Social capital and the shadow economy: a Bayesian analysis of the BRICS

Purpose – The paper aims to examine the impact of social capital on the size of the shadow economy in the BIRCS countries over the period 1995 – 2014. Design/methodology/approach – The authors employ the Bayesian linear regression method to uncover the relationship between social capital and the shadow economy. The methodapplies a normal distributionfor the prior probability distribution while the posterior distribution is determined using the Markov chain Monte Carlo technique. Findings – The results indicate that the unemployment rate and tax burden positively affect the size of the shadow economy. By contrast, corruption control and trade openness are negatively associated with the developmentofthisinformalsector.Moreover,thepaper ’ sprimaryfindingisthatsocialcapitalrepresentedby social trust and tax morale can hinder the size of the shadow economy. Researchlimitations/implications – Thisstudyislimitedto the caseofthe BRICScountriesfor the period 1995 – 2014. The determinants of the shadow economy in different groups of countries can be heterogeneous. Moreover, social capital is a multidimensional concept that may consist of various components. This difficulty of measuringthe socialcapital calls for further research on the relationship between other dimensions of social capital and the shadow economy. Originality/value – Many studies investigate the effect of economic factors on the size of the shadow economy.Thispaperappliesanewapproachtodiscovertheissue.Notably,theauthorsusetheBayesianlinearregressionmethodtoanalyzetherelationshipbetweensocialcapitalandtheshadoweconomyintheBRICScountries.


Introduction
The shadow economy happens in many countries and can cause significant impacts on the economic performance. Notably, findings by Medina and Schneider (2019) indicate that the average shadow economy size of 157 countries between 1991 and 2017 is 30.9% of gross domestic product (GDP). Overall, the size of the informality is significant in countries such as Bolivia and Georgia with 62.9 and 61.7% of GDP, respectively. By contrast, the shadow economy is much lower in countries such as Switzerland and the USA with an average value of 6.4 and 7.6% of GDP, respectively. Moreover, previous literature suggests that the informal sector is associated with the economic growth, although authors disagree whether the relationship is positive or negative (Asea, 1996;Goel et al., 2019;Baklouti and Boujelbene, 2019;Nguyen and Duong, 2021). Other studies reveal that the shadow economy can harm the tax revenue collection (Kodila-Tedika and Mutascu, 2014;Mazhar and M eon, 2017;Ishak and Farzanegan, 2020).
Since the shadow economy plays a crucial role in various countries worldwide, it is essential to identify factors that affect the informal sector. Findings from studies on this issue can help provide recommendations for governments to adopt appropriate policies. Many authors indicate that economic factors such as the unemployment rate, corruption, tax burden and trade openness can be associated with the size of the shadow economy (Schneider, 2005;Fugazza and Fiess, 2010;Buehn and Schneider, 2012; Maule on and Sard a, 2017).
Meanwhile, other authors suggest that it is vital to examine the impact of the psychological and social dimensions on economic phenomena (Arrow, 1970(Arrow, , 1973Torgler, 2003;Feld and Frey, 2007). Notably, Arrow (1970, p. 20) argues that there exists "a whole set of customs and norms which might be similarly interpreted as agreements to improve the efficiency of the economic system (in the broad sense of satisfaction of individual values) by providing commodities to which the price system is inapplicable." Indeed, some studies show that this dimension can significantly affect the size of the shadow economy in many countries, even after controlling different influences (Torgler and Schneider, 2009;D'Hernoncourt and M eon, 2012;Lee, 2013;Williams and Horodnic, 2015).
The study, therefore, aims to examine the effect of social capital on the shadow economy using a sample of the BRICS countries between 1995 and 2014. In addition to the primary explanatory variable, we also discover the impact of control variables, including the unemployment rate, corruption control, tax burden and trade openness. The Bayesian linear regression will be applied to investigate this relationship. The BRICS was formalized in 2006 and included five countries: Brazil, Russia, India, China and South Africa. Together, they make up more than 40% of the world's population and are important drivers for global economic growth. According to Medina and Schneider (2019), the average shadow economy size of the BRICS is approximately 27.98% of GDP during this period.
The remainder of the paper is constructed as follow. Section 2 presents the literature review. Section 3 describes the models, data, methodology and correlation analysis. In Section 4, we perform a robust check for our findings and report the empirical results. The paper ends with some concluding remarks.

