1 Introduction

Following the seminal paper by Hart (1973) on the working unemployed in Ghana, substantial effort has been invested to conceptualise and control economic activities that are inherently legitimate but remain hidden from the authorities.Footnote 1 While there is still a lot of work to be done in other parts of this research field (e.g., the task of quantification), the quest for the factors underlying the decisions of workers and companies to go into the shadows seems to have reached saturation point (Dularif et al. 2019; Hofmann et al. 2017). The last 50 years have witnessed a stream of research studies either upgrading the existing theories on the drivers of informality or complementing them with new ones (Castells and Portes 1989; de Soto 1989; Feld and Frey 2007; Maloney 2004; Moser 1978; Round 2009; Sethuraman 1976). This went in parallel with discoveries of novel forms of noncompliance, which enabled continual refinements with respect to definitions and scope of activities in the focusFootnote 2 (Franic 2020a; Pfau-Effinger 2009; Williams and Horodnic 2017a).

A cursory insight into the publicly available literature exposes hundreds of research papers striving to explain why some market participants consciously circumvent legislation on taxes, employment relations, and/or various administrative obligations (see Hofmann et al. 2017). The vast amount of qualitative, quantitative, and experimental evidence from all parts of the world has, in fact, stimulated the emergence of a novel research paradigm grounded on re-usage of the results from available studies in yet another round of quantitative pursuit for undisputable determinants of concealed economic activities. The so-called meta-analytic approach assumes a systematic evaluation of the significance, direction, and (where possible) the magnitude of effect for a set of chosen indicators so as to understand whether, and in what circumstances, they indeed influence the behaviour of economic agents (Blackwell 2010; Dularif et al. 2019; Hofmann et al. 2017).

However, in spite of being invaluable for broadening horizons about the roots of noncompliance and the most common offenders, neither the existing core nor meta-studies have managed to provide an answer to the fundamental question: have we really gained a holistic picture of the forces behind unregistered activities or are there still some missing pieces of the puzzle? In other words, it is unclear whether the list of factors identified so far as drivers of undeclared work is complete or not.

This paper seeks to fill the gap by addressing an equivalent problem. Suppose that for a certain person/company, we possess all the information on the features hitherto known to shape the decision on (non)declaration of activities. If the list of causal factors is exhaustive, there must exist a statistical model (or a group of models) able to determine with extremely high exactness if this agent participates in the undeclared economy. Conversely, if such a model cannot be constructed, this would imply that there are some other drivers of informality which we are not aware of at this point.

This research problem was insolvable until recently due to well-known limitations of the traditional econometric techniques (Athey and Imbens 2019; Boulesteix and Schmid 2014; Di Franco and Santurro 2021; Mullainathan and Spiess 2017). However, recent developments in the area of machine learning not only have enabled the realisation of this sanity check, but also have opened the prospect for enhancements in other niches of this research field (e.g., detection and quantification). That being said, this paper intends to have both theoretical and practical contributions. Besides adding to the ongoing debate on the determinants of unregistered activities, it will also reveal the full potential of artificial intelligence in the fight against this deleterious practice. Above and beyond, the presented methodology will be highly relevant for the government officials around the world who will face the need to substantially increase public budget revenues in the post-COVID era.

To achieve these aims, the rest of the paper is organised as follows: alongside summarising the most important findings on the matter over the last 5 decades, Sect. 2 also provides a further discussion on the gaps in the literature and specifies how this paper is going to address them. Section 3 introduces the datasets and methods employed to answer the posed research question. Particular attention will be paid to deficiencies of the traditional econometric methods and the supremacy of machine learning in this respect. After presenting and discussing the most important results of the conducted analysis in Sect. 4, the paper ends with concluding remarks and suggestions for future research.

Before moving forward, it ought to be said that the focus of this paper is strictly on undeclared work, i.e., all market-oriented activities which, despite being legitimate per se, remain deliberately hidden from the authorities to evade taxes and social security contributions, to avoid compliance with labour legislation, and/or to circumvent any other administrative requirement (European Commission 1998). This would say that prohibited activities (human trafficking, prostitution, drug-smuggling, etc.), as well as self-provisioning, neighbour help, volunteering, and alike forms of unpaid work remain out of the scope of this study. The same is true for tax evasion related to activities that do not result in any added value (e.g., frauds with capital gains) and tax avoidance.

