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<reference>
Erasmus University Rotterdam
PwC Risk Advisory
Master Thesis

Interpretable Machine Learning in Asset
Price Forecasting

Sam Rigter (428904)
University Supervisor: S.H.L.C.G. Vermeulen
Company supervisor: C. van Loon
Second Reader: dr. A. Pick

July 1, 2021

Abstract
We perform a comparative empirical analysis on the performance of Machine Learning models in
the forecasting of risk premia on Equity, Government Bond and Commodity return classes using
a set of predictor variables. Out-of-sample forecasting performance is compared for regularized
linear models, tree ensemble methods and a Neural Network over an expanding window
framework. Boosted Regression Trees outperform most models for all asset classes and the
models best capture the return dynamics of the Government Bonds class. In addition, we adopt
the SHapley Additive exPlanations (SHAP) methodology proposed by Lundberg and Lee (2017)
to increase the models’ transparency and interpret the return forecasts. In our analysis, we recognize partial relations between forecasts and predictors that align with macroeconomic theory.

The content of this thesis is the sole responsibility of the author and does not reflect the view
of the supervisors, second assessor, Erasmus School of Economics or Erasmus University.

Contents
1 Introduction

1

2 Data

4

2.1

Asset Returns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

2.2

Predictor Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

3 Methodology
3.1

6

Model Specifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

3.1.1

Linear Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

3.1.2

Tree-based Models . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

3.1.3

Neural Network Model . . . . . . . . . . . . . . . . . . . . . . . . .

9

3.2

Expanding Window Framework . . . . . . . . . . . . . . . . . . . . . . . . 11

3.3

Forecasting Performance Measures

3.4

Return Forecast Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . 15

. . . . . . . . . . . . . . . . . . . . . . 13

4 Results

17

4.1

Hyperparameter Configurations . . . . . . . . . . . . . . . . . . . . . . . . 18

4.2

Forecasting Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

4.3

Crisis vs. Non-Crisis Robustness Analysis

4.4

Model Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

. . . . . . . . . . . . . . . . . . 24

5 Conclusion

28

References

30

Appendix A Data lags table

33

Appendix B Crisis Robustness Results

34

Appendix C Hyperparameter Timeseries Plots

36

1

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Introduction

Asset pricing models describe the relation between systematic risk and expected returns
on assets. Traditional macroeconomic asset pricing models are built on the assumption
that the relation between asset returns and macroeconomic variables is exclusively linear.
The non-linear behaviour found in returns (Scheinkman and LeBaron (1989) and Hsieh
(1991)) therefore makes traditional asset pricing models restrictive. Currently, with the
rise of “Big Data” and the exponential increase in computing power (Schaller (1997)),
asset pricing literature is adopting Machine Learning (ML) models, which provides various means to capture unobserved non-linear relations between returns and risk factors.
We apply ML models to the forecasting of risk premia on Equity, Government Bond and
Commodity classes and produce forecasts that outperform traditional methods. Additionally, we increase the models’ transparency and interpret the forecasts produced by the
models using the Shapley Additive exPlanations (SHAP), proposed by Lundberg and Lee
(2017). We recognize multiple partial relations between the models’ output and predictor
variables that are in line with macroeconomic theory.
Accurate forecasting of asset returns is valuable to asset managers and market speculators and has an extensive history in literature. Sharpe (1964) introduces the Capital
Asset Pricing Model (CAPM), which connects excess returns on equity to a shared market risk factor. Ross (1976) makes a key contribution to asset pricing by introducing the
Arbitrage Pricing Theory, which expresses excess returns as a linear function of macroeconomic state variables. The risk associated with multiple macroeconomic indicators is
significantly priced in asset returns, shown by Chen et al. (1986). It was later found that
multiple factors are needed in asset pricing models to adequately describe the cross section of returns (Ferson and Harvey (1991), Fama and French (1993)). More recently, ML
estimation methods are starting to be integrated into financial applications in literature.
Due to their non-parametric structure, ML models are suited to capture potential nonlinear relations (Atsalakis and Valavanis (2009)). Lee (2009) finds that Support Vector
Machines outperform linear model specifications in forecasting financial time-series. Lasso
regularization can be used to construct global equity returns with the lagged returns on
equity across the globe (Rapach et al. (2013)). Heaton et al. (2016) show multiple financial
applications using Deep Learning and demonstrate an application in portfolio selection.
The fundamental no-arbitrage condition is used by Chen et al. (2019) as criterion function to construct a Neural Network asset pricing model that outperforms all benchmark
approaches in terms of Sharpe ratio. Gan et al. (2020) use deep learning to produce fast
and accurate approximations in derivative pricing. Gu et al. (2020) perform a comparative analysis between ML methods in return modelling and demonstrate economic gains

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to investors using ML predictions.
ML tools are known for being able to achieve high forecast precision (Ahmed et al.
(2010)). However, due to their non-parametric structure, most methods suffer from a lack
of interpretability. ML models can be difficult to interpret and are generally considered
“black-box“ models. A lack of understanding and transparency of ML models can have
severe consequences in finance as well as other domains (Rudin (2019)). Financial managers must be able to justify financial decisions to stakeholders, and therefore cannot fully
rely on models that are not transparent. For this reason many financial institutions are
skeptical of its potential (Doshi-Velez and Kim (2017)). Recently, more light is shed on
ML interpretability, uncovering “black-box“ models (Du et al. (2019)). Model agnostic
methods that can uncover the “black-boxes“ and explain patterns in the model predictions
are evolving. Ribeiro et al. (2016) propose model-agnostic approaches for explaining ML
predictions and review the local model agnostic Local Interpretable Model-Agnostic Explanation (LIME) method. Lundberg and Lee (2017) propose the SHAP method, which
measures the contributions of feature training data to prediction output in ML models.
SHAP is based on Shapley values, introduced by Shapley (1953) to compute the payout
distribution across players in coalitional game theory. Layer-wise Relevance Propagation
can be used to interpret factor contributions to return forecasts produced by a Neural
Network specification (Nakagawa et al. (2018)). Bracke et al. (2019) compute Shapley
values to address the black box problem that is present in the default risk modelling done
by the Bank of England.
This research attempt to explain if ML techniques can help fulfill the demand for both
accurate and explainable asset pricing models in the financial industry. In this paper,
we perform an empirical study to investigate the out-of-sample forecasting performance
and we interpret the forecasts produced by the models. Using a set of predictor variables
that describe the current state of the U.S. economy, we model returns on U.S. equity,
government bonds and commodity indices. We implement different ML solutions that are
commonly used in practice, which can be categorized as regularized linear models, tree
(ensemble) methods and a Neural Network architecture. The return forecasts produced by
the models are compared on forecasting precision with forecasts produced by a standard
linear asset pricing model. Furthermore, we adopt the SHAP methodology proposed by
Lundberg and Lee (2017), to uncover the contributions of predictor variables to return
forecasts in the ML models. We construct variable importance measures and partial
dependence plots to gain a better understanding of the return forecasts of the estimated
models.
We find that the performance of the ML models differs per asset class. Boosted Re-

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gression Trees (BRTs) outperform most models in all asset classes in terms of Mean
Absolute Scaled Error (MASE). A naive model that produces forecasts by averaging over
the training sample is difficult to outperform in the equity and commodity classes. The
best forecast results are achieved for the Government Bonds, yielding Out-of-Sample R2
values of above sixty percent. Furthermore, we construct feature importance and partial
dependence plots for the return forecasts produced by the Random Forest (RF), BRT and
Neural Network (NN) models. We find that the models capture non-linear influences of
the predictor variables on the asset returns. We recognize multiple macroeconomic phenomena in the partial dependence relations between the return forecasts and the predictor
variables.
The contribution of this research is two-fold: (1) we examine return forecast accuracy
of the different ML asset pricing models, and (2) we interpret the forecasts produced
by the models. We do this by explaining how the return forecasts of non-transparent
ML models depend on the predictor variables. The return modelling of equity funds,
government bonds and commodity price levels is of interest to mutual funds, pension
funds, banks and other financial institutions which hold large positions in these asset
classes. The exponential growth in computing power over the last decades combined with
the vast amounts of (financial) data provides interesting forecasting opportunities using
ML. Achieving superior investment results using asset pricing models based on ML can
be of great economic value for these asset managers. But it is often desirable to be able
to explain how financial models produce their output. By interpreting the model’s output
using SHAP, we aim to recognize relations between returns and predictor variables that
are in line with well accepted macroeconomic theory. Through this analysis, we aim to
give insight into the applicability of ML solutions in the finance sector. The attribution
to interpretability and model transparency is not limited to asset managers only, but also
for risk managers and other types of business.
The remainder of this paper is organized as follows: In Section 2 we discuss the financial
return data and predictor variables that we use in our empirical analysis. We introduce the
ML models, forecast accuracy measures and SHAP interpretation framework in Section 3.
Section 4 presents and discusses the results of our empirical analysis. Finally, we conclude
our research in Section 5.