Related literature 2.1 Definition and theoretical considerations
Before analyzing the effect of social capital on the shadow economy, it is helpful to mention the definition of social capital and the shadow economy. Insightful thoughts about social capital are addressed in Arrow (1999), a collection of papers on the definitions and applications of social capital. Arrow (1999, p. 4) claims that "The term 'capital' implies three aspects: (1) extension in time; (2) deliberate sacrifice in the present for future benefit; and (3) alienability." On the other hand, Leonardi et al. (2001) argue that social capital is the characteristics of social organization, including trust, norms and networks that can enhance social efficiency.
Meanwhile, although many studies have investigated this informal sector, there are disagreements about the definition of the shadow economy. According to Smith (1994, p. 18), the shadow economy can be defined as "market-based production of goods and services, whether legal or illegal, that escapes detection in the official estimates of GDP." Meanwhile, AJEB Dell'Anno (2007) suggests that the shadow economy is a nonobserved sector and contains underground production, informal production and illegal production. Medina and Schneider (2019) posit that the shadow or informal economy includes "all economic activities which are hidden from official authorities for monetary, regulatory, and institutional reasons" (Medina and Schneider, 2019, p. 4). This paper employs the definition suggested by Medina and Schneider (2019).
Moreover, to investigate the research objective, we follow a principal-agent framework based on Aidt (2003), Bjørnskov (2010) and Lee (2013). Specifically, Aidt (2003) constructs an agency model of corruption for collecting taxes. The author states that a tax collector can be willing to accept a bribe if the expected benefit is higher than the return to honest reporting. Meanwhile, Bjørnskov (2010) introduces the bureaucratic mechanism that examines the tradeoff among wages w, bribes B, an outside option, w out , and a moral cost m under an assumption that decision-makers are risk-neutral. Accordingly, bribes are likely to occur if: ð1 À λÞ ðw þ BÞ þ λw out À m > w Hence, the amount of bribe given a level of m is: Based on previous frameworks, Lee (2013) where: S is the estimated size of the shadow economy C is the intrinsic moral cost related to social capital F is the probability that an economic activity, being corrupted, is in the shadow economy The author shows that vS vF is positive and vF vC is negative, and thus vS vC is negative. This implies that if other things equal, an increase in social capital can reduce the size of the shadow economy in a country.
The above studies suggest that moral factors, including social capital, can be associated with the size of the shadow economy. We, therefore, explore this relationship for the BRICS countries in the following sections.

