Causality by Vote: Aggregating Evidence on
Causal Relations in Economic Growth
Processes

Manuel de Mier (UNS)
Fernando Delbianco (UNS/INMABB-CONICET)
Fernando Tohmé (UNS/INMABB-CONICET)
Luisina Patrizio (UNS)
Facundo Rodriguez (UNS)
Mauro Romero Stéfani (UNS)

DOCUMENTO DE TRABAJO N° 260

Julio de 2023

Los documentos de trabajo de la RedNIE se difunden con el propósito degenerar comentarios y debate, no habiendo estado sujetos a revisión de pares.
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Citar como:de Mier, Manuel, Fernando Delbianco, Fernando Tohmé, Luisina Patrizio,
Facundo Rodriguez y Mauro Romero Stéfani (2023). Causality by Vote:
Aggregating Evidence on Causal Relations in Economic Growth Processes.
Documento de trabajo RedNIE N°260.

Causality by Vote: Aggregating Evidence on Causal
Relations in Economic Growth Processes
Manuel de Mier1 , Fernando Delbianco1,2,3 , Fernando Tohmé1,2 , Luisina Patrizio1 ,Facundo Rodriguez1 , and Mauro Romero Stéfani1
1 Department of Economics, Universidad Nacional del Sur
2 INMABB-CONICET, Bahía Blanca, Argentina* Corresponding author: fernando.delbianco@uns.edu.ar, ORCID 0000-0002-1560-2587

July 2023

Abstract
In this paper we investigate the performance of five causality-detection methods and how
their results can be aggregated when multiple units are considered in a panel data setting. The
aggregation procedure employs voting rules for determining which causal paths are identified
for the sample population. Using simulated and real-world panel data, we show the performance of this methods in detecting the correct causal paths in comparison to a benchmarkthat comprises a standard representation of growth processes as ground truth model. We findthat the results may be better when only simulated, instead of real-world, data are analyzed.
While this may suggest that the methods presented here are are currently incapable of detecting causal links, it is plausible that the ground “truth” may incorporate false relations.
Keywords: Granger causality, Transfer Entropy, Stochastic Causality, LiNGAM, Ground Truth,
Economic Growth.
JEL Classification: C18, C43, O47.

1

Introduction

The notion of causal relation has been, until recently, only used in heuristic terms in empiricalstudies. That is, it has been applied solely to offer intuitions behind mathematical or statistical
relationships within models and datasets. However, in the last decades, there has been a shift as
researchers seek to provide precise and formal characterizations of the previously vague conception. The literature now presents a plethora of approaches to both the treatment of causality in1already available networks of relations among variables, as well as the detection of causal relations in data (Peters et al., 2017; Schölkopf et al., 2021). But this diversity comes at the price of alack of consensus among the different methods when no agreed-on ground truth model exists.
In the context of a panel data structure, when the same causal detection method is appliedon each individual time series, it could produce dissimilar results. This is particularly worrisome
when the phenomenon under study is assumed to be common to all the units in the panel, despite
the differences that might arise when each corresponding time series is analyzed in isolation. In
more precise terms, the question is, given variables X and Y, to detect whether a causal relation
between them exists for all the panel, by checking its validity for each individual i. But a problem
arises if this is true for some individuals and not for others.
In Econometrics, the standard concept of causality was introduced by Granger (1969), definedin terms of a time series X and another Y. Holtz-Eakin et al. (1988) presented the first version ofGranger causality for panel data. But the assertion of the existence of a causal relation between
X and Y was predicated under the assumption of the homogeneity of the relation, i.e., that the
specifics of the causal model was the same for all the individuals in the panel. Dumitrescu andHurlin (2012) lifted this assumption by allowing heterogeneity among the units and the causal
relations. The test statistic they developed is obtained by averaging the Wald statistics of theindividual Granger estimates. Juodis et al. (2021) notice that the asymptotic properties of thisestimator make it prone to the emergence of a correlation between the estimates and the errors
when the number of periods is small while the number of individuals is large. They address it
by carrying out the averaging of the Wald statistics based on bias-free estimators.
The key idea behind these extensions of Granger causality to panel data is that the information drawn from running the causality detection method on each individual is then aggregated.
In this paper we adopt this idea and extend it to other techniques of causality detection. The
difference resides in the method in which the individual estimations is aggregated to find aconsensus among them.
A possible way of achieving some degree of consensus among these results is by voting.
That is, by including a causal relation among variables only when it is detected in a majority(or supermajority) of time series. In this paper, we explore the aggregation of the outcomes of
causality detection procedures on economic growth data. The idea is to obtain a network of
causally related variables reflecting the main features of growth processes.