2 Literature review

The pioneering studies on the matter, based primarily on qualitative research in developing countries, described undeclared work as a leftover of the pre-capitalist period which “would disappear once these countries achieved sufficient levels of economic growth or modern industrial development” (Chen et al. 2004, p. 16). This view regarded low employment prospects and pervasive poverty among the population of fast-growing cities as the sole drivers of unregulated activities (Hart 1973; Sethuraman 1976; Tokman 1978). In line with that, the name informal sector was used to reflect the idea of two distinct and autonomous realms, namely “a dynamic, profit-making modern sector and everything else—a vast sponge of surplus labor” (Peattie 1987, p. 852). The latter was believed to consist mostly of low-skilled individuals who emigrated from rural parts of the country in search of any income opportunity (Hart 1973).

Following the increasing interest in this topic, it was soon realised that reality is actually far more complex. Not only was the existence of undeclared work evidenced in developed countries, as well, but it quickly became apparent that this harmful practice was going to stay ingrained in economies around the world (Moser 1978; Rakowski 1994). Explicitly, a number of studies conducted during the late 1970s found that a considerable part of concealed market-oriented activities in advanced economies was, in fact, a by-product of capitalism (see for instance Moser 1978). Owing to the combination of economic turbulences and accelerated globalisation, many formal firms were forced to decrease production costs to survive on the market. Alongside the automation of production processes and transfers of plants to less-developed countries, subcontracting of work to small unregistered firms and hiring workers off-the-books also emerged as pleasing strategies in the race to the bottom (Castells and Portes 1989; Davis 2006).

These findings gave rise to the so-called structuralist school of thought, whose proponents saw the roots of undeclared work in weak and inefficient state institutions unable to prevent the exploitation of the impoverished masses. Structuralists thus called for better protection of workers’ rights, more stringent regulation of businesses, and strengthening the rule of law (Portes and Sassen-Koob 1987; Rakowski 1994).

The turbulent 1980s put the state once again in the centre of discussion on the flourishing informalities. However, this time, the focus shifted to raising tax burdens, overcomplicated registration procedures, and amplified income inequalities (Annis and Franks 1989; de Soto 1989). Introducing the concept of emotional agents, research studies from the period showed that many individuals and firms decided to operate on an undeclared basis out of defiance. This legalist interpretation of the state of affairs portrayed tax evaders as the democratic force that openly stood against an unfair and intrusive state (Rakowski 1994).

The emotional agent approach was complemented during the 1990s and 2000s with the theories of rational and quasi-rational agents. The concept of rational voluntarism arose from studies which revealed that some workers and companies freely chose to operate off-the-books after assessing the costs and benefits of such behaviour (Fields 1990; Maloney 2004).Footnote 3 While acknowledging the driving forces identified by the legalists, this view on the roots of noncompliance introduced additional elements to the equation, namely the risk of being detected, plausible sanctions, the quality of pension and welfare systems, and the difference in pay rates between declared and undeclared work. However, the most important novelty brought by this school of thought was the recognition of the upper tier informal economy—part of the undeclared sphere attracting affluent individuals eager to increase their wealth (Fields 1990).

The mismatch between the compliance rates implied by the rational-agent theory and actual compliance inspired a stream of research on the role of personal and social norms in the process (Alm et al. 2017; Frey and Torgler 2007; Torgler 2004). In the quest for the reasons why some people always comply, while others seek evasion strategies even when the potential cost outweighs the benefits, the academic community has recently put a greater emphasis on a latent construct known as tax morale. Defined as “individual’s willingness to pay taxes, in other words, the moral obligation to pay taxes or the belief that paying taxes contributes to society” (Frey and Torgler 2007, p. 140), this attribute was found to be a compound outcome of numerous socio-economic, psychological, and demographic peculiarities. Although still not fully understood (which particularly applies to the hereditary context), evidence suggests that tax morale is a dynamic feature heavily influenced by vertical trust (i.e., trust in the state institutions), horizontal trust (trust in other taxpayers), and various personal characteristics. For instance, it was shown that men generally express lower intrinsic readiness to pay taxes than women, and the same applies to younger individuals compared to more experienced ones (Alm and Torgler 2006; Lago-Peñas and Lago-Peñas 2010). The importance of religion in one’s life, marital status, size of the family, the level of education, and occupation are also some of the key facets in this respect (Benk et al. 2016; Lago-Peñas and Lago-Peñas 2010; Strielkowski and Čábelková 2015).