The determinants of the shadow economy
Various authors believe that the psychological and social dimension can have a significant impact on individual behavior as well as economic phenomena. Different empirical studies have also shown that agents care about positive/negative reciprocity, trust, fairness and distribution. For example, Fehr and Schmidt (1999) and Houser and McCabe (2014) indicate that many ultimatum agents offer between 40 and 50% of the endowment and the smaller the offer, the higher the likelihood that the proposers will reject. Likewise, Engel (2011) reports that the average offer in the dictator game experiment is approximately 28.35% of the total sum. Indeed, Thaler (2000, p. 140) claims that "As economists become more sophisticated, their ability to incorporate the findings of other disciplines such as psychology improves." The study of the determinants of the shadow economy should be no exception from the above arguments. Previous literature analyzes the effect of economic factors such as the unemployment rate, corruption, tax burden and trade openness. For example, Buehn and Schneider (2012) analyze the effect of corruption on the size of the shadow economy using data from 51 countries over the 2000-2005 period. The authors report that corruption can increase the size of the shadow economy because bureaucrats may receive a bribe to ignore activities in the informal sector. Maule on and Sard a (2017) investigate the relationship between the unemployment rates and the size of the shadow economy in different countries.
The authors indicate that this relationship is significantly relevant in Greece and Spain, where the unemployment rates are high. For countries with moderate unemployment rates, such as Germany and Italy, they are less relevant. Fugazza and Fiess (2010) reveal that while macrofounded data supports the conventional view, namely that trade liberalization can lead to an increase in the informality, microfounded data seems not to support such a view. Similarly, Birinci (2013) examines a sample of 12 advanced economies for the period from 1964:1 to 2010:4. The author indicates that the relationship between trade openness and the size of the informality is inconclusive. Schneider (2005) investigates a sample of 110 developing, transition and OECD countries. The author claims that the main drivers for the size and growth of the shadow economy are the tax burden, social security payments and the regulatory activities of the state. Likewise, Schneider et al. (2015) analyze data from the European Union countries between 2003 and 2014 and report that one major driving force of the shadow economy is the indirect taxes.
Regarding this issue, some studies have examined the motivation for citizens to comply or evade taxes. Notably, Torgler (2003) analyzes the effect of trust using data from the World Values Survey (WVS) and the Taxpayers Opinion Survey. The author indicates that trust in the officials and the legal system can enhance the tax compliance of citizens. Feld and Frey (2007, p. 104) state that individuals may consider the tax payments "as contributions to the bonum commune." Hence, if they believe that the political process is fair and the policy outcomes are legitimate, they are more willing to comply with taxes, even if their benefits from the public good are not equivalent to their tax payments. In fact, Arrow (1973) posits that there exist situations where economic agents may forgo their own profit or other benefits to achieve certain social goals, especially to avoid detrimental to other people.
On the other side, Torgler and Schneider (2009) investigate the relationship between tax morale, institutional quality and the shadow economy. The authors reveal that tax morale and institutional quality are negatively associated with the shadow economy, even after controlling various potential factors. Meanwhile, D'Hernoncourt and M eon (2012) examine the impact of social trust on the size of the shadow economy using a sample of 62 countries. They claim that social trust is adversely related to the shadow economy, and the results are robust even when controlling various influences such as economic, policy and institutional variables. The authors also show that this connection seems more significant for developing countries than for developed countries.
Likewise, Lee (2013) analyzes a sample of 65 countries between 1999 and 2007 to uncover the relationship between social capital and the shadow economy. The author posits that two important indicators of social capital are social trust and social norms (i.e. tax morale). Particularly, Lee (2013) reveals that both social trust and tax morale have a detrimental effect on the size of the shadow economy. Moreover, the study indicates that social networks are not significantly related to the size of the shadow economy. Williams and Horodnic (2015) discover the relationship between tax morale and the shadow economy in the Baltic states, namely Estonia, Lithuania and Latvia. They suggest that a decline in tax morale increases the likelihood that individuals participate in the shadow economy.
Based on the results of previous studies, we construct the following research hypotheses: H1. Social capital is negatively associated with the size of the shadow economy.
H2. Unemployment is positively associated with the size of the shadow economy.
H3. Control of corruption is negatively associated with the size of the shadow economy.
H4. Tax burden is positively associated with the size of the shadow economy.
H5. Trade openness is positively associated with the size of the shadow economy.
3. Methodology and model specification 3.1 Model To test the hypotheses, we propose the following model: In this paper, social capital is the primary explanatory variable. As suggested by Lee (2013), we use two measures of social capital, which are social trust and tax morale. Thus, model 1 is written into models 2 and 3 as follows: Social trust is the respondents' response rate who answered that "most people can be trusted". D'Hernoncourt and M eon (2012), Lee (2013) use social trust in their studies. Meanwhile, tax morale is measured by societal attitudes toward tax evasion, which is the proportion of respondents who answered that cheating on taxes could not be justifiable. Torgler and Schneider (2009), Enste (2010) and Lee (2013) use tax morale in their empirical analysis. We collect social capital data from the WVS, which provides data on cultural, social and political changes worldwide. In these surveys, the survey question about social trust is "Generally speaking, would you say that most people can be trusted or that you need to be very careful in dealing with people?" The possible answers are "Most people can be trusted", "Don't know" and "Can't be too careful." In this paper, we use the percentage of people who agree with the statement "Most people can be trusted" to measure the BRICS countries' social trust.
Meanwhile, the social attitudes toward cheating on taxes are measured from 1 to 10, where 1 indicates that cheating on taxes is never justifiable and 10 is always justifiable. A general question to assess tax morale is "Please tell me for each of the following statements whether you think it can always be justified, never be justified, or something in between:. . . Cheating on tax if you have the chance." As Torgler and Schneider (2009) suggested, we will re-encode the scale for tax morale variables to facilitate discussion (i.e. the value of 10 becomes 1 and vice versa). Thus, the value of 1 means that cheating on tax is always justified, and 10 indicates that tax fraud is never justifiable (see Table 1).
The shadow economy represents a percentage of the official GDP. We collect data on the size of the shadow economy from Medina and Schneider (2019). In this study, Medina and Schneider (2019) use multiple indicator-multiple causes (MIMIC) to estimate the size of the shadow economy.
Moreover, we consider a broad set of control variables commonly used in the shadow economy literature, namely unemployment, control of corruption, tax burden and trade Social capital and the shadow economy openness (Schneider, 2005;Torgler and Schneider, 2009;Fugazza and Fiess, 2010;Buehn and Schneider, 2012;Lee, 2013; Maule on and Sard a, 2017). Notably, unemployment is the unemployment rate, calculated as a percentage of the total labor force. We obtain this index from the World Bank's World Development Indicators (WDI). Control of corruption is measured by the Corruption Perceptions Index (CPI). The CPI defines corruption as "the abuse of public power for self-interest." (Heywood, 2014). The CPI is collected from Transparency International's annual report.
Meanwhile, the tax burden is a measure of the tax burden imposed by the governments and calculated as a percentage of GDP. We collect tax burden data from the Heritage Foundation. Finally, trade openness is the sum of imports and exports normalized by GDP. Data on trade openness is obtained from the WDI of the World Bank.

