$\overline{R}\cap\partial{\sigma}_{\rho,x}(A)\xrightarrow{\mathcal{K}}\overline{R}\cap
T_{x,\nu_{A}\left(x\right)}$ and
$\overline{R}\cap\partial{\sigma}_{\rho,x}({S_{h,K}})\xrightarrow{\mathcal{K}}\overline{R}\cap
T_{x,\nu_{A}\left(x\right)}$ as $\rho\to 0$, where
$T_{x,\nu_{A}\left(x\right)}$ is the approximate tangent line at $x$ of
$\partial A$ (or of $\partial{S_{h,K}}$),

  4. (h4)

$\frac{d\mu_{0}}{d\mathcal{H}^{1}\mathbin{\vrule
height=4.82224pt,depth=0.0pt,width=0.39182pt\vrule
height=0.39182pt,depth=0.0pt,width=3.91806pt}(\partial^{*}{S_{h,K}}\cap\partial
A\cap A^{(1)})}\left(x\right)$ exists and it is finite,

is $\mathcal{H}^{1}$-negligible. Therefore, we prove (5.125) for any fixed
$x\in\partial^{*}{S_{h,K}}\cap\partial A\cap A^{(1)}$ satisfying (h1)-(h4).
Without loss of generality we assume that $x=0$ and we denote
$T_{0}=T_{0,\nu_{A}(0)}$. Furthermore, by using (h2) we choose in (h3) the
rectangle $R_{\nu_{A}}:=Q_{1}$ if $\nu_{A}(0)=\textbf{e}_{i}$ for
$i=\textbf{1},\textbf{2}$, or
$R_{\nu_{A}}:=(-\cos\theta_{\nu_{A}},\cos\theta_{\nu_{A}})\times_{\mathbb{R}^{2}}(-\sin\theta_{\nu_{A}},\sin\theta_{\nu_{A}})$,
where $\theta_{\nu_{A}}$ is the angle formed between the tangent line $T_{0}$
and the $x_{1}$-axis, otherwise. For any $\rho>0$, we write $R_{\rho}:=\rho
R_{\nu_{A}}$.

In view of the definition of $R_{\nu_{A}}$ and again by using also the
Besicovitch Derivation Theorem (see [28, Theorem 1.153]) there exists a
subsequence $\rho_{n}\searrow 0$ such that

$\mu_{0}(\partial
R_{\rho_{n}})=0,\quad\lim_{k\rightarrow+\infty}{\mu_{k}(\overline{R_{\rho_{n}}})}=\mu_{0}(R_{\rho_{n}})$
(5.166)

and

$\frac{d\mu_{0}}{d\mathcal{H}^{1}\mathbin{\vrule
height=6.88889pt,depth=0.0pt,width=0.55974pt\vrule
height=0.55974pt,depth=0.0pt,width=5.59721pt}(\partial^{*}{S_{h,K}}\cap\partial
A\cap
A^{(1)})}(0)=\lim_{n\rightarrow\infty}{\frac{\mu_{0}(R_{\rho_{n}})}{2\rho_{n}}}.$
(5.167)

We now claim that

${\rm{sdist}}(\cdot,\partial\sigma_{\rho_{n}}(A))\rightarrow-{\rm
dist}(\cdot,T_{0})\quad\text{and}\quad{\rm{sdist}}(\cdot,\partial\sigma_{\rho_{n}}({S_{h,K}}))\rightarrow{\rm{sdist}}(\cdot,\partial
H_{0})$ (5.168)

uniformly in $\overline{R_{\nu_{A}}}$ as $n\to\infty$. To prove the claim we
can for example observe that by [36, Proposition A.4] we have (not only (h3),
but also) that
$\overline{Q_{r}}\cap\partial{\sigma}_{\rho,x}(A)\xrightarrow{\mathcal{K}}\overline{Q_{r}}\cap
T_{x}\quad\text{and}\quad\overline{Q_{r}}\cap\partial{\sigma}_{\rho,x}({S_{h,K}})\xrightarrow{\mathcal{K}}\overline{Q_{r}}\cap
T_{x}\quad\text{as }\rho\to 0$ for any square $Q_{r}$ such that
$R_{\nu_{A}}\subset Q_{r}$ and hence, by Proposition 5.3 Items (a) and (c),
applied to $Q_{r}$,
${\rm{sdist}}(\cdot,\partial\sigma_{\rho_{n}}(A))\rightarrow-{\rm
dist}(\cdot,T_{0})\quad\text{and}\quad{\rm{sdist}}(\cdot,\partial\sigma_{\rho_{n}}({S_{h,K}}))\rightarrow{\rm{sdist}}(\cdot,\partial
H_{0})$ uniformly in $\overline{Q_{r}}\supset R_{\nu_{A}}$.

