0.99 0.80 0.70 0.32 1.67 1.42 1.40

Big Value -1.05 -1.88 -3.14 -0.03 0.70 1.41 1.04 0.78 1.20 1.57 1.17 1.42

Small Growth 0.35 1.54 0.72 -0.03 0.95 0.54 0.62 1.68 1.26 1.48 1.53 1.44

Small Value -0.06 0.40 -0.12 -0.57 0.02 0.71 0.90 0.00 0.47 0.46 0.41 0.53

Big Conservative -0.24 -0.17 -1.97 0.19 0.69 0.96 0.78 1.08 0.67 1.68 1.46 1.56

Big Aggressive -0.12 -0.77 -2.00 -0.91 0.68 1.83 1.45 1.14 1.65 1.87 1.55 1.69

Small Conservative 0.02 0.75 0.48 -0.46 0.55 0.59 0.60 0.94 0.91 0.93 0.99 0.88

Small Aggressive 0.14 0.97 0.06 -0.54 0.19 0.86 1.04 0.25 0.66 0.75 0.67 0.79

Big Robust -0.58 -0.22 -2.89 -0.27 1.54 1.41 0.70 0.60 0.84 1.14 1.05 1.21

Big Weak -0.24 -1.47 -1.95 -0.40 -0.26 0.67 0.83 0.24 0.60 1.21 0.95 1.07

Small Robust -0.77 0.77 0.18 -0.32 0.41 0.27 -0.06 -0.06 -0.02 0.06 0.13 0.15

Small Weak 0.02 0.32 -0.28 -0.25 0.17 0.90 1.31 0.84 0.85 1.09 0.96 1.08

Big Up -1.53 -2.54 -3.93 -0.21 0.40 1.12 0.68 0.46 0.85 1.28 0.99 1.05

Big Down -0.10 -1.20 -2.05 -0.26 0.36 1.09 0.77 0.48 0.89 1.34 1.17 1.36

Small Up -0.79 0.42 -0.36 -0.33 -0.33 0.31 0.40 0.23 0.60 0.67 0.55 0.61

Small Down 0.40 1.16 0.47 -0.46 0.62 0.93 1.20 0.80 0.97 0.97 0.97 0.96

Variable ImportanceNN5

NN4

NN3

NN2

NN1

GBRT+

HRF

GLM

+H

ENet

+HPCR

PLS

mom

1mm

vel1

chm

omm

axre

t

mom

12m

indm

omdo

lvol

secu

redi

nd spre

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turn

ninc

rid

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mom

36m

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turn

basp

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mom

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nvin

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csho rd

depr

zero

trade

beta

age

beta

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gcap

cash

debt lgr

bmlev

cash

prch

inv

inve

stbm

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rd_s

ale

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sale

inv

oper

prof

roav

olegr

priced

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herf

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hire

cash

roaq

sale

rec

sale

cash

mve

_ia

gma

grca

pxcu

rrat

absa

ccacc

quick

roeq

pchc

apx_

iagr

ltnoa

cfp_

iata

ngse

cure

dpc

tacc

std_

dolvol tb

chem

pia

aeav

olch

pmia

chtx

rsup

pchs

ale_

pchi

nvt

cinv

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real

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ise5

pchs

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pchx

sga

pchs

alei

nv

pchg

m_p

chsa

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pchd

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pchq

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chat

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noise4

stda

ccst

dcf

divi

pchc

urra

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ise1

pchs

ale_

pchr

ect

noise3

noise2sin

Marginal Relation: Characteristics and Et(ri ,t+1)

mom1m

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

×10-3

-5

0

5

ENet+H

GLM+H

GBRT+H

RF

NN3

retvol

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

×10-3

-5

0

5

mvel1

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

×10-3

-5

0

5

acc

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

×10-3

-5

0

5

Machine Learning Long-Short PortfoliosOLS-3+H PLS PCR

Pred Avg Std SR Pred Avg Std SR Pred Avg Std SR

Low -0.41 0.19 6.52 0.10 -0.85 -0.05 6.73 -0.03 -0.92 -0.49 7.02 -0.242 -0.08 0.40 5.28 0.26 -0.26 0.31 6.02 0.18 -0.29 0.15 6.20 0.089 1.44 0.76 7.10 0.37 1.50 1.10 5.34 0.71 1.45 1.34 5.44 0.85High 1.81 1.84 8.52 0.75 2.15 1.51 5.79 0.90 2.09 1.91 6.08 1.09