Methodology
This paper employs the Bayesian linear regression to estimate the effect of social capital on the size of the shadow economy in the BRICS countries. Unlike the frequentist approach, the Bayesian approach assumes that all parameters are random (Kruschke et al., 2012;Thach, 2021). The Bayesian inference is performed based on the Bayes' theorem.
pðθjDÞ ¼ pðDjθÞ pðθÞ= pðDÞ where D is the observed data and θ is a vector of parameters in the descriptive model. The posterior distribution, pðθjDÞ, specifies the relative credibility of every combination of parameters given the data. pðDjθÞ is the marginal likelihood function, pðθÞ is the prior distribution of the hypothesis and pðDÞ is the probability of the data. Because we are facing a small sample of data (the observations is 100), the use of uninformative a priori can cause a Type I error or a Type M error. Therefore, to obtain a balanced posterior regression model, Lemoine (2019) suggests using weak information a priori. Specifically, we perform simulations with a generic weakly informative prior: normal (0, 1). Normal prior distribution has been used in previous studies Duong, 2021, 2022;Oanh et al., 2022). Furthermore, the Markov chain Monte Carlo (MCMC) method using the Gibbs sampling algorithm will determine the posterior distribution. Finally, the Bayesian inference is robust if the MCMC converges (Cowles and Carlin, 1996) and the sampling is effective Duong, 2021, 2022;Thach, 2021

Descriptive statistics
We collect panel data of five BRICS countries from 1995 to 2014. Table 2 presents the descriptive statistics of the shadow economy and the primary independent variables (social capital), that is, social trust and tax morale, for all countries in the sample. Among the BRICS countries, the size of the shadow economy in China is the lowest, that is, 13.94% of GDP, while Russia has the largest size of the shadow economy (39.59% of GDP). For social capital variables, China is a country with the highest social trust (53.13%) and the highest tax morale (8.17). By contrast, Brazil has the lowest social trust (6.37%) as well as the lowest tax morale (6.75). Table 3 describes the Pearson Bayesian correlation matrix among the variables in the research models. Moreover, we present the 95% credible intervals of the correlation coefficient in parentheses. The results indicate that the size of the shadow economy and social capital are inversely correlated. The correlation coefficient between the shadow economy and social trust is À0.717, while the correlation coefficient between the shadow economy and tax morale is À0.642. Also, Table 3 shows a positive correlation between the size of the shadow economy and the unemployment rate and between the size of the shadow economy and tax burden. In contrast, the relationships between the size of the shadow economy and control of corruption and between the size of the shadow economy and trade openness are negative.

Convergence tests
In the Bayesian analysis, the main concern to consider with MCMC sampling is convergence (Gelman and Rubin, 1992;Kruschke et al., 2012;Thach, 2021 Table 3.

Bayesian correlation matrix
Social capital and the shadow economy convergence diagnostic using Gelman-Rubin's Rc statistics (Gelman and Rubin, 1992), a standard test that applies multiple chains. It compares variance between chains and variance within chains. Table 4 reports that Gelman-Rubin's Rc statistics of all parameters in models 2 and 3 are lower than 1.2; therefore, the Markov chains converge to the target distribution.

Effective sample size
Along with convergence, we also consider stability when performing the Bayes regression analysis. The stability of the estimates is determined by the effective sample size (ESS). It reports the total number of independent MCMC samples from the total number of MCMC samples. As this value approaches the sum, the autocorrelation will become lower, which indicates the estimates' stability and accuracy. Table 5 shows the lowest ESS of 26,333 and the highest ESS of 30,000 for the posterior parameters. Simultaneously, all parameters' efficiency indexes are above 0.01. Hence, the MCMC algorithm achieves sampling efficiency (Roberts and Rosenthal, 2001).
The above results on the convergence diagnostics and the ESS test reveal that the Bayesian inference is stable. Table 6 presents the posterior simulation results of models 2 and 3 using the Bayesian linear regression method. Notably, we find a strong relationship between social capital, represented by social trust and tax morale, and the size of the shadow economy. Hence, the core hypothesis H1 is validated for the BRICS countries.

