Explaining central government’s tax revenue categories through the Bradley-Terry Regression Trunk model

Abstract

The Bradley-Terry Regression Trunk (BTRT) model combines the log-linear Bradley-Terry model, including subject-specific covariates, with a particular tree-based model, the so-called regression trunk. It aims to consider simultaneously the main effects and the interaction effects of covariates on data expressed as paired comparisons. We apply this model to financial data expressed as rankings and then transformed into paired comparisons. Tax revenues differentiated by category represent the statistical units of the analysis (i.e., taxes on income, social security contributions, taxes on property, and taxes on goods and services). We combine data from OECD, World Bank, and IMF databases for the year 2018 to investigate the effect size of socio-economic covariates and their interaction on the composition of tax revenues for a set of 100 countries worldwide. We also present a comparison with a more established method proposed in tax determinants literature and with two alternative models used for matched pairs. Finally, we discuss the implications of reported results for stakeholders and policymakers.

Keywords:

• Public finance
• cross-country analysis
• tax determinants
• rankings
• ordinal data
• paired comparisons
• regression tree
• STIMA

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1 Introduction

Public finance studies the government’s role in the economic system (Gruber, 2004), emphasizing the importance of maintaining a balanced budget by managing the gap between public revenues and expenditures. This balance became crucial after the 2007 financial crisis in the US, which later escalated into a sovereign debt crisis. The objective of public authorities is to achieve a balance between revenues and expenditures to ensure positive outcomes and prevent negative ones (Jain, 1989). Examining revenues and expenditures separately shows their effect on a country’s Gross Domestic Product (GDP): government spending directly affects GDP, whereas taxation has a contrasting effect. Nonetheless, the complex interaction between tax revenues and public spending with economic factors is not fully understood. Some studies have investigated the relationships between taxation and government expenditure. Manage and Marlow (1986) demonstrated that taxation leads to expenditure at the state level, with this causality becoming bidirectional in the short term. Anderson et al. (1986) found that government spending leads to increased taxation.

The marked interest in understanding the determinants of tax revenues led to the pivotal question of the sources of variance in tax revenues among countries. Kaldor (1963) observed that tax revenues are lower in underdeveloped countries than in developed ones, a disparity attributed to the capability of populations to pay taxes from their income surplus beyond basic subsistence needs (Boukbech et al., 2019). Emerging economies, therefore, face limitations in converting national income into tax revenues to fulfill community needs without inciting severe social unrest. It follows logically that a country’s ability to generate tax resources is positively correlated with its level of development, suggesting that more developed nations have a higher capacity to levy taxes (Brun and Diakite, 2016).

Within the discussed context, we present a modeling approach to analyze the relationship between tax revenues and government expenditure, focusing on tax revenue components and the diverse characteristics of countries. We employ the Bradley-Terry Regression Trunk (BTRT) model (Baldassarre et al., 2022) to examine the impact of socio-economic covariates and their interactions on tax revenue composition across 100 countries globally. The most significant covariates are identified using the Bradley-Terry-Luce Lasso method and are utilized in the BTRT model to group countries based on the BTRT’s output. BTRT facilitates the incorporation into the model of country-specific fiscal system heterogeneity and covariate interactions. Notably, the Environmental Performance Index (EPI) is employed as an indicator of a country’s development level and economic growth. Our findings reveal that EPI significantly influences tax revenue composition by interacting with government gross debt, health expenditure, and employment rate. The BTRT algorithm selects EPI as the primary splitting covariate. This interaction notably enhances the model’s fit, highlighting the critical role of EPI in delineating tax revenue structures.

The paper is organized as follows. Section 2 presents previous literature. Section 3 presents the original data set. Section 4 reports the key features of both the BTRT model and algorithm. Section 5 focuses on variable selection. Empirical results are shown in Section 6. Next, the BTRT model is compared with a more established model in tax determinants literature (Section 6.1), as well as with both the basic log-linear Bradley-Terry model without subject-specific covariates and the Bradley-Terry tree model (Section 6.2). Finally, Section 7 ends the paper with final remarks and advices to policymakers.

2 Background and motivations

The relationship between socioeconomic explanatory variables and tax revenues (response variable) has been investigated by analyzing cross-sectional or panel data. The goal has been to find the determinants of tax revenues. Main findings suggest the most important explanatory variables are the per capita GDP (Gupta, 2007; Pessino and Fenochietto, 2010), the peculiarities of the industries reproduced in the sectoral composition of GDP (Chelliah, 1971; Chelliah et al., 1975; Tait et al., 1979; Piancastelli, 2001; Karagoz, 2013), external factors like the levels of Foreign Direct Investment (FDI), trade (Cassou, 1997; Gupta, 2007; Bird et al., 2008), and public debt (Teera and Hudson, 2004), as well as policy makers’ choices like exchange rate, inflation rules (a.k.a. Keynes-Oliveira-Tanzi effect) and financial-fiscal policies (Tanzi, 1989; Brun and Diakite, 2016). Other studies analyze the role of government efficiency and institutional factors like political stability and political and civil rights (Bird et al., 2008; Martin and Uribe Teran, 2010). On the social side, the impact of the educational level (as a share of public expenditure on education), illiteracy rate, and population growth on tax revenues have been studied (Wallace and Bahl, 2005; Piancastelli, 2001; Pessino and Fenochietto, 2010). Accountability and civil and political rights are also considered determinants of tax revenues (Bird et al., 2008; Martin and Uribe Teran, 2010). Lastly, corruption, entry regulations, and the rule of law play an important role in explaining differences in tax revenues (Bird et al., 2014).

Different methodological approaches are utilized for similar purposes. Feltenstein and Cyan (2012) apply the dynamic general equilibrium model, whilst other studies use econometric techniques. Lotz and Morss (1967) proposed the first cross-sectional study on international tax ratios: the tax effort concept is introduced and it is demonstrated that per capita income and trade share are determinants of the tax share. Studies analyzing panel data utilize static fixed and random effect models based on the generalized method of moments for dynamic panels. Apart from these, Pessino and Fenochietto (2010) introduces a panel version of a stochastic tax frontier model.

As for the sample size, Teera and Hudson (2004) and Pessino and Fenochietto (2010) consider large samples of countries selected for the geographical location or the income level. Other studies are based on a restricted sample of countries. Based on the idea that the research for tax determinants may be inadequate if a heterogeneous group of countries is observed, as the composition of tax revenues can be different in low, middle, and high-income countries, Castro and Camarillo (2014) analyze the 34 countries that are part of the Organization for Economic Co-operation and Development (OECD) only and, for them, consider lagged values of the tax revenues over the 10 years 2001-2011.

Our empirical analysis is aimed at finding the tax revenue determinants taking into account the heterogeneity of countries, with a focus on the relationship between tax revenues and government expenditure. The design of the empirical study is peculiar and differs from those previously presented. Following the OECD classification, we study the determinants of tax revenues by decomposing them into four categories: taxes on income, social security contributions - SSC, taxes on property, and taxes on goods. This decomposition-based approach represents one of the major advantages of this analysis: the disaggregation of tax revenues allows the identification of different effects of the same covariate on different tax revenue categories. A single socio-economic characteristic could affect in opposite direction the size of two or more different tax revenue categories. In addition, the tax categories are paired compared on their size, which means that the tax revenue categories are first ordered according to their size and next compared to each other. To quantify for each country the effects of socioeconomic covariates on the size of each specific tax revenues category, the variable tax revenues is re-coded into a set of categorical variables (i.e., tax revenues are transformed into rankings and next into paired comparisons) to apply the Bradley-Terry model for matched pairs (Bradley and Terry, 1952) using the log-linear formulation with subject-specific covariates introduced in Dittrich et al. (1998).

The use of paired comparisons is reasonable when comparing countries with different fiscal systems. The OECD tax revenues classification defines a common ground where data from different countries are compared. In line with this, paired comparisons work on sizes, and not on numerical values of tax revenues, thus facilitating the comparability between countries. Most of the studies on tax revenue determinants use panel data since they assume that tax revenues change over time. Since the order of tax revenues by categories is more stable, our study utilize cross-sectional data and estimates tax determinants through paired comparisons in a cross-country dataset for a single year. We integrate data from OECD, International Monetary Fund (IMF), and the World Bank, and focus on 2018. The heterogeneity among countries is included in the BTRT model also concerning the covariates: their values are expressed as a percentage of GDP. Government expenditure is decomposed following the Classification Of the Functions Of Government (COFOG) to capture the effect of each type of government expenditure on each category of tax revenues.

To model possible interaction effects in addition to the main effects we resort to BTRT. It allows us to define interaction effects when no a-priori hypotheses are available, and produces a small tree, called a trunk, representing a fair compromise between a straightforward interpretation of the interaction effects and an easy-to-read partition of countries based on their socioeconomic characteristics and the order of their tax revenues. Thus, our study focuses on finding interaction effects on tax determinants in a data-driven manner.