When it comes to trust, a number of studies revealed that some taxpayers tend to comply with tax legislation as long as they think that the authorities respect an imperceptible psychological contract between the state and citizens (Franic 2019; van Dijke and Verboon 2010). Conversely, if they believe the public funds are not spent fairly and efficiently, such people will seek strategies to reduce their tax duties (Barone and Mocetti 2011). The efficacy of the state apparatus, the quality of the services received, and the perceived prevalence of corruption in public institutions are the most important factors shaping the views of taxpayers on this matter (Alm et al. 2010; van Dijke and Verboon 2010).

Horizontal trust, on the other hand, is embodied in the concept of conditional cooperation. As explained by Frey and Torgler (2007), an individual’s willingness to pay own taxes is strongly affected by the perception regarding the behaviour of their counterparts in this regard. If they think that others are not respecting the implicit social deal, some of the honest taxpayers will shift to the undeclared sphere simply because they feel fooled. Furthermore, the pervasive informality usually signals tacit approval of this practice in society, thus reducing the moral cost of the wrongdoing and igniting further noncompliance (Alm et al. 2017; Torgler 2004).

Even though each subsequent view on the mechanisms underlying undeclared work was grounded on criticism of the existing theories, time has shown that all of them are, in fact, valid. Indeed, the latest stream of quantitative inquiry, based on large-scale questionnaire surveys and experimental studies, has revealed that these schools of thought are complementing rather than contesting each other (Chen 2012; Williams and Round 2007). For instance, a number of recent studies have underlined limited employment prospects in the formal sector as being responsible for a large portion of modern-day undeclared work, both in developed and developing countries (Elek and Köllő 2019; Williams and Efendic 2021; Williams and Horodnic 2015b). Likewise, numerous low-skilled workers are still being pulled into this sphere by reckless employers not hesitating to resort to exploitative practices for their own gain (Franic 2020b; Palumbo 2017). Finally, it appears that more market participants than ever before nowadays eagerly embrace undeclared work: while some do so because they believe that the benefits exceed perceived cost, others decide to go into the shadows as a rebellion against the inefficient and over-intrusive state and/or due to feeling deceived by their fellow taxpayers (Alm et al. 2010, 2017; Kogler et al. 2013; van Dijke and Verboon 2010). For the majority of taxpayers, their (non)participation in the undeclared economy is, however, a complex outcome of a number of intertwined factors (Franic 2020b; Franic and Cichocki 2021).

Since each of the aforementioned theorizations represents one piece of the key for the decision-making riddle on the part of economic agents, the term holistic approach now dominates the literature on this phenomenon (see Chen 2012; Williams 2016). However, while academics and professionals do agree that a holistic view is needed, it is not clear whether all parts of the puzzle are already on the table. In the rest of the paper, we seek to provide an answer precisely to this question. This will be done by applying the latest machine learning techniques on an all-inclusive dataset derived from various sources, as explained in the following section.

3 Data and methods

A careful reader might ask themselves why this essential matter has not been evaluated so far. To understand the reasons, one must be familiar with the limitations of the traditional econometric approach. First of all, the research questions and hypotheses were until recently driven strictly by data availability, given that quantitative researchers interested in the determinants of undeclared work were forced to rely solely on the variables available as part of a particular questionnaire survey or experimental study (Grimmer et al. 2021). In line with that, most research articles published before the 2010s provide quite a narrow insight into the matter.

This issue was somewhat mitigated by the emergence of multilevel techniques, which made it possible to combine the data from questionnaire surveys with external macroeconomic and akin figures. However, analyses based on a multilevel approach commonly follow the sequential modelling strategy, meaning that in practice researchers are not able to test all the required variables in parallel (see for instance Franic and Cichocki 2021; Williams and Horodnic 2016). This is a direct consequence of the susceptibility of traditional econometric models to multicollinearity, as well as of requirements with respect to the ratio of a sample size to the number of covariates (Athey and Imbens 2019; Grimmer et al. 2021).