H-L 2.22 1.65 6.41 0.89 3.01 1.56 4.45 1.22 3.01 2.40 4.65 1.79

Enet+H GLM+H RFLow 0.06 -0.31 7.01 -0.15 -0.43 -0.51 6.94 -0.25 0.26 -0.33 7.13 -0.162 0.33 0.39 6.03 0.22 0.02 0.33 6.08 0.19 0.41 0.32 5.68 0.199 1.34 1.03 5.92 0.60 1.37 1.29 5.59 0.80 0.89 1.17 5.64 0.72High 1.65 1.80 7.31 0.86 1.84 1.64 6.34 0.90 1.07 1.83 6.78 0.93

H-L 1.59 2.12 5.48 1.34 2.27 2.15 4.39 1.70 0.81 2.16 5.34 1.40

GBRT+H NN1 NN2Low -0.03 -0.38 6.67 -0.20 -0.47 -0.80 7.47 -0.37 -0.37 -0.82 7.97 -0.362 0.16 0.43 5.66 0.26 0.14 0.21 6.24 0.12 0.19 0.17 6.44 0.099 0.81 1.11 5.26 0.73 1.64 1.21 5.49 0.76 1.40 1.13 5.46 0.72High 1.02 1.70 6.57 0.90 2.46 2.13 7.30 1.01 2.32 2.36 8.03 1.02

H-L 1.04 2.08 4.25 1.70 2.93 2.93 4.81 2.11 2.69 3.18 4.90 2.25

NN3 NN4 NN5Low -0.39 -0.96 7.77 -0.43 -0.28 -0.90 7.87 -0.40 -0.21 -0.76 7.93 -0.332 0.17 0.13 6.42 0.07 0.25 0.18 6.57 0.09 0.25 0.24 6.58 0.139 1.44 1.16 5.50 0.73 1.32 1.22 5.60 0.75 1.30 1.24 5.54 0.77High 2.30 2.23 7.78 0.99 2.28 2.35 7.95 1.02 2.19 2.21 7.78 0.98

H-L 2.69 3.19 4.77 2.32 2.56 3.25 4.79 2.35 2.39 2.97 5.05 2.03

Machine Learning (Value-Weighted) Long-Short PortfoliosOLS-3+H PLS PCR

Pred Avg Std SR Pred Avg Std SR Pred Avg Std SR

Low -0.42 0.39 5.22 0.26 -0.86 0.27 5.57 0.17 -0.90 0.04 5.92 0.032 -0.08 0.60 4.47 0.46 -0.27 0.49 5.10 0.33 -0.29 0.43 5.33 0.289 1.42 0.56 7.50 0.26 1.49 0.94 5.01 0.65 1.47 1.18 4.98 0.82High 1.72 0.90 8.18 0.38 2.03 0.96 5.45 0.61 2.02 1.36 5.61 0.84

H-L 2.14 0.51 6.46 0.27 2.89 0.70 4.35 0.56 2.92 1.32 4.72 0.97

ENet+H GLM+H RFLow 0.05 0.08 5.64 0.05 -0.42 0.11 5.43 0.07 0.28 0.09 6.09 0.052 0.33 0.52 5.07 0.35 0.02 0.46 4.67 0.34 0.41 0.39 5.17 0.269 1.33 0.88 5.59 0.55 1.36 0.97 5.49 0.61 0.89 1.20 5.88 0.70High 1.59 0.80 6.83 0.40 1.76 1.18 6.30 0.65 1.01 1.49 7.18 0.72

H-L 1.54 0.72 5.49 0.45 2.17 1.08 4.52 0.83 0.73 1.40 5.54 0.87

GBRT+H NN1 NN2Low 0.00 0.03 5.76 0.02 -0.40 -0.37 7.16 -0.18 -0.30 -0.50 7.89 -0.222 0.16 0.50 5.00 0.34 0.15 0.40 6.03 0.23 0.18 0.36 6.13 0.219 0.81 0.99 5.08 0.67 1.60 0.94 5.09 0.64 1.40 1.01 5.52 0.63High 0.97 1.20 5.81 0.71 2.18 1.37 6.31 0.75 2.03 1.43 6.95 0.72

H-L 0.97 1.16 4.27 0.94 2.58 1.73 5.62 1.07 2.32 1.94 5.68 1.18

NN3 NN4 NN5Low -0.21 -0.51 7.83 -0.23 -0.29 -0.43 7.74 -0.19 -0.15 -0.36 7.63 -0.162 0.26 0.32 6.39 0.18 0.20 0.39 6.15 0.22 0.26 0.29 6.36 0.169 1.28 1.20 5.79 0.72 1.36 1.07 5.87 0.63 1.26 1.31 5.77 0.79High 1.99 1.58 7.33 0.74 2.02 1.47 7.11 0.72 1.91 1.55 6.90 0.78

H-L 2.20 2.09 5.78 1.25 2.30 1.90 5.83 1.13 2.06 1.91 6.01 1.10

Conclusion

Machine learning holds promise for empirical asset pricing

1. Vast predictor sets viable in linear prediction when penalization used

2. Non-linearities substantially improve predictions

3. Shallow learning outperforms deeper learning

4. Distance between non-linear methods and benchmark widens when predicting portfolios

5. Gains from machine learning forecasts are economically large

6. Most successful predictors: price trends, liquidity, and volatility

Can Machines Learn Finance?