Furthermore, from the $\tau_{\mathcal{B}}$-convergence it follows that

${\rm{sdist}}(\cdot,\partial A_{k})\rightarrow{\rm{sdist}}(\cdot,\partial
A)\quad\text{and}\quad{\rm{sdist}}(\cdot,\partial{S_{k}})\rightarrow{\rm{sdist}}(\cdot,\partial{S_{h,K}})$
(5.169)

uniformly in $\overline{R_{\nu_{A}}}$ as $k\rightarrow\infty$ and hence, by
(5.168) and (5.169), a standard diagonalization argument yields that there
exists a subsequence $\left\\{(A_{k_{n}},h_{k_{n}},K_{k_{n}})\right\\}$ such
that

${\rm{sdist}}(\cdot,\partial\sigma_{\rho_{n}}(A_{k_{n}}))\rightarrow-{\rm
dist}(\cdot,T_{0})\quad\text{and}\quad{\rm{sdist}}(\cdot,\partial\sigma_{\rho_{n}}({S_{{k_{n}}}}))\rightarrow{\rm{sdist}}(\cdot,\partial
H_{0})$ (5.170)

uniformly in $\overline{R_{\nu_{A}}}$ as $n\to\infty$ and by (5.166) such that

$\mu_{k_{n}}(\overline{R_{\rho_{n}}})\leq\mu_{0}(R_{\rho_{n}})+\rho_{n}^{2},$
(5.171)

for any $n\in\mathbb{N}$. By (5.166), (5.171) and arguing as in (5.140) we
infer that

$\begin{split}\limsup_{n\to\infty}{\frac{\mathcal{H}^{1}(\overline{R_{\rho_{n}}}\cap\partial
A_{k_{n}})+\mathcal{H}^{1}(\overline{R_{\rho_{n}}}\cap\partial
S_{k_{n}}\setminus\partial A_{k_{n}})}{2\rho_{n}}}\leq
c_{1}^{-1}\frac{d\mu_{0}}{d\mathcal{H}^{1}\mathbin{\vrule
height=6.88889pt,depth=0.0pt,width=0.55974pt\vrule
height=0.55974pt,depth=0.0pt,width=5.59721pt}(\partial^{*}{S_{h,K}}\cap\partial
A\cap A^{(1)})}(0).\end{split}$ (5.172)

By applying Lemma 5.2, we have that
$\overline{R_{\nu_{A}}}\setminus\sigma_{\rho_{n}}(A_{k_{n}})\xrightarrow{\mathcal{K}}T_{0}$
and in view of Remark 3.8-(i) we have that
$\overline{R_{\nu_{A}}}\cap\partial\sigma_{\rho_{n}}(S_{k_{n}})\xrightarrow{\mathcal{K}}T_{0}$.
Since
$\left\\{(A_{k_{n}},S_{h_{k_{n}},K_{k_{n}}})\right\\}\subset\mathcal{B}_{\mathbf{m}}$,
we know that
$(\sigma_{\rho_{n}}({A_{k_{n}}}),S_{(1/{\rho_{n}})h_{k_{n}}(\rho_{n}\cdot),\sigma_{\rho_{n}}({K_{k_{n}}})})\in\mathcal{B}_{\textbf{m}}(\sigma_{\rho_{n}}(\Omega))$.
By applying Lemma 5.8 with $\phi_{\alpha}(\cdot)=\varphi_{\alpha}(0,\cdot)$
for $\alpha=\text{F,S,FS}$ and $\delta=\varepsilon$, there exists
$n^{1}_{\varepsilon}\in\mathbb{N}$ such that for every $n\geq
n^{1}_{\varepsilon}$,

$\begin{split}\mathcal{S}_{L}(\sigma_{\rho_{n}}({A_{k_{n}}}),S_{\sigma_{\rho_{n}}({S_{k_{n}}}),\sigma_{\rho_{n}}({K_{k_{n}}})},R_{\nu_{A}})&\geq\int_{\overline{R_{\nu_{A}}}\cap
T_{0}}{\varphi_{\mathrm{F}}(0,\nu_{T_{0}}))+\varphi(0,\nu_{T_{0}})\,d\mathcal{H}^{1}}-\varepsilon\\\
&=2(\varphi_{\mathrm{F}}(0,\nu_{T_{0}})+\varphi(0,\nu_{T_{0}}))-\varepsilon.\end{split}$
(5.173)

By definition of $\mu_{k}$, the non-negativeness of
$\varphi_{\mathrm{F}},\varphi$ and $\varphi_{\mathrm{FS}}$, (5.127), (5.173)
and arguing as in (5.143) we deduce that

$\mu_{k_{n}}(\overline{Q_{\rho_{n}}})\geq
2(\varphi_{\mathrm{F}}(0,\nu_{T_{0}})+\varphi(0,\nu_{T_{0}}))-\varepsilon\rho_{n}-\varepsilon\left(\mathcal{H}^{1}(\overline{Q_{\rho_{n}}}\cap\partial
A_{k_{n}})+\mathcal{H}^{1}(\overline{Q_{\rho_{n}}}\cap\partial
S_{k_{n}}\setminus\partial A_{k_{n}})\right),$ (5.174)

for every $n>n^{\prime}_{\varepsilon}$. By (5.167), (5.171), (5.174) and by
arguing as in (5.144) we have that

$\begin{split}\frac{d\mu_{0}}{\,d\mathcal{H}^{1}\mathbin{\vrule
height=6.88889pt,depth=0.0pt,width=0.55974pt\vrule
height=0.55974pt,depth=0.0pt,width=5.59721pt}(\partial A\cap\partial^{*}S\cap
A^{(1)})}(0)&\geq\varphi_{\mathrm{F}}(0,\nu_{T_{0}})+\varphi(0,\nu_{T_{0}})-\frac{\varepsilon}{2}\\\
&\qquad\qquad-\varepsilon
c_{1}^{-1}\frac{d\mu_{0}}{\,d\mathcal{H}^{1}\mathbin{\vrule
height=6.88889pt,depth=0.0pt,width=0.55974pt\vrule
height=0.55974pt,depth=0.0pt,width=5.59721pt}((\partial{S_{h,K}}\cap\partial
A)\cap S^{(1)}_{h,K})}(0).\end{split}$

Finally, by (h4) and by taking $\varepsilon\to 0^{+}$ in the inequality above,
we deduce (5.123). ∎

###### Proof of (5.126).

We repeat the arguments of (5.120) by using Proposition 5.3-(a) for the sets
$A$ and $A_{k}$, and Proposition 5.3-(b) for the sets $S_{h,K}$ and $S_{k}$,
and by replacing Lemma 5.7 with Lemma 5.12. ∎

∎

We are finally in position to prove Theorem 3.12.