Few works in public finance studies focus on paired comparison data. Dittrich et al. (2000) analyze rank-ordered preference through Bradley-Terry type models with categorical subject-specific covariates. The (complete) rankings are transformed into paired comparisons on which a tree-based log-linear model is estimated. Data partitioning considers model deviance and a forward selection and backward elimination procedure. This approach is well suited for hypothesis-based modeling. However, when no a priori hypotheses are known it requires the arbitrary introduction of higher-order interactions. Instead, Strobl et al. (2011) propose a tree-based classifier where paired comparisons are treated as response variables in a Bradley-Terry tree (BTtree) model. BTtree defines interactions when no a priori hypothesis is known based on a model-based recursive partitioning approach (Zeileis et al., 2008): Splits are selected assessing the instability of the parameters of the basic Bradley-Terry model. The tree returns in each terminal node the preference scales deriving from the object-related parameters. Nevertheless, BTtree lacks of information about how the subject-specific covariates affect the judges’ preferences. This semi-parametric model does not return beta coefficients neither for the main effects nor for the interaction effects.

BTRT is preferable to BTtree and Dittrich’s model as it accounts for the data-driven definition of interaction effects between subject-specific covariates. It results in a completely parametric model: each covariate that enters the main effect part of the model and each interaction effect has an associated regression parameter. Thus, the interpretation of results is highly facilitated as it derives both from the inspection of the tree and the interpretation of the estimated regression parameters.

3 Data

We consider 100 countries and their associated tax revenues by category for the year 2018. To ensure consistency across countries and granularity of tax revenue categories (OECD, 2018a), data are taken from the Global Revenue Statistics Database (OECD, 2018b), which utilizes the OECD classification of taxes. Taxes are classified by their base in: income and profits (heading 1000), compulsory Social Security Contribution (SSC, head. 2000), payroll and workforce (head. 3000), property (head. 4000), goods and services (head. 5000), other taxes (head. 6000). All categories are expressed in terms of the level of taxation through the tax-to-GDP ratio, computed by the ratio of the nominal tax revenue of a country and its nominal GDP for the year. This ratio is used in cross-country research studies where the aim is to compare tax levels across countries with different degrees of development.

The numerical variables of tax revenue categories are transformed into rankings assigning values from 1 to 6 to each tax category: the category generating the highest (lowest) tax revenue is assigned a value of 1 (6). Rankings are first transformed into paired comparisons between tax categories and next into consensus ranking, which represents the best compromise between a set of rankings and corresponds to that ranking in the permutation space that maximizes the sum of correlations between itself and all the other rankings. Operationally, consensus rankings maximize the extended correlation coefficient τ_x (Emond and Mason, 2002). The R package $\mathrm{ConsRank}$ (D’Ambrosio et al., 2019) is used to compute τ_x , and the result obtained for the tax categories is: (goods and services > income and profits > compulsory SSCs > property > workforce $=$ other taxes). The categories taxes on the workforce and other taxes are both ranked in the last position. In the original data, in most cases, they present values equal to 0. For this reason, we do not consider them. Thus, our analysis covers 100 countries and 4 tax revenue categories.

Initial data on tax revenues are integrated with other socio-economic covariates, in prevalence expressed in terms of GDP, that explain variations in tax revenues and are obtained by combining data from IMF, OECD, and World Bank databases. These covariates are (see Table 9 in the Appendix): current account balance; employment and unemployment rate; government gross debt; government net lending/borrowing; gross fixed capital formation; change in the gross domestic product (in %); gross national savings; the volume of exports and imports of goods and services; account ownership at a financial institution; subsidies and other transfers; interest payments; compensation of employees; value-added of agriculture, services, industry, and manufacturing; population density; final consumption expenditure; banking non-performing loans; claims on central government; households consumption; government consumption; trade volume; environmental performance index; government expenditure for military, education, health, and other (as a residual category). The last covariate is obtained by subtracting military expenses, education expenses, and healthcare expenses from the total government expenditure and is intended as a proxy of the other public expenditure items included in the COFOG classification.

The variable is location (OECD database) is used to assign the H = 100 countries to four geo-economic areas: OECD countries (36), Africa (29), Asia (15), and South America (21). This variable is used to impute missing values for the other covariates. Missing values are replaced with the median value of the covariate computed in the specific area (see Table 9 in the Appendix). Observed countries for each area are displayed in the Appendix (Fig. A1 to A4) together with the tax revenues composition for each country.

4 Road to the Bradley-Terry Regression Trunk (BTRT)

Bradley and Terry (1952) proposed a method to derive a latent preference scale from paired comparison data when no natural measuring scale is available. This model has been successfully used in Dittrich et al. (2006), Choisel and Wickelmaier (2007) and Rodriguez Montequin et al. (2020).

Hereinafter, the terms rankings and orderings are used interchangeably. Rankings are numerical vectors that assign values to objects based on their size. Usually, they represent preferences expressed by H individuals (or judges). In our study, they are intended as orders of magnitude of tax revenues differentiated by tax category for H countries. Instead, orderings are nominal lists of objects that express the same concept as rankings.

The paired comparison method splits the ordering process into a series of evaluations carried out on two objects at a time. To address the problem of determining the scale values of a set of objects on a preference continuum that is not directly observable, each pair is compared and a decision is made based on which of the two objects is preferred. When the number of objects is n_o , the number of paired comparisons is equal to $\left(\begin{array}{c}{n}_o\\ 2\end{array}\right)$.

The basic Bradley-Terry (BT) model computes the probability ${\pi }_{(ij)i}$ that an object i is ranked above another object j (Agresti, 1990) as(1) $${\pi }_{(ij)i}=\frac{{\pi }_i}{{\pi }_i+{\pi }_j},$$(1) where π_i and π_j are non-negative worth parameters describing the location of objects on the overall ranking scale. When ${\sum }_{i=1}^{n_o}{\pi }_i=1$ is imposed, π_i and π_j can be regarded as probabilities: for example, π_i is the probability that the object i is ranked above all the other objects by considering all the rankings expressed by or related to H individuals.

The BT model is expressed as a quasi-symmetry logistic model with parameters ${\lambda }_i^O$ (2) $$\mathit{logit}({\pi }_{(ij)i})=\hspace{0.17em}\log \hspace{0.17em}\left(\frac{{\pi }_{(ij)i}}{{\pi }_{(ij)j}}\right)={\lambda }_i^O-{\lambda }_j^O,$$(2) where ${\lambda }_i^O$ and ${\lambda }_j^O$ are object-parameters related to π_i ’s in Eq. (1) by ${\lambda }_i^O=\frac{1}{2}\ln \hspace{0.17em}({\pi }_i)$. The superscript O refers to object-specific parameters. The estimated probability in Eq. (2) is computed as ${\widehat{\pi }}_{(ij)i}=\frac{\hspace{0.17em}\exp \hspace{0.17em}({\widehat{\lambda }}_i^O-{\widehat{\lambda }}_j^O)}{1+\hspace{0.17em}\exp \hspace{0.17em}({\widehat{\lambda }}_i^O-{\widehat{\lambda }}_j^O)}$, and ${\widehat{\pi }}_{(ij)i}=\frac{1}{2}$ if ${\lambda }_i^O={\lambda }_j^O$ (the two objects are tied). Eq. (2) is equivalent to Eq. (1), and the identifiability of these two models requires a restriction on the parameters related to the last object n_o , such as ${\lambda }_{n_o}^O=0$ and ${\sum }_i^{n_o}{\pi }_i$ = 1.

Sinclair (1982) extended Eq.(1) introducing a log-linear version (LLBT): in comparing object i with object j, the Poisson-distributed random variables $y_{(ij)i}$ and $y_{(ij)j}$ are the number of times in which the objects i and j are ranked above the other objects, respectively. In total, $2\left(\begin{array}{c}{n}_o\\ 2\end{array}\right)$ expected counts are estimated in a design matrix that is used to synthesize the information about all the considered rankings. Sometimes, it is reasonable to assume that the rankings associated with judges or individuals depend on their features. For example, the preferences expressed by judges usually depend on their characteristics. This issue characterizes the present study as well: the tax revenues composition of a country h and their order is associated with the country’s socio-economic structure. In the BT model literature, these features are called subject-specific covariates to differentiate them from the object-specific covariates and subject-object-specific covariates (Schauberger, 2015).