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2

July 1, 2021

Data

In this section, we discuss the data that is used in this research to model the asset returns,
and the predictor variables that are used to compute the predictions. Section 2.1 describes
the return data of the Equity, Government Bonds and Commodity asset classes. Section
2.2 describes the considered predictor variables in detail.

2.1

Asset Returns

We study the excess returns of N = 3 asset classes: Equity (EQ), U.S. Government
Bonds (GOVBO) and Commodities (COM), in the United States over the period April
1987 - December 2019. We compute the monthly excess returns that are earned by
these classes, henceforth asset class returns. This results in T = 392 monthly time-series
observations for the asset class returns. We use the S&P500 index to measure returns
on equity, which we obtain from the Yahoo Finance database. The S&P500 Index tracks
500 leading companies in leading sectors that are publicly held on U.S. stock exchanges,
and hence provides a diversified representation of the overall equity return level in the
U.S. market. For the Government Bonds asset class we use the returns on the one-month
Treasury bills, which we obtained from the Kenneth R. French Data Library1 . Lastly, we
measure the returns on the Commodities asset class by calculating monthly returns on
the Producer Price Index by Commodity: All Commodities (PPIACO), which is obtained
from the Federal Reserve Economic Data (FRED)2 . The PPIACO index indicates the
selling price levels received by domestic producers of commodities for their output, thereby
representing the average price level of U.S. commodities. For the Equity and Commodities
classes, we are interested in the excess returns. For these classes, we construct excess
returns by subtracting the one-month U.S. short rate from the monthly returns.
Table 1: Descriptive Statistics of the Asset Classes Excess Returns
Asset Class

Mean

Median

Min

Max

Std Dev

Sharpe Ratio

Equity
Government bonds
Commodities

0.45
0.25
-0.07

0.91
0.235
-0.11

-22.13
0.00
-5.41

10.83
0.79
2.80

4.27
0.20
0.96

0.11
1.23
-0.07

NOTE: This table presents descriptive statistics for monthly percentage excess returns per asset class,
calculated from the monthly percentage change over index values over the sample period April 1987 December 2019.

Table 1 presents summary statistics of the excess returns on the asset classes. As
1
2

https://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html
https://fred.stlouisfed.org/series/PPIACO

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expected, the Equity class has the highest mean returns, but these are paired with the
highest standard deviation which results in a lower Sharpe Ratio than that of the Fixed
Income class.

2.2

Predictor Variables

We aim to predict and explain the returns of the asset classes, described in section 2.1,
by including a collection K = 19 predictor variables including macroeconomic indicators,
treasury yield rates, a U.S. climate index and Fama-French (Fama and French (1996))
factors in the asset pricing models.
Inspired by Bianchi et al. (2017) and Gu et al. (2020), we have selected a set of eleven
macroeconomic indicators. The first one is the U.S. Personal Consumption Expenditure
(PCE) index, which tracks the average consumption patterns of U.S. households. Secondly, we use the U.S. Consumer Price Index (CPI), that measures the average price level
of goods and services purchased by U.S. households. The third and the fourth indicator
are the Industrial Production (IP) and Gross Domestic Product (GDP), which are different measures of U.S. economic output. Furthermore, we consider the Term Spread (TS)
on bonds, which is constructed as the difference between the ten-year and three-month
U.S. Treasury yields. The sixth macroeconomic indicator is the Default Risk Premium
(DRP) on bonds, which is calculated as the difference between Moody’s Baa corporate
bond yield averages, and the ten-year U.S. Treasury yields. For the seventh indicator we
use the CBOE S&P-100 Volatility Index (VXO), which measures the expected volatility
on the S&P-100 index.
Furthermore, we use the S&P/Case-Shiller U.S. National Home Price Index (RE) as a
proxy for U.S. residential real estate prices. Additionally, we include a U.S. Unemployment
Rate (UR) index which represents the number of unemployed U.S. citizens as a percentage
of the total labor force. We also include a Trade Weighted U.S. Dollar Index, which
represents a weighted average of the FX value of the U.S. Dollar against a set of the other
leading index currencies. Finally, we include the market value of the gross federal debt
(GFD) which represents the debt burden faced by the U.S. government. We obtained all
macroeconomic indicators from the FRED database.
Interest rates can have predictive value on economic activity in the U.S (Kozlowski
(1995)). Hence, besides the macroeconomic indicators, we include the U.S. treasury rates
of the 3-month, 10-year and 30-year maturities in our set of predictors. (Dell et al.
(2014) show the current interest in literature on the effect of climate change on economic
output. They highlight the direct effects and risks that climate change has on the economy.
This interest is directly reflected by the increasing importance of risk associated with

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Environmental, Social & Governance (ESG) to financial institutions (Kotsantonis et al.
(2016)). Hence, in our research, we include the U.S. Climate Extremes Index (CEI),
obtained from the National Centers for Environmental Information3 to capture the effects
of the observed change the U.S. climate on the excess returns of the assets. Finally, we
include the well-known Market, Size and Value factors of the 3 Factor Fama-French model
by Fama and French (1996) which are known for their ability to explain and predict excess
returns on Equity.

3

Methodology

To predict the asset returns we examine multiple ML approaches, introduced in Section
3.1. These are supervised ML methods which are trained using the asset returns as dependent variable and the predictor variables as explanatory variables. In Section 3.2, we
describe the expanding window framework used to estimate the ML models and their
hyperparameters. We compare the return forecasts by adopting the comparison methods described in Section 3.3. Finally, we interpret the return forecasts using variable
importance and partial dependence plots in Section 3.4.

3.1

Model Specifications

We define r1:T = (r1 , r2 , . . . , rT ) to be the (3 × T ) asset return data matrix, where rt =
(r1,t , r2,t , r3,t )0 is the vector of excess returns on the asset classes at time t. Furthermore,
we define f1:T = (f1 , f2 , . . . , fT ) to be the (K × T ) matrix of predictor data, where ft =
(f1,t , f2,t , . . . , fK,t )0 is the vector of predictors at time t.
Using asset pricing models, we aim to accurately predict the returns of asset class i on
time t + 1, given the information available at time t. Hence, following Gu et al. (2020),
we model the excess returns ri,t+1 as:
ri,t+1 = Et [ri,t+1 ] + i,t+1 ,

Et [ri,t+1 ] = gi (ft )

for time t = 1, . . . , T and class i = {1, 2, 3}. Et [ri,t+1 ] denotes the conditional expectation
at time t of the returns of asset class i during time t + 1 and i,t+1 the prediction error at
time t + 1. We express the conditional return expectation as a flexible function gi (·) of
the (K × 1) vector ft of the macroeconomic predictor variables. We now specify different
functional forms of gi (·).
3

https://www.ncdc.noaa.gov/extremes/cei/definition

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3.1.1

July 1, 2021

Linear Models

We first set the baseline model to a simple linear regression representation:
gi (ft ) = f0t βi .

(1)

Equation 1 represents a linear combination of the macroeconomic indicator data with the
(K × 1) parameter vector βi .
To prevent overfitting in the linear model due to too many risk factors, we examine
regularization methods that can shrink certain predictor coefficients to zero. This is done
by including a penalty term to the OLS loss function, as described in Equation 2. These
methods include Ridge (α = 0), Lasso (α = 1) and Elastic Net (EN) for (0 < α < 1),
where the EN penalty is a convex combination of Ridge and Lasso.

βi∗ = arg min
βi

K
T
X
X
0
2
2
((1 − α)βk,i
+ α|βk,i |)
(ri,t − β0,i − ft βi ) + λ
t=1

(2)

k=1

The penalty term in the loss function forces certain predictor coefficients to zero, allowing the model to prevent overfitting in the data. The hyperparameter λ > 0 represents
the penalization magnitude. Increasing λ will increase the penalty and thereby force more
coefficients towards zero. Setting λ to zero results in the linear baseline model from Equation 1. We will optimize for α and λ using timeseries cross-validation, as explained in
Section 3.2. This results into the following (shrunk) functional form of gi (·):
gi (ft ) = f0t βi∗ .