Given these constraints, classical methods commonly require data selection procedure and/or feature engineering to be applied, which frequently leads to biased results (Athey and Imbens 2019; Boulesteix and Schmid 2014). Nevertheless, even if all steps are taken to obtain unbiased estimates, not much can be done to address the overfitting to the sample (Molina and Garip 2019; Mullainathan and Spiess 2017). Consequentially, the still-prevailing statistical methods are in practice mainly used for the exploration of causality, as they seldom show robustness in predictive modelling.

These shortcomings led to the development of a range of machine learning techniques, which were designed specifically for predictive purposes. As a result of tremendous enhancement in this area, many supervised machine learning models are nowadays able to reach predictive accuracy close to 100% (see for instance Li et al. 2020).Footnote 4 The secret of success resides in a combination of iterative estimation approach (which eliminates the concern with multicollinearity and selection bias), flexible model architectures, ability to detect latent mechanisms, and, consequently, a huge number of model coefficients (Ghoddusi et al. 2019; Mullainathan and Spiess 2017). These features make machine learning a perfect choice for our task.

Before proceeding with the details on the exact models employed in the analysis, it is first necessary to introduce the datasets used for this purpose. The starting point in this respect was the latest wave of the Special Eurobarometer on undeclared work. Being one of the most comprehensive sources of data on this topic, this survey conducted in September 2019 provides an insight into the experience, views, and attitudes of 26,514 EU citizens regarding undeclared activities.Footnote 5 Akin to the previous two waves (from 2007 and 2013), this one also contained the following question:

Have you yourself carried out any undeclared paid activities in the last 12 months, either on your own account or for an employer?

The resulting variable is binary, which means we are faced with a standard classification task.Footnote 6 In line with the findings of existing studies on the drivers of undeclared work and reflecting the discussion from earlier parts of the paper, the following three sets of features directly available from the survey are used as input variables in the modelling:

Socio-demographic: gender, age, marital status, and household size.

Socio-economic: type of community (urban/rural), country of residence, migrant status, education, occupation, size of the company, financial difficulties, and social class.

Perceptions and attitudes: perceived detection risk, expected sanctions, having undeclared workers among friends and relatives, estimated percentage of the population engaged in undeclared activities, trust in tax authorities, trust in labour inspectorate, and tax morale index.

Details on coding and research papers finding a significant effect of these features are given in Table 1. Since these 19 variables do not represent a complete list of the factors so far known to influence the behaviour of economic agents, we also add a range of country-level determinants compiled from various sources. These variables can be roughly divided into four groups as follows:

Table 1 Overview of individual-level features.
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Economic constraints: unemployment rate, the implicit tax rate on labour, income inequality index, and relative median income ratio for persons above the age of 65 (a proxy for the quality of pension systems).

State intervention: the size of the government, stringency of labour market regulations, and stringency of business regulations.

Quality of formal institutions: government effectiveness, rule of law, regulatory quality, and corruption perceptions index.

Informal institutions: trust in government, trust in other people, religiosity, and the average level of tax morale.

With the exception of the average level of tax morale, which was devised directly from the Special Eurobarometer on undeclared work, all other figures were collected from credible international institutions (see Table 2 for details). To ensure compatibility with the baseline dataset, values for 2019 were taken whenever possible. In case of data for 2019 not being available, the most recent figures were considered. These variables were then incorporated into the Eurobarometer dataset, which gave a total of 34 features whose predictive power was to be tested simultaneously.

Table 2 Overview of country-level variables.
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At this point, it is important to underline yet another difference between the traditional econometric approach and machine learning. While the former requires special techniques to deal with variables collected at different levels, the latter is robust to the violation of the independence of observations assumption. That is to say, since they only care about whether and to what extent a certain feature can help in predicting the modelled variable, machine learning methods do not make any distinction between individual-level and country-level variables.

Given a relatively small number of input variables, a total of 26,514 sampled units would in most cases be sufficient to construct and train well-performing machine learning models. However, things are slightly complicated in our case due to the imbalanced nature of the target variable. Explicitly, only 950 Eurobarometer survey participants admitted their participation in the undeclared economy from the supply side, which introduces a substantial risk of models being biased towards the negative outcome.Footnote 7 To address this issue, during the training phase, we applied the random oversampling scheme with weights inversely proportional to class frequencies (see Fernández Hilario et al. 2018; Viloria et al. 2020).