###### Proof of Theorem 3.12.

Without loss of generality, we assume that the _liminf_ in the right side of
(3.22) is reached and finite in $\mathbb{R}$. From Theorem 5.13 it follows
that

$\mathcal{S}(A,S_{h,K})\leq\liminf_{k\rightarrow\infty}{\mathcal{S}(A_{k},S_{h_{k},K_{k}})}.$
(5.175)

In view of the definition of $\mathcal{F}$, in order to reach the assertion,
it suffices to establish the lower semicontinuity of $\mathcal{W}$, to which
the rest of the proof is devoted.

Let $D\subset\subset\textrm{Int}(A)$, by the fact that
${\rm{Int}}{(A_{k})}\xrightarrow{\mathcal{K}}{\rm{Int}}{(A)}$, we deduce that
$D\subset\subset{\rm{Int}}(A_{k})$ for $k$ large enough. As $u_{k}\rightarrow
u$ a.e. in ${\rm{Int}}(A)$, then, $u_{k}\rightarrow u$ a.e. in $D$.
Furthermore, since $e(u_{k})$ are bounded in the $L^{2}(D)$ norm, we have that
$e(u_{k})\rightharpoonup e(u)$ in $L^{2}(D)$. By convexity of
$\mathcal{W}(D,\cdot)$ we obtain that

$\displaystyle\mathcal{W}(D,u)\leq\liminf_{k\rightarrow+\infty}{\mathcal{W}(D,u_{k})}\leq\liminf_{k\rightarrow+\infty}{\mathcal{W}(A_{k},u_{k})}.$

To conclude it is now enough to let $D\nearrow{\rm{Int}}(A)$. ∎

##  6\. Existence

In view of Theorems 3.11 and 3.12 we are in position to prove Theorem 3.10 by
employing the _direct method of the calculus of variations_.

###### Proof of Theorem 3.10.

Fix $m\in\mathbb{N}$ and let
$\\{(A_{k},S_{h_{k},K_{k}},u_{k})\\}\subset\mathcal{C}_{\mathbf{m}}$ be a
minimizing sequence of $\mathcal{F}$ such that
$\mathcal{L}^{2}({A_{k}})=\mathbbm{v}_{1}$,
$\mathcal{L}^{2}({S_{h_{k},K_{k}}})=\mathbbm{v}_{0}$, and

$\sup_{k\in\mathbb{N}}{\mathcal{F}(A_{k},S_{h_{k},K_{k}},u_{k})}<\infty.$

By Theorem 3.11 there exist a subsequence
$\\{(A_{k_{l}},S_{h_{k_{l}},K_{k_{l}}},u_{k_{l}})\\}$, a sequence
$\\{(\widetilde{A}_{l},S_{h_{k_{l}},\widetilde{K}_{k_{l}}},v_{l})\\}\subset\mathcal{C}_{\mathbf{m}}$,
and $(A,S_{h,K},u)\in\mathcal{C}_{\mathbf{m}}$ such that
$(\widetilde{A}_{l},S_{h_{k_{l}},\widetilde{K}_{k_{l}}},v_{l})\xrightarrow{\tau_{\mathcal{C}}}(A,S_{h,K},u)$
as $l\to\infty$ and

$\liminf_{l\to\infty}{\mathcal{F}(\widetilde{A}_{l},S_{h_{k_{l}},\widetilde{K}_{k_{l}}},v_{l})}=\liminf_{l\to\infty}{\mathcal{F}(A_{k_{l}},S_{h_{k_{l}},K_{k_{l}}},u_{k_{l}})}.$
(6.1)

By Theorem 3.12, we have that

$\mathcal{F}(A,S_{h,K},u)\leq\liminf_{l\to\infty}{\mathcal{F}(\widetilde{A}_{l},S_{h_{k_{l}},\widetilde{K}_{k_{l}}},v_{l})}.$
(6.2)

We claim that
$\\{(\widetilde{A}_{l},S_{h_{k_{l}},\widetilde{K}_{k_{l}}},v_{l})\\}$ and
$(A,S_{h,K},u)$ satisfy the volume constraints of (3.18). Indeed, by Theorem
3.11, for any $l\geq 1$,
$\mathbbm{v}_{1}=\mathcal{L}^{2}\left({A_{k_{l}}}\right)=\mathcal{L}^{2}({\widetilde{A}_{l}})$
and
$\mathbbm{v}_{0}=\mathcal{L}^{2}({S_{h_{k_{l}},{K}_{k_{l}}}})=\mathcal{L}^{2}\left({S_{h_{l},\widetilde{K}_{l}}}\right)$,
where ${S_{h_{l},\widetilde{K}_{l}}}\in{\rm AS}(\Omega)$. Thanks to the fact
that
$(\widetilde{A}_{l},S_{h_{k_{l}},K_{k_{l}}})\xrightarrow{\tau_{\mathcal{\mathcal{B}}}}(A,S_{h,K})$
as $l\to\infty$, by applying [36, Lemma 3.2] we infer that
$\widetilde{A}_{l}\to A$ in $L^{1}(\mathbb{R}^{2})$ as $l\to\infty$, and thus
$\mathcal{L}^{2}({A})=\mathbbm{v}_{1}$, and, similarly, we deduce that
$\mathcal{L}^{2}({S_{h,K}})=\mathbbm{v}_{0}$. Finally, from (6.1) and (6.2) we
deduce that

