# Forecasting the Stability and Growth Pact compliance using Machine Learning

Kea Baret, Amélie Barbier-Gauchard, Theophilos Papadimitriou

<figure><img src="image_1.png" /></figure>

## To cite this version:

Kea Baret, Amélie Barbier-Gauchard, Theophilos Papadimitriou. Forecasting the Stability and Growth Pact compliance using Machine Learning. The World Economy, 2023, 47 (1), pp.188-216. 10.1111/twec.13518. hal-03121966v2

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Submitted on 3 May 2024

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Distributed under a Creative Commons Attribution 4.0 International License
# Forecasting the Stability and Growth Pact compliance using Machine Learning

Kea Baret, Amélie Barbier-Gauchard, Theophilos Papadimitriou

► To cite this version:

Kea Baret, Amélie Barbier-Gauchard, Theophilos Papadimitriou. Forecasting the Stability and Growth Pact compliance using Machine Learning. The World Economy, 2023, 47 (1), pp.188-216. 10.1111/twec.13518. hal-03121966v2

HAL Id: hal-03121966

https://hal.science/hal-03121966v2

Submitted on 3 May 2024

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
<figure><img src="image_3.png" /></figure>

# Forecasting stability and growth pact compliance using machine learning

Kéa Baret<fnref n="1" /> ID

Amélie Barbier-Gauchard<fnref n="2" /> ID

Théophilos Papadimitriou<fnref n="3" />

## Correspondence

Kéa Baret, Banque de France, Paris and BETA, Université de Strasbourg, Paris, France.

Email: k.baret@unistra.fr

## Funding information

European Union, Grant/Award Number: Chair Jean Monnet

## Abstract

The 2011 reform of the Stability and Growth Pact (1996) strengthened the European Commission's monitoring of EU member states' public finance. Failure to comply with the 3% limit on public deficit triggers an audit. In this paper, we present a machine learning based forecasting model for compliance with the 3% limit. We use data from 2006 to 2018 (a turbulent period including the Global Financial Crisis and the Sovereign Debt Crisis) for the 28 EU member states. After identifying 8 features as predictors among 138 variables, forecasting is performed using a support vector machine (SVM) algorithm. The proposed model achieved a forecasting accuracy of nearly 92% and outperformed the logit model used as a benchmark.

## KEYWORDS

fiscal rules, fiscal compliance, machine learning, stability and growth pact

The views expressed in this document are those of the author and do not necessarily reflect the position of the Banque de France.

This is an open access article under the terms of the [Creative Commons Attribution License](https://creativecommons.org/licenses/by/4.0/), which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

© 2023 The Authors. *The World Economy* published by John Wiley & Sons Ltd.

<figure><img src="image_4.png" /></figure>

<footnote n="1">¹Banque de France, Paris and BETA, Université de Strasbourg, Paris, France</footnote>

<footnote n="2">²BETA, Université de Strasbourg, Strasbourg, France</footnote>

<footnote n="3">³Department of Economics, Democritus University of Thrace, Komotini, Greece</footnote>
<figure><img src="image_2.png" /></figure>

# Forecasting stability and growth pact compliance using machine learning

Kéa Baret<fnref n="1" /> ID

Amélie Barbier-Gauchard<fnref n="2" /> ID

Théophilos Papadimitriou<fnref n="3" />

**Correspondence**

Kéa Baret, Banque de France, Paris and BETA, Université de Strasbourg, Paris, France.

Email: k.baret@unistra.fr

**Funding information**

European Union, Grant/Award Number: Chair Jean Monnet

**Abstract**

The 2011 reform of the Stability and Growth Pact (1996) strengthened the European Commission's monitoring of EU member states' public finance. Failure to comply with the 3% limit on public deficit triggers an audit. In this paper, we present a machine learning based forecasting model for compliance with the 3% limit. We use data from 2006 to 2018 (a turbulent period including the Global Financial Crisis and the Sovereign Debt Crisis) for the 28 EU member states. After identifying 8 features as predictors among 138 variables, forecasting is performed using a support vector machine (SVM) algorithm. The proposed model achieved a forecasting accuracy of nearly 92% and outperformed the logit model used as a benchmark.

**KEYWORDS**

fiscal rules, fiscal compliance, machine learning, stability and
growth pact

The views expressed in this document are those of the author and do not necessarily reflect the position of the Banque de France.

This is an open access article under the terms of the [Creative Commons Attribution License](https://creativecommons.org/licenses/by/4.0/), which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

© 2023 The Authors. *The World Economy* published by John Wiley & Sons Ltd.