2

An important proviso is that we apply this only on pairs of variables. The reason is that, asshown by Papana et al. (2021), causality detection methods like those to be considered here are
hard to apply to multivariate time series. If the relations involve more than two variables, thesemethods are hampered by the existence of mediators, confounders and colliders (Glymour et al.,2016). Thus, it is not convenient to consider more than two variables at a time.
The voting approach to the aggregation of causal evidence is scalable. That is, once a method
agrees on some causal link between two variables, we can see this as casting a vote for it in anelection towards the construction of an ensemble model. In this way, the agreement obtainedin the specific methods can be extended to a consensus among them when a plurality of themethods endorse a link. The causal links in the ensemble model can be deemed as more robust
than those just agreed on by the specific methods.
Another advantage of this aggregation procedure can be seen in the light of the meta-analysis
conducted by Bruns and Stern (2019) that shows that, in the case of Granger causality, the lagsin the specification of the causal relations are obtained by overfitting the model. This indicatesthat the specification is usually cherry-picked. We address this issue by considering a numberof different lags in a relation. Then, the existence of a causal relationship obtains by a weighted
voting process where each unit casts votes for the different lags and the total number of its votes
it gets is weighed by L1 , where L is the number of lags under consideration. This gives us anumber for the relation, independently of the number of lags. Then, aggregating these numbers,
if their sum passes the criterion of acceptance under voting, the relation among the variables is
included in the causal graph.
One of the main reasons justifying the adoption of this methodology is the possibility ofobtaining empirical evidence upon which future theoretical models can be developed, inverting
the usual top-down approach to modeling in Economics. Nevertheless, this, as all bottom-up
approaches, has a downside: its sensitivity to the quality of the data. That is, if the datasets are
scarce or somehow imprecise, the relations detected may be erroneous while the correct ones
may be missed.
The main contribution of this paper resides in the introduction of voting as a method of
aggregation of causal information. This methodology facilitates assessing both the positive andnegative aspects of the different methods. We also discuss the convenience of using ensemble
methods obtained by aggregating the results of the individual procedures of causality detection.

3

The remainder of the paper is organized as follows. Section 2 describes the causality detection
techniques, the vote aggregation procedure, and the data on which they are applied. Section 3
presents the results over synthetic and real-world data. Finally, Section 4 concludes. Additional
information is contained in Appendix A.

2

Data and methodology

2.1

Data

The dataset used in this study is extracted from the Penn World Table (PWT) database (Feenstra
et al., 2015). Our sample covers 62 developed and developing countries for the period 1961–2019.
We work with 11 time series of annual frequency. The variables with their corresponding description are detailed in Table 1. The list of countries from which they were drawn can be foundin Appendix A.
Table 1: Variables used in this study (drawn from PWT)
Variable
GDP per capita growth rate (GROWTH)
Population growth rate (POP)
Human capital (HK)
Total factor productivity (TFP)
Real interest rate (IRR)
Depreciation (DELTA)
Exchange rate (XR)
Investment (INV)
Government spending (GOV)
Openness (OPEN)
Terms of trade (TOT)

Description
Expenditure-side real GDP at chained PPPs growth rate
Population (in millions) growth rate
Human capital index
TFP level at current PPPs (USA=1)
Real internal rate of return
Average depreciation rate of the capital stock
National currency/USD (market+estimated)
Gross capital formation as a percentage of GDP
Government consumption as a percentage of GDP
Total merchandise trade as a percentage of GDP
Price level of exports/Price level of imports (USA 2017=1)