The LLBT model accounts for multiple subject-specific covariates. The BT model can include categorical or continuous covariates. Considering numerical covariates only, as in our application, the LLBT for the h-th judge and objects i and j is(3) $$\hspace{0.17em}\log \hspace{0.17em}m\left(y_{(ij)i,h}\right)={\mu }_{ij,h}+y_{(ij)i,h}({\lambda }_{i,h}^O-{\lambda }_{j,h}^O).$$(3)

LLBT with subject-specific covariates requires a contingency table for each judge, or individual, and each covariate value; ${\lambda }_{i,h}^O$ can be re-parameterized through a linear formulation(4) $${\lambda }_{i,h}^O={\lambda }_i^O+\sum _{p=1}^P{\beta }_{ip}{x}_{p,h},$$(4) where $x_{p,h}$ is the continuous covariate x_p $(p=1,\dots ,P)$ observed for individual h, and β_ip is the effect of x_p on the location of the object i in a ranking h. The intercept ${\lambda }_i^O$ is the location of the object i in the overall consensus ranking, which includes the rankings associated with all the H individuals.

The deviance of the model is computed as the deviance of a fitted Poisson regression(5) $$D=2\sum _{h=1}^H{y}_{ij,h}\times \hspace{0.17em}\log \hspace{0.17em}\left(\frac{y_{ij,h}}{{\widehat{y}}_{ij,h}}\right),$$(5) where $y_{ij,h}$ is the value of the comparison ij for a judge h, and ${\widehat{y}}_{ij,h}$ is the predicted value based on the estimated model parameters.

4.1 Combining STIMA, regression trunk, and BTRT

The LLBT model with subject-specific covariates is expressed for ordinal data through a regression model for paired comparisons. The aim is to estimate in a data-driven way the main effects part of the model and, if present, the interaction effects part by implementing the Simultaneous Threshold Interaction Modeling Algorithm (STIMA) (Dusseldorp et al., 2010; Conversano and Dusseldorp, 2017) within the BTRT model. STIMA utilizes the regression trunk methodology (Dusseldorp and Meulman, 2004). Interaction effects are considered when two or more covariates do not combine additively (de González and Cox, 2007), or when they have a combined effect over and above their additive combination (Cohen et al., 2022, p.257). STIMA simultaneously estimates both main and interaction effects by growing a small tree called a regression trunk, which is next pruned to avoid overfitting. The covariates can interact with each other so that the interaction effects are considered as a particular kind of non-additivity. The STIMA’s recursive partitioning algorithm is based on the integration of Generalized Linear Models (GLM) (McCullagh and Nelder, 1989) and Classification And Regression Trees (CART) (Breiman et al., 1984).

The BTRT model combines the LLBT model with subject-specific covariates (Eq. 4) with STIMA, when the response derives from the outcome of paired comparisons. The estimated object-parameters ${\widehat{\lambda }}_{i,h}$ for the object i and the individual h in the node t are¹ (6) $${\widehat{\lambda }}_{i,h}={\widehat{\lambda }}_i+\sum _{p=1}^P{\widehat{\beta }}_{i,p}{x}_{p,h}+\sum _{t=1}^{T-1}{\widehat{\beta }}_{i,P+t}I\{(x_{1,h},\dots ,x_{P,h})\in t\}$$(6)

The main effect part is ${\sum }_{p=1}^P{\widehat{\beta }}_{i,p}{x}_{p,h}$, whilst ${\sum }_{t=1}^{T-1}{\widehat{\beta }}_{i,P+t}I\{(x_{1,h},\dots ,x_{P,h})\in t\}$ are the interaction effects estimated for individuals in each terminal node T. The interaction terms are T – 1 since one terminal node is treated as the reference group. The intercept ${\widehat{\lambda }}_i$ quantifies the overall location of the object i for all the individuals of the trunk. To estimate the intercept in each terminal node it is necessary to add ${\widehat{\beta }}_{i,P+t}$ to ${\widehat{\lambda }}_i$. The overall number of parameters includes the number of intercepts λ_i , plus all the main effects ${\beta }_{i,P}$ and the interaction effects ${\beta }_{i,P+t}$. Since one object and one terminal node are considered as reference categories, the number of model parameters is $(n_o-1)+[P\times (n_o-1)]+[(T-1)\times (n_o-1)]$.

4.2 The BTRT algorithm

The BTRT algorithm, described in Algorithm 1, follows the STIMA algorithm with some adjustments due to pairwise comparisons.

**** Algorithm 1 about here ***

First, a design matrix composed of $n=n_o\times (n_o-1)\times H$ rows is defined. Indeed, the total number of rows is $2\times (n_o\times (n_o-1)/2)\times H$, i.e., the product between the number of comparing objects, that is 2 when ties are not allowed, the number of paired comparisons $(n_o\times (n_o-1)/2)$, and the number of individuals H. If ties are considered, the model can be extended by incorporating undecidedness parameters. Table 1 is an example of a design matrix with two subjects, three objects, and one continuous subject-specific covariate.

**** Table 1 about here ***

A generalized linear model with a log link and Poisson distribution is fitted to the root node to compute the main effects β_p . The splitting procedure is applied to the root node, and the resulting child nodes, by searching for a threshold interaction between the P covariates assuming they have a combined effect on the objects’ order besides their individual (main) effect. The best candidate split maximizes the decrease in deviance of a Poisson regression model when moving from a parent node to the two possible child nodes. Once found, the best split defines a dichotomous variable $z_{\mathit{ijp},t}^{*}$ that updates the indicator function $I(·)$ in Eq. (6). Next, all regression coefficients are re-estimated. The number of dichotomous variables $z_{\mathit{ijp},t}^{*}$ is equal to the number of splits leading to node t.

Two main approaches to growing the BTRT trunk are presented in Baldassarre et al. (2022): One Split Only (OSO) and Multiple Splitting (MS). In OSO, all the covariates are used as splitting covariates only once. In MS, each covariate can be used more than once to split a node. Generally, MS performs better than OSO in reducing deviance, although OSO is more appropriate when the goal is obtaining an easier-to-read partition. For our analysis, results suggest MS represents a fair compromise between goodness-of-fit and interpretation. The selected trunk shown in Sec. 6 presents few splits and fits the data satisfactorily.

The BTRT stopping criterion is based on the prior definition of a minimum number of individuals (minimum bucket size) in a node. In the default BTRT implementation, it is set to 5 but it can be changed based on the desired depth of the trunk.

Once the entire trunk is grown, pruning based on V-fold cross-validation and the $c·SE$ rule is applied. The cross-validation deviance D^cv for a Poisson distribution and its associated standard error SE^cv are computed in each step of the splitting procedure². D^cv initially follows a decreasing trend and then begins to grow after a certain number of splits. The $c·SE$ rule works as follows: Let $t^{*}\in [1,T]$ be the size of the regression trunk with the lowest D^cv , say $D_{t^{*}}^{cv}$. The best size of the BTRT trunk $t^{**}$ corresponds to the minimum value of t such that $D_{t^{**}}^{cv}\le D_{t^{*}}^{cv}+c·S{E}_{t^{*}}^{cv}$, where $c\in [0,1]$. The optimal choice of the pruning parameter c has been investigated in Baldassarre et al. (2022): the proportion of times BTRT captures a true interaction effect is higher than 0.8 when $0.5<c\le 1$, and it is advised to increase c as the number of objects, rankings, and the effect size of the covariates on the orderings increase. In general, the higher the value of c, the more weight is given to the standard error so that the selected trunk has a reduced size. In this case, since BTRT includes main effects and interaction effects within the same model, it would be advisable to choose conservative values of c (i.e., $0.5<c\le 1$).

5 Bradley-Terry-Luce Lasso for covariates selection

We consider a wide set of covariates as initial candidate variables for BTRT. However, one of the main drawbacks of applying the BT model with subject-specific covariates is the number of parameters to estimate. This drawback is accentuated when the BTRT model is used: in addition to the intercepts for each tax category, also main effects and interaction effects between covariates are included. This issue is solved by following a penalized likelihood approach instead of ordinary ML estimation. In this way, it is possible to first consider a high number of candidate covariates, and next, reduce the number of involved parameters by selecting the relevant ones only.