<figure><img src="image_3.png" /></figure>

<footnote n="1">¹Banque de France, Paris and BETA, Université de Strasbourg, Paris, France</footnote>

<footnote n="2">²BETA, Université de Strasbourg, Strasbourg, France</footnote>

<footnote n="3">³Department of Economics, Democritus University of Thrace, Komotini, Greece</footnote>
<figure><img src="image_5.png" /></figure>

# 1 | INTRODUCTION

For almost 25 years, enhancing fiscal discipline in the eurozone has been a bone of contention between European authorities and EU member states. European supranational fiscal discipline in the form of the Stability and Growth Pact (1996) -SGP hereafter¹has been reformed several times and remains under debate. Since its inception, this set of fiscal rules aims two complementary objectives: ‘stability’ of public finances promoting sound management of public finances on the one hand, and ‘economic growth’ in the EMU, to ensure that national governments have the leeway to intervene when necessary (especially if a cyclical shock occurs), on the other hand. To achieve these two objectives, the SGP includes two instruments: a ‘dissuasive’ arm intended to ensure strict compliance with the rules<fnref n="2" /> and a ‘preventive’ arm designed to encourage member states to maintain balanced and sound public finances in the medium term.<fnref n="3" />

The eurozone has nevertheless experienced several periods of turbulence⁴that have fundamentally challenged fiscal discipline. In each case, the SGP was deemed imperfect and was reformed. The reforms of 2005, 2011 and 2013, replaced the fiscal rule with a catalogue of indicators that member states are expected to monitor without any in-depth reflection on the underlying causes of fiscal indiscipline in the euro area and the absence of any real disciplinary power over member states.

Kopits and Symansky (1998) seminal paper on the features of an ideal fiscal rule is essential to better understand the strengths and weaknesses of fiscal discipline in the euro area. The authors put forward eight properties of a ‘good’ fiscal rule, one of which, credibility (or enforceability), is a focal point of the debate on fiscal rules efficiency and also appears in the ‘fiscal rules trilemma’.⁵It is expressed in practice by levels of compliance with the fiscal rule. Fiscal rule compliance can be defined as the proportion of the considered fiscal aggregates (the budget balance, the debt-to-GDP ratio or government expenditure) that meet the targets set by the fiscal rules.<fnref n="6" />

The purpose of this paper is not to question whether fiscal rules are justified or not<fnref n="7" /> or to discuss the thresholds used to control compliance. Nor is it to assess the effectiveness of fiscal rules<fnref n="8" /> or to propose new fiscal rules.⁹Rather, this paper aims to fill a gap in the fiscal rule compliance

<footnote n="1">¹The Stability and Growth Pact (1996) succeeded the public finance criteria (public debt below 60% of GDP and public deficit below 3% of GDP) introduced by the Maastricht Treaty (1992) as one of the eligibility conditions for the European monetary union.</footnote>

<footnote n="2">²The dissuasive arm consists in a public deficit ceiling with, barring exceptional economic circumstances, sanctions imposed in the case of non-compliance.</footnote>

<footnote n="3">³The preventive arm consists of a multilateral surveillance procedure with ‘stability programs’, multiannual programs setting fiscal guidelines over 3 years and providing visibility on public finances for the following 3 years, with the objective of achieving a balanced budget in the medium term.</footnote>

<footnote n="4">⁴A first crisis in 2004, the Great Recession from 2007 to 2009 with the subprime mortgage crisis followed by the Sovereign Debt Crisis, then the COVID-19 pandemic and the energy crisis since 2022.</footnote>

<footnote n="5">⁵Debrun and Jonung (2019)</footnote>

<footnote n="6">⁶In most cases, this assessment does not take into account escape clauses.</footnote>

<footnote n="7">⁷See for instance Debrun et al. (2008) or Wyplosz (2012) for a general overview of the main reasons for introducing fiscal rules.</footnote>

<footnote n="8">⁸See instead Foremny (2014), Sacchi and Salotti (2015), Bergman et al. (2016), Barbier-Gauchard et al. (2021) or Barbier-Gauchard et al. (2023) for instance.</footnote>

<footnote n="9">⁹See papers on the second generation of fiscal rules by Eyraud et al. (2018), Caselli et al. (2018), Darvas et al. (2018), Hauptmeier and Kamps (2020), Debrun and Jonung (2019) and Blanchard et al. (2020).</footnote>
<figure><img src="image_4.png" /></figure>

# 1 | INTRODUCTION

For almost 25 years, enhancing fiscal discipline in the eurozone has been a bone of contention between European authorities and EU member states. European supranational fiscal discipline in the form of the Stability and Growth Pact (1996) -SGP hereafter¹has been reformed several times and remains under debate. Since its inception, this set of fiscal rules aims two complementary objectives: ‘stability’ of public finances promoting sound management of public finances on the one hand, and ‘economic growth’ in the EMU, to ensure that national governments have the leeway to intervene when necessary (especially if a cyclical shock occurs), on the other hand. To achieve these two objectives, the SGP includes two instruments: a ‘dissuasive’ arm intended to ensure strict compliance with the rules<fnref n="2" /> and a ‘preventive’ arm designed to encourage member states to maintain balanced and sound public finances in the medium term.<fnref n="3" />

The eurozone has nevertheless experienced several periods of turbulence⁴that have fundamentally challenged fiscal discipline. In each case, the SGP was deemed imperfect and was reformed. The reforms of 2005, 2011 and 2013, replaced the fiscal rule with a catalogue of indicators that member states are expected to monitor without any in-depth reflection on the underlying causes of fiscal indiscipline in the euro area and the absence of any real disciplinary power over member states.