The choice of this panel for our analysis is justified by the need to consider a phenomenon
like economic growth, which is usually conceived country-wise, with reduced across-borderinfluences. Accordingly, we can analyze the causal relations among variables for the individual
countries in the panel, which we later aggregate via voting, constituting an appropriate arena forthe evaluation of the performance of our proposal.1
1 The result of this exercise can be depicted as a causal graph As shown by Raunig (2023), this is particularly usefulfor the analysis of economic policies.

4

2.2

Causality detection techniques

In this paper we focus on two classes of methods of causality detection.2 One is that of methods
based on autoregressive models. The other is that of those that rely on structural causal learningtechniques. The former methods are specific for time series analyses where a causal relation
between variables X and Y is understood as indicating that past values of X provide uniqueinformation to predict or explain future values of variable Y. Conversely, any structural causal
learning method makes assumptions about the functional form, breaking the symmetry between
the variables, and enabling the identification of causal effects.
In this paper, we adopt two autoregressive-based methods, namely Granger Causality andTransfer Entropy. In turn, we apply three structural causal learning methods. One is Stochastic
Causality and the other two are based on the linear non-Gaussian acyclic model (LiNGAM):
ICA-LiNGAM and Direct-LiNGAM.
Granger causality is based on the premise that the causes precede the effects (Granger, 1969).
To claim that one variable causes another, past values of an independent variable in a regression
must have a significant effect on the dependent variable, given the past values of the latter. Thisestimation method not only allows us to detect causality but also its direction, which can be
unidirectional or bidirectional. It should be emphasized that Granger causality is only applicableto linear structures. Therefore, if there are nonlinearities in the data, the true causal relationships
between variables might not be accurately captured.
Since the Granger estimation requires that the time series are stationary, we proceed by differentiating the entire dataset if the Dickey-Fuller unit root test rejects the hypothesis of stationarityof the original series. In the case of simulated data, we run the Granger test with one lag foreach pair of variables, while for the real-world growth dataset, we obtain estimations for one tofour-time lags. We use for this task the R package lmtest. In the vote aggregation procedure, eachof the lagged specifications casts a vote if the p-value is less than 0.05. Each of these votes, fourin total, counts as 1/4 of a single vote.
Transfer Entropy is a non-parametric method based on Information Theory that uses theconcept of information transfer to detect causal relationships (Schreiber, 2000). The entropy of avariable measures its degree of uncertainty or randomness. Thus, Transfer Entropy from X to Y
2 We do not consider methods based on the independences among variables, like PC or PCMCI (Runge et al., 2019;

Spirtes and Glymour, 1991).