$\begin{split}\inf_{(A,S_{h,K},u)\in\mathcal{C}_{\mathbf{m}},\,\mathcal{L}^{2}({A})=\mathbbm{v}_{1},\,\mathcal{L}^{2}({S_{h,K}})=\mathbbm{v}_{0}}\mathcal{F}(A,S_{h,K},u)&=\lim_{k\to\infty}{\mathcal{F}(\widetilde{A}_{l},S_{h_{k_{l}},\widetilde{K}_{k_{l}}},v_{l})}\\\
&\geq\liminf_{l\to\infty}{\mathcal{F}\left(\widetilde{A}_{l},S_{h_{k_{l}},\widetilde{K}_{k_{l}}},v_{l}\right)}\\\
&\geq\mathcal{F}(A,S_{h,K},u)\end{split}$

and hence, $(A,S_{h,K},u)$ is a solution of the minimum problem (3.18). By
observing that (6.1) and (6.2) hold true also by replacing $\mathcal{F}$ with
$\mathcal{F}^{\lambda}$ for ${\lambda}:=(\lambda_{0},\lambda_{1}),$ with
$\lambda_{0},\lambda_{1}>0$, we deduce that also the unconstrained minimum
problem (3.19) can be solved by employing the same method. This concludes the
proof. ∎

## Acknowledgement

The authors thank Shokhrukh Kholmatov for the various discussions on the
topic. The authors acknowledge the support received from the Austrian Science
Fund (FWF) projects P 29681 and TAI 293, from the Vienna Science and
Technology Fund (WWTF) together with the City of Vienna and Berndorf
Privatstiftung through Project MA16-005, and from BMBWF through the OeAD-WTZ
project HR 08/2020. R. Llerena thanks for the support obtained by a French
public grant as part of the “Investissement d’avenir” project
ANR-11-LABX-0056-LMH, LabEx LMH. P. Piovano is member of the Italian “Gruppo
Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni”
(GNAMPA) and has received funding from the GNAMPA-INdAM 2022 project CUP:
E55F22000270001 and 2023 Project CUP: E53C22001930001. P. Piovano also
acknowledges the support obtained by the Italian Ministry of University and
Research (MUR) through the PRIN Project “Partial differential equations and
related geometric-functional inequalities”. Finally, P. Piovano is also
grateful for the support received as _Visiting Professor and Excellence Chair_
and from the _Visitor in the Theoretical Visiting Sciences Program_ (TSVP) at
the Okinawa Institute of Science and Technology (OIST), Japan.

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# Machine Learning for Economics Research:
When What and How?††thanks: The opinions here are solely those of the authors
and do not necessarily reflect those of the Bank of Canada. We thank Jacob
Sharples for his assistance on the project. We also thank Andreas Joseph, Dave
Campbell, Jonathan Chiu, Narayan Bulusu, and Stenio Fernandes for their
suggestions and comments on the article.

Ajit Desai
Bank of Canada

###### Abstract

This article provides a curated review of selected papers published in
prominent economics journals that use machine learning (ML) tools for research
and policy analysis. The review focuses on three key questions: (1) when ML is
used in economics, (2) what ML models are commonly preferred, and (3) how they
are used for economic applications. The review highlights that ML is
particularly used to process nontraditional and unstructured data, capture
strong nonlinearity, and improve prediction accuracy. Deep learning models are
suitable for nontraditional data, whereas ensemble learning models are
preferred for traditional datasets. While traditional econometric models may
suffice for analyzing low-complexity data, the increasing complexity of
economic data due to rapid digitalization and the growing literature suggests
that ML is becoming an essential addition to the econometrician’s toolbox.

_K_ eywords Economics $\cdot$ Econometrics $\cdot$ Machine learning

JEL Codes: A10, B23, C45, C55

##  1 Introduction

The economy is becoming increasingly digital, and as a result, the size and
complexity of economic data are growing rapidly. This presents both
opportunities and challenges for analysts who want to process and interpret
this data to gain insights into economic phenomena. Machine learning (ML)
which has emerged as a powerful tool for analyzing large and complex datasets
across disciplines, has the potential to mitigate some of the challenges posed
by the digitization of the economy for economics research and analysis. The
ability of ML models to effectively process large volumes of diverse data
could allow us to build more complex models. As a result, the use of ML has
expanded in economics research.

Figure 1: The number of publications over five years (between 2018-2022) in
the leading economics journals that use ML. The data includes articles from
the following ten journals: American Economic Review (AER), Econometrica,
Journal of Economic Perspectives (JEP), Journal of Monetary Economics (JME),
Journal of Political Economy (JPE), Journal of Econometrics (JoE), Quarterly
Journal of Economics (QJE), Review of Economic Studies (RES), American
Economic Journal (AJE): Macroeconomics and Microeconomics. The relevant papers
are identified using the following search terms: Machine learning, Ensemble
learning, Deep learning, Statistical learning, Reinforcement learning, and
Natural language processing.