Kopits and Symansky (1998) seminal paper on the features of an ideal fiscal rule is essential to better understand the strengths and weaknesses of fiscal discipline in the euro area. The authors put forward eight properties of a ‘good’ fiscal rule, one of which, credibility (or enforceability), is a focal point of the debate on fiscal rules efficiency and also appears in the ‘fiscal rules trilemma’.⁵It is expressed in practice by levels of compliance with the fiscal rule. Fiscal rule compliance can be defined as the proportion of the considered fiscal aggregates (the budget balance, the debt-to-GDP ratio or government expenditure) that meet the targets set by the fiscal rules.<fnref n="6" />

The purpose of this paper is not to question whether fiscal rules are justified or not<fnref n="7" /> or to discuss the thresholds used to control compliance. Nor is it to assess the effectiveness of fiscal rules<fnref n="8" /> or to propose new fiscal rules.⁹Rather, this paper aims to fill a gap in the fiscal rule compliance

<footnote n="1">¹The Stability and Growth Pact (1996) succeeded the public finance criteria (public debt below 60% of GDP and public deficit below 3% of GDP) introduced by the Maastricht Treaty (1992) as one of the eligibility conditions for the European monetary union.</footnote>

<footnote n="2">²The dissuasive arm consists in a public deficit ceiling with, barring exceptional economic circumstances, sanctions imposed in the case of non-compliance.</footnote>

<footnote n="3">³The preventive arm consists of a multilateral surveillance procedure with ‘stability programs’, multiannual programs setting fiscal guidelines over 3 years and providing visibility on public finances for the following 3 years, with the objective of achieving a balanced budget in the medium term.</footnote>

<footnote n="4">⁴A first crisis in 2004, the Great Recession from 2007 to 2009 with the subprime mortgage crisis followed by the Sovereign Debt Crisis, then the COVID-19 pandemic and the energy crisis since 2022.</footnote>

<footnote n="5">⁵Debrun and Jonung (2019)</footnote>

<footnote n="6">⁶In most cases, this assessment does not take into account escape clauses.</footnote>

<footnote n="7">⁷See for instance Debrun et al. (2008) or Wyplosz (2012) for a general overview of the main reasons for introducing fiscal rules.</footnote>

<footnote n="8">⁸See instead Foremny (2014), Sacchi and Salotti (2015), Bergman et al. (2016), Barbier-Gauchard et al. (2021) or Barbier-Gauchard et al. (2023) for instance.</footnote>

<footnote n="9">⁹See papers on the second generation of fiscal rules by Eyraud et al. (2018), Caselli et al. (2018), Darvas et al. (2018), Hauptmeier and Kamps (2020), Debrun and Jonung (2019) and Blanchard et al. (2020).</footnote>
literature by using the groundwork on compliance to strengthen the preventive arm of the SGP in the euro area.

Indeed, a large part of the fiscal rule compliance literature has focused on compliance scores and the determinants of fiscal rule compliance. Delgado-Téllez et al. (2017) for Spanish regions, Reuter (2019) for EU member states, and Nandelenga and Ellyne (2020) for sub-Saharan African countries all used logistic regression models and found that the determinants were mainly rule-related.¹⁰Larch and Santacroce (2020) highlight correlations between SGP fiscal rules and various macroeconomic variables such as the market volatility index, the output gap, nominal GDP growth and governance quality. Nandelenga and Ellyne (2020) and Reuter (2019) seek causal relationships whereas Larch and Santacroce (2020) simply look at correlations. These studies are all based on contemporaneous data and cannot be used for forecasting. Moreover, Nandelenga and Ellyne (2020) and Reuter (2019) use a universal model to investigate compliance with different rules: in some countries the limit is on structural balance while in others it concerns the overall balance or the balance excluding public investment.¹¹In this paper, we focus on the ‘3% limit on public deficit’ which applies to all European Union member states.

Furthermore, the Macroeconomic Imbalance Procedure (MIP) scoreboard introduced as part of the ‘Six-Pack’ reform (2011) fundamentally modified the preventive arm of the SGP. The purpose of this scoreboard is to monitor a wide range of indicators to identify risks of internal and external imbalances that could destabilise public finance. The implementation of the European Semester, since 2010, was considered, at the time, a milestone towards efficient monitoring of public finance. While SGP reforms have indeed introduced powerful tools for close and thorough monitoring of public finance, these tools have created new constraints for member states and created an excessively complex framework. A more viable solution may be to simplify the rules in the monitoring process.¹²Improving the monitoring procedure could be interpreted as making the preventive arm more efficient, while simultaneously strengthening the dissuasive arm. In other words, strengthening the preventive arm may reduce the number of cases of excessive deficit and thus strengthen the dissuasive arm. Following this idea, in this paper, we propose a model to forecast compliance with the 3% limit on public deficit. The model is created using machine learning, a method that has seldom been employed to date in macroeconomics but that often outperforms traditional econometric approaches, especially in term of forecasting accuracy (see Ince and Trafalis, 2006; Plakandaras et al., 2013).