Lasso regression (Tibshirani, 1996) is commonly used for variable selection in presence of numerical covariates. In case of paired comparisons the penalty terms used in standard Lasso have to adapt the specific structure of the model (Schauberger and Tutz, 2017), and the method for variable selection that adapts to paired comparison is the Bradley-Terry-Luce Lasso (BTLL) (Schauberger, 2015). We use BTLL to select the most influential country-specific covariates to be included in BTRT. In BTLL, the BT model is reinterpreted according to Luce’s axiom (Luce, 1959) of independence from irrelevant alternatives. It states that other objects do not influence the decision between two objects. BTLL is based on the maximization of the penalized log-likelihood:(7) $$l_p(\xi )=l(\xi )-\Lambda J(\xi ),$$(7) where $l(\xi )$ is the log-likelihood of the vector of model parameters $(\xi ),J(\xi )$ is a penalty term and Λ is a tuning parameter that quantifies the penalty degree. If Λ = 0, $l_p(\xi )$ reduces to the standard maximum likelihood estimate. In BTLL, the penalty for the set x_p of P subject-specific covariates that share the same effect size on the response is(8) $${\varphi }_P({\beta }_{i,1},\dots ,{\beta }_{i,P})=\sum _{p=1}^P\sum _{i<j}|{\beta }_{i,p}-{\beta }_{j,p}|$$(8)

If $\Lambda \to \infty$, all the effect sizes of x_p are merged into a single one, and x_p is removed from the model as all its effect sizes tend to zero. Instead, when Λ = 0 BTLL reduces to BT with subject-specific covariates. In general, for a finite value of Λ some of the covariates are removed, whilst the remaining are still identified in the model. The penalty terms ${\varphi }_P$ can be weighted based on the adaptive lasso principle (Zou, 2006), and next combined as(9) $$J(\xi )=\sum _{l=1}^L{\varphi }_l{P}_l,$$(9) where ψ_l represents penalty-specific weights. Thus, all penalty terms P_l are combined into a unique penalty $J(\xi )$ controlled by the tuning parameter Λ. The comparability of different penalty terms requires two conditions: 1) all subject-specific covariates have to be scaled to allow the comparison of their effect sizes; 2) the weights ψ_l have to be assigned according to the number of penalties and free parameters they include (Bondell and Reich, 2009; Oelker et al., 2015). The penalty is interpreted as a Lasso-type fusion penalty rather than a simple Lasso (Schauberger and Tutz, 2017). Similar approaches have been used in GLM (Bondell and Reich, 2009; Gertheiss and Tutz, 2010), and in paired comparison models (Masarotto and Varin, 2012; Tutz and Schauberger, 2015). Since the information matrix used in penalized likelihood approaches does not lead to standard errors or confidence intervals, we apply bootstrap to the cross-validated model to compute confidence intervals that account for the additional variance originating from the model selection procedure.

To select covariates for BTRT we use the $\mathrm{BTLLasso}$ R package (Schauberger and Tutz, 2019), which penalizes the Fisher scoring with a L₁ penalty. The optimal tuning parameters Λ is selected through 10-fold cross-validation. For the main effect part of the BTRT model (root node), BTLL returns the cross-validated Λ as the optimal value for the size of penalties associated with the ${\beta }_{i,p}$ s; 13 out of 30 subject-specific covariates with an effect size equal to zero at least on two tax revenue categories are removed from the model. Descriptive statistics for the remaining covariates are reported in Table 2.

**** Table 2 about here ***

The final dataset includes H = 100 countries, n_o = 4 tax revenue categories, and 17 covariates. Interestingly, the covariates related to public expenditure remain in the model as the size of their main effect on the composition of tax revenues can not be overlooked.

The major drawback of our approach is the computational cost of the bootstrap procedure. In addition, BTLL selects the best covariates by only considering their main effects on each tax revenues category. Nevertheless, it still constitutes the more appropriate covariate selection procedure when dealing with the Bradley-Terry model.

6 Results

Figure 1 and Table 3 describe the BTRT pruned regression trunk. The CART’s node numbering criterion is used: each terminal node R_t is the region of the predictor space obtained from the recursive partitioning procedure. For each R_t , the number of countries H, the consensus ranking C and the correlation coefficient τ_x between rankings are reported. C is computed maximizing τ_x inside the nodes. In Figure 1, it represents a summary measure for each R_t that results from the minimization of distances between preferences without considering the characteristics of the countries (the subject-specific covariates).

**** Table 3 about here ***

**** Figure 1 about here ***

BTRT confirms EPI (x₁₃) is a key covariate as it is selected to split the first node: 76 countries are assigned to node 2 $(\text{EPI}\le 0.7)$ and 24 to node 3 $(\text{EPI}>0.7)$. The second splitting covariate is the government health spending (x₄) leading to nodes 4 and 5 that are further split according to EPI and Employment rate (x₁), respectively, to obtain terminal nodes R₂ to R₅. The employment rate is a useful indicator of current labour market conditions as it is unaffected by voluntary changes in labour force participation. The unemployment rate is influenced by the labour force size, decreasing when workers stop job-seeking and potentially increasing during labour market recovery as more individuals join the labour force. The third splitting covariate is Gross debt (x₂) which splits node 3 into terminal nodes R₁ and R₆. Thus, BTRT finds a first-order interaction between EPI and Gross debt (R₁ and R₆), another first-order interaction between EPI and Health spending (R₂ and R₃), and a third-order interaction between EPI, Health spending and Employment rate (R₄ and R₅). Overall, in 5 terminal nodes ${\tau }_x\ge 0.7$ indicates a broad consensus. R₂, R₃, and R₅ show a similar consensus ranking: for them, taxes on goods have a substantial effect on the composition of tax revenues. In R₁, R₄ and R₆ taxes on income are the most influential item of tax revenues. Ties are observed in R₁ and R₄.

Table 4 summarizes the BTRT pruning procedure. Results of model6 (mod6) are reported even if the pruning indicates model5 (mod5) as the final model. Adding another split to the mod5 trunk causes the deviance D and the cross-validation deviance D^cv to decrease. The D^cv values are mean values of deviances computed on each row of the design matrix, thus they are much smaller than the D values, computed as a sum (Eq. 5). The $c·SE$ rule with c = 0.5 leads to prune mod6 since $0.3565+0.5\times 0.0014=0.3572>0.3567$. Thus, the final trunk corresponds to mod5 with five splits and T = 6 terminal nodes.

**** Table 4 about here ***

Figure 2 shows the world map with the final breakdown of the countries in the regions $R_1,\dots ,R_6$. Most of the OECD countries with higher EPI and gross public debt are in R₁ (reference region). R₂ includes 31 countries with lower EPI and health expenditure. Besides Turkey, they are all located in Asia and Africa. R₃ includes 15 countries located on different continents (e.g. Egypt, Morocco, Mongolia, and Peru) that have poor EPI but spend more than 6% of GDP on health. R₄ consists of 5 countries, all located in South Africa except Tunisia, characterized by low EPI and employment rates, but with health spending larger than 6% of GDP. Countries in R₅, in prevalence located in South America and Central Africa, share the same characteristics of those in R₄ but have a higher labour force participation rate. Finally, R₆ includes the most virtuous countries in terms of EPI and public budget management.

**** Figure 2 about here ***

In general, Figure 2 provides a proxy for groups of countries that follow a common fiscal policy. For example, R₁ includes all OECD member countries with fairly similar fiscal policies (e.g. high levels of taxation on income, goods and services, and social security contribution; lower taxation on property). As discussed in Caselli and Reynaud (2020), Buera et al. (2011) and Giuliano et al. (2013), the intuition is that tax reforms in neighbouring countries affect the adoption of reforms through equal tax burdens and imitation effects.

**** Table 5 about here ***

Besides the inspection of terminal nodes, BTRT provides the estimated regression coefficients for each tax revenue. These are reported in Table 5 for taxes on income (o₁), compulsory SSC (o₂), and taxes on property (o₃). For the reference category taxes on goods (o₄) they are set to zero. Some interesting considerations emerge:

$•$ as for o₁ (taxes on income), military spending (x₁₄) and exports (x₄) have a strong impact on its size. The more is the level of x₁₄ or x₄, the lower the probability that taxes on income are higher than taxes on goods (o₄, reference category). On the contrary, imports (x₅) and health expenditures (x₁₆) have a positive effect on o₁. Then, for taxes on income, all the coefficients of the interaction effects except for R₆ (OECD members) have a negative sign.

$•$ As for SSC (o₂), savings (x₃), EPI (x₁₃), and other spending (x₁₇) have a positive and strong impact on its size. For instance, the higher x₁₃, the lower the log odds that SSC is higher than taxes on goods. EPI has a positive and high impact on o₂, despite the negative effect observed for o₁. In most developed countries SSCs are likely higher than taxes on goods. In addition, government consumption (x₁₂) has a strong and negative effect on SSC. Finally, the interaction terms for R₄ (R₆) has a positive (negative) effect on o₂;

$•$ Taxes on property (o₃) has a strong tendency to be the last ranked object as it shows the lowest value for the estimated intercept. As for the main effects, export levels (x₄) and government consumption (x₁₂) have a strong and negative effect on the size of taxes on the property. On the contrary, military spending (x₁₄) and other spending (x₁₇) have a positive impact on o₃. All the interaction effects have a positive impact on the comparison between taxes on property and taxes on goods and services.