(3)

The models described in Equation 1 and 2 can be regarded as restrictive due to their
linearity (Atsalakis and Valavanis (2009)). Hence, in addition to the (shrunk) linear
models, we use Regression Trees (RT), Random Forests (RF), Boosted Regression Trees
(BRT) in Section 3.1.2 and Neural Networks (NN) in Section 3.1.3 to capture potential
nonlinear dynamics in the asset returns.
3.1.2

Tree-based Models

RTs (Breiman et al., 1984) are a fully non-parametric estimation technique that is used
to group sample returns of the training set into nodes/branches, based on their similarity
in predictor values. We estimate the RT by minimizing the following cost complexity
function:

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Interpretable Machine Learning in Asset Pricing

|T |
X
X

(rt − r̂Rm )2 + |T |.

July 1, 2021

(4)

m=1 t:ft ∈Rm

In Equation 4, T represents the RT, |T | corresponds to the number of terminal nodes
in tree T and Rm the region in the sample space of the macroeconomic indicators corresponding to the mth terminal node. The variable r̂Rm corresponds to the return prediction
for returns in a region Rm , computed by the average of the sample points that lay in that
region. The additive term |T | in the optimization function is used to prevent overfitting
in the data. This happens when the leafs of the tree grow too “deep” into the data and
out-of-sample forecasts will become too variable.
Forecasts for returns can be made by classifying a test sample point to a specific
terminal node m of the regression tree and then, using the average of the sample points
in that node denoted by cm , to predict the return:

gi (ft ) =

|T |
X

cm I{ft ∈Rm } .

(5)

m=1

Ensemble methods, which combine the output of multiple models, can often improve
forecasting precision (Dietterich (2000)). Multiple RTs can also be combined, giving rise to
the use of RFs (Breiman (2001)). To estimate a RF, B RTs are trained individually using
only random subsets of the predictor variables. These random subsets of the predictors
decorrelate the individual trees, ensuring variance reduction in the return forecasts. If we
denote the return forecast of an individual tree b ∈ {1, . . . , B} as gib (ft ) , a RF forecast
can be constructed by averaging over the individual tree forecasts:
B

gi (ft ) =

1 X b
g (ft ).
B b=1 i

(6)

RTs can also be combined sequentially to boost their collective performance through
weighted data sampling. When iteratively estimating new trees, more weight can be given
to sample return forecasts that are inaccurate. In this way, a new tree can be estimated
on an adjusted version of the training set, prioritizing on previous errors. Subsequently,
errors of previous models are minimized while increasing (or “boosting”) the performance
of high-performing trees.
Friedman (2001) introduce Gradient Boosting Machines by employing gradient descent
algorithms to minimize errors in sequential models, yielding a competitive and robust
forecasting procedure. The Extreme Gradient Boosting (XGB) algorithm is introduced

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by Chen and Guestrin (2016), which is optimized for memory efficiency and speed, using
parallel and distributed computing. We apply the XGB algorithm to compute return
predictions by producing an average of the gib (ft ) predictions made by the sequentially
“boosted” regression trees as described in Equation 6. The collection of boosted RTs
forms the BRT model. The process is visualized in Figure 1:
Figure 1: Regression Tree Boosting

NOTE: This figure illustrates the sequential boosting of regression trees, which is performed by the
XGBoost algorithm. After fitting a new tree, more weight is given to previously inaccurately forecasted
samples.

Because tree boosting algorithms produce trees to correct for the errors of previous
trees on the training data, they are prone to overfitting quickly. A key hyperparameter in
the XGB algorithm to control overfitting is the learning rate, denoted by η. The learning
rate functions as a weighting factor for the corrections made in the new trees.
3.1.3

Neural Network Model

Finally, we use NNs to model potential non-linear risk-return specifications. NN models
are a powerful ML tool due to their ability to approximately fit any functional form that
is hidden in the data (Stinchcombe et al. (1989)). This flexibility is achieved from the
network architecture of nodes that can pass signals through (multiple) hidden layers. This
ML network is popularized through its biological interpretation, because it mimics the
neurological connections found in biological brains.
NN models can provide an approximation of our return prediction function gi (ft ),
using a very flexible and layered structure. We illustrate the model by explaining the
components of a Single Layer NN. This example is illustrated in Figure 2. The structure
of such a network can be generalized to a NN that contains multiple middle layers.
The Single Layer NN consists of an input layer, a hidden layer, and an output layer.
The three network layers each contain a specified number of nodes. In our context,
(1)

(1)

the input layer contains K nodes {n1 , . . . , nK }, the middle layer contains M nodes
(2)

(2)

(3)

{n1 , . . . , nM }, and the output layer contains 1 output node {n1 }. The superscripts
refer to the input layer (1), middle layer (2) and the output layer (3).

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Figure 2: Single Layer Neural Network

NOTE: This figure illustrates the Single Layer NN example. The input layer consists of K nodes,
corresponding to the K predictor variables. The middle layer contains M nodes, and the output layer
consists of one terminal node. Signals are passed and combined through the architecture of nodes to
produce a return forecast in the output layer.

All nodes in the network receive an activation signal, which is then used to update the node value. The input nodes receive the prediction variable data of time t:
{f1,t , f2,t , . . . , fK,t } as activation signals and these are set as their node value. These input node values are then send as signals to the M nodes of the hidden layer. Each node
(2)

(2)

nm of the hidden layer receives an activation signal am,t from the input layer which is a
linear combination of the values of the input nodes:
(2)
am,t =

K
X

(1)

βm,k fk,t ,

(7)

k=1
(1)

where βm,k is the scaling coefficient from the k th input node value to the mth node of
(2)

the hidden layer. This node in the hidden layer processes the activation signal am,t in
(2)

(2)

a nonlinear activation function h(am,t ), to compute the value of hidden layer node nm .
Subsequently, the values of the hidden nodes are sent to the output node as a linear
(3)

(3)

combination, producing the activation signal a1,t for the output node n1 :
(3)

a1,t =

M
X

(2)

(2)
βm
h(am,t ).

(8)

m=1

The output node has a distinct output activation function σ(·). We use a linear output
function to map the node signals back to the real number scale R. Therefore, we choose
(3)

(3)

function σ(·) as the identity function: σ(a1,t ) = a1,t .

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We can now express the NN return forecast as:

(3)
gi (ft ) = a1,t =

M
X

(2)
(2)
βm
h(am,t ) =

m=1

M
X
m=1

K
X

(2)
βm
h(

(1)

βm,k fk,t ),

(9)

k=1

which we obtain by substituting Equation 7 and Equation 8 recursively in the identity
output activation function σ(·). This simple example of 1 hidden layer can be extended
to a NN of multiple hidden layers, which interact with subsequent streams of activation
signals before reaching the output layer.
The key in this network structure is the recursive use of linear combinations and nonlinear

Figure 3: ReLU

transformations of the signals passing through the
nodes. There are several activation functions, each
influencing the model’s performance. The activation function can be considered as a hyper parameter of the model. We use the popular Rectified
Linear Unit (ReLU) hReLU (a) = max(0, a) activation function, graphed in Figure 3. The ReLU

NOTE: This figure graphs the Rectified

activation function has a nonzero derivative which

Linear Unit activation function used for

allows gradient based learning, resulting in quick

the node activation in the proposed NN.

convergence.
We compare the forecasting performance of the ML models with the performance of
a naive asset pricing model. The naive model does not use the predictor variables and
produces return forecasts by simply averaging over the return training sample.

3.2

Expanding Window Framework

We are interested in the performance of ML based asset pricing models. To measure
the performance in an out-of-sample setting, we adopt an expanding window framework
that iterates over our dataset. From this point, we differentiate between the end of
our starting training sample and the end of the forecasting sample, which we denote
by T1 and Te respectively. Ranging from 1987 until 2019, we have approximately 33
years of return and predictor data, consisting of 392 time series observations. From this
follows that Te = 392. Figure 4 visualizes the expanding window framework used to
re-estimate the ML models and to produce one-month-ahead forecasts. The first 196
monthly observations of the sample are used as starting training sample, which includes
observations from t = 1 to t = T1 − 1, so T1 = 196. The forecast sample ranges from

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t = T1 to t = Te . In each iteration, the training set is expanded with the next observation
from the forecast sample and then used to refit the ML models. In this way, we simulate
a progression through time and the real-time monthly arrival of new data points. The
updated models are used to produce the next one-month-ahead forecasts, until the end of
the forecast period is reached. This results in Te − T1 = 196 expanding window iterations.
As shown in Section 4.1, the optimal hyperparameter specifications can show variation
while progressing through the expanding window. On the other hand, cross-validating all
models in each iteration is computationally infeasible. Hence, we re-select the optimal
hyperparameters for the models every 12 months. However, it is important to note that
the models are still retrained every month with the new data points that enter through
the expanding window.
Figure 4: Expanding Window Framework

NOTE: This figure displays the expanding window structure for estimating the models and producing
the forecasts. We set T1 = 196 and Te = 392 as the bounds of the forecasting sample.