This paper makes several original contributions. First, it focuses on compliance with the European supranational fiscal rules of the SGP (1996) using machine learning. Macroeconomic studies involving machine learning methods are still rare in the literature but are becoming increasingly popular (see e.g. Medeiros et al., 2021; Gogas et al., 2022; Yoon, 2021). The use of machine learning to study SGP compliance is original. Second, the proposed method can handle the size limitation of datasets often encountered in macroeconomics studies. While early machine

<footnote n="10">¹⁰Features that tended to strengthen fiscal rule compliance were registration in the law, level of rigor, degree of public finance coverage, etc.</footnote>

<footnote n="11">¹¹For example, Denmark and the United Kingdom both had a budget balance rule in 2005 but while the UK had a Golden rule, in Denmark the rule imposed a general 2% GDP threshold on general government surplus (see e.g. Schaechter et al. (2016)).</footnote>

<footnote n="12">¹²A simplification of the fiscal framework is advocated in the European European Commission (2020) or the European Fiscal European Fiscal Board (2020).</footnote>
literature by using the groundwork on compliance to strengthen the preventive arm of the SGP in the euro area.

Indeed, a large part of the fiscal rule compliance literature has focused on compliance scores and the determinants of fiscal rule compliance. Delgado-Téllez et al. (2017) for Spanish regions, Reuter (2019) for EU member states, and Nandelenga and Ellyne (2020) for sub-Saharan African countries all used logistic regression models and found that the determinants were mainly rule-related.¹⁰Larch and Santacroce (2020) highlight correlations between SGP fiscal rules and various macroeconomic variables such as the market volatility index, the output gap, nominal GDP growth and governance quality. Nandelenga and Ellyne (2020) and Reuter (2019) seek causal relationships whereas Larch and Santacroce (2020) simply look at correlations. These studies are all based on contemporaneous data and cannot be used for forecasting. Moreover, Nandelenga and Ellyne (2020) and Reuter (2019) use a universal model to investigate compliance with different rules: in some countries the limit is on structural balance while in others it concerns the overall balance or the balance excluding public investment.¹¹In this paper, we focus on the ‘3% limit on public deficit’ which applies to all European Union member states.

Furthermore, the Macroeconomic Imbalance Procedure (MIP) scoreboard introduced as part of the ‘Six-Pack’ reform (2011) fundamentally modified the preventive arm of the SGP. The purpose of this scoreboard is to monitor a wide range of indicators to identify risks of internal and external imbalances that could destabilise public finance. The implementation of the European Semester, since 2010, was considered, at the time, a milestone towards efficient monitoring of public finance. While SGP reforms have indeed introduced powerful tools for close and thorough monitoring of public finance, these tools have created new constraints for member states and created an excessively complex framework. A more viable solution may be to simplify the rules in the monitoring process.¹²Improving the monitoring procedure could be interpreted as making the preventive arm more efficient, while simultaneously strengthening the dissuasive arm. In other words, strengthening the preventive arm may reduce the number of cases of excessive deficit and thus strengthen the dissuasive arm. Following this idea, in this paper, we propose a model to forecast compliance with the 3% limit on public deficit. The model is created using machine learning, a method that has seldom been employed to date in macroeconomics but that often outperforms traditional econometric approaches, especially in term of forecasting accuracy (see Ince and Trafalis, 2006; Plakandaras et al., 2013).

This paper makes several original contributions. First, it focuses on compliance with the European supranational fiscal rules of the SGP (1996) using machine learning. Macroeconomic studies involving machine learning methods are still rare in the literature but are becoming increasingly popular (see e.g. Medeiros et al., 2021; Gogas et al., 2022; Yoon, 2021). The use of machine learning to study SGP compliance is original. Second, the proposed method can handle the size limitation of datasets often encountered in macroeconomics studies. While early machine

<footnote n="10">¹⁰Features that tended to strengthen fiscal rule compliance were registration in the law, level of rigor, degree of public finance coverage, etc.</footnote>

<footnote n="11">¹¹For example, Denmark and the United Kingdom both had a budget balance rule in 2005 but while the UK had a Golden rule, in Denmark the rule imposed a general 2% GDP threshold on general government surplus (see e.g. Schaechter et al. (2016)).</footnote>

<footnote n="12">¹²A simplification of the fiscal framework is advocated in the European European Commission (2020) or the European Fiscal European Fiscal Board (2020).</footnote>
<figure><img src="image_6.png" /></figure>

learning algorithms required extensive datasets that are typically unavailable in economics (with the exception of finance), many newer machine learning architectures (such as support vector machines (SVM), Random Forests, and various boosting-based methods) have more accessible data requirements and offer an interesting avenue in economic forecasting. Third, this study provides a geometric representation of the input set with a separation line as a valuable by-product. As well as accurately forecasting compliance with the 3% limit, the model also measures their distance from the separation line. This distance can be used (a) to estimate the confidence of the forecast (the further from the line, the higher the confidence in the classification is), and (b) to outline the recommendations and policies most likely to change a country's forecasted negative outcome.

This analysis focuses on the 28 EU member states from 2006 to 2018 and offers a new perspective in the debate on the relative weights of the dissuasive vs preventive arm of the SGP. We argue for a simpler but more powerful preventive arm, to minimise the number of cases in which dissuasion is required. Indeed, this approach can be used to optimise recommendations to 'non-compliant' countries. Instead of sanctions, EU authorities could preemptively propose a set of well-targeted interventions, one year in advance, to alter the outcomes of 'non-compliers' (i.e. the interventions most likely move the country from the non-compliant subspace to the compliant subspace of the model).