**** Table 6 about here ***

Table 6 reports for each terminal node the worth parameters π_i (Eq. 6) and the mean value of the incidence of each tax category for all the countries inside the node. The worth parameters are computed based on the relationship between π_i and λ_i . They are more informative than the consensus ranking C. For instance, the regions R₂ and R₃ in Figure 1 show the same consensus ranking. These orderings are in line with the order of the π_i s in Table 6, but with a relevant difference: in R₂, SSC and taxes on property are very close to each other, while in R₃ the difference among them is larger. The π_i s provide additional information about the composition of the estimated ordering given the characteristics of the countries falling in a terminal node. Note that OECD countries (most of them in green in Fig. 2 and belonging to R₁) are characterized by the highest EPI and government gross debt. Compared to countries in other terminal nodes, they show a more balanced composition of tax revenues, with direct income taxes and social contributions dominating. Taxes on goods and services play a secondary role, whilst property taxes make up a residual share of government revenues. This interpretation is useful for researchers and policy-makers: a country with high gross debt and high environmental performances is likely to have tax revenue compositions similar to that of OECD countries.

6.1 Comparing BTRT results with tax determinants literature

Results presented in Section 6 are compared with those obtainable from the model specified in Gupta (2007), whose study is characterized by the same objectives and structure of data and represents a milestone in the tax determinants literature, and is based on the study of Lotz and Morss (1967), one of the first attempts to estimate tax determinants.

Lotz and Morss (1967) estimate the tax ratio for 52 developing countries using Ordinary Least Squares (OLS) and including the per capita income and openness to foreign trade as covariates. Their dataset is cross-sectional since tax ratios are computed as the average ratio for the years 1963-1965. The estimated tax ratios are compared, to create groups of countries with similar tax performance. Similarly, Gupta (2007) examines the main determinants of central government tax revenue using panel data for 105 developing countries over 25 years. The dependent variable is the total central government tax revenue (as a percentage of GDP). As in our analysis, Gupta focuses on the impact of debt, agricultural value-added, and trade on the determination of tax revenues. In addition, he utilizes the disaggregation of tax revenue categories as covariates, whereas in BTRT these are transformed them into paired comparisons. Moreover, in line with most of the studies on tax determinants, Gupta investigates the effect of socio-economic covariates on total tax revenue through a panel regression with fixed and random effects that include some institutional factors as additional covariates. Instead, BTRT is orientated towards the investigation of the relationship between tax revenue category and government expenditures by function (COFOG). To achieve comparability of results, the model proposed in Gupta (2007) is applied as follows: a) The dependent variable is the logarithm of the total central government tax revenue instead of tax revenue categories. The logarithmic term accounts for the heterogeneity of total tax revenues across countries. As confirmed by BTRT, low-income countries typically have tax revenues between 10% and 20% of GDP, while for high-income countries it rises to 40% (see also Besley and Persson (2014)); b) OLS is estimated on the same set of countries used for BTRT; c) All the data refer to the year 2018; d) The OLS model includes the same set of covariates used for BTRT (Table 2).

Beside the main effects, an exhaustive search of all the possible cross-product interaction effects up to the third order is performed. The selected OLS model is the one minimizing the Bayesian Information Criterion (BIC). It is compared with BTRT (Table 5) to highlight the additional information supplied by BTRT when interpreting the interaction effects.

**** Table 7 about here ***

Table 7 shows the OLS estimated coefficients. To understand how disaggregation of tax revenues by category improves interpretation of the interaction effects, values reported in Table 7 is compared with Table 5. For both OLS and BTRT the null hypothesis that the correct model includes only the main effects part is rejected in favor of the model with interaction effects (for OLS the test statistic is ${\chi }_4^2=36.09$; p < .000001, whilst for BTRT we get ${\chi }_{15}^2=141.81$; p < .000001). Important differences leading to different findings arise when interpreting the estimated coefficients. Gupta (2007) reports that central government gross debt has a positive impact on total tax revenue. This result is confirmed in the estimated OLS (Table 7, ${\widehat{\beta }}_3=0.09$; p = 0.04) but ${\widehat{\beta }}_3$ is borderline significant. This positive effect is due to the behaviour of countries with the burden of future loan repayments, which induces policymakers to raise tax rates. BTRT reports the same result in terms of the direction of the causal effect for all tax categories. For x₃, the increase in gross debt has a positive and significant effect on social security contributions (o₂).

Consistent with Lotz and Morss (1967), Gupta (2007) outlines the positive relationship between openness and revenue performance. Having an efficient tax system is critical, especially for countries noncompetitive in exports and natural resources (Karagoz, 2013). The OLS estimated coefficients for exports and imports are not significant. On the contrary, BTRT confirms the findings of Karagoz (2013): there is a negative (positive) relationship between exports (imports) and some tax categories: exports (x₄) influence negatively taxes on income and property, whilst imports (x₅) influence positively taxes on income only.

As for the effect of agriculture value-added on tax revenue, Gupta (2007) reports a strong negative impact (in line with Leuthold (1991); Stotsky and WoldeMariam (1997), and Eltony et al. (2001)). Likewise, Agbeyegbe et al. (2006) find a negative association between tax revenue and agricultural share, but not in Sub-Saharan countries where agricultural products are extensively exported and in turn increase tax revenues. However, also for these countries, the sign of the agricultural sector share is negative when income tax revenue is used as the response variable. OLS results in Table 7 show this negative association ( ${\widehat{\beta }}_8=-4.58$; p = 0.00). The unique cross-product interaction of the third order found by OLS involves agriculture value added together with government consumption and military spending. It indicates a relevant decrease in tax revenues ( ${\widehat{\beta }}_{21}=-1334.55$; p = 0.00), at the net of the positive effect on tax revenues of agriculture combined with either government consumption or military spending (see estimated coefficient of the second-order interactions). The same finding is not reported by BTRT: agriculture (x₇) hurts the taxes on income only. On the supply side, BTRT shows that for a country with a large agricultural sector, it is difficult or politically infeasible to raise taxes, especially if the agricultural sector is mostly subsistence or if it is organized on a small scale basis (Agbeyegbe et al., 2006). On the demand side, if the agricultural sector is leading, there is no need for high expenditures for the country. The need for public goods and services, which in most cases leads to raised tax rates, tends to be relatively urban-based (Gupta, 2007).

The EPI index, used as a proxy for the country and human development, has a positive effect in OLS ( ${\widehat{\beta }}_{13}=0.35$; p = 0.00). In Gupta (2007), the GDP per capita is used as a proxy for development and has a positive effect on tax revenue. This finding is consistent with Chelliah (1971) and demonstrates that a high level of development corresponds to a higher capacity to collect and pay taxes. BTRT shows similar results, with one unique difference: the proxy for the degree of development (x₁₃) has a positive impact on every tax revenue category except for income taxes. This result demonstrates that taxes from broad-based sources such as VAT and income taxes are lower in case of greater access to other forms of revenue (Besley and Persson, 2014). BTRT provides additional findings from threshold interactions induced by EPI (from ${\widehat{\beta }}_{i,R_2}$ to ${\widehat{\beta }}_{i,R_6}$ in Table 5). In R₂ and R₃, low values of EPI interact with health spending to reduce taxes on income. Conversely, in R₄ and R₅ the average value of EPI interacts with health spending and employment rate to increase taxes on social security contributions (and reduce taxes on income when the employment rate is high). In R₆, countries with high levels of EPI and reduced gross debt experience a reduction in taxes on social security contributions.

As for government expenditure by categories, OLS reports a positive association for education spending, health spending and other spending. In the main effects part their $\widehat{\beta }$ have the largest positive impact on the total tax revenue. For example, an increase in health expenditure is compensated by an increase in total tax revenue of about 2.42%. BTRT does not always estimate the same positive effect: in the main effects part, a consistent increase in taxes on property is caused by an increase in military spending (x₁₄). Interestingly, the interaction effects induced by ${\widehat{\beta }}_{i,R_4}$ to ${\widehat{\beta }}_{i,R_5}$ indicate that higher health spending induces an increase in taxes on social security contributions (and a decrease in taxes on income) in countries with reduced (increased) employment rate and EPI below 0.70. The positive associations involving health spending indicate that high public health spending leads to lower household consumption of health services, which in turn leads to higher disposable income and, consequently, to a wider tax base for income tax (Jiang, 2021).

It is possible to explore the BTRT results from a different perspective by focusing on the threshold interaction effects that lead to the terminal nodes of the trunk. Additional information derived from BTRT is obtained from the worth parameters estimated in each terminal node, which are conditioned on a specific country-specific covariate. The worth parameters allow us to investigate how the composition of tax revenues varies as the magnitude of a covariate changes in a group of countries with similar socioeconomic characteristics. To demonstrate that this additional information is useful for policymakers, we fix all the covariates at the mean value observed in each terminal node and compute the worth parameters for each tax category as the size of agricultural value-added to GDP varies. This corresponds to simulating agricultural sector size values within the range $[0,1]$.