2011: Google Brain launches. Uncertain if deep neural

networks would identify a cat, let alone drive a car

Answer: Yes, but there is much more to learn

I Most anecdotes in low capacity settings (HFT, OTC, etc.)

I We provide first large scale evidence of value for long-term asset management

I These are early days (2011 cat recognition)

Comparing Predictions: Diebold-Mariano TestsPositive numbers = column model outperforms row model


+H +H +H +H

OLS+H 3.81 3.82 3.85 3.81 3.83 3.91 3.94 3.96 3.96 3.98 3.97 3.96

OLS-3+H 0.23 1.72 -0.80 0.63 1.55 1.93 1.98 2.83 3.01 2.61 2.63

PLS 1.58 -0.71 0.08 1.39 1.61 1.52 2.29 2.43 2.18 2.15

PCR -1.51 -1.62 0.06 0.48 0.54 1.13 1.20 0.94 0.85

ENet+H 1.00 1.59 1.79 2.09 2.02 2.19 1.92 1.94

GLM+H 1.21 1.59 1.70 2.55 2.76 2.44 2.33

RF 0.66 0.66 1.12 1.30 0.94 0.90

GBRT+H 0.24 0.73 0.83 0.53 0.46

NN1 0.87 1.11 0.49 0.31

NN2 0.10 -1.09 -1.20

NN3 -1.03 -1.92

NN4 -0.47

Predicting Pre-specified Portfolios (Annual R2)


S&P 500 -3.31 0.43 -7.17 0.26 2.07 8.80 7.28 9.99 12.02 15.68 15.30 13.15

Big Growth 3.36 4.88 -4.04 3.62 0.49 9.50 5.86 8.76 8.54 12.42 9.95 7.56

Big Value -11.82 -6.92 -10.22 -2.13 2.44 7.14 6.93 7.47 11.06 11.67 13.37 10.03

Small Growth 6.11 10.81 8.94 8.41 4.31 8.05 3.75 7.24 6.37 7.48 6.60 4.81

Small Value 4.25 2.87 3.19 0.21 0.03 6.20 2.13 3.96 5.52 6.84 2.60 7.23

Big Conservative -8.34 -2.42 -9.77 -3.77 5.17 8.44 5.26 -1.31 8.64 9.65 12.47 6.09

Big Aggressive -0.92 1.89 -4.72 1.36 2.00 7.42 6.67 11.00 11.74 13.08 11.27 10.67

Small Conservative 1.30 6.36 5.01 3.19 2.35 4.60 0.62 5.31 5.39 5.97 4.22 4.71

Small Aggressive 5.53 5.12 2.88 1.04 0.37 6.43 3.23 2.50 4.50 5.50 1.47 6.56

Big Robust -7.17 -2.55 -9.18 1.33 5.42 7.61 6.60 12.55 12.04 13.92 15.29 13.35

Big Weak -1.81 3.09 -7.15 -1.02 -1.12 9.62 7.62 4.41 9.95 11.39 11.73 8.40

Small Robust -2.33 0.93 -0.20 0.76 3.72 0.41 -0.87 2.92 3.67 4.47 0.86 4.19

Small Weak 4.72 9.89 5.68 2.15 -1.11 7.53 3.10 -0.48 1.53 2.96 1.61 1.08

Big Up -24.02 -11.77 -19.16 -5.11 0.52 6.15 6.21 4.26 11.44 11.11 14.48 10.62

Big Down -2.32 0.39 -2.79 -0.15 0.71 7.64 5.53 3.58 8.78 9.54 10.32 6.79

Small Up -5.47 3.82 0.71 -2.83 1.57 1.84 -0.19 -4.22 0.70 1.12 -1.42 2.83

Small Down 4.72 5.59 4.84 2.87 0.50 7.23 3.49 3.24 4.63 5.90 3.28 5.22

Cumulative Returns of Long-Short ML Portfolios

1987 1990 1993 1996 1999 2002 2005 2008 2011 2014 2016

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Lo

ng

Po

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4

2

0

2

4

6

8

OLS-3+H PLS PCR ENet+H GLM+H RF GBRT+H NN3 SP500-Rf solid = long dash = short

Drawdowns, Turnover, and Risk Adjusted Alpha

OLS-3 PLS PCR ENet GLM RF GBRT NN1 NN2 NN3 NN4 NN5 MOM1m

+H +H +H +H

Drawdowns and Turnover

Max DD 64.74 35.51 34.17 35.25 27.05 50.22 35.24 19.70 23.72 14.84 18.66 21.73 68.91