The rest of the paper is structured as follows. Section 2 reviews the literature. Section 3 presents the data and descriptive statistics. Section 4 describes the empirical strategy. Section 5 presents the results and validation procedures and Section 6 concludes.

## 2 | LITERATURE REVIEW

Fiscal rule performance has been widely studied. Fiscal rules are indeed one of the main methods of controlling fiscal discipline and their assessment usually begins with Kopits & Symansky's, 1998 'ideal fiscal rule'.<fnref n="13" /> Since fiscal rules are heterogeneous in design and application, they also have heterogeneous effects and achieve different levels of compliance. Following Kopits and Symansky (1998), some studies rank fiscal policy rules in terms of these ideal properties,<fnref n="14" /> and many others such as Debrun et al. (2008) have used empirical approaches to show that national fiscal rules seem to be correlated with government fiscal performance (see also IMF, 2009). This theory is supported by other empirical studies, such as Reuter (2015) based on least square dummy variable regression, Bergman et al. (2016) based on system-GMM, and Tapsoba (2012) and Barbier-Gauchard et al. (2021) which use the propensity-score matching method. The performance of fiscal rules also depends on their macro-stabilising power. For instance, Sacchi and Salotti (2015) found that national fiscal rules contribute to GDP stabilisation. Guerguil et al. (2017) showed that flexible budget balance rules support public expenditure stabilisation (for the standard definition of flexible rules see Dabán (2011), Schick (2010) or Caselli et al. (2018)). Many studies have investigated the impact of the supranational fiscal rules in the

<footnote n="13">¹³For Kopits and Symansky (1998), the 'ideal fiscal rule' has the following eight properties: (1) Suitability for the intended objective, (2) Clear definition, (3) General consistency, (4) Robust analytical foundations, (5) Transparency, (6) Simplicity, (7) Flexibility, (8) Credibility.</footnote>

<footnote n="14">¹⁴See Creel (2003) for an attempt at evaluating these properties.</footnote>
<figure><img src="image_5.png" /></figure>

learning algorithms required extensive datasets that are typically unavailable in economics (with the exception of finance), many newer machine learning architectures (such as support vector machines (SVM), Random Forests, and various boosting-based methods) have more accessible data requirements and offer an interesting avenue in economic forecasting. Third, this study provides a geometric representation of the input set with a separation line as a valuable by-product. As well as accurately forecasting compliance with the 3% limit, the model also measures their distance from the separation line. This distance can be used (a) to estimate the confidence of the forecast (the further from the line, the higher the confidence in the classification is), and (b) to outline the recommendations and policies most likely to change a country's forecasted negative outcome.

This analysis focuses on the 28 EU member states from 2006 to 2018 and offers a new perspective in the debate on the relative weights of the dissuasive vs preventive arm of the SGP. We argue for a simpler but more powerful preventive arm, to minimise the number of cases in which dissuasion is required. Indeed, this approach can be used to optimise recommendations to 'non-compliant' countries. Instead of sanctions, EU authorities could preemptively propose a set of well-targeted interventions, one year in advance, to alter the outcomes of 'non-compliers' (i.e. the interventions most likely move the country from the non-compliant subspace to the compliant subspace of the model).

The rest of the paper is structured as follows. Section 2 reviews the literature. Section 3 presents the data and descriptive statistics. Section 4 describes the empirical strategy. Section 5 presents the results and validation procedures and Section 6 concludes.

## 2 | LITERATURE REVIEW

Fiscal rule performance has been widely studied. Fiscal rules are indeed one of the main methods of controlling fiscal discipline and their assessment usually begins with Kopits & Symansky's, 1998 'ideal fiscal rule'.<fnref n="13" /> Since fiscal rules are heterogeneous in design and application, they also have heterogeneous effects and achieve different levels of compliance. Following Kopits and Symansky (1998), some studies rank fiscal policy rules in terms of these ideal properties,<fnref n="14" /> and many others such as Debrun et al. (2008) have used empirical approaches to show that national fiscal rules seem to be correlated with government fiscal performance (see also IMF, 2009). This theory is supported by other empirical studies, such as Reuter (2015) based on least square dummy variable regression, Bergman et al. (2016) based on system-GMM, and Tapsoba (2012) and Barbier-Gauchard et al. (2021) which use the propensity-score matching method. The performance of fiscal rules also depends on their macro-stabilising power. For instance, Sacchi and Salotti (2015) found that national fiscal rules contribute to GDP stabilisation. Guerguil et al. (2017) showed that flexible budget balance rules support public expenditure stabilisation (for the standard definition of flexible rules see Dabán (2011), Schick (2010) or Caselli et al. (2018)). Many studies have investigated the impact of the supranational fiscal rules in the

<footnote n="13">¹³For Kopits and Symansky (1998), the 'ideal fiscal rule' has the following eight properties: (1) Suitability for the intended objective, (2) Clear definition, (3) General consistency, (4) Robust analytical foundations, (5) Transparency, (6) Simplicity, (7) Flexibility, (8) Credibility.</footnote>

<footnote n="14">¹⁴See Creel (2003) for an attempt at evaluating these properties.</footnote>
SGP on the counter-cyclical features of national fiscal policy, as recently reported by Larch et al. (2021).