The number of academic publications that use ML tools has increased
significantly in the last few years, as depicted in Figure 1, which shows the
number of articles published in ten leading economics journals that use ML.
This trend is expected to continue as researchers explore new ways to apply ML
techniques to a wide range of economic problems. Nevertheless, the suitability
and applicability of these tools is not widely understood among economists and
data scientists. To bridge this gap, in this article, we provide a curated
review of selected papers published in prominent economics journals that
employ ML tools. The objective of this review is to assist economists
interested in leveraging ML tools for their research and analysis, as well as
data scientists seeking to apply their skills to economic applications.111The
article is motivated from the American Economic Association’s (AEA) continuing
education session on Machine Learning and Big Data at the 2023 ASSA annual
meeting [1].

The article aims to showcase the potential benefits of utilizing ML in
economics research and policy analysis, while also offering suggestions on
where, what and how to effectively apply ML models. It should be noted that
the article takes a suggestive approach, rather than an explanatory one, with
a particular focus on supervised learning methods commonly used for prediction
problems such as regression and classification. The article is organized into
three main sections, each focusing on one key question: 1) When ML is used in
economics? 2) What ML models are commonly preferred?, and 3) How ML is used
for economic applications? Finally, we briefly discuss the limitations of
machine learning in its current state.

The key lessons of the review are summarized as follows: First, ML models are
used to process nontraditional and unstructured data such as text and images,
to capture strong nonlinearity that is difficult to capture using traditional
econometric models, and to improve prediction accuracy, extract new
information, or automate feature extraction when dealing with large but
traditional datasets. ML tools are probably not useful for cases with small
data complexity, and the traditional econometric models will likely to
suffice.

Second, the choice of ML model depends on the type of application and
underlying data characteristics. For example, when conducting textual
analysis, the latent Dirichlet allocation (LDA), which is an ML algorithm for
probabilistic topic modelling, is commonly preferred. However, deep learning
models such as Transformers are also employed for handling large text or audio
data. Convolutional neural network models such as ConvNext are preferred for
image or video datasets. Ensemble learning models are commonly employed for
traditional datasets, and causal ML models are utilized when analysis focuses
on causal inference.

Third, the usefulness of ML models can be greatly improved by tailoring them
to the specific application, especially when dealing with large and complex
but traditional data. This approach is more effective than using off-the-shelf
tools. Moreover, pre-trained models with transfer learning can be advantageous
when dealing with nontraditional but limited data and deep learning.222See
Appendix A for more details on LDA model, Appendix B for Transformer model,
Appendix C for ConvNext model, Appendix D for ensemble learning models, and
Appendix E for transfer learning.

This review highlights the potential benefits of using new tools provided by
ML in various areas of economics research, but also acknowledges the
challenges that needed to overcome to effectively use it for economic analysis
and research. For instance, ML models require large amounts of data and ample
computational resources, which can be a limitation for some researchers
because it can be difficult to obtain high-quality data, and data may be
incomplete or biased. Additionally, ML models are prone to overfitting and can
be challenging to interpret, which limits their utility. Moreover, most ML
models do not have standard errors, and other statistical properties have not
yet been well-defined, which can make it difficult to draw conclusions from
the results. Therefore, caution is recommended when using ML models. Despite
these limitations, our review suggests, ML is successfully employed alongside
traditional ecnometrics tools to advance our understanding of economic
systems. By combining the strengths of both fields, researchers can improve
the accuracy and reliability of economic analyses to better inform policy
decisions.

##  2 When ML is used in economics?

The literature suggest that the following three cases where the ML models
could add value to economic research and analysis.

  * •

To process non-traditional data, such as images, texts, audios, and videos.

  * •

To capture nonlinearity which is difficult to capture using traditional
models.

  * •

To process traditional data at scale to improve prediction accuracy, extract
new information, or automate feature extraction.

For instance, article [2] suggest that the big data, due to its sheer size and
complexity, may require more powerful manipulation tools, which ML can offer.
Also, we may have more potential predictors (or features) than appropriate for
estimation in some cases, so we need to make some variable selections, where
ML can help. Lastly, large datasets allow for more flexible relationships than
simple linear models can capture. ML techniques are handy in those cases due
to their ability to model intricate and nonlinear relationships potentially
offering new insights.

Similarly, article [3] argue that ML not only provides new tools but also
solves a different problem. They assert ML’s success is largely due to its
ability to discover the complex structure that was not specified in advance.
They suggest applying ML to economics requires finding relevant tasks, for
instance, where the focus is on increasing prediction accuracy or uncovering
generalizable patterns from complex datasets. Also, article [4] point that the
methods developed in the ML have been particularly successful in big data
settings, where we observe information on a large number of units, many pieces
of information on each unit, or both. The authors suggest that for using ML
tools for economics research and analysis, researchers should clearly
articulate their goals and why certain properties of ML algorithms may or may
not be important.

###  2.1 ML is used for processing non-traditional data.

Non-traditional datasets, such as images, text, and audio, can be difficult to
process using traditional econometric models. In such cases, ML models can be
used to extract valuable information that can be incorporated into traditional
models to address economic questions.

For instance, article [5] published in QJE, the authors assess how
transparency, a key feature of central bank design, affects monetary policy
makers’ deliberations, using an ML algorithm for probabilistic topic
modelling. Similarly, article [6] published at JME, they use a large news
corpus and ML algorithms to investigate the role played by the media in the
expectations formation process of households, and in [7] published in JoE,
uses the twitter data and ML model to measure inflation expectation. Likewise,
artciel [8] published in AER, the authors use satellite data to measure GDP
growth at the sub and supranational regions, in [9] also published in AER, the
authors employed a computer vision algorithm that measures the perceived
safety of streetscapes, and how strongly it is correlated with population
density and household income, and in [10] published in AER, the authors use
deep learning to detect emotions embedded in press conferences after the
Federal Open Market Committee meeting and examine the influence of the
detected emotions on financial markets.