As our emphasis is first and foremost on the training set accuracy (owing to the nature of the research question), this procedure elegantly resolves the problem. However, it does not help much when it comes to the generalisability of the results.Footnote 8 Given the complexity of the researched matter, information on as few as 950 undeclared individuals from across the EU is simply not enough to achieve high classification accuracy on unseen cases with a single model, regardless of its achievement on the training set.

To mitigate this problem, the decision was made to train seven different models from a wide palette of supervised machine learning techniques and combine their results through the ensemble voting approach. The first and the simplest of models chosen for this purpose was the decision tree which is, despite being useful for grasping the relative importance of examined causal factors, highly prone to overfitting (Hastie et al. 2008; Mitchell 1997). For this reason, it was complemented with the Adaptive Boosting (AdaBoost) model, which combines a sequence of weak classifiers into a single strong classifier in an iterative ensemble-learning fashion (Hastie et al. 2008; Wyner et al. 2017). To compensate for the limitations of these two methods, we further constructed a random forest model. Also belonging to a family of ensemble-learning techniques, random forest combines the results of multiple decision trees constructed using randomly drawn subsamples of the training set (Breiman 2001; Genuer and Poggi 2020).

Turning to the linear classifiers, which are commonly less prone to a high variance problem, the two most obvious choices were support vector machine (SVM) and logistic regression. However, while expected to outperform the nonlinear classifiers on out-of-sample cases, these two methods usually exhibit higher bias (i.e., lower accuracy on the train set). Their limited efficiency in situations where data are not linearly separable is envisaged to specifically come to the fore in our case, given the likely existence of multiple latent mechanisms underlying decisions of labour suppliers with respect to (non)declaration of activities.

To account for the presence of multifaceted interactions between causal factors, two artificial neural networks were also devised. A deep learning approach commonly beats other classifications techniques, but it is not entirely immune to overfitting, especially when handling imbalanced data (Dabare et al. 2018; Johnson and Khoshgoftaar 2019; Mitchell 1997). For this reason, alongside a deep artificial neural network with five hidden layers, we also defined a shallow version, which incorporated only one hidden layer. The exact details on the design of these two artificial neural networks, as well as of the remaining five models are given in Appendix 1.

Before moving to the results, it should be mentioned that the data preparation phase was done in STATA, while Python modules TensorFlow and Scikit-learn were used to construct, train, and test the models introduced above. To make hyperparameter tuning and evaluation of performance possible, the original dataset was split into the training, validation, and test sets, whereby 20,000 individuals (approximately 75% of the sample) were randomly allocated to the training set, and the remaining ones were evenly split into the other two sets.

4 Findings and discussion

Results of the conducted analysis, presented in Table 3, provide strong evidence for the completeness of the list of causal factors. Five out of seven models managed to classify training examples with an accuracy above 99.5%, which indicates that all the building blocks for the holistic approach are already there.

Table 3 Performance metrics of the trained machine learning models.
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Nonetheless, a more thorough insight into the key performance metrics reveals that it is not individual drivers, but rather complex interactions of theirs what shapes the behaviour of labour suppliers. This can be easily concluded by inspecting the figures for support vector machine and logistic regression, which were strongly outperformed by the remaining models on the training set. Being able to assess only a direct effect of the examined features on the target variable, these two models managed to correctly classify just 8 out of 10 individuals from the training set.

On the other hand, the shallow artificial neural network with a total of 625,001 parameters capturing latent interconnections between the explanatory variables achieved a remarkable accuracy of 99.97%. A similar result was obtained with the five-layer network, which comprised 749,901 parameters.Footnote 9 The importance of interdependence between causal factors is further evidenced by evaluation statistics of the remaining three models. Decision tree and AdaBoost managed to correctly classify all training examples, while random forest made only a few mistakes (see Table 3).