Another well-studied question in fiscal rule research is how to solve the ‘fiscal rules tri-lemma’. This literature has inspired a second generation of fiscal rules, as highlighted by Caselli et al. (2018) and Eyraud et al. (2018), based on fiscal frameworks and stronger incentives to achieve compliance. The ‘fiscal Taylor rule’ proposal by Debrun and Jonung (2019) is an illustration of what a second generation fiscal rule could be. In the same vein, Blanchard et al. (2020) propose a shift from fiscal rules to ‘enforceable fiscal standards.’

Other studies of existing fiscal rules have investigated their enforceability (which also appears in the ‘fiscal rules trilemma’), which ultimately translates into credibility and influences the level of compliance with the rules. This paper fits into the latter strand of literature. Thus, we assume that: (i) the existence of fiscal rules is justified (see for instance Debrun et al. (2008) or Wyplosz (2012) for a general overview of the main reasons for introducing fiscal rules) and (ii) that the applied quantitative limits are optimal. It is not the purpose of this paper to question these assumptions.

There have been many studies of fiscal rule compliance. Some assess levels of compliance with fiscal rules based on fiscal rule databases maintained by the EU Commission (2017) or the IMF (2016).¹⁵These databases provide information on the fiscal rule and its coverage, its statutory base, monitoring bodies, correction mechanisms in cases of deviation, and records of compliance with the rule.

Composite indicators have been defined based on these data to assess the potential coercive power of fiscal rules (see for instance, the European Commission's or the IMF's Fiscal Rule Index (FRI)<fnref n="16" />). However, to assess the effective coercive power of a fiscal rule, the level of relevant fiscal aggregates needs to be compared with the limits set by the rule. Reuter (2015) and Larch and Santacroce (2020) found that numerical fiscal rules are complied with in 50% of cases. In the same vein, Delgado-Téllez et al. (2017) have analysed compliance at the subnational level in Spain and Cordes et al. (2015) have investigated compliance with public expenditure rules in advanced and emerging countries.

Other studies have focused on the key determinants of fiscal rule compliance. Reuter (2019) found that in the European Union from 1995 to 2015, certain features of the rules (in particular their legal basis and the existence of independent monitoring and enforcement authorities), the degree of government fragmentation and the political cycle had a significant effect on whether or not national fiscal rules were followed. Nandelenga and Ellyne (2020) implemented a similar analysis for sub-Saharan African countries but found that countries' economic status (output gap, inflation rate, public debt or interest payments) and the position in the economic cycle were not significantly associated with national fiscal rule compliance, and that combining fiscal rules (at the national level or by adding to regional or local fiscal rules) did not significantly affect compliance. Larch and Santacroce (2020) studied the determinants of compliance with SGP (1996) fiscal rules in the European Union. These fiscal rules set different targets in terms of fiscal aggregates (deficit rule, debt rule, structural balance and expenditure). Their study of European Union countries from 1998 to 2019 reveals stark and persistent differences between countries and highlights noteworthy links between numerical compliance on the one hand and key macroeconomic variables (especially episodes

<footnote n="15">¹⁵Schaechter et al. (2016).</footnote>

<footnote n="16">¹⁶The European Commission's FRI is based on five parameters: (1) legal basis, (2) binding character, (3) compliance monitoring bodies, (4) correction mechanisms, and (5) resilience to shocks.</footnote>
SGP on the counter-cyclical features of national fiscal policy, as recently reported by Larch et al. (2021).

Another well-studied question in fiscal rule research is how to solve the ‘fiscal rules tri-lemma’. This literature has inspired a second generation of fiscal rules, as highlighted by Caselli et al. (2018) and Eyraud et al. (2018), based on fiscal frameworks and stronger incentives to achieve compliance. The ‘fiscal Taylor rule’ proposal by Debrun and Jonung (2019) is an illustration of what a second generation fiscal rule could be. In the same vein, Blanchard et al. (2020) propose a shift from fiscal rules to ‘enforceable fiscal standards.’

Other studies of existing fiscal rules have investigated their enforceability (which also appears in the ‘fiscal rules trilemma’), which ultimately translates into credibility and influences the level of compliance with the rules. This paper fits into the latter strand of literature. Thus, we assume that: (i) the existence of fiscal rules is justified (see for instance Debrun et al. (2008) or Wyplosz (2012) for a general overview of the main reasons for introducing fiscal rules) and (ii) that the applied quantitative limits are optimal. It is not the purpose of this paper to question these assumptions.

There have been many studies of fiscal rule compliance. Some assess levels of compliance with fiscal rules based on fiscal rule databases maintained by the EU Commission (2017) or the IMF (2016).¹⁵These databases provide information on the fiscal rule and its coverage, its statutory base, monitoring bodies, correction mechanisms in cases of deviation, and records of compliance with the rule.