The starting training sample is used to tune the hyperparameters of the ML specifications. Because we use timeseries data, the sample points are not independent and thus we
cannot use regular k-fold cross validation. Intertemporal dependence structures like autocovariance or seasonality would make random splits invalid. The situation would arise
that data from the (i + 1)th fold would be used to forecasts of the iht fold. We adopt an
adjusted cross-validation scheme that is suited for timeseries analysis, illustrated by Figure 5. The training sample is split into 5 folds. Again, an expanding window framework
is used to construct subsequent training and validation samples. In the training samples,
the ML models are estimated with different hyperparameter configurations and tested
in the validation samples. This results in an expanding 5 − 1 = 4 -fold cross validation
scheme. We select the hyperparameter specifications that on average achieve the lowest
Mean Squared Prediction Errors (MSPE) on the validation sets.
To create a realistic out-of-sample setting while forecasting, data snooping must be

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Figure 5: Timeseries Cross-Validation

NOTE: This figure displays the timeseries cross-validation scheme that is used for the hyperparameter
tuning of the ML algorithms in the first iteration of the expanding window framework.

avoided. For a forecast made at time t, only information that would be available prior to
and at time t may be used. The majority of the predictor variables exhibit a publication
lag, meaning that their realizations are published later than the time that they correspond
to. We account for publication lags by lagging the predictor variables appropriately. We
refer to Table 6 in Appendix A for the lag orders that are used.

3.3

Forecasting Performance Measures

We are interested in the predictive performance of the different asset pricing models on
the test data set. As displayed in Table 1, the returns of the asset classes differ in scale
and standard deviation. Hence, to compare the performance of the models across the
different asset classes, we adopt the scale invariant Mean Absolute Scaled Error (MASE)
measure proposed by Hyndman and Koehler (2006). MASE is constructed by scaling
the forecast errors based on the in-sample Mean Absolute Error from the random walk
forecast method. For a one-month-ahead return forecast r̂T +1 , produced using a training
sample t = 1, . . . , T , the scaled forecast error is constructed as:
qT +1 =

rT +1 − r̂T +1
.
PT
1
|r
−
r
|
t
t−1
t=2
T −1

(10)

Then, we construct the MASE by averaging the scaled forecast errors of the forecast
period {T1 , . . . , Te }:
MASE =

Te
X
1
qt .
Te − T1 t=T

(11)

1

The MASE measure allows us to compare the performance of the ML models across the
different asset classes. A MASE score smaller than one indicates that the model produces,
on average, more accurate forecasts than a naive (random walk) forecast.
2
Additionally, we examine the Out-of-Sample ROOS
measure. For a return forecast

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2
series r̂t+1 , with t = T1 , . . . , Te we compute the ROOS
measure as

PTe

t=T1 (rt+1 − r̂t+1 )
PTe
2
t=T1 (rt+1 )

2
ROOS
=1−

2

(12)

2
The ROOS
measure indicates how much of the variance in the returns is explained by
2
the return forecasts. Negative ROOS
values signal that a horizontal line would better
2
explain the out-of-sample variation than the corresponding model. A ROOS
measure of

one occurs when all forecast errors are zero and thus the corresponding model describes
all out-of-sample variation in the returns.
To formally test for a difference in predictive accuracy, we use the Diebold-Mariano
(DM) Test (Diebold and Mariano (2002)). We evaluate the test statistic for the return
forecasts over the forecast sample of our dataset, consisting of Te − T1 = 196 data points.
(1)

(2)

Using the DM-test we compare the return forecasts r̂i,t and r̂i,t , for t = T1 , . . . , Te ,
resulting from ML model (1) and (2) for asset class i. The residuals of the forecasts can
be written as:
(1)

(1)

(2)

ei,t = ri,t − r̂i,t

(2)

and ei,t = ri,t − r̂i,t

for

t = T1 , . . . , Te .

From the forecast residuals the loss differentials can be constructed as:
di,t =



(1)
ei,t

2

−



(2)
ei,t

2

for

t = T1 , . . . , Te ,

which are related to the Mean Squared Prediction Error (MSPE) statistic. We are interested in the expectation of this loss differential, defined as µ = E(di,t ). This µ can be
approximated by the sample mean. We define the sample mean loss differential d¯i , and
the k’th order autocorrelation γi,k as:
d¯i =

Te
X
1
di,t
Te − T1 t=T
1

and

Te
X
1
γi,k =
(di,t − d¯i )(di,t−k − d¯i )
Te − T1 t=T

for

k ≤ 1.

1

1

Then, for h > 0, where h = (Te −T1 ) 3 +1 should be sufficient, because we assume the auto
1

correlations at lags of order (Te − T1 ) 3 + 1 and higher to be close to zero and insignificant.
We define the DM test statistic as:
DM = q

d¯i
γ0,i +2

Ph−1

.

(13)

k=1 γk,i

n

Essentially, the DM test is a regular t-test where we test for the equality in means for

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two series of prediction errors. We use the statistic to test the H0 hypothesis of µ = 0
vs the alternative hypothesis Ha : µ 6= 0. Under the null hypothesis the DM statistic has
a Standard Normal distribution. Hence, we reject H0 when |DM | > zz−α/2 where zz−α/2
is the two-sided critical value of the standard normal distribution for a α% significance
level.

3.4

Return Forecast Interpretation

In this subsection we present the outline of the interpretation techniques we use to better
understand the forecasts produced by the models presented in Section 3.1. Doshi-Velez
and Kim (2017) define interpretability in the context of ML as the ability to explain or to
present in understandable terms to a human. The models that we examine vary in interpretability, for example, the (regularized) linear model specifications and the regression
trees are visually comprehensible. For the linear models, one can inspect the sign, magnitude and significance of the estimated regression coefficients to understand the effect
of the macroeconomic indicators on the return forecasts of the data. The regression tree
model estimates can be visualized by a graphical representation, which displays the path
splits towards the leaf nodes of the tree. On the other hand, RF, BRT and NN models
are less transparent. Their high flexibility comes at the price of lower transparency and
interpretability. We refer to the last three models as the non-transparent models.
While interpreting the ML model forecasts, we aim to answer two sub-questions:
1. Which predictor variables contribute to the return forecasts most and do these differ
per model?
2. What are the partial relations between predictor variables and return forecasts?
To answer these two sub-questions we use the SHAP method introduced by Lundberg
and Lee (2017) to construct variable importance measures and plot partial dependence
plots.
SHAP is based on Shapley Values, which are introduced by Shapley (1953). Shapley
values are used in coalitional game theory and tell how in a game the payout should be
distributed fairly across the players. In SHAP, Lundberg and Lee (2017) apply this idea
to ML explainability. Shapley quantifies the contribution of each player to the payout,
while SHAP quantifies feature variable contributions to predictions produced by a model.
These contributions can be interpreted as importance values of the predictor variables
in producing the individual return forecasts in the test sample. This makes SHAP a
local interpretation method, providing explanations of specific forecasts produced by our
models. Its ability to explain specific return forecasts instances makes SHAP an attractive

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method to use in practise. Furthermore, SHAP is the only method proven to be efficient,
meaning that the sum of all Shapley values of a return forecast add up to the forecast
itself (Lundberg and Lee (2017)). Among other properties, SHAP is also proven to be
symmetric, meaning that two features will get the same Shapley values if their marginal
importance to the return forecast is equal.
For a return forecast g(ft ), produced by a model at time t in our test sample, SHAP
specifies the output explanation as an additive model:
g(ft ) = φ0 +

K
X

φk,t ,

(14)

k=1

which we can refer to as the “explanation model”. In the explanation model, the Shapley
Value φk,t is the feature attribution of feature k to the return forecast at time t. The value
φ0 corresponds to the expected return forecast over all test sample return predictions,
E(g(f)). Hence, we can rewrite Equation 14 and express the summation over the Shapley
values φk,t as the deviation of a specific return forecast from the mean forecast:
K
X

φk,t = g(ft ) − E(g(f)).