**** Figure 3 about here ***

Figure 3 shows the results for each terminal node. It highlights some common features of all the groups identified by the terminal nodes. In each node, as the value added by the agricultural sector increases relative to GDP, income tax revenue (black line) decreases. This decrease is consistent with the BTRT estimated coefficients for the share of the agricultural sector. Moreover, in all terminal nodes except for R₁ (third row, bottom right), when the contribution of the agricultural sector to the economy is high, countries prefer taxes on goods and services (blue line), whilst resources from property taxes and social contributions are lower (red and green lines, respectively). The only case when this situation does not occur is for countries belonging to R₁ (countries belonging to OECD). Here, when the agricultural sector is important, other things being equal, governments prefer to raise resources through social contributions, whilst goods and services and property taxes have approximately the same degree of preference. Finally, countries in R₆ (third line, bottom left), which are those more virtuous in terms of debt management and environmental performance, raise their resources mainly through income taxes until the share of the agricultural sector is lower than about 80% of its maximum value. If this share exceeds 80% most of the fiscal resources of central governments come from taxes on goods and services.

The analysis of the composition of tax revenues for different groups of countries based on the size of their agricultural sector can be repeated for any covariate. Once the final trunk and the groups of countries with similar tax revenues have been identified, BTRT further investigates how the composition of government budget revenues varies as long as a country-specific covariate varies within the same group. This investigation can be done by comparing the groups of countries in the terminal nodes of the tree against each other.

To conclude, BTRT is preferable to OLS because of its tree-based structure: an OLS coefficient is not case-specific, as it refers to a conditional expectation. Thus, it is more interesting to estimate specific coefficients for different subsets of observations. In this respect, BTRT identifies automatically a homogeneous subset of cases corresponding to groups of countries with which the same set of coefficients and similar orders of tax revenues are associated. This homogeneous subset identification helps in explaining the structure of, and the differences between, the fiscal and economic systems of the subsets of countries.

6.2 Comparing BTRT with LLBT and BTtree

BTRT is compared with LLBT (Sinclair, 1982) without country-specific covariates and BTtree (Strobl et al., 2011). LLBT is similar to the BT model of Dittrich et al. (2000).

Both models are implemented in the $\mathrm{prefmod}$ R package (Hatzinger and Dittrich, 2012), which does not allow for the inclusion of numerical subject-specific covariates. Thus, these models assume the covariates do not affect the order of magnitude of the tax revenue categories. Thus, their main aim is to find a consensus ranking based on a probabilistic approach. Moreover, LLBT is not based on recursive partitioning, thus different results based on the characteristics of individuals are not available and only the average value of λ for each individual is computed. The LLBT’s estimated worth parameters (Table 8) are: ${\pi }_1=0.33,{\pi }_2=0.11,{\pi }_3=0.04$, and ${\pi }_4=0.50$. The consensus ranking in the root node of the BTRT trunk reproduces the same ordering of the items: Table 8 ( $4^{th}$ object) in the first position; taxes on income ( $1^{st}o\mathit{bject}$) in the second position; taxes on property ( $3^{rd}$ object) in the third position and the SSC ( $2^{nd}$ object) in the last position.

**** Table 8 about here ***

The BTtree model, implemented in the $\mathrm{psychotree}$ R package (Strobl et al., 2011), is instead capable of integrating recursive partitioning and the BT model: splits are found semi-parametrically searching for instability of the basic BT model object parameters.

**** Figure 4 about here ***

The tree obtained from BTtree (Figure 4) consists of two terminal nodes, with EPI being the splitting covariate. Results align with those derived from BTRT. BTtree does not identify interactions among covariates. After fitting a BT model, it verifies whether the item order established in the root node remains consistent across all observations. Covariates that introduce instability into the item order, and thus into the $\widehat{\lambda }$, are prioritized. Employing EPI as the initial split covariate allows for the generation of two distinct orders: For nations where $\mathit{EPI}\le 0.661$, income tax revenues are inferior to those from goods. In contrast, for EPI > 0.661, the situation inversely applies. Additionally, in the left child node, income taxes occupy the second position, while taxes on goods take precedence. Conversely, in the right child node, this order is reversed.

Interaction effects in BTRT allow to further differentiation of the subgroups based on the ordering of their income by taxation. Thus, the final output of BTtree contains less information than those contained in BTRT. BTtree provides the preference scales in each group of the partition that derive from the order of object-related parameters, but it does not specify how the subject-specific covariates affect the judges’ preferences. Therefore, this semi-parametric model returns beta coefficients neither for the main effects nor for the interaction effects. In addition, as pointed out in Strobl et al. (2011), the testing procedure for the split search is challenging (Wiedermann et al., 2021). BTtree is based on the M-fluctuation test which is a score-based procedure (Zeileis and Hornik, 2007), whereas BTRT is based on the easy-to-compute decrease in deviance.

7 Discussion

This study innovatively examined global tax revenue categories, diverging from traditional analyses, to assess how countries’ characteristics influence their tax revenue composition. A specialized dataset was created by merging information from the OECD, IMF, and World Bank for 2018. Tax revenues for 100 countries were classified into four categories and ranked. The Bradley-Terry Regression Trunk (BTRT) model, a novel probabilistic method for analyzing preference rankings, was applied. BTRT allows for an assessment of tax categories across a broad range of countries through pairwise comparisons. The BTRT model segments individuals by considering their tax revenue rankings and the causal link between these rankings and individual characteristics. It operates on the premise that the hierarchy of tax revenues within a country is influenced by its socio-economic traits. Thus, for each country, determinants of tax revenue were modeled as covariates. Our findings added to the discourse on the interplay between tax revenues and government spending by examining four key government expenditure categories: military, education, health, and a residual group encompassing all other types of government spending.

In our application, BTRT partitioned 100 countries into six groups based on several socio-economic covariates. It identified important interactions between these covariates, including both first-order interactions and complex interactions. Moreover, BTRT computed coefficients that helped in ordering tax revenues, shedding light on the impact of significant covariates and their interactions. Unlike traditional Bradley-Terry models which focus on preference data, BTRT offers a more nuanced understanding by probabilistically assessing the causal relationships between covariates and the outcomes in paired comparisons, facilitating the grouping of data into homogenous clusters from the model’s terminal nodes. This approach provides a more actionable analysis compared to standard Bradley-Terry models. Compared to Strobl’s semi-parametric BTtree, BTRT provides precise estimates of the impact of subject-specific covariates in paired comparisons. Our empirical results revealed several advantages of BTRT: it accommodates a diverse array of countries and fiscal systems thanks to its paired comparison structure; it enables the identification of varied effects of covariates on different tax revenue categories; its fully parametric nature facilitates the detection of threshold interactions, enhancing model fit; and its tree structure offers a clear visualization of countries grouped by similar fiscal revenue characteristics.

In conclusion, this paper presents a public finance application that provides valuable insights for policymakers, exploring the impact of economic shocks on tax revenue structures and offering guidance for countries looking to modify their tax compositions. The study suggests that changes in a country’s socio-economic characteristics can influence tax revenue amounts without necessitating new tax policy regulations, which are often slow to implement. The paper also discusses the limitations of discretionary fiscal policy due to recognition and implementation delays, advocating for the effectiveness of automatic stabilizers that respond immediately to economic changes. Using the BTRT model, the study identifies comparable fiscal policies across diverse fiscal systems by analyzing socioeconomic traits.

Disclaimer

The authors declare they have no conflict of interest and the findings, interpretations, and conclusions expressed herein do not necessarily reflect the view of the institutions of affiliation.

Algorithm

Algorithm 1: Bradley-Terry Regression Trunkinput: X (matrix of covariates), Y paired comparisons;

1 initialize: estimate the main effects model in the root node of the trunk: ${\widehat{\lambda }}_{i,h}={\widehat{\lambda }}_i+{\sum }_{p=1}^P{\widehat{\beta }}_{i,p}{x}_{p,h}$;

2 while splitting rule causes a decrease in model deviance or sample size within each child is greater or equal to 5 do

3 Split search for each current node t_c ;

4 for $p=1\to P$ do

5 find the dichotomous variable $z_{\mathit{ijp},t}$ that minimizes the local loglikelihood deviance of the model;

6 end

7 detect $z_{\mathit{ijp},t}^{*}$ as the quantity that minimizes the global loglikelihood deviance of the model;

8 create child nodes $t_{c+1}$ and $t_{c+2}$;

9 $z_{\mathit{ijp},t}^{*}$ updates the indicator function $I\{(x_{1,h},\dots ,x_{P,h})\}\in t$ and the model including the last threshold interaction effect is re-estimated:$${\widehat{\lambda }}_{i,h}={\widehat{\lambda }}_i+\sum _{i=1}^P{\widehat{\beta }}_{i,p}{x}_{p,h}+\sum _{t=1}^{T-1}{\widehat{\beta }}_{i,P+t}I\{(x_{1,h},\dots ,x_{P,h})\in t\}$$

10 endoutput: Bradley-Terry Regression Trunk

Notes

¹ The superscript O used in Eq. 4 is intentionally left out to lighten notation.