Composite indicators have been defined based on these data to assess the potential coercive power of fiscal rules (see for instance, the European Commission's or the IMF's Fiscal Rule Index (FRI)<fnref n="16" />). However, to assess the effective coercive power of a fiscal rule, the level of relevant fiscal aggregates needs to be compared with the limits set by the rule. Reuter (2015) and Larch and Santacroce (2020) found that numerical fiscal rules are complied with in 50% of cases. In the same vein, Delgado-Téllez et al. (2017) have analysed compliance at the subnational level in Spain and Cordes et al. (2015) have investigated compliance with public expenditure rules in advanced and emerging countries.

Other studies have focused on the key determinants of fiscal rule compliance. Reuter (2019) found that in the European Union from 1995 to 2015, certain features of the rules (in particular their legal basis and the existence of independent monitoring and enforcement authorities), the degree of government fragmentation and the political cycle had a significant effect on whether or not national fiscal rules were followed. Nandelenga and Ellyne (2020) implemented a similar analysis for sub-Saharan African countries but found that countries' economic status (output gap, inflation rate, public debt or interest payments) and the position in the economic cycle were not significantly associated with national fiscal rule compliance, and that combining fiscal rules (at the national level or by adding to regional or local fiscal rules) did not significantly affect compliance. Larch and Santacroce (2020) studied the determinants of compliance with SGP (1996) fiscal rules in the European Union. These fiscal rules set different targets in terms of fiscal aggregates (deficit rule, debt rule, structural balance and expenditure). Their study of European Union countries from 1998 to 2019 reveals stark and persistent differences between countries and highlights noteworthy links between numerical compliance on the one hand and key macroeconomic variables (especially episodes

<footnote n="15">¹⁵Schaechter et al. (2016).</footnote>

<footnote n="16">¹⁶The European Commission's FRI is based on five parameters: (1) legal basis, (2) binding character, (3) compliance monitoring bodies, (4) correction mechanisms, and (5) resilience to shocks.</footnote>
<figure><img src="image_7.png" /></figure>

of pro-cyclical fiscal policy) and the quality of governing institutions on the other (countries with ‘watchdogs’ (Debrun et al., 2019), i.e. national independent fiscal institutions). Nevertheless, as suggested by Reuter (2015), fiscal rules can be seen as a tool ‘to force governments to adjust their budgetary plans in such a way that the constrained variable is moving in the direction of the constraint’. In this case, levels of compliance and the factors that explain levels of compliance could enable the forecasting of fiscal rule compliance.

The aim of this paper is to offer additional insight into how the preventive arm of fiscal rules can prevent the occurrence of excessive deficits. We focus on the eurozone but our approach could also be applied to future eurozone candidates (in the sense that most of the countries in the EU but not in the EMU are destined to join the EMU in the long term). We propose to deepen the analysis in this direction by identifying the determinants of SGP compliance and use these to forecast future compliance. The focus is therefore on the preventive arm of the SGP. This article presents a new machine learning based forecasting model for supranational fiscal rule compliance.

Machine learning methodologies for classification and forecasting are increasingly being employed in economics. Gogas et al. (2015) investigated the ability of the yield curve to forecast economic activity. They forecasted positive and negative deviations of real US GDP from long-run trends from 1976 to 2014 and found that the best support vector machine (SVM) model outperformed the econometric probit model. Gogas et al. (2018) subsequently used SVM to forecast U.S. bank failures and obtained a striking 99.22% overall forecasting accuracy, outperforming the well-established Ohlson score. Härdle et al. (2009) studied the default risk of companies with SVM and Huang et al. (2004) used SVM to forecast corporate credit ratings in the U.S. and Taiwan. They compared SVM to back propagation neural networks, with linear SVM performing best in all cases. Our study extends the application of machine learning to public policy issues.

# 3 | DATA AND DESCRIPTIVE STATISTICS

## 3.1 | Compliance with supranational fiscal rules in the European union and public finance statistics

In the European Union, the concept of supranational fiscal rules first appears in the Maastricht Treaty (1992), which launched the monetary union project and set the conditions that had to be satisfied to achieve it. Some of these relate to the public finance stability that candidate countries need to achieve to become members of the eurozone: (a) public deficit should not exceed 3% of GDP and (b) public debt should not exceed 60% of GDP. As soon as candidate countries are admitted to the eurozone they become subject to the rules of the SGP (1996)<fnref n="17" /> which initially only referred to the 3% deficit threshold.<fnref n="18" /> In the early 2000s, some countries did not respect the fiscal

<footnote n="17">¹⁷The term ‘Stability and Growth Pact’ only appears in the Amsterdam European Council Resolution of June 1997 (from p. 18) but has been the established term for the budgetary discipline rules in force in EMU since 1 January 1999 (See e.g EU Council Regulation (1997a, 1997b)).</footnote>

<footnote n="18">¹⁸Nevertheless, all EU-members (both eurozone and non-eurozone members) are monitored as part of the SGP since the European Commission requests multiannual programs on public finance that forecast the level and nature of public finances for the next 3 years. EU countries that do not belong to the eurozone are expected to provide ‘stability programs’ every year; eurozone countries must provide ‘convergence programs’. In the event of adverse public finance trajectories, the European Commission provides recommendations for to rectify the deficiencies. No deadline was initially imposed for these programs and monitoring was not as thorough as it is in the latest version of the SGP.</footnote>
<figure><img src="image_6.png" /></figure>

of pro-cyclical fiscal policy) and the quality of governing institutions on the other (countries with ‘watchdogs’ (Debrun et al., 2019), i.e. national independent fiscal institutions). Nevertheless, as suggested by Reuter (2015), fiscal rules can be seen as a tool ‘to force governments to adjust their budgetary plans in such a way that the constrained variable is moving in the direction of the constraint’. In this case, levels of compliance and the factors that explain levels of compliance could enable the forecasting of fiscal rule compliance.