(15)

k=1

When for a return forecast g(ft ) the Shapley values φ1,t , . . . φK,t are estimated, we can
use the additive model from Equation 14 to explain how much the value of each predictor
variable has contributed to the return forecast g(ft ). We illustrate this using an example
with four predictor variables to compute a forecast in Figure 6.
Figure 6: SHAP illustration

NOTE: This figure displays the contribution of individual predictor variables to the final forecast output
r̂t+1 . This example uses K = 4 predictor variables to compute the forecast.

The Shapley values are computed by simulating cases where only subsets of all predictors are “present”. We define F as the set containing all predictor variables. A Shapley
value φk,t is estimated by averaging forecast differences resulting from in- and excluding
the k th predictor to all subsets S of F. Thus, for each subset S, we estimate the model

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using predictor variable sets S ∪ {k} and S, and denote these models by gS∪{k} (·) and
gS (·) respectively. A difference term (gS∪{k} (fS∪{k},t ) − gS (fS,t )) can be constructed, which
denotes the marginal contribution of predictor variable k to the forecasts. Because the
effect of a predictor variable k depends on the other predictor variables, we compute the
difference term for all possible subsets S. To compute a weighted average, the terms
are scaled by the probability of the ordered combination of features derived from set S,
resulting in a weighted average expression:

φk,t =

X
S⊆F \{k}

|S|!(|F| − |S| − 1)!
(gS∪{k} (fS∪{k},t ) − gS (fS,t )).
|F|!

(16)

Estimating models on all possible subsets S would require estimating 2|F | models, which
is practically infeasible. Lundberg and Lee (2017) propose efficient model-specific approximation methods for our linear models (linear SHAP), tree based methods (Tree
SHAP) and NN (Deep SHAP). By restricting on specific model types, they develop faster
model-specific approximations, based on sampling. These approximation techniques are
implemented in the SHAP4 package and used in this analysis.
Note that the Shapley values are estimated for all predictor variables, and for all instances in the forecasting sample. This results in K sets of (Te − T1 ) Shapley values.
To measure the overall importance of predictor variables in the production of the forecasts, we average over the Shapley values of the individual return forecasts. The average
Shapley values from this analysis help us answer sub-question (1) of our interpretation
research, because they measure the contributions of the different predictor variables to
return forecasts. Additionally, we are interested in the partial relations between the predictor variables and the return forecasts produced by the models. For this reason, we
produce scatter plots for the Shapley values per predictor variable. This results in partial
dependence plots between the indicator values on the x-axis and the effect on the return
prediction on the y-axis, which helps us answering sub-question (2) of our interpretation
research.

4

Results

In this section, we forecast the asset returns introduced in Section 2.1 using the predictor
variables of Section 2.2 in multiple ML models using the expanding window framework.
Section 4.1 presents the hyperparameter configurations that are used to train the ML
models. In Section 4.2 we present and discuss the forecasting performance of the models.
4

https://github.com/slundberg/shap

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The forecasting performance of the models is compared between crisis and non-crisis
periods in Section 4.3. We interpret the forecasts produced by the models in Section 4.4
using the SHAP methodology.

4.1

Hyperparameter Configurations

The models as described in Section 3.1 are estimated for the asset classes using the
expanding window framework as described in Section 3.2. We find that the optimal
hyperparameter configurations differ per asset class and also vary over time. We illustrate
this with timeseries plots of the λ-penalty parameter used in the Lasso models and the
learning rate parameter η in the XGB algorithm in Figure 7 for the estimation of the
BRT model. We refer to Appendix C for the other hyperparameter timeseries plots.
Figure 7: Time-variation in Hyperparameters

(a) λ-penalty in Lasso

(b) η in XGBoost

NOTE: This figure shows how optimal hyperparameter values λ (penalty parameter of Lasso) and η
(learning rate parameter of XGBoost) vary over the 196 iterations of the expanding window framework
through the test period. The hyperparameters are displayed for the Equity (EQ), Government Bonds
(GOV) and Commodities (COM) asset classes.

To summarize the hyperparameter configurations of the models for the different asset
classes, we present the mode values of the selected hyperparameters in Table 2. We
use the mode statistics to maintain nominal hyperparameter values. The mode values
for the λ parameter in the Lasso model and the η from the XGB algorithm that are
reported in Table 2 can be directly linked to the timeseries plots. For the Ridge and
Lasso regression, the α parameter is fixed at 1 and 0 respectively, in line with Equation
2. For all asset classes, the Ridge model acquires high penalty values λ, compared to the
Lasso and Elastic Net. When observing the sizes of the NN layers, lower # trees and lower
max depth hyperparameters of the commodities class, the ML models appear to prefer
shallower learning for the Commodities class compared to the other classes. This could
indicate that the models are prone to overfitting in the Commodities return data and
prevent this by selecting shallower settings in the cross-validation steps. For the Equity

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and Government Bonds class, the mixing parameter α is smaller than 0.5, indicating a
model preference for the Ridge penalty term in the optimization function.
Table 2: Hyperparameter Settings Equity Class
Model

α

λ

max depth

# trees

max features

η

layers

Equity
Linear Regression ×
Ridge
1
Lasso
0
ElasticNet
0.3
Regression Tree ×
Random Forest ×
Boosted RT
×
Neural Network ×

×
10
10
1
×
×
×
×

×
×
×
×
6
6
4
×

×
×
×
×
1
12
10
×

×
×
×
×
×
8
×
×

×
×
×
×
×
×
0.01
×

×
×
×
×
×
×
×
(200, 200)

Government Bonds
Linear Regression × ×
Ridge
1
1
Lasso
0 0.1
ElasticNet
0.1 0.1
Regression Tree × ×
Random Forest × ×
Boosted RT
× ×
Neural Network × ×

×
×
×
×
8
8
4
×

×
×
×
×
1
10
30
×

×
×
×
×
×
10
×
×

×
×
×
×
×
×
0.3
×

×
×
×
×
×
×
×
(200, 200)

Commodities
Linear Regression × ×
Ridge
1
10
Lasso
0 0.1
ElasticNet
0.2 0.5
Regression Tree × ×
Random Forest × ×
Boosted RT
× ×
Neural Network × ×

×
×
×
×
6
10
4
×

×
×
×
×
1
8
10
×

×
×
×
×
×
6
×
×

×
×
×
×
×
×
×
×
×
×
×
×
0.01
×
× (20, ... , 20)5

NOTE: This table presents the mode values of hyperparameter settings used to train the ML models.
We use the mode to maintain nominal values of the hyperparameters, which are often defined as integers.
In the expanding window the hyperparameter settings are re-chosen every 12 months through (k-1)-fold
timeseries cross validation with k = 5.

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July 1, 2021

Forecasting Performance

Using the hyperparameter configurations presented in Table 2, we iteratively re-estimate
the models in our expanding window framework. While iterating through the forecasting
sample we compute the one-step-ahead forecasts of the asset returns for the next months.
Using the realized returns, we construct the forecast errors and use these to compute the
MASE values in Table 3.
Table 3: Mean Absolute Scaled Error (MASE)

Transparent models
Naive Average Model
Linear regression (baseline)
Ridge
Lasso
ElasticNet
Regression Tree
Non-transparent models
Random Forest
Boosted RT
Neural Network

Equity

Government Bonds

Commodities

0.637
0.780
0.671
0.631
0.633
0.814

7.118
0.566
0.579
3.906
0.745
0.478

1.227
1.398
1.268
1.231
1.257
1.724

0.672
0.631
0.897

0.519
0.494
0.663

1.386
1.229
1.860

NOTE: This table presents the MASE values of the return forecasts produced by the different ML
specifications over the test sample January 2003 - December 2019.