² See Eq. 14 and Eq. 15 in Baldassarre et al. (2022).

Fig. A1 OECD: tax revenues composition by country

Fig. A2 Africa: tax revenues composition by country

Fig. A3 Asia: tax revenues composition by country

Fig. A4 South America: tax revenues composition by country

Table 1 Design matrix with two judges/individuals, three objects, and one continuous subject-specific covariate.

Response	μ	${\lambda }_A^O$	${\lambda }_B^O$	${\lambda }_C^O$	x1
yAB = 1	1	−1	1	0	23
yAB = 0	1	1	−1	0	23
yAC = 1	2	−1	0	1	23
yAC = 0	2	1	0	−1	23
yBC = 1	3	0	1	−1	23
yBC = 0	3	0	−1	1	23
yAB = 0	1	−1	1	0	24
yAB = 1	1	1	−1	0	24
yAC = 0	2	−1	0	1	24
yAC = 1	2	1	0	−1	24
yBC = 0	3	0	1	−1	24
yBC = 1	3	0	−1	1	24

Notes: The first column indicates if the object i is preferred (y_ij = 1) or not (y_ij = 0) in a certain preference for each pair of objects ij. The second column serves as an index for the $n\times (n-1)/2$ comparisons. Preferences are expressed in the next three columns, and finally, the covariate x₁ is shown in the last column. In this example, the two individuals present opposite rankings, BCA and ACB respectively.

Table 2 Descriptive statistics for the subject-specific covariates selected by the BTLL model.

Covariates	xp	# missing	mean	sd	median	min	max	range	skew	kurtosis	se
Employment rate	x1	2	0.59	0.09	0.59	0.39	0.85	0.45	0.05	0.27	0.01
Government gross debt	x2	1	0.59	0.35	0.50	0.09	2.38	2.29	2.12	6.96	0.03
Savings	x3	5	0.21	0.08	0.21	0.02	0.46	0.44	0.53	0.48	0.01
Exports	x4	13	0.04	0.06	0.05	−0.36	0.17	0.54	−3.59	22.69	0.01
Imports	x5	11	0.04	0.06	0.05	−0.23	0.18	0.41	−1.66	6.12	0.01
Interests	x6	21	0.08	0.05	0.07	0.00	0.29	0.29	1.34	2.98	0.01
Agric added-value	x7	2	0.09	0.09	0.05	0.00	0.45	0.45	1.69	2.79	0.01
Services added-value	x8	21	0.57	0.10	0.57	0.29	0.79	0.50	−0.40	−0.20	0.01
Final consumption	x9	8	0.78	0.11	0.79	0.43	1.12	0.69	−0.49	1.88	0.01
Bank NPloans	x10	16	0.04	0.06	0.03	0.00	0.42	0.42	4.77	25.75	0.01
Claims centr gov	x11	4	0.12	0.18	0.08	−0.14	1.42	1.56	3.85	23.62	0.02
Gov consumption	x12	8	0.16	0.05	0.16	0.04	0.39	0.35	0.63	2.42	0.01
EPI	x13	0	0.59	0.16	0.59	0.00	0.87	0.87	−0.93	2.21	0.02
Military spending	x14	2	0.01	0.01	0.01	0.00	0.05	0.05	1.44	3.72	0.00
Education spending	x15	33	0.04	0.01	0.04	0.01	0.08	0.07	0.72	0.25	0.00
Health spending	x16	0	0.07	0.03	0.07	0.02	0.17	0.15	0.74	0.96	0.00
Other spending	x17	9	0.20	0.10	0.20	0.03	0.76	0.73	1.77	7.89	0.01

Notes: # missing = n. of missing values; sd = standard deviation; min = minimum; max = maximum; skew = skewness; se = standard error.

Table 3 Pruned BTRT regression trunk.

	Node n.	$\mathit{Splitting}\mathit{covariate}$	$\mathit{Split}\mathit{Point}$	$\mathit{Model}\mathit{Deviance}$
		main effects (no splits)		313
bestsplit1	1	x15 (Environmental Performance Index)	0.70	271
bestsplit3	2	x4 (Health spending)	0.06	213
bestsplit5	4	x8 (Environmental Performance Index)	0.56	171
bestsplit4	5	x8 (Employment rate)	0.51	190
bestsplit2	3	x2 (GDP % change)	0.40	238

Table 4 Pruned BTRT regression trunk: 10-fold cross-validation results.

	D	D^cv	SE^cv
mod0	313.0671	0.3775	0.0010
mod1	271.9198	0.3682	0.0010
mod2	238.8955	0.3659	0.0011
mod3	213.5469	0.3617	0.0011
mod4	190.8353	0.3594	0.0013
mod5	171.2580	0.3567	0.0013
mod6	136.5742	0.3565	0.0014

Notes: $D=$ model deviance for a Poisson regression; $D^{cv}=$ casewise cross-validation deviance (Eq. 14 Baldassarre et al., 2022); $S{E}^{cv}=$ standard error of D^cv (Eq. 15 Baldassarre et al., 2022)

Table 5 BTRT estimated coefficients.

	${\widehat{\lambda }}_{o_1,h}$	${\widehat{\lambda }}_{o_2,h}$	${\widehat{\lambda }}_{o_3,h}$	Interaction effect
${\widehat{\lambda }}_i$	13.15${}^{*}$ (5.33)	−33.63${}^{**}$ (11.83)	−50.09 (115.79)
${\widehat{\beta }}_{i,x1}$	−4.16 (2.77)	1.24 (4.62)	8.73 (4.54)
${\widehat{\beta }}_{i,x2}$	0.71 (0.93)	−5.28${}^{**}$ (1.94)	−2.99 (1.98)
${\widehat{\beta }}_{i,x3}$	5.45 (3.68)	20.95${}^{**}$ (7.56)	15.78 (8.15)
${\widehat{\beta }}_{i,x4}$	−16.68${}^{**}$ (5.58)	−5.02 (9.13)	−32.94${}^{*}$ (12.96)
${\widehat{\beta }}_{i,x5}$	14.89${}^{*}$ (5.43)	−11.30 (7.38)	5.96 (8.51)
${\widehat{\beta }}_{i,x6}$	−15.22${}^{*}$ (7.38)	−2.61 (10.94)	4.06 (12.10)
${\widehat{\beta }}_{i,x7}$	−10.12${}^{*}$ (4.48)	2.40 (10.70)	−15.69 (12.72)
${\widehat{\beta }}_{i,x8}$	−8.05${}^{*}$ (3.78)	16.25${}^{*}$ (6.72)	11.85 (7.51)
${\widehat{\beta }}_{i,x9}$	2.82 (3.39)	5.23 (5.50)	7.94 (6.69)
${\widehat{\beta }}_{i,x10}$	−4.54 (4.09)	16.92${}^{*}$ (7.60)	−0.19 (10.47)
${\widehat{\beta }}_{i,x11}$	0.33 (1.60)	7.91${}^{**}$ (3.05)	9.32${}^{*}$ (4.04)
${\widehat{\beta }}_{i,x12}$	−11.06 (7.49)	−28.28${}^{*}$ (12.56)	−51.68${}^{**}$ (16.17)
${\widehat{\beta }}_{i,x13}$	−6.54${}^{*}$ (2.65)	22.30${}^{***}$ (6.52)	24.17${}^{**}$ (7.61)
${\widehat{\beta }}_{i,x14}$	−25.63 (25.38)	6.32 (33.37)	137.84${}^{**}$ (53.16)
${\widehat{\beta }}_{i,x15}$	−3.98 (21.42)	−65.68 (38.44)	−45.98 (43.92)
${\widehat{\beta }}_{i,x16}$	10.26 (13.88)	20.80 (17.44)	5.51 (25.97)
${\widehat{\beta }}_{i,x17}$	−4.15 (2.63)	22.81${}^{**}$ (7.01)	33.23${}^{***}$ (8.64)
${\widehat{\beta }}_{i,R_2}$	−3.21${}^{*}$ (1.32)	0.83 (1.88)	14.30 (115.09)	$R_2=I(\text{EPI}\le 0.56,\text{Healthsp}.\le 0.06)$
${\widehat{\beta }}_{i,R_3}$	−2.25${}^{*}$ (1.06)	0.50 (1.42)	11.58 (115.09)	$R_3=I(0.56<\text{EPI}\le 0.7,\text{Healthsp}.\le 0.06)$
${\widehat{\beta }}_{i,R_4}$	−0.77 (1.32)	4.87${}^{*}$ (2.22)	19.19 (115.10)	$R_4=I(\text{EPI}\le .7,\text{Healthsp}.>.06,\text{Empl}.\text{rate}\le .51)$
${\widehat{\beta }}_{i,R_5}$	−3.24${}^{***}$ (0.96)	3.31${}^{*}$ (1.29)	6.00 (153.15)	$R_5=I(\text{EPI}\le .7,\text{Healthsp}.>.06,\text{Empl}.\text{rate}>.51)$
${\widehat{\beta }}_{i,R_6}$	7.93 (201.22)	−4.26${}^{*}$ (1.73)	8.35 (115.10)	$R_6=I(\text{EPI}>0.7,\text{Gross}\mathrm{ }\text{debt}\le 0.4)$

Notes: Estimated coefficients associated to the objects taxes on income o₁, social security contributions o₂, taxes on property o₃, and taxes on goods and services o₄ (reference category, i.e., coefficients equal to zero). Standard errors are in parenthesis and the stars $"*"$ associated with some estimated coefficients indicate they are different from zero with a p-value $\le 0.001$ $("***"),\le 0.01$ $("**")$, and $\le 0.05("*")$, respectively.