Parameter estimates
For model 2, the mean coefficient of social trust is À0.21657, and the negative probability effect of social trust on the shadow economy is 100%. These figures suggest robust evidence of the negative relationship between social trust and the shadow economy. In other words, higher social trust will decrease the size of the shadow economy. This finding is in accordance with studies by D'Hernoncourt and M eon (2012) and Lee (2013). For model 3, the posterior simulation results report that the mean coefficient of tax morale is À0.71465, and the negative probability impact of tax morale on the shadow economy is 90.87%. These figures indicate a strong negative relationship between tax morale and the shadow economy. They imply that when the social norms (i.e. the tax morale) become higher, the size of the shadow economy will be smaller. This finding is consistent with the submissions of Torgler and Schneider (2009), Lee (2013), andWilliams andHorodnic (2015).
In this regard, Torgler (2003) and Feld and Frey (2007) claim that increased trust can enhance tax compliance. Meanwhile, Lee (2013) posits that social trust and tax morale have a more significant impact on the shadow economy than taxes and regulations. Consequently, to reduce the shadow economy, there should be improvements in individuals' tax morale using education and enhancing consciousness about the benefits of paying taxes. Moreover, we suggest that governments should ensure that tax laws and the political process are implemented fairly and reliably. As citizens' confidence in the legal system improves, they are more likely to reduce tax avoidance.
Meanwhile, the mean coefficients of the unemployment rate in models 2 and 3 are 0.66870 and 0.89245, respectively. The positive effect probability of unemployment on the size of the shadow economy is 100%. The results uncover that the unemployment rate significantly affects the size of the shadow economy. This finding supports the hypothesis H2 suggested in Section 2.
Likewise, the posterior simulation results in models 2 and 3 indicate a strong positive relationship between the tax burden and the size of the shadow economy. Notably, their mean coefficients are 0.65582 and 0.71253, respectively, while the posterior probability indicators in both models are 100%. The results posit that the higher tax burden will increase the size of the shadow economy in the BRICS countries. Thus, the hypothesis H4 is also valid for the countries examined. Note(s): Dependent variable: Shadow economy; As priors for the effects of the independent variables, we use normal distributions with a mean of 0 and a standard deviation of 1; 95% credible intervals in parentheses; * Probability of coefficient > 0; ** Probability of coefficient < 0 Social capital and the shadow economy By contrast, good corruption control reduces the size of the shadow economy in the BRICS. Specifically, the mean coefficients of control of corruption in models 2 and 3 are À0.45054 and À0.49607, respectively. This negative impact is strong since the posterior probabilities in both models are 100%. Therefore, we document that for these countries, the hypothesis H3 is also validated. Moreover, we suggest that the governments should focus more on controlling corruption because good corruption control can increase citizens' trust and help decrease the size of the shadow economy.
Finally, the mean coefficients of trade openness in models 2 and 3 are negative (À0.17005 and À0.29451, respectively). The posterior probability of this factor in model 2 and 3 is 99.94% and 100%, respectively. These numbers reveal that the negative relationship between trade openness and the size of the shadow economy is strong. In other words, an increase in trade openness will reduce the size of the shadow economy in the BRICS countries. Hence, the hypothesis H5 is rejected. As mentioned in the literature review, there is still controversy about this relationship. This implies that further research is needed to assess the impact of trade openness on the shadow economy.

Concluding remarks
The paper aims to examine the impact of social capital on the shadow economy in the BRICS countries over 1995-2014. The models also contain control variables: unemployment rate, corruption control, tax burden and trade openness. In particular, we use the Bayesian linear regression method with the prior distribution as the normal distribution and the posterior distribution through the MCMC technique.
The study measures social capital through social trust and tax morale, as Lee (2013) suggested. The results indicate that social trust and tax morale have a negative effect on the size of the shadow economy. These relationships are strong since their posterior probability indicators are 100% and 90.87%, respectively. Based on the findings, we suggest that governments concentrate more on these social dimensions when controlling the shadow economy.
Furthermore, we find that the unemployment rate and tax burden can promote the development of the shadow economy in the BRICS countries. Their positive effect probabilities of these factors are 100% in both research models. On the contrary, the control of corruption and trade openness adversely influence the size of the shadow economy. These negative relationships are strong in both research models.
Finally, the study is not free from limitation. This paper concentrates on the impact of social capital on the size of the shadow economy in the BRICS. Nevertheless, a comparison between groups of countries provides a more comprehensive picture of the shadow economy worldwide. Moreover, social capital is a multidimensional concept that may contain different components. This difficulty in measuring social capital calls for further research. Analyzing the impact of other dimensions of social capital on the shadow economy can be seen as a complement to shed light on this relationship.