Max 1M Loss 38.69 25.05 22.22 34.11 16.92 34.94 22.87 16.98 23.72 10.15 18.66 21.71 43.67

Turnover 156.78 76.92 106.79 143.61 129.22 113.91 136.82 110.87 112.08 113.07 114.08 113.57 172.56

Risk-adjusted Performance

Mean Ret. 1.65 1.56 2.40 2.12 2.15 2.16 2.08 2.93 3.18 3.19 3.25 2.97 1.80

FF3 α 1.43 1.39 2.44 1.96 2.15 2.09 2.08 2.89 3.20 3.20 3.24 2.98 1.54

R2 4.92 12.75 11.59 4.35 3.85 8.64 0.31 8.89 7.28 8.45 8.34 8.38 5.92

FF5 α 1.64 0.96 2.01 1.58 1.74 1.82 1.95 2.66 2.97 3.00 3.03 2.73 1.88

R2 7.05 25.10 23.50 10.96 15.28 14.97 4.09 12.59 10.03 10.74 11.11 11.43 10.60

FF5+Mom α 1.88 0.82 1.76 1.34 1.54 1.75 1.78 2.62 2.93 2.98 2.95 2.68 2.14

R2 17.46 32.91 43.77 26.27 31.80 16.59 16.81 13.19 10.59 10.88 13.61 12.25 20.53

Note: All numbers are in percentage.

Which Covariates Predict Returns?

PLS PCR ENet+H GLM+H

levoperprof

retvolchcsho

nincrcashpr

agrrd_mve

dolvolep

mvel1mom6m

spturn

maxretstd_turn

mom12mindmomchmom

mom1m

0.0 0.1 0.2 0.3mom36m

bm_iachinv

lgrbm

deprcashpr

mom6mep

investagr

rd_mvechcshomvel1

spmaxret

indmommom12m

chmommom1m

0.0 0.1 0.2 0.3mom36m

mom6mretvolmvel1

epturn

chinvchmom

nincrps

std_turnsp

chcshoinvest

rd_mvedolvol

agrindmom

mom12mmom1m

0.0 0.2 0.4 0.6mom36m

securedindsic2

chinvlgrep

turnchmom

dolvolcashprchcsho

illinvest

agrrd_mvemaxretmvel1

indmommom12m

mom1m

0.0 0.1 0.2 0.3 0.4 0.5

RF GBRT+H NN2 NN3

mom36mdolvol

illbetasq

betasp

idiovolconvindmom6m

baspreadmom12m

chmomretvolnincr

securedindindmommaxretmvel1

dymom1m

0.00 0.05 0.10rd_mve

turnmom36m

agebeta

mom6mbaspread

idiovolsp

convindchmom

mvel1indmom

retvolnincr

maxretmom12m

securedindmom1m

dy

0.00 0.05 0.10 0.15 0.20betasq

spmom36m

securedindzerotrade

nincrindmomstd_turn

idiovolill

baspreadmom6m

mom12mturn

dolvolmaxretchmom

retvolmvel1

mom1m

0.00 0.05 0.10 0.15 0.20beta

spsecuredind

mom36mzerotrade

nincrindmomstd_turn

illmom12mbaspread

mom6midiovol

turndolvol

chmommaxret

retvolmvel1

mom1m

0.00 0.05 0.10 0.15 0.20

Characteristic Importance over Time by NN319

87198819

89199019

91199219

93199419

95199619

97199819

99200020

01200220

03200420

05200620

07200820

09201020

11201220

13201420

152016

mom

1mm

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mom

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chin

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ms

quic

khe

rfhi

re

pric

edel

aysa

lere

cro

avol

roeq

grca

pxcu

rrat

cash

std_

dolvol

acc

cfp_

iagr

ltnoa

gma

pcta

ccab

sacc

sale

cash

secu

red

pchd

epr

tang

pchc

apx_

iach

empi

aea

r

pchs

ale_

pchi

nvt

pchs

alei

nvch

txch

pmia

chat

oia tb

aeav

olrs

up

pchg

m_p

chsa

le

pchs

ale_

pchx

sga

cinv

est

pchq

uick

pchs

ale_

pchr

ect

real

esta

te

pchc

urra

tst

dacc

stdc

fdi

vidi

vosin

empirical asset pricing via machine learning

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