The aim of this paper is to offer additional insight into how the preventive arm of fiscal rules can prevent the occurrence of excessive deficits. We focus on the eurozone but our approach could also be applied to future eurozone candidates (in the sense that most of the countries in the EU but not in the EMU are destined to join the EMU in the long term). We propose to deepen the analysis in this direction by identifying the determinants of SGP compliance and use these to forecast future compliance. The focus is therefore on the preventive arm of the SGP. This article presents a new machine learning based forecasting model for supranational fiscal rule compliance.

Machine learning methodologies for classification and forecasting are increasingly being employed in economics. Gogas et al. (2015) investigated the ability of the yield curve to forecast economic activity. They forecasted positive and negative deviations of real US GDP from long-run trends from 1976 to 2014 and found that the best support vector machine (SVM) model outperformed the econometric probit model. Gogas et al. (2018) subsequently used SVM to forecast U.S. bank failures and obtained a striking 99.22% overall forecasting accuracy, outperforming the well-established Ohlson score. Härdle et al. (2009) studied the default risk of companies with SVM and Huang et al. (2004) used SVM to forecast corporate credit ratings in the U.S. and Taiwan. They compared SVM to back propagation neural networks, with linear SVM performing best in all cases. Our study extends the application of machine learning to public policy issues.

# 3 | DATA AND DESCRIPTIVE STATISTICS

## 3.1 | Compliance with supranational fiscal rules in the European union and public finance statistics

In the European Union, the concept of supranational fiscal rules first appears in the Maastricht Treaty (1992), which launched the monetary union project and set the conditions that had to be satisfied to achieve it. Some of these relate to the public finance stability that candidate countries need to achieve to become members of the eurozone: (a) public deficit should not exceed 3% of GDP and (b) public debt should not exceed 60% of GDP. As soon as candidate countries are admitted to the eurozone they become subject to the rules of the SGP (1996)<fnref n="17" /> which initially only referred to the 3% deficit threshold.<fnref n="18" /> In the early 2000s, some countries did not respect the fiscal

<footnote n="17">¹⁷The term ‘Stability and Growth Pact’ only appears in the Amsterdam European Council Resolution of June 1997 (from p. 18) but has been the established term for the budgetary discipline rules in force in EMU since 1 January 1999 (See e.g EU Council Regulation (1997a, 1997b)).</footnote>

<footnote n="18">¹⁸Nevertheless, all EU-members (both eurozone and non-eurozone members) are monitored as part of the SGP since the European Commission requests multiannual programs on public finance that forecast the level and nature of public finances for the next 3 years. EU countries that do not belong to the eurozone are expected to provide ‘stability programs’ every year; eurozone countries must provide ‘convergence programs’. In the event of adverse public finance trajectories, the European Commission provides recommendations for to rectify the deficiencies. No deadline was initially imposed for these programs and monitoring was not as thorough as it is in the latest version of the SGP.</footnote>
<figure><img src="image_8.png" /><figcaption>FIGURE 1 Overview of SGP compliance for the 28 EU countries between 2006 and 2018 ('0' means non-compliance and '1' means compliance). [Colour figure can be viewed at wileyonlinelibrary.com]</figcaption></figure>

rule and excessive deficit procedures were launched against Portugal (in 2002), and France and Germany (in 2003), but sanctions were never implemented.

The SGP has been reformed several times (in 2005, the Six Pack<fnref n="19" /> in 2011, the Two Pack<fnref n="20" /> in 2013, the next reform is expected in 2024) to attempt to strengthen the preventive arm (to balance public finance in the medium term) and the dissuasive arm (defining the rules to respect and the sanctions in case of non-compliance<fnref n="21" />) of the SGP. The main idea is to foster balanced public budgets in the medium term to ensure compliance with the 3% rule should the economic

<footnote n="19">¹⁹The Six Pack was adopted in October 2011 by the European Parliament and the Council and came into force on 13 December 2011. It is a set of six new legislative texts: five regulations and one directive (hence the name six pack). See e.g. EU Regulation (2011a, 2011b, 2011c, 2011d, 2011e), EU Council Regulation (2011), EU Council Directive (2011).</footnote>

<footnote n="20">²⁰The Two Pack was adopted in May 2013 by the European Parliament and the Council and came into force immediately in all eurozone states. It consists of two regulations (hence the name Two Pack). See e.g. EU Regulation (2013a, 2013b).</footnote>

<footnote n="21">²¹Ranging from 0.5% to 2% of GDP.</footnote>
<figure><img src="image_7.png" /><figcaption>FIGURE 1 Overview of SGP compliance for the 28 EU countries between 2006 and 2018 ('0' means non-compliance and '1' means compliance). [Colour figure can be viewed at wileyonlinelibrary