Nevertheless, an extraordinary performance on the training set does not imply that these models can be straightforwardly applied in practice. As a matter of fact, results on the unseen data are also quite misleading in this respect. Although the performance metrics for the test set, which are also presented in Table 3, reveal that five models were able to correctly classify at least nine out of ten previously unseen individuals (with accuracy rates ranging from 93.58% for decision tree and 96.38% in case of the random forest), this is not as satisfying as one might assume on a first glance. Given that only 3.58% of the survey respondents admitted working on an undeclared basis, a naive model predicting a negative outcome for each and every worker would achieve higher accuracy than any of our seven models (it would be correct in 96.42% of cases). This implies that accuracy is not a particularly informative performance measure for models dealing with rare events if the primary interest lies in the correct identification of positive cases.

Recall, F1 score, and area under the curve (AUC), which are better indicators of the classification power in our situation, shed completely new light on the models. In spite of much lower overall accuracy, logistic regression and support vector machine would actually perform best in practice, as each of them was able to correctly flag three out of four workers receiving undeclared income. The remaining models exhibit extremely low ability to identify true violators, with recall rates ranging from 0.0732 to 0.1870.

Although seeming contradictory, these results are fully reasonable and in line with the aforementioned caveat regarding a small number of undeclared workers in the sample. Besides modelling an imbalanced target variable, we also deal with the situation in which hidden interrelations of explanatory factors are vital for the segregation of positive and negative cases. At the same time, our models utilised information from 20,000 training examples, whereby only 728 of them were engaged in undeclared work (the remaining ones were allocated to the validation and test sets). Given such a small number of positive cases, five models suffer from high variance, meaning that they overfit to the training set.

This implies that practical applications of machine learning models on the identification of undeclared workers would require far larger datasets. It is hard to speculate on the exact size at this point, but no model would likely be able to generalise well unless there is at least a four-digit number of positive cases in the sample.

However, not everything is lost even with the existing dataset. To exemplify this, we modified hyperparameters of the models so as to maximise the accurateness in identifying true positives.Footnote 10 As can be seen from Fig. 1, which presents the accompanying confusion matrices, the majority of these adjusted models were able to correctly identify at least three out of four out-of-sample violators. AdaBoost and artificial neural network with one hidden layer were most successful in this respect, with a hit rate of 80%. The worst achievers, on the other hand, are decision tree and logistic regression, which managed to recognise 69.9% of violators.

Fig. 1
figure 1

Source: Author’s own work

Confusion matrices on the test set for models maximising recall.

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To further increase recall, an additional logistic regression model was constructed and trained with input variables being predictions of individual models (see Appendix 2). The resulting coefficients, which are essentially the importance weights attached to each model based on their credibility, were then applied to the test set. This ensemble voting strategy indeed provided more accurate estimates, as can be seen from the last panel of Fig. 1. Specifically, the seven models were collectively able to spot 102 out of 123 previously unseen undeclared workers from the test set. Of course, this came with the cost of a substantial number of false positives, i.e., fully compliant workers who were incorrectly labelled as offenders. Adding more models to the ensemble voting scheme would certainly reduce this number and yield better results on recall. Nevertheless, due to the insufficient size of the training set, the final product would still fall below the level required for practical application.

Before finalising our discussion, it is beneficial to scrutinise the relative importance of individual drivers. This can be easily done by inspecting the results of the decision tree model. Due to the greedy variable selection approach and clear visualisation, decision trees provide a straightforward insight into the hierarchy of explanatory variables in terms of their contribution to the segregation of the target concept. In line with this, Fig. 2 unfolds the structure of the first four layers of the original decision tree as defined in Appendix 1 and elaborated in Table 1.

Fig. 2
figure 2

Source: Author’s own work

Results of the decision tree. (i) Numbers in parentheses denote the split for target variable.

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The visualisation exposes socio-psychological factors as the dominant forces driving one’s decision whether or not to participate in the undeclared economy. Specifically, the existence of direct social contact with undeclared workers was identified as the best discriminant between the two cohorts of taxpayers. According to the results, individuals knowing somebody operating off-the-books are almost 2.5 times more likely to do the same. A note of caution is, however, necessary here due to a bidirectional causality: from the very nature of their activities, it follows that people who participate in the undeclared economy are more likely to have friends and acquaintances who also work in this sphere.

Further endorsing the findings from the recent stream of research on this matter, the decision tree highlighted tax morale as the second most important causal factor (see Franic 2020a; Frey and Torgler 2007; Williams and Bezeredi 2017). This variable, which emerged as the best separator in both branches of the second layer, also appears to be correlated with the previously mentioned determinant. This follows directly from the cut-off points of the two sub-trees, which are somewhat lower on the right side.