We compare naive model forecasts with the forecasts produced by the ML models.
The naive forecasts are constructed by predicting the average returns value of the current
training sample. For the Equity and Commodity returns, not all ML models outperform
the naive model in terms of MASE. For the Government Bonds class, the forecasts produced by all ML models score lower MASE values than the naive model. As explained
in Section 3.3, MASE values lower than one indicate that the model produces, on average, more accurate forecasts than the naive on-step-ahead random walk forecast, which
is achieved by all ML models of the Equity class. For Equity, the ElasticNet and BRT
models reach the lowest MASE values. For the Government Bonds class, the NN and
Lasso are outperformed by a naive random walk model. The tree (ensemble) methods
perform best for the Government Bonds class, where the BRT and RT models produce
the lowest MASE values. For the Commodities class, all models produce worse forecasts
than the naive one step ahead random walk model. The regularized linear models result
in the lowers MASE values of the ML methods for the Commodities class.
For all asset classes, we graph the return forecasts with the lowest MASE in Figure 8.
Complemented by the descriptive statistics Table 1 of the asset returns, we see that the

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Equity returns are most volatile followed by the Commodity returns. In comparison to the
Equity and Commodity classes, the Government bonds class returns are more persistent
and also more accurately predicted. For the Equity return class, the randomness in the
returns seems to force the BRT model forecasts to zero. This results in conservative
forecasts that approximate a constant forecast series to minimize prediction error. The
commodity return forecasts occasionally follow the trend of the real returns, but this is
frequently accompanied by a lag. This can be observed around the Credit Crisis around
2008.
Figure 8: Out-of-sample Return Forecasts

(a) Equity return forecasts by Boosted Regres- (b) Gov. bonds return forecasts by Regression
sion Trees
Trees

(c) Commodity return forecasts by Boosted Regression Trees
NOTE: This figure presents the out-of-sample return forecasts and realizations of the three asset classes.
We select the best performing model in terms of MASE which are the BRT, RT and BRT models for
the Equity, Government bonds and Commodity classes respectively. The plots range from January 2003
- December 2019 including 196 observations.
2
The ROOS
values per model for the different asset classes are presented in Table 4.

These are generally in line with the MASE results. For the Equity and Commodities class,
2
we generally find negative ROOS
values, indicating that a small part of the variance in the

out-of-sample returns is explained by the predictor variables through the ML models. The
2
negative ROOS
values indicate that the corresponding models do not follow the trend of the

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returns, and thereby fit worse than a constant model. For the Equity asset class, the Lasso
2
and BRT models perform best in terms of MASE and in ROOS
. For the Commodities
2
class, the Naive model and Lasso also achieve the highest ROOS
. Within the Government
2
Bonds class, the BRT has the highest ROOS
value of 0.614.

Table 4: Out-Of-Sample R2

Transparent models
Naive Average Model
Linear Regression (baseline)
Ridge
Lasso
ElasticNet
Regression Tree
Non-transparent models
Random Forest
Boosted RT
Neural Network

Equity

Government bonds

Commodities

0.014
-0.386
-0.138
0.004
-0.002
-0.998

-1.504
0.612
0.613
-0.043
0.602
0.613

-0.047
-0.411
-0.239
-0.079
-0.157
-1.248

-0.153
0.010
-0.927

0.610
0.614
0.605

-0.347
-0.261
-1.442

2
of the return forecasts produced by the different ML specifications
NOTE: This table presents the ROOS
over the test sample January 2003 - December 2019.

Table 5 shows the results of the DM-test. It tests for significant differences between
the forecasting performance of the models. For the Equity class we find that the naive
average forecast model significantly outperforms the linear model, RF and NN. The linear
model is outperformed by all linear models and the BRT model. The NN has produced
forecasts that are significantly less accurate than the forecasts of all other models, except
the RT. The regularization methods Lasso, Ridge and Elastic Net improve return forecast
accuracy significantly for the Equity class.
For the Government Bonds return class, we see mixed DM-test results between the
models. The Naive model is outperformed by all ML models. This is in line with the
MASE scores from Table 3. For this class, the return forecasts produced by the Lasso regression are significantly outperformed by all other models. The NN produces significantly
lower forecast errors than the Naive and the Lasso model.
For the Commodities class, the naive model produces more accurate forecasts than all
ML models. According to the DM test, these differences are significant for the RT, RF
and NN model. The RT model is significantly outperformed by all models except the NN.
Combining our results from Tables 3, 4 and 5 we see that the relative accuracy performance differs per asset class. The variation in the Government Bonds class returns

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Table 5: Diebold-Mariano Test Statistics
Equity Returns
Linear
Ridge
Naive 3.231∗
1.679
Linear 0.000
-5.588∗
Ridge
0.000
Lasso
EN
RT
RF
BRT

Lasso
0.981
-3.233∗
-1.596
0.000

Government Bond Returns
Linear
Ridge
Lasso
Naive -6.802∗
-6.803∗
-6.960∗
Linear 0.000
-0.367
5.947∗
Ridge
0.000
5.948∗
Lasso
0.000
EN
RT
RF
BRT

Commodity Returns
Linear
Ridge
Naive 1.601
0.967
Linear 0.000
-4.069∗
Ridge
0.000
Lasso
EN
RT
RF
BRT

Lasso
0.683
-1.790
-1.018
0.000

EN
1.315
-3.418∗
-1.687
0.628
0.000

RT
1.832
1.250
1.746
1.819
1.818
0.000

RF
2.063∗
-1.908
0.187
2.006∗
2.010∗
-1.743
0.000

BRT
0.281
-3.084∗
-1.550
-0.353
-0.581
-1.834
-2.044∗
0.000

NN
4.378∗
4.145∗
5.859∗
4.391∗
4.637∗
-0.158
4.409∗
4.292∗

EN
-6.808∗
1.733
2.036∗
-5.975∗
0.000

RT
-6.769∗
-0.183
-0.079
-5.875∗
-1.497
0.000

RF
-6.762
0.356
0.512
-5.870∗
-1.459
0.895
0.000

BRT
-6.789∗
-0.388
-0.278
-5.931∗
-2.205∗
-0.208
-1.586
0.000

NN
-6.784∗
1.124
1.417
-5.936∗
-0.465
1.172
0.827
1.439

EN
1.039
-1.893
-0.797
1.218
0.000

RT
3.405∗
2.615∗
3.434∗
3.501∗
3.497∗
0.000

RF
2.083∗
-0.564
1.284
2.597∗
2.461∗
-2.880∗
0.000

BRT
1.210
-1.468
0.328
1.379
1.279
-3.054∗
-0.941
0.000

NN
2.891∗
3.389∗
3.734∗
3.067∗
3.260∗
0.461
2.814∗
3.187∗

NOTE: This table contains the DM test statistics of the pairwise forecast comparisons between all
models, for all three asset classes. A positive number indicates that the model in the row produces more
accurate forecasts than the model in the column. DM statistics that are significant at a 5% confidence
level the number are marked with an asterisk (*). The DM statistics are computed using the forecasts
constructed on the test sample January 2003 - December 2019.
2
seems to be explained best by the models according to the ROOS
measure. A potential

explanation could be that returns on Government Bonds exhibit less randomness and are
potentially subject to less external influences. For the Equity and Commodity return
classes we observe similar accuracy performances, where the linear model is outperformed

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by most ML methods.

4.3

Crisis vs. Non-Crisis Robustness Analysis

The test sample contains data points sampled from the Global Financial Crisis during
the period of 2007 - 2008. We compare the performance of the models during crisis and
the non-crisis periods. We define the data samples from 2007 and 2008 as the crisis
period, resulting in 24 crisis observations. The remaining 172 observations are selected
2
as non-crisis observations. We display the MASE and ROOS
measures for both periods

in Appendix B. For all asset classes, we observe higher MASE values for the 2007-2008
period. This shows that the models produce better forecasts during non-crisis periods.
Furthermore, the model-ranking based on MASE values does not show much variation
between the crisis and non-crisis periods. The relative forecasting performance can thus
be considered robust to being in crisis or non-crisis periods.
2
values differ between the crisis and non-crisis period.
Like the MASE values, the ROOS
2
For the Government Bonds class, we see a strong increase in ROOS
, while for the Equity

and Commodities classes we observe small declines. For the Government Bonds class, the
increase could partially be explained by examining the returns and forecasts produced by
the RT in Figure 8. By visual inspection, the dynamics of the forecasts of the Government
Bonds better follow the real returns during the non-crisis period. We cannot directly
observe this for the Equity and Commodities classes.

4.4

Model Interpretation

We have analyzed the forecast accuracy of the models for the different asset classes. In
order to increase transparency in the forecasting of the asset returns, we use the SHAP
method described in Section 3.4. We compute the additive contributions of the predictor
variables to the return forecasts of the asset classes. As described in Section 3.4, we
estimate the models with and without each predictor variable and thereby measure the
contribution of each of the predictors to the return forecasts.
To increase the transparency of the models, we are interested in which predictor variables contribute most to the return forecasts produced by the models. For each asset class
and for each model, we report the average Shapley value over the data points in the test
sample. We report bar plots of the average feature importance value in Figure 9 for the
RF, BRT and NN models. These models are considered as least transparent because we
cannot visually inspect the model results like we can for a linear regression or RT models.
Figure 9 shows that for all three asset classes, the RF and the BRT algorithm use

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Figure 9: Feature Importance Barplots

(a) RF - EQ

(b) RF - GOVBO

(c) RF - COM

(d) BRT - EQ

(e) BRT - GOVBO

(f) BRT - COM

(g) NN - EQ

(h) NN - GOVBO

(i) NN - COM

NOTE: This figure displays Feature Importance Barplots for the Random Forest, Boosted Regression
Tree and Neural Network models. They are constructed by averaging over the absolute Shapley values.

similar most influential predictor variables. These are the Fama-French factors: Excess
Market return (EM) and the High-Minus-Low (HML) factor for the Equity class returns,
according to the RF and BRT models. Fama and French (1996) have shown that these factors can price excess returns on equity fairly well, and our results appear to be in line with
that finding. For the Government Bonds returns, the 3-Month Treasury Rate (3MTR)
and the Domestic Federal Funds Rate (DFF) are the most important predictor variables,
which also logically connect to Government Bonds returns. For the Commodities index,
we see that Industrial Production (IP) and Real Estate (RE) are important predictor
variables according to the RF and BRT models. Intuitively, there should be a connection
between movements in industrial production levels and demands for commodities. The
same reasoning holds for real estate price movements, as the prices of real estate will partially depend on the prices of construction materials. For the NN models, the importance
of the features is spread more evenly over the predictor variables. The Climate factor
appears to be the most important factor for all three asset classes according to the NN.
The following important features are VXO for Equity, UR for Government Bonds and

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UR for the Commodity class. Due to their extreme flexibility, NN models can capture
any hidden patterns found in data. The feature importance levels of the NN models are
less intuitive and not directly in line with macroeconomic theory. VXO, however, is also
among the more important predictors in the RF and BRT model for the Equity class.
We continue increasing transparency by analyzing scatter plots of the Shapley values
for the predictor variable realisations. This results in partial dependence plots of the
test sample values between the predictor variables and their contributions to the forecasts
produced by the model. The plots show the marginal contributions of a predictor variable
on the model’s output. Figure 10 displays scatter plots of the Shapley values of the most
important predictor variables according to Figure 9. To illustrate the dependence patterns
we display the Shapley values derived from the linear model first. As expected, these
values are perfectly aligned.
Where in a Linear Regression model, the Shapley values appear on a linear line, we
observe that the ML models attribute the predictor variables to the return forecast in a
non-linear way. These results are in line with our assumption of non-linear dependencies that exist between the asset returns and the predictor variables. All models show a
negative relation between Equity returns and the HML predictor variable. For the Government Bonds return class, however, the linear model shows a negative relation between
the 3MTR rate and the returns on the Government Bonds class which contradicts to the
signs captured by the ML models. An explanation could be that the predictor set consists
of, among others, multiple yield variables. This can result in multicollinearity in the linear
model and hence cause a bias in the sign of the coefficients.
For the RF model we observe s-curved dependence patterns between the predictor
variables and the return forecasts. We observe a negative relation between the Equity
returns and the HML variable. Especially low HML values lead to positive contributions
to the next month’s return forecasts. A potential explanation could be that this the
model picks up on the short term reversal effect in stocks (Jegadeesh (1990)). For the
RF models we observe positive relations between the variables 3MTR and IP and return
predictions on Government Bonds and Commodities respectively. Considering that we use
the returns on 1-month Treasury Bonds as Government Bonds Class returns, we expect a
positive relation between the return forecasts and the 3-month Treasury yields according
to the term structure of interest rates. Furthermore, we also expect a positive relation
between IP levels and return forecasts on Commodities.
Compared to the RF model, the BRT shows step-wise shaped relations between the
predictor values and their contribution to the return forecasts. While the RF and the
BRT find similar directions of the effect of 3MTR and IP on the Government Bonds and

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Figure 10: Shapley Scatter Plots

(a) Linear HML - EQ

(b) Linear 3MTR - GOVBO

(c) Linear IP - COM

(d) RF HML - EQ

(e) RF 3MTR - GOVBO

(f) RF IP - COM

(g) BRT HML - EQ

(h) BRT 3MTR - GOVBO

(i) BRT IP - COM

(j) NN HML - EQ

(k) NN 3MTR - GOVBO

(l) NN IP - COM

NOTE: This figure displays Shapely Scatter Plots for the Random Forest (RF), XGBoost (XGB) and
Neural Network (NN) models, for the High-minus-low (HML), Three Month Treasury Rate (3MTR) and
the Industrial Production (IP) predictor variables.

Commodity returns, they show opposite effects of HML predictor on the Equity Class.
The NN model appears to pick up on linear marginal relations rather than on s-shaped
or step-wise relations. The directions of the relations are in line with those produced by
the RF model. In the scatter plots for the NN we observe a dispersion of the Shapley

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Interpretable Machine Learning in Asset Pricing

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values for extreme values of the predictor variables. This suggests that the NN shows
“certain” behaviour for normal situations of the economy, but becomes more “uncertain”
when extreme situations of the economy occur. This indicates that the NN model performs
well with normal data, but worse in case of outliers.
In this section we performed a comparison study between the ML models by obtaining
variable importance and feature dependence plots using the SHAP methodology. For the
RF, BRT and NN models we inspect which variables are most influential to the forecast
output. We also show the partial relations between the predictor variables and the asset
return forecasts. We generally find similarities in the sign of the relation between the
predictors and the forecasts, and find explanations connecting the relations to expectations
according to macroeconomic theory.

5

Conclusion

This research aims to identify whether ML models are valuable in predictive asset price
modelling of different asset classes. Specifically, we are interested if ML models can
outperform traditional linear models in accuracy, while still maintaining explainable model
results. To measure the forecasting performance of the ML models in producing onemonth-ahead return predictions, we use an expanding window framework. Furthermore,
we implement the SHAP methodology proposed by Lundberg and Lee (2017) to obtain
variable importance measures and partial dependence plots of the predictor variables and
the model forecasts in order to increase model transparency.
The methods are applied to the forecasting of excess returns on an Equity, Government bonds and Commodities asset class over the sample period of April 1987 - December
2019, using a set of predictor variables. We show that the performance of the ML models
strongly differs across the different classes. The Boosted Regression Tree model overall
attains high Mean Absolute Scaled Error accuracy over the three classes. The return
dynamics of Government bonds are best captured by the ML models, which could be
explained by their high timeseries persistence. The models generally show similar partial
relationships between the predictor variables and the return forecasts. We can link most
dependence structures to economic theory, which attributes to the interpretability of the
models. The ability to explain the magnitude of the return forecasts make the ML applications valuable to institutional investors and risk managers. The forecasting performance
ranking between the models is robust to being in crisis and non-crisis periods. However,
the overall performance of the models decreases significantly during the Global Financial
Crisis in the period 2007-2008 in our test sample.

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Interpretable Machine Learning in Asset Pricing

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In general, the SHAP methodology can be used for explaining individual model output values produced by any ML model, making it a local model agnostic method. We
use SHAP for quantifying the contribution that each predictor variable adds to the return forecasts made by the ML models. In this way, we increase model transparency by
uncovering the partial relations and feature importance values captured by the models.
However, the SHAP results refer to how the models perceive relations between predictor
variables and returns after training them on our specific dataset. The return data can
contain noise and is subject to external forces that are not incorporated in our predictor
variable set. Furthermore, besides the uncertainty resulting from model estimation, an
additional level of uncertainty arises by approximating the Shapley values over our model
forecasts and interpreting them. Hence, it is important to note that the relations found
using SHAP can differ from actual relations in return dynamics. We do, however, find
similarities between the partial relations and variable importance measures between the
different models. Furthermore, the relations that are found seem understandable and are
generally in line with macroeconomic beliefs and theory. This is a positive result that
supports our choice for adopting the SHAP method.
Our analysis leaves several directions for future research. One of the restricting elements in our analysis is that the forecast accuracy results are a quantitative measure,
while the interpretation section consist of qualitative results. Combining accuracy and
interpretability into a single quantitative metric would result in a more efficient model
ranking and comparison technique. Furthermore, while new ML algorithms and forecasting techniques are emerging quickly, our research focuses on the performance of only a
subset of these models. Centering focus on specific models and investing more time in
hyperparameter tuning could increase their relative performance. Furthermore, in our
analysis on the Equity and Commodity classes, we considered portfolio indices. Our
methodology could be applied to the forecasting of excess returns on individual stocks
or commodity types. Additionally, one could simplify the forecasting exercise by solely
focusing on the direction of change in asset prices, making it a classification problem
instead of forecasting the magnitude.

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Interpretable Machine Learning in Asset Pricing

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Appendix A

July 1, 2021

Data lags table
Table 6: Predictor Variables Overview

Data
Economic Variables
PCE
CPI
IP
GDP
DRP
TS
VXO
RE
GFD
UR
FXD
Yields/rates
DFF
3MTR
10YTR
30YTR
Climate variables
CEI
Fama- French factors
EM
SMB
HML

Lag

Source

×
1 month
1 month
3 months
×
×
×
1 month
1 month
1 month
×

Federal Reserve Economic Data
Federal Reserve Economic Data
Federal Reserve Economic Data
Federal Reserve Economic Data
Federal Reserve Economic Data
Federal Reserve Economic Data
Federal Reserve Economic Data
Federal Reserve Economic Data
Federal Reserve Economic Data
Federal Reserve Economic Data
Federal Reserve Economic Data

×
×
×
×

Federal Reserve Economic Data
Federal Reserve Economic Data
Federal Reserve Economic Data
Federal Reserve Economic Data

12 months

National Center for Environmental Information

1 month
1 month
1 month

Kenneth R. French Data Library
Kenneth R. French Data Library
Kenneth R. French Data Library

NOTE: This table presents the lag orders and sources of the predictor values used to train the ML
models and to compute return forecasts. We classify the predictor variables among groups of Economic
Variables, Yield/rates, Climate variables and Fama-French factors.

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Appendix B

July 1, 2021

Crisis Robustness Results

Table 7: Mean Absolute Scaled Error (MASE) for Crisis and Non-Crisis Periods

Crisis Period
Transparent models
Naive Average Model
Linear regression (baseline)
Ridge
Lasso
ElasticNet
Regression Tree
Non-transparent models
Random Forest
XGBoost
Neural Network
Non-Crisis Period
Transparent models
Naive Average Model
Linear regression (baseline)
Ridge
Lasso
ElasticNet
Regression Tree
Non-transparent models
Random Forest
XGBoost
Neural Network

Equity

Government Bonds

Commodities

1.088
1.168
1.018
1.090
1.069
1.488

5.585
1.277
1.254
3.759
1.404
1.101

2.520
2.988
2.714
2.522
2.624
3.189

1.067
1.106
1.067

1.284
1.249
1.353

2.595
2.800
3.861

0.577
0.728
0.625
0.570
0.571
0.694

7.322
0.471
0.490
3.925
0.658
0.416

1.055
1.186
1.076
1.059
1.075
1.563

0.624
0.568
0.874

0.431
0.394
0.572

1.134
1.100
1.594

NOTE: This table presents the MASE values of the return forecasts produced by the different ML models
over a crisis and a non crisis period. We select the return samples in our dataset from 2007 and 2008 as
the crisis period, based on the Global Financial Crisis from 2007-2008. The remaining observations are
selected as non-crisis sample points, resulting in 24 crisis observations and 172 non-crisis observations.

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Table 8: Out-Of-Sample R2 for Crisis and Non-Crisis Periods

Crisis Period
Transparent models
Naive Average Model
Linear Regression (baseline)
Ridge
Lasso
ElasticNet
Regression Tree
Non-transparent models
Random Forest
XGBoost
Neural Network
Non-Crisis Period
Transparent models
Naive Average Model
Linear Regression (baseline)
Ridge
Lasso
ElasticNet
Regression Tree
Non-transparent models
Random Forest
XGBoost
Neural Network

Equity

Government bonds

Commodities

-0.042
-0.030
0.097
-0.045
0.036
-0.919

-1.793
-0.811
-0.808
-1.215
-0.832
-0.804

-0.010
-0.400
-0.295
-0.068
-0.189
-0.560

-0.015
-0.080
0.057

-0.822
-0.816
-0.817

-0.280
-0.327
-1.236

0.037
-0.540
-0.240
0.025
-0.019
-0.750

-1.886
0.527
0.526
-0.194
0.520
0.524

-0.093
-0.434
-0.206
-0.102
-0.144
-2.052

-0.191
0.048
-1.349

0.527
0.530
0.519

-0.283
-0.218
-1.632

2
NOTE: This table presents the ROOS
of the return forecasts produced by the different ML specifications
over the test sample January 2003 - December 2019. We select the return samples in our dataset from
2007 and 2008 as the crisis period, based on the Global Financial Crisis from 2007-2008. The remaining
observations are selected as non-crisis sample points, resulting in 24 crisis observations and 172 non-crisis
observations.

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Appendix C

July 1, 2021

Hyperparameter Timeseries Plots

Figure 11: Ridge hyperparameter λ timeseries plot

NOTE: This figure shows how the optimal hyperparameter values of λ vary over the 196 iterations
of the expanding window framework through the test period for the Ridge model.

Figure 12: Elastic Net hyperparameter α timeseries plot

NOTE: This figure shows how the optimal hyperparameter values of α vary over the 196 iterations
of the expanding window framework through the test period for the Elastic Net model.

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Figure 13: Elastic Net hyperparameter λ timeseries plot

NOTE: This figure shows how the optimal hyperparameter values of λ vary over the 196 iterations
of the expanding window framework through the test period for the Elastic Net model.

Figure 14: Regression Tree hyperparameter max depth timeseries plot

NOTE: This figure shows how the optimal hyperparameter values of max depth vary over the 196
iterations of the expanding window framework through the test period for the Regression Tree model.

Figure 15: Random Forest hyperparameter #Trees timeseries plot

NOTE: This figure shows how the optimal hyperparameter values of #Trees vary over the 196 iterations of the expanding window framework through the test period for the Random Forest model.

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Figure 16: Random Forest hyperparameter max depth timeseries plot

NOTE: This figure shows how the optimal hyperparameter values of max depth vary over the 196
iterations of the expanding window framework through the test period for the Random Forest model.

Figure 17: Random Forest hyperparameter max f eatures timeseries plot

NOTE: This figure shows how the optimal hyperparameter values of max f eatures vary over the
196 iterations of the expanding window framework through the test period for the Random Forest
model.

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Figure 18: XGBoost hyperparameter max depth timeseries plot

NOTE: This figure shows how the optimal hyperparameter values of max depth vary over the 196
iterations of the expanding window framework through the test period for the XGBoost model.

Figure 19: XGBoost hyperparameter #Trees timeseries plot

NOTE: This figure shows how the optimal hyperparameter values of #Trees vary over the 196 iterations of the expanding window framework through the test period for the XGBoost model.
</reference>

<statements>
1. Risk is often modeled indirectly: models predict conditional expected returns or risk premia, and downstream portfolio construction may still use familiar risk measures (variance, VaR, CVaR) in an optimization step.
2. Performance: Empirical studies show that such models can explain complex cross-sectional return structures and outperform classical factor models in predictive accuracy.
3. Interpretability challenge: These models are typically black boxes; their lack of economic interpretability raises concerns in high-stakes finance and regulatory contexts.
4. Post-hoc XAI: Techniques like SHAP and LIME attribute predictions to features, providing local and global explanations of complex ML asset pricing models.
5. Forecast-then-optimize pipelines: ML models predict expected returns and sometimes risk; those predictions then serve as inputs to a downstream optimizer (mean-variance, mean-CVaR, etc.).
6. Supervised deep / ML: Risk measure / handling: Often uses standard risk measures in downstream optimization; risk indirectly via predicted distributions.
7. Supervised deep / ML: Return prediction / prior: Non‑linear, high‑dimensional prediction of risk premia or returns; can outperform linear factor models.
8. Supervised deep / ML: Main strengths: Captures complex structures and interactions; improved predictive accuracy and flexibility.
9. Supervised deep / ML: Main limitations: Black‑box behavior; fragile under regime shifts; interpretability and economic grounding often weak.
10. This layer is agnostic about the higher-level model but is designed to support both interpretable factors (value, momentum, quality) and high-dimensional signals used by deep models.
11. This yields a traceable decision record that can be explained to stakeholders, addressing the regulatory and practical interpretability concerns around black-box models.
12. Deep and AutoML models improve predictive accuracy for asset pricing relative to classic linear models.
13. XAI methods like SHAP/LIME make complex models more transparent and economically interpretable.
</statements>

Begin the assessment now. Output only the JSON list, without any conversational text or explanations.