Table 6 Mean worth parameters π_i for each tax revenue category in each terminal node of the pruned regression trunk.

	πincome	πsocial	πproperty	πgoods
R2	0.04713	0.00000	0.00000	0.95287
R3	0.12403	0.00011	0.00001	0.87585
R4	0.96217	0.00000	0.00000	0.03783
R5	0.02498	0.00276	0.00000	0.97226
R6	1.00000	0.00000	0.00000	0.00000
R1	0.40127	0.39267	0.00000	0.20605

Notes: Each π_i corresponds to the average value of the worth parameters associated with each country in the specific terminal node

Table 7 OLS estimated coefficients with interaction effects

	Estimate	Std. Error	t value	Pr( $>|$ t $|$)
Intercepts	−2.10	0.21	−9.85	0.00
Employment rate	0.01	0.10	0.10	0.92
Government gross debt	0.03	0.04	2.06	0.04
Savings	−0.11	0.16	−0.68	0.50
Exports	0.11	0.21	0.52	0.60
Imports	−0.03	0.20	−0.42	0.68
Interests	−0.29	0.21	−1.39	0.17
Agriculture value added	−4.58	0.81	−5.62	0.00
Services	−0.22	0.14	−1.54	0.13
Final consumption	−0.20	0.14	−1.40	0.17
Bank NP loans	−0.27	0.20	−1.40	0.17
Claims centr gov	−0.16	0.07	−2.19	0.03
Gov consumption	0.23	0.57	0.41	0.68
EPI	0.35	0.08	4.32	0.00
Military spending	6.08	5.14	1.18	0.24
Education spending	2.84	0.83	3.42	0.00
Health spending	2.42	0.45	5.32	0.00
Other spending	2.80	0.12	24.17	0.00
Agriculture VA:Gov consumption	30.15	5.84	5.16	0.00
Agriculture VA:Military sp	190.34	36.34	5.24	0.00
Gov cons:Military sp	−8.01	27.54	−0.29	0.77
Agriculture VA:Gov cons:Military sp	−1313.86	260.58	−5.04	0.00

Table 8 Log-linear Bradley-Terry model without subject-specific covariates: results

	Estimate	Std. Error	z value	Pr($>|$z$|$)
(Intercept)	3.61	0.05	67.24	0.00
o1	−0.20	0.07	−2.72	0.00
o2	−0.73	0.08	−9.24	0.00
o3	−1.26	0.08	−15.37	0.00
o4	0.00	0.00	0.00	0.00

Table 9 Descriptive statistics for the subject-specific covariates.

Covariates	Source	missing	mean	sd	median	min	max	range	skew	kurtosis	se
Current account balance	OECD	13	−0.02	0.07	−0.03	−0.23	0.23	0.46	0.38	1.72	0.01
employment rate	OECD	2	0.59	0.09	0.59	0.39	0.85	0.45	0.03	0.32	0.01
Government gross debt	OECD	1	0.59	0.35	0.50	0.09	2.38	2.29	2.11	6.91	0.03
government net lend/borrow	OECD	21	−0.03	0.04	−0.03	−0.11	0.24	0.36	3.68	26.28	0.00
gross fixed capital form	OECD	0	0.24	0.07	0.23	0.11	0.48	0.37	1.29	2.50	0.01
GDP savings	OECD	5	0.21	0.08	0.21	0.02	0.46	0.44	0.53	0.48	0.01
investment	OECD	4	0.24	0.08	0.23	0.11	0.64	0.53	1.74	6.31	0.01
unemployment rate	OECD	21	0.06	0.04	0.06	0.00	0.28	0.28	1.72	6.63	0.00
exports	OECD	13	0.04	0.06	0.05	−0.36	0.17	0.54	−3.58	22.58	0.01
imports	OECD	11	0.04	0.06	0.05	−0.23	0.18	0.41	−1.65	6.07	0.01
financial account ownership	WORLD BANK	18	0.70	0.25	0.70	0.18	1.00	0.82	−0.23	−1.38	0.03
subsidies and other transfers	WORLD BANK	20	0.39	0.20	0.35	0.04	0.84	0.80	0.41	−0.85	0.02
interest payments	WORLD BANK	21	0.08	0.05	0.07	0.00	0.29	0.29	1.35	2.96	0.01
compensation of employees	WORLD BANK	19	0.26	0.13	0.22	0.06	0.53	0.47	0.27	−1.13	0.01
agriculture value added	WORLD BANK	2	0.09	0.09	0.05	0.00	0.45	0.45	1.68	2.76	0.01
manufacturing value added	WORLD BANK	10	0.13	0.06	0.13	0.03	0.32	0.30	0.66	0.34	0.01
services value added	WORLD BANK	21	0.57	0.10	0.57	0.29	0.79	0.50	−0.39	−0.23	0.01
industry value added	WORLD BANK	2	0.25	0.08	0.25	0.10	0.58	0.47	0.77	1.46	0.01
pop density	WORLD BANK	0	207.94	796.28	79.78	2.04	7953.00	7950.96	9.20	86.51	79.63
final consumption expenditure	WORLD BANK	8	0.78	0.11	0.79	0.43	1.12	0.69	−0.49	1.87	0.01
access to electricity	WORLD BANK	0	0.87	0.24	1.00	0.12	1.00	0.88	−1.75	1.91	0.02
bank NPloans	WORLD BANK	16	0.04	0.06	0.03	0.00	0.42	0.42	4.77	25.73	0.01
claims centr gov	WORLD BANK	4	0.12	0.18	0.08	−0.14	1.42	1.56	3.85	23.60	0.02
Gov consumption	WORLD BANK	8	0.16	0.05	0.16	0.04	0.39	0.35	0.63	2.37	0.01
Households’ consumption	WORLD BANK	0	0.62	0.11	0.64	0.30	0.86	0.56	−0.33	0.04	0.01
trade	WORLD BANK	0	0.89	0.54	0.79	0.28	3.87	3.60	2.76	10.89	0.05
EPI	EPI	0	0.59	0.16	0.58	0.00	0.87	0.87	−0.91	2.19	0.02
military spending	IMF	2	0.01	0.01	0.01	0.00	0.05	0.05	1.44	3.69	0.00
education spending	IMF	33	0.04	0.01	0.04	0.01	0.08	0.07	0.72	0.25	0.00
health spending	IMF	0	0.07	0.03	0.07	0.02	0.17	0.15	0.73	0.98	0.00
other spending	AS RESIDUAL	9	0.20	0.10	0.20	0.03	0.76	0.73	1.76	7.81	0.01

Notes: # missing = n. of missing values; sd = standard deviation; min = minimum; max = maximum; skew = skewness; se = standard error.

Figure 1 Pruned regression trunk.

Notes: In each terminal node $R_1,\dots ,R_6$ and in the root note, the number of countries H, the consensus ordering C, and the correlation coefficient τ_x are shown. Taxes are named as follows: I - Taxes on income, S - Social security contributions, P - Taxes on property, and G - Taxes on goods. When two tax categories are in brackets, it means they are in a tie.

Figure 2 Countries’ distribution map in each region (terminal node) $R_1,\dots ,R_6$, where R₁ is the reference region in the BTRT model.

Figure 3 Mean worth parameters π_i conditioned to the share of agricultural sector to GDP.

Notes: The worth parameters are computed for $R_1,\dots ,R_6$ by fixing all covariates (except for the share of the agricultural sector) to the average level of the group. The order of graphs within the figure follows the trunk in Figure 1: R₂ top left, R₃ top right, R₄ center left, R₅ center right, R₆ bottom left, R₁ bottom right. The x-axis shows simulated values of the covariate agricultural share within the standardized range $[0,1]$. Finally, for each plot, the mean values of the covariates resulting from the splitting procedure of the regression trunk are reported. For the sake of clarity, the mean values of all the other country-specific covariates are not reported.

Figure 4 Output of BTtree.

Notes: H is the number of countries, I = taxes on income, S = social security contributions, P = taxes on property, and G = taxes on goods and services. The plot shows the estimated worth parameters in both terminal nodes.