Things get more interesting when going deeper into the tree. In most cases, the third-best choice for filtering wrongdoers was their age, which exerts a substantial negative effect on the propensity to participate in the undeclared sphere (as already evidenced by, e.g., Arendt et al. 2020; Hofmann et al. 2017; Popescu et al. 2016). The only exception was individuals with low tax morale who have an undeclared worker in their surroundings (the rightmost rectangle in layer 3). For them, it is the perception regarding the pervasiveness of undeclared work in a society that matters most. Those convinced that at least every fifth citizen is hiding activities were found to have a 15% higher probability of doing the same than their counterparts who are more optimistic regarding the prevalence of undeclared work.

The subsequent layer of the tree further accentuates the relevance of subjective perceptions, given that trust in other people and religiosity were identified as the next best discriminant variables for a great many taxpayers. On the other hand, economic factors, such as government effectiveness, the stringency of business regulations, and the migrant status of a worker only at this stage do come to the fore.

5 Conclusion

A 5-decade-long endeavour to comprehend the fundamentals of undeclared work has brought to light a range of economic and socio-psychological factors influencing the behaviour of taxpayers in this respect. To underpin recent calls for an all-inclusive approach towards the eradication of the phenomenon (see Franic 2020b; Williams 2016), in this paper, we assessed the exhaustiveness of the list of known driving forces. The conducted analysis, which represents one of the very first applications of up-to-date machine learning techniques to this research field, verified that all components necessary for a holistic approach are already there.

Specifically, five out of seven supervised machine learning models exhibited remarkable classification accuracy (99.5% or more) on the training set. Alongside demonstrating that the existing causal factors jointly provide sufficient information to fully segregate wrongdoers, our findings also support a growing body of research on the dominance of personal norms, beliefs, and values over mere economic constraints in the modern-day reasoning of workers (e.g., Alm et al. 2017; Franic 2020b; Williams and Yang 2018). As shown, intrinsic willingness to pay taxes, exposure to information about concealed activities, and their perception about the pervasiveness of such activities in society are nowadays crucial for one’s decision whether to choose the same path. The quality of the state institutions and rigidness of regulations, on the other hand, are less important from the perspective of labour suppliers.

This, however, does not mean that the same is true for companies, buyers of undeclared goods and services, afternoon moonlighters, and akin offenders. Furthermore, given the economic, cultural, and political specificities of the European Union, it would be overly optimistic to claim that the same hierarchy of causal factors applies to workers from other parts of the world. If this study encourages similar research on the motives of other players within this realm and on other geographical areas, then it will have fulfilled one of its broader aims.

Besides the apparent theoretical contribution, the paper also paves the way for a wide usage of artificial intelligence in the fight against illegitimate economic activities. Yet, successful practical applications of machine learning models require not only much larger but also more credible datasets. This brings us to the main limitation of our study, which is closely linked to the nature of the information collected during the fieldwork. Due to the sensitivity of the researched matter, it is very likely that many survey respondents consciously denied their involvement in undeclared work. While imputation techniques helped to reduce the bias arising from a considerable number of missing responses, not much could be done in case of deliberate misreporting.

Given this, official records of labour inspectorates and tax administrations appear to be a much better source of information for training efficient violator-detecting machines. Although inferior to questionnaire surveys from the standpoints of the traditional econometric approach, data held by enforcement authorities are quite appealing in our case given the robustness of machine learning to the non-randomness of the sample. Typical examples of the existing sources that can be used for this purpose are the large-scale audit campaigns by the Internal Revenue Service (IRS 2016) and Her Majesty's Revenue and Customs (HMRC 2018). These and alike datasets already make it possible to train models which not only could detect noncompliance, but also would be able to provide more details on the type of offense in place and the magnitude of the violation.

That being said, machine learning might be a promising avenue for solving the most challenging task within this research field, which is the issue of quantification. As a matter of fact, recent advances in the fields of unsupervised learning and reinforcement learning, coupled with the introduction of sophisticated transaction-monitoring procedures on the part of surveillance bodies, open the prospects for real-time detection of noncompliance in a near future. If this paper encourages research studies heading in this direction, then it will have fulfilled its main goal.