5

measures the reduction in uncertainty (entropy) about the values of Y gained from the knowledgeof the values of X, given the past values of Y. It is based on the same general principles ofGranger Causality, but Transfer Entropy does not require any assumptions about the underlying
structure of the data (Barnett et al., 2009). Therefore, we use Transfer Entropy to capture possible
nonlinear causal relations that could not be detected with Granger Causality. Specifically, weapply the Shannon version of Transfer Entropy, using the R package RTransferEntropy. In theaggregation stage, a causal relation gets one vote if the p-value of the estimation is less than 0.05.
The Stochastic Causality approach is based on the information obtained by running a kernel
regression of Y against some variables, including X to obtain a model Y = f ( X ) and a kernel
regression to get X = g(Y ). The results are evaluated according to three empirical criteria andare summarized in a unanimity index that determines whether a causal path exists among X
and Y (Vinod, 2017, 2019). The first criterion is based on the application of the Wu-Hausman
endogeneity test to choose between Y = f ( X ) and X = g( X ). The second criterion selects
the model with the smallest absolute value of the residuals. Finally, the third criterion chooses
the model with the higher prediction accuracy, using the general measure of correlation forasymmetry of Zheng et al. (2012). Then, a unanimity index, ranging from -100 to 100, measuresthe degree of consensus among the three criteria regarding the direction of causality. The index
equals 100 if all the criteria agree on the same causal direction. We adopt here a conservative
threshold of 90 in absolute value and a significant p-value to indicate the presence of a causal
relationship. A shortcoming of this method is that it does not detect bidirectional causal paths.
Therefore, the directed acyclic graph (DAG) representing the outcome of the method tends tohave less directed edges than those generated by other techniques. In our analyses, we use the R
package generalCorr to produce the estimations in this approach.
LiNGAM (Linear Non-Gaussian Model) is a statistical model that assumes a linear relationship between variables with a non-normally distributed error term. In this setting, the asymmetry
between the residuals in the regressions of X on Y and Y on X allows to distinguish cause from
effect. The reason is that the regression residuals are independent of the predictor only for thecorrect causal direction. This approach was first introduced by Shimizu et al. (2006) with the
ICA-LiNGAM algorithm, which uses independent component analysis (ICA) for the estimationof the model. Afterward, Shimizu et al. (2011) developed the Direct-LiNGAM algorithm, which
improves upon ICA-LiNGAM by ensuring the convergence of the procedure of inference in a
specific number of steps and with a known level of complexity. The R package rlingam was used6

to implement both versions of the method.

2.3

Vote aggregation

We run each method at the country level, disregarding the panel structure of the dataset. This is,on one hand, to detect patterns that correspond to single economies. But more importantly, withthe exception of Granger causality, the remaining causality detection techniques to be applied
are not defined for panel data and thus force us to run our analysis country-wise. Consequently,
we obtain a large class of results for each method that we seek to aggregate into a unified causal
structure valid for a relevant subset of the sample of countries.
A useful way to do this is through a voting system, which involves considering each country
as a voter who casts a vote on the presence or absence of a causal relationship between a pair ofvariables. Then, the existence of a causal path between these variables is determined according
to a voting rule that indicates how the votes are aggregated. Among the large class of voting
rules, the most commonly used are unanimity, the simple majority rule, and the two-thirds rule.
Each of these procedures implies a certain degree of robustness in the final results obtained.
The unanimity rule yields a causal path only if that relation is detected in all the countries ofthe sample. The two-thirds rule, instead, requires that at least two-thirds of the sample share acausal link to accept it, while the simple majority rule requires an agreement of more than halfof the sample.
Thus, the application of the simple majority rule mandates that a causal path is included only
if it is detected in at least 32 countries among the 62 in the sample. The two-thirds rule, in turn,accepts a causal relationship only if it receives at least 42 votes.
Figure 1 exemplifies the aggregation procedure for three individuals and three variables.
Each individual runs the causal detection procedure, obtaining a causal graph, represented as an
adjacency matrix, where the rows represent the effects and the columns indicate the causes. In
this matrix, a causal relationship from variable A to variable B corresponds to an entry 1 in the( B, A) position, implying that the individual will cast a vote in favor of this relationship. The
matrices of the three individuals are then added, yielding an adjacency matrix with the total
number of votes for each causal path. A voting rule is then applied on each cell of this matrix.
Only the directed edges that get approved by the voting rule are kept in the aggregate causal
DAG.

7Figure 1: Diagram of the voting aggregation

Let us note that some variables have a constant value in some countries of the sample. It isthus not feasible to apply any of the causality detection methods discussed above to find relations
among those variables. This is the case of the exchange rate in Ecuador, the United States, andZimbabwe, as well as the total factor productivity in the United States. This reduces the size ofthe sample that can be used to study the relations among these variables, lowering the numberof votes required to accept a causal effect between them.
In the next section, we present the results obtained by applying this methodology to simulated
and real-world data.

8

3

Results

3.1

Synthetic data

As a first step in our study, we start by evaluating the quality of the results obtained by eachmethod. We proceed by simulating a growth dataset with the same variables and length as
the one obtained from the Penn World Table. This involves creating a causal direct acyclicalgraph (DAG) in which the same variables are the nodes while the arrows among them represent
the causal relations assumed in economic growth theory. We intend this graph to represent the
ground truth against which we can compare the evidence found by the different methods of causal
detection.
Building the ground truth model poses a challenge since there exist different models of economic growth, in which the assumptions, functional forms and variables involved differ widely.3
For our exercise, we disregard as much the specific details of the different models and include
all the variables in Table 1, establishing pairwise links whenever such relation is favored in theliterature, checking for the consistency of the resulting graph, depicted in Figure 2.
Figure 2: Ground truth DAG
TOT

XR

IRR

DELTA

POP

OPEN

INV

GOV

TFP

HK

GROWTHWe proceed by generating time series based on the ground truth model. We do this forvarying lengths (T = 59, 200, and 2000). For simplicity, we assume that the independent variables
are defined by a stationary AR(1) process with a white-noise error term:β < 1 , ε t ∼ N 0, σ2

X t = α + β X t −1 + ε t ;

3 To have a taste of this variety of models, see textbooks like Barro and Sala-i-Martin (2004) or surveys like Acemoglu(2012).

9

The three dependent variables, in turn, are defined by the following linear specifications:
OPENt = f ( TOTt , XRt ) + ηt
I NVt = f ( IRRt , DELTAt ) + υt
GROWTHt = f (OPENt , I NVt , POPt , GOVt , TFPt , HKt ) + ξ t
Each method generates a causal graph, which can be compared to the graph of the groundtruth. We also apply two ensembling techniques, combining the predictions of the different
methods. On one hand, ensemble∩ incorporates a causal path only if all the methods agree on its
inclusion, while ensemble∪ includes a causal path if at least one of the methods detects it.
We evaluate the results according to four performance measures: recall, precision, specificity,and the F1 score. We define TP, TN, FP, and FN as, respectively, the quantities of true positive,
true negative, false positive, and false negative edges obtained by a method, compared to theground truth model. That is, TP is the number of causal links in the ground truth graph that are
correctly detected while TN is the number of potential links absent in the ground truth that arerightly discarded by the method. In turn, FP is the number of non-causal links wrongly deemed
to be causal while FN represents the correct links that are missed by the method. Then:
TP
TN
2TP
recall = TPTP
+ FN , precision = TP+ FP , specificity = TN + FP , F1 -score = 2TP+ FP+ FN .

A method that yields a high recall recovers most of the correct causal links of the groundtruth, while higher levels of precision indicate that the method returns more correct results thanwrong ones. Specificity measures how well the method can identify links absent in the groundtruth. The F1 measure is the harmonic mean of recall and precision and indicates the accuracy ofthe method.
The results are presented in Table 2, where values are expressed in percentage terms. We
can see that Direct-LiNGAM yields, in general, the best results, although no method is perfectly
accurate.

10

Table 2: Performance metrics on ground truth
MethodGranger (lags=1)
Transfer Entropy
Stochastic Causality
ICA-LiNGAM
Direct-LiNGAM
ensemble∩ensemble∪
Granger (lags=1)
Transfer Entropy
Stochastic Causality
ICA-LiNGAM
Direct-LiNGAM
ensemble∩ensemble∪
Granger (lags=1)
Transfer Entropy
Stochastic Causality
ICA-LiNGAM
Direct-LiNGAM
ensemble∩ensemble∪

3.2

T
59
59
59
59
59
59
59
200
200
200
200
200
200
200
2000
2000
2000
2000
2000
2000
2000

Recall
10.0
0.0
0.0
20.0
40.0
0.0
50.0
20.0
30.0
10.0
30.0
80.0
0.0
100.0
10.0
30.0
0.0
30.0
30.0
0.0
70.0

Precision
7.7
0.0
0.0
15.4
28.6
13.2
14.3
23.1
33.3
25.0
72.7
25.6
5.6
30.0
0.0
23.1
18.8
16.3

Specificity
88.0
94.0
95.0
89.0
90.0
100.0
67.0
88.0
90.0
98.0
91.0
97.0
100.0
71.0
83.0
93.0
99.0
90.0
87.0
100.0
64.0

F1-score
8.7
0.0
0.0
17.4
33.3
0.0
20.8
16.7
26.1
15.4
27.3
76.2
0.0
40.8
7.1
30.0
0.0
26.1
23.1
0.0
26.4

Real-world data

When we turn to the real-world data drawn from the Penn World Table dataset, the first result isthat no method obtains causal relations unanimously detected in each country. Therefore, all the
DAGs presented in this section are the ones that pass either the simple majority or the two-thirds
rule. In the DAG corresponding to any method, causal paths are tagged with the number ofvotes that it gets under that method.
The results obtained according to each method are the following:• Granger: no DAG can be generated by even the majority rule.
• Transfer Entropy: two different DAGs are generated under the majority and the two-thirds
rules. In both human capital is the source of all the causal links.

11

Figure 3: Simple majority rule (Transfer Entropy)
HK

57

GROWTH

POP

42

OPEN

54

54

INV

52

45

GOV

51

TFP

42

53

TOT

XR

34

IRR

DELTA

Figure 4: Two-thirds rule (Transfer Entropy)
HK

57

GROWTH

OPEN

54

54

INV

52

POP

45

GOV

51

DELTA

42

TFP

53

TOT

XR

IRR

• Stochastic Causality: we find similar results to those obtained by the Transfer Entropy
approach, with the human capital as the cause of most of the variables. Unlike with Transfer
Entropy, we do not find a causal relationship between human capital and economic growth.
Figure 5: Simple majority and two-thirds rule (Stochastic Causality)
GROWTH

50

POP

OPEN

INV

51

42

GOV

HK

43

45

TFP

46

48

TOT

51

XR

49

IRR

DELTA

• The LiNGAM methods: economic growth is causally independent of the rest of the variables in all the DAGs generated under the two versions of this approach.
– ICA-LiNGAM: the growth rate of the population plays a key role, by initiating all thecausal paths in the simple majority DAG, while in the case of the two-thirds DAG, it
cedes part of its role. In that case, all the causal paths are initiated either by the growthrate of the population or the rate of depreciation of capital.
12Figure 6: Simple majority rule (ICA-LiNGAM)GROWTH

POP

32

37

55

46

INV

43

36

48

GOV

58

34

DELTA

37

41

42

38

45

IRR

49

35

HK

44

38

52

49

TOT

52

34

51

34

TFP

33

42

OPEN

42

31

XR

Figure 7: Two-thirds rule (ICA-LiNGAM)GROWTH

46

INV

DELTA

POP

43

GOV

49

52

58

45

48

TFP

TOT

42

52

49

OPEN

42

IRR

51

42

XR

55

44

HK

– Direct-LiNGAM: the two-thirds rule does not yield a DAG, while many variables arecausally independent of the others in the simple majority DAG. The rate of depreciation of capital is one of the sources of causal paths in the graph. The other source isthe real interest rate.
13

Figure 8: Simple majority rule (Direct-LiNGAM)GROWTH

DELTA

35

IRR

33

POP

35

GOV

OPEN

INV

TFP

TOT

31

41

XR

33

HK

One aspect worth highlighting in the results is that, except for Stochastic Causality, whichlacks the ability to detect it, no method maintained a bidirectional relationship between thevariables post-voting. In other words, in this case the methods did not find it for a sufficientmajority of countries.
• ensemble∪ 4 : the two graphs inherit from Transfer Entropy the only causal link involving thevariable Growth, namely as caused by Human Capital. The graphs are the following:
4 ensemble∩ is empty both under the majority and the two-thirds rules, even disregarding votes for Granger (thatyields an empty graph in both cases).

14

Figure 9: Simple majority rule (ensemble∪ )HK

GROWTH

POP

DELTA

IRR

TOT

GOV

INV

OPEN

TFP

XR

Figure 10: Two-thirds rule (ensemble∪ )HK

GROWTH

INV

DELTA

GOV

POP

OPEN

TFP

TOT

IRR

XR

The comparison to the ground truth model of subsection 3.1 is summarized in the following
table (notice that Granger and ensemble∩ are not included since these two methods do not yieldgraphs):15

Table 3: Performance metrics with real-world data (majority rule)
MethodTransfer Entropy
Stochastic Causality
ICA-LiNGAM
Direct-LiNGAM
ensemble∪

Recall
10.0
0.0
10.0
0.0
20.0

Precision
10.0
0.0
3.6
0.0
5.1

Specificity
91.0
91.0
73.0
94.0
63.0

F1-score
10.0
0.0
5.3
0.0
8.2

Table 4: Performance metrics with real-world data (two-thirds rule)
MethodTransfer Entropy
Stochastic Causality
ICA-LiNGAM
Direct-LiNGAM
ensemble∪

Recall
10.0
0.0
10.0
0.0
20.0

Precision
12.5
0.0
6.7
8.0

Specificity
93.0
91.0
86.0
100.0
77.0

F1-score
11.1
0.0
8.0
0.0
11.4

We can see that scores are consistently equal or higher under the two-thirds rule, although
they are (unsurprisingly) worse than with simulated data. We also can see that Transfer Entropy
is the method that, under the majority rule, seems to yield better scores than Direct-LiNGAM. In
the case of the two-thirds rule, the best results are obtained by ensemble∪ .

4Discussion

In this paper, we have proposed a novel approach based on voting rules for determining causal
relationships in panel data settings. This method allows to obtain a causal structure that is
supported by several independent sources. Because of the voting structure used, this method
could also perform a more comprehensive analysis in terms of country subgroups, in cases
where it is relevant. This is because it is possible to disaggregate the votes obtained for a causal
relationship into the subgroups identified in the data.
The results reported in Tables 3 and 4 could be seen as indicating that the methods appliedin this study may not be too efficient in detecting the causal links present (and absent) in the
ground truth. This idea may get reinforced by the results obtained with synthetic data (Table 2).
But while it may be the case that the methods presented here are still too crude to provide
causal inferences, an alternative explanation can be put forward. Namely, that the ground truth
itself is inaccurate. While the former argument cannot be discarded, the latter shows why the16exploration of causal detection methods matters.
The causal links in the ground truth should not be taken as having indisputable validity. On
the contrary, the ground truth model should only be taken provisionally and fully accepted if itgets validated by data. Therefore, the real power of the methods explored here is the possibilityof detecting the strongest causal relations in the datasets.
In our analysis, we found that real-world data under the ensemble∪ method supports, as the
only relevant cause of growth, the human capital in the economy. Notice that this result canbe understood in terms of the crucial difference between correlation and causation. So, while theground truth model is consistent with the theory and evidence of economic growth, most of thelinks in it may just represent strong correlations, with human capital as the main underlying
common cause.
To examine if this is the case, we may need to apply the approach of Pearl (2000) to detect
confounders. Natural experiments in which some variables are intervened may provide the information needed to implement such exploration. In the meanwhile, the exploration of alternative
Machine Learning methods seems worthwhile.

Statements and Declarations
Conflict of interest: Authors declare that they have no conflict of interest.

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A

Appendix
Table 5: Countries in the sample
Argentina
Australia
Austria
Belgium
Bolivia
BrazilCanadaSwitzerland
Chile
China
Colombia
Costa Rica
CyprusGermany
Denmark
Dominican Republic
Ecuador
Egypt
Spain
FinlandFranceUnited Kingdom
GreeceGuatemalaIndonesia
India
Ireland
Iran
IcelandIsraelItaly
Jamaica
JordanJapan
Kenya
Republic of Korea
Sri Lanka
Luxembourg
MoroccoMexicoMalta
Mauritius

19

Malaysia
Nigeria
Netherlands
NorwayNew Zealand
Peru
Philippines
Portugal
Paraguay
SwedenThailandTrinidad and Tobago
TurkeyTaiwanUruguay
United States
Venezuela
South Africa
ZambiaZimbabwe