###  2.2 ML is used for capturing strong nonlinearity

The ML could be useful if the data and application contain strong
nonlinearity, which is hard to capture using traditional approaches. For
instance, in the article [11], published in QJE, the authors evaluate whether
ML models can help improve judges’ decisions on bail or no bail. Although the
outcome is binary, this is a highly complex problem that demands processing
complex data to make prudent decisions.

Similarly, in the article [12], published in JME, the authors use ML to solve
dynamic economic models by casting them into nonlinear regression equations.
Here ML is used to deal with multicollinearity and to perform the model
reduction. Likewise, article [13], published in QJE, uses ML to test how
effective physicians are at diagnosing heart attacks. Using a large and
complex dataset available at the time of the physician’s decision, they
estimate the model to predict the outcome of testing to uncover potential
sources of error in decisions.

###  2.3 ML is used for processing traditional data at scale to improve
prediction accuracy or extract new information

ML could be useful for processing large and complex but traditional data sets
with many variables. In such cases, the ML models can help to 1. improve
prediction accuracy, 2. extract new information or 3. automate feature
extraction. For instance, in the article [14], published in AER, the authors
combine a data-rich environment with an ML model to provide new estimates of
time-varying expectational errors embedded in survey responses and show that
the ML can be productively deployed to correct errors in human judgment and
improve predictive accuracy.

Similarly, in the article [15], published in JPE, the authors measure CEO
behaviour types using high-frequency, high-dimensional diary data and an ML
algorithm. In [16], published in JoE, the author uses ML for fraud detection
in insurance claims using unstructured data comprising inputs of varying
lengths and variables with many categories. They argue that ML alleviates
these challenges that are otherwise hard for traditional methods. Likewise, in
[17] published in RES, the authors suggest that the accuracy of ML-based data
driven decision rule for consumer lending could be more accurate than
examiner-based decisions.

ML is probably not useful for the cases where data complexity (which could be
related to shape, size, collinearity, nonlinearity, etc.) is small, and
traditional econometric models would likely suffice. However, if the data
complexity increases, i.e., when dealing with big data, the value added by ML
models could be higher after a certain threshold, as shown by the dotted line
in Figure 2.

Figure 2: Schematic diagram representing the relative merits of ML and
traditional econometric methods. The plot is adapted from [1, 18].

##  3 What ML models are commonly preferred?

With a variety of ML tools available, different models are better suited for
different types of applications and data characteristics. This section will
discuss which models are most effective for a given type of economic
application.

###  3.1 Deep learning models are used when dealing with nontraditional data

Natural language processing (or NLP) primarily relies upon processing textual
data and has many applications in economics. For instance, it could be used
for topic modelling or sentiment analysis. The model of choice for the topic
modelling to quantify text is LDA proposed in [19]. It is an ML algorithm for
probabilistic topic modelling that decomposes documents in terms of the
fraction of time spent covering a variety of topics [5, 6].

An review of the use of text as data and ML methods for various economics
applications is presented in [20]. The use of deep learning models for NLP is
evolving rapidly, and various large language models (LLMs) could be used to
process text data; However, transformer models [21] are proven to be more
useful to efficiently extract useful information from the textual data [1].
For instance, in [10], they use transformer to detect emotions embedded in
press conferences after the Federal Open Market Committee meeting for
sentiment analysis. Moreover, almost all large general-purpose LLMs, including
GPT-3 and chatGPT, are trained using Generative Pre-trained Transformer [22].

An interesting application of computer vision models in economics is using a
broad set of satellite images or remote sensing data for analysis. For
instance, in the article [8], the authors use satellite data to measure GDP
growth at the sub and supranational regions. Similarly, in [23], using
satellite data and machine learning, they develop high-resolution predictors
of household income, wealth, and poverty rates in five African countries.

A review of the literature, presenting opportunities and challenges in using
satellite data and ML in economics, is documented in [24]. It concludes that
such models have the potential to perform economic analysis at spatial and
temporal frequencies that are an order of magnitude higher than those that are
commonly available. Various deep learning models can be employed to extract
useful information from images, but the ConvNext model [25] is evolving to be
more successful in efficiently processing image data sets [1]. However, the
transformers can also be effectively employed for image processing [26].

###  3.2 Ensemble learning models are used when dealing with traditional data

Ensemble learning models could be useful if the data size is small but
includes many features and if there is collinearity or nonlinearity, which is
hard to model. For instance, the article [3] compares the performance of
different ML models in predicting house prices and demonstrates that the
nonparametric ML algorithms, such as random forests, can do significantly
better than ordinary least squares, even at moderate sample sizes and with a
limited number of covariates.

Similarly, article [13] use ensemble learning models which combine gradient-
boosted trees and LASSO to study physicians’ decision-making to uncover the
potential sources of errors. Also, in [27], they propose generalized random
forests, a method for nonparametric statistical estimation based on random
forests. This can be used for three statistical tasks: nonparametric quantile
regression, conditional average partial effect estimation and heterogeneous
treatment effect estimation via instrumental variables. Moreover, the ensemble
learning models are popular in macroeconomic prediction, as demonstrated by
many articles [28, 29, 30, 31, 32]).

###  3.3 Causal ML models are used when the focus is on causal inference

The causal ML could be helpful when the primary objective is to make causal
inferences, but the dataset is big and complex. For instance, in [33], the
authors develop a nonparametric causal forest for estimating heterogeneous
treatment effects that extends the random forest, an ML algorithm. They
demonstrate that any type of random forest, including classification and
regression forests, can be used to provide valid statistical inferences. In
experimenting with these models, they find causal forests to be substantially
more powerful than classical methods—especially in the presence of irrelevant
covariates.

Similarly, the article [34] use ML for estimating heterogeneity in causal
effects in experimental and observational studies. Their approach is tailored
for applications where there may be many attributes of a unit relative to the
number of units observed and where the functional form of the relationship
between treatment effects and the attributes is unknown. It enables the
construction of valid confidence intervals for treatment effects, even with
many covariates relative to the sample size, and without sparsity assumptions.
The applicability of these methods is demonstrated in [35] for predicting
treatment heterogeneity for summer jobs application.

##  4 How ML is used for economic applications

Pre-trained models and transfer learning are recommended, especially while
using deep-learning models [1]. Deep Learning based approaches are state-of-
the-art for processing non-traditional data; However, there are many
difficulties in using them for economic applications. For instance, large data
and ample computational resources are often necessary to train these models.
Both of which are scarce resources in many economic applications. Moreover,
these models are notoriously convoluted, and many economics researchers would
benefit from using these methods but lack the technical background to
implement them from scratch ([36]). Therefore, transfer learning, i.e., large
models pre-trained on a similar application (for instance, large language or
computer vision models), could be adapted for similar economic applications.

Of-the-shelf ensemble learning models are useful when dealing with panel data
with strong collinearity or nonlinearity, but it is recommended to adapt these
models to suit the task. For instance, in [27], they adapt the popular random
forests algorithm, which then can be used for nonparametric quantile
regression or estimating average treatment effects. Similarly, in [16] propose
an explainable attention network for fraud detection in claims management by
adapting a standard neural network and demonstrate that the adapted model
performs better than off-the-shelf. Likewise, in [37], the author proposed
macroeconomic random forest, an algorithm adapting the canonical ML tool. They
show that the adapted model forecasts better than off-the-shelf ML algorithms
and traditional econometric approaches, and it can be interpreted.

There are many other examples where the model or the procedure to train the
model is adapted to improve the performance. For instance, [34] adopt the
standard cross-validation approach, which then enables the construction of
valid confidence intervals for treatment effects, even with many covariates
relative to the sample size. Similarly, a variation of cross-validation
approaches is proposed in [30] to improve the prediction performance of the
macroeconomic nowcasting model during economic crisis periods.

A few other practical recommendations discussed in the AEA’s course are: 1.
Bigger, both in terms of data and model size, is better when using deep
learning models for processing non-traditional data. 2. Due to its speed and
community support, the Python programming language is preferred over languages
for applied ML. 3. When training large ML models, it is suggested to use Unix-
based operating systems over windows systems.

##  5 Other emerging applications

Other types of ML approaches, such as unsupervised learning and reinforcement
learning, are yet to make a notable impact in economics research. However,
there are some initial applications of these approaches in the literature. For
instance, in [38] uses an unsupervised dimension reduction model called
autoencoder neural network for asset pricing models. Similarly, in [39]
autoencoder-based unsupervised model is used for anomaly detection in high-
value payments systems. Also in [40] using data on nearly 40 million Google
keyword auctions and unsupervised machine learning algorithms to cluster
keywords into thematic groups serving as relevant markets.

Reinforcement learning (RL) models could be employed to model complex
strategic decisions arising in many economic applications. For instance, in
[41], the authors use RL to estimate optimal decision rules of banks
interacting in high-value payment systems. Similarly, in [42], deep RL is used
to solve dynamic stochastic general equilibrium models for adaptive learning
at the interaction of monetary and fiscal policy, and [43], the authors use RL
to optimize monetary policy decisions. Likewise, in [44], the authors use a RL
approach to learn dynamic tax policies.

##  6 Limitations

The key limitations of ML for economics research and analysis are outlined
below:

  * •

Large data sets and ample computational resources are often necessary to train
ML models—especially the deep learning models.

  * •

The ML models, owing to their flexibility, are easy to overfit, and their
complexity makes them hard to interpret—which is crucial for many economic
applications.

  * •

Most ML models have no standard errors and asymptotic properties—which could
be essential for many economic applications.

  * •

ML models can be biased if the data used to train these models is of low
quality and biased.

The literature is evolving to address these challenges; however, some are hard
and could take longer to mitigate. For instance, we have limited data in many
economic applications, which restricts the applicability of large ML models.
This could be potentially mitigated in certain applications as the economy
becomes more digital, allowing us to gather more data at a much higher
frequency than traditional economics data sets.

The interpretability or explainability of models is another major challenge in
using ML in economics. The researchers are making progress toward overcoming
these challenges. For instance, one approach recently developed to mitigate
interpretability issues is by using the Shapley-value-based methodologies such
as developed in [45], and [46]. These methods are useful for macroeconomic
prediction models in [46, 30, 47, 32]. However, note that, although such
methods are based on game theory, they do not provide any optimal statistical
criterion, and asymptotics for such approaches are not available yet. To
overcome that, for instance, in the recent papers [48, 49], the authors
propose ML-based mixed data sampling and develop the asymptotics in the
context of linear regularized regressions. However, much progress needs to be
made to use such asymptotic analysis for popular nonlinear ML approaches.

##  7 Conclusions

This concise review highlights that ML is increasingly used for economics
research and policy analysis, particularly for analyzing non-traditional data,
capturing nonlinearity, and improving prediction accuracy. Importantly, ML can
complement traditional econometric tools by identifying complex relationships
and patterns in data that can be incorporated into econometric models. As the
digital economy and economic data continue to grow in complexity, ML remains a
valuable tool for economic analysis. However, a few limitations need to be
addressed to improve the utility of ML models, and the literature is
progressing toward mitigating those challenges.

Lastly, in Figure 3, we present the word clouds generated from the titles and
abstracts of the articles in our dataset. These word clouds illustrate the
frequency of certain terms, with larger font sizes indicating more frequent
usage. For example, as shown in the word clouds, the terms “machine" and
“learning" are prominently featured in both titles and abstracts, highlighting
its relevance in those articles. This is followed by words such as “data,"
“effect," and “decision".

(a) Word cloud for titles

(b) Word cloud for the abstracts

Figure 3: Our dataset of articles collected from prominent economics journals
using ML is visualized through word clouds of their titles and abstracts.

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##  Appendix A Latent Dirichlet Allocation

Latent Dirichlet Allocation is a probabilistic ML model commonly used for
topic modelling. It assumes that each text document in a corpus contains a
mixture of various topics which reside within a latent layer and that each
word in a document is associated with one of these topics. The model infers
those topics based on the distribution of words across the entire corpus. The
output of the model is a set of topic probabilities for each document and a
set of word probabilities for each topic. It has many practical applications,
including text classification, information retrieval, and social network data
analysis. Refer to [19] for more details on the LDA model and its formulation.

##  Appendix B Transformers

The Transformers are a deep learning model architecture commonly used in text
and image processing tasks. The key feature of transformers is self-attention
mechanism to process sequential input data, such as words in a sentence. This
self-attention allows the model to identify the most relevant parts of the
input sequence for each output. Refer to [21] for more details of the model.

Generally, the transformer architecture comprises an encoder and a decoder,
consisting of multiple self-attention layers and feed-forward neural networks.
The encoder processes the input sequence, such as a sentence in one language,
and produces a sequence of context vectors. The decoder uses these context
vectors to generate a sequence of outputs, such as translating into another
language. The key benefit of transformer models is their ability to handle
long-range dependencies in input sequences, making them particularly effective
for NLP tasks that require understanding the context of words or phrases
within a longer sentence or paragraph.

##  Appendix C ConvNext

The ConvNeXT is a convolutional neural network (CNN) model inspired by the
design of the type of transformers model architecture. It is a deep learning
model commonly used for processing image and video data for various tasks, for
instance, object detection, image classification, and facial recognition.
Refer to [25] for more details on the model and its formulation.

Generally, CNN models comprise multiple layers sandwiched between input and
output layers. It includes convolutional, pooling, and fully connected layers.
In the convolutional layers, the model performs a set of mathematical
operations called convolution on the input image to extract high-level
features such as edges, corners, and textures. The pooling layers are used to
downsample the convolutional layers’ output, reducing the feature maps’ size
for the subsequent layers. Finally, the fully connected layers are used to
classify the image based on the extracted features. One of the key benefits of
CNNs is their ability to learn spatial hierarchies of features, i.e., the
model can identify complex patterns and objects in images by learning to
recognize simpler features first and gradually build up to more complex ones.

##  Appendix D Ensemble Learning

Ensemble learning is a popular ML technique that combines multiple individual
models, called base models or weak learners, to improve the accuracy and
robustness of the overall prediction. Each base model is trained on randomly
sampled subsets of the training data, and their predictions are then combined
in some way to produce a final prediction. Ensemble learning using decision
trees (DTs) is popular among others. By combining multiple DTs trained on
different subsets of the data and using different parameters, ensemble
learning can capture a broader range of patterns and complex relationships in
the data to produce more accurate and robust predictions.

Bagging and boosting are two commonly used ensemble learning approaches. The
bagging involves creating multiple decision trees, each trained on a randomly
sampled subset of the training data. The final prediction is made by combining
the predictions of all the individual decision trees, such as by taking the
average or majority vote of the individual tree predictions. Random Forest
[50] is a popular example of this approach. On the other hand, boosting
involves training DTs sequentially, with each new tree attempting to correct
the errors of the previous tree by using the modified version of the original
training data. The final prediction is made by combining the predictions of
all the individual decision trees, with greater weight given to the
predictions of the more accurate trees. A popular example is gradient boosting
[51]. The advanced version of boosting methods such as XGBoost [52] and
LightGBM [53] are popular among others. Check out [54], a review article that
details ensemble learning, its types, applications and limitations.

##  Appendix E Transfer Learning

Transfer learning is an ML technique that involves leveraging knowledge gained
from training a model on one task to improve the performance of a model on a
different but related task. In transfer learning, a pre-trained model is used
as a starting point for a new task, hoping that the knowledge and patterns
learned by the model in the pre-training stage will be useful for the new
task. The pre-trained model can be trained on a similar or different dataset.

Transfer learning is a powerful technique particularly useful in deep learning
because it can help improve the accuracy of models and reduce the amount of
data required for training. By using a pre-trained model as a starting point
for a new task, the model can leverage the knowledge and patterns learned from
the previous task to improve its performance on the new task. This can lead to
faster convergence, better generalization, and improved accuracy, especially
in situations where the new task has limited data available for training.
Generally, it is effective in cases for large models with millions of
parameters and requires significant amounts of data for training. Transfer
learning has been successfully applied in a wide range of tasks related to
nontraditional data processing. Check out [23], a review article which
provides further details on transfer learning, its types, applications and
limitations.