6 Appendix 1: Details on the design of supervised machine learning models maximising accuracy

Decision tree

Criterion: Gini; splitter: best; max depth: 187; minimum samples at a leaf node: 1; maximum number of features: none; class weights: balanced; random state: 1

Random forest

Number of trees: 5000; criterion: Gini; maximum depth: 20; minimum samples at a leaf node: 2; minimum samples to split: 2; maximum number of features: auto; class weights: balanced; random state: 1

AdaBoost

Maximum number of estimators: 5000; base estimator: decision tree (class weights: balanced); learning rate: 1; random state: 1

Support vector machine

Regularization parameter: 2; kernel: linear; class weights: {0:1,1:26}; random state: 0

Logistic regression

Loss: binary cross-entropy; kernel initializer: Glorot uniform; kernel optimiser: Adam; kernel regularizer: l2 (λ = 1e−2); learning rate: 1e−4 for epochs 1–150, 1e−5 for epochs 151–200, 1e−9 for epochs > 200; class weights: {0: 1, 1: 26}; batch size: 64; number of epochs: 210

Neural network (1 hidden layer)

Number of neurons: 5000; activation: relu; loss: binary cross-entropy; kernel initializer: Glorot uniform; kernel optimiser: Adam; kernel regularizer: l2 (λ = 1e−4); learning rate: 1e−4 for epochs 1–180, 1e−5 for epochs > 180; class weights: {0: 1, 1: 26}; batch size: 64; number of epochs: 300

Neural network (5 hidden layers)

Number of neurons: 1000–500–200–100–50; activation: relu; loss: binary cross-entropy; kernel initializer: Glorot uniform; kernel optimiser: Adam; kernel regularizer: dropout (rate = 1e−1); learning rate: 1e−4 for epochs 1–100, 1e−5 for epochs 101–150, 1e−6 for epochs > 150; class weights: {0: 1, 1: 26}; batch size: 64; number of epochs: 200

  1. Source: Author’s own work

7 Appendix 2: Details on the design of supervised machine learning models maximising recall

Decision tree

Criterion: Gini; splitter: best; max depth: 8; minimum samples at a leaf node: 2; maximum number of features: none; class weights: balanced; random state: 1

Random forest

Number of trees: 500; criterion: Gini; maximum depth: 5; minimum samples at a leaf node: 2; minimum samples to split: 2; maximum number of features: auto; class weights: balanced; random state: 1

AdaBoost

Maximum number of estimators: 350; base estimator: decision tree (class weights: balanced; max depth: 1); learning rate: 1; random state: 1

Support vector machine

Regularization parameter: 2; kernel: polynomial; degree: 2; class weights: {0:1,1:26}; random state: 0

Logistic regression

Loss: binary cross-entropy; kernel initializer: Glorot uniform; kernel optimiser: Adam; kernel regularizer: l2 (λ = 1e−5); learning rate: 1e−4 for epochs 1–50, 1e−5 for epochs 51–100, 1e−6 for epochs > 100; class weights: {0: 1, 1: 26}; batch size: 64; number of epochs: 700; callbacks: early stopping (monitor: validation loss; patience: 4; restore best weights: true)

Neural network (one hidden layer)

Number of neurons: 5000; activation: relu; loss: binary cross-entropy; kernel initializer: Glorot uniform; kernel optimiser: Adam; kernel regularizer: l2 (λ = 1e−4); learning rate: 1e−5 for epochs 1–180, 1e−6 for epochs > 180; class weights: {0: 1, 1: 26}; batch size: 64; number of epochs: 300; callbacks: early stopping (monitor: validation loss; patience: 4; restore best weights: true)

Neural network (five hidden layers)

Number of neurons: 1000–500–200–100–50; activation: relu; loss: binary cross-entropy; kernel initializer: Glorot uniform; kernel optimiser: Adam; kernel regularizer: dropout (rate = 1.5e−1); learning rate: 1e−5 for epochs 1–100, 1e−6 for epochs 101–150, 1e−7 for epochs > 150; class weights: {0: 1, 1: 26}; batch size: 64; number of epochs: 200; callbacks: early stopping (monitor: validation loss; patience: 4; restore best weights: true)

  1. Source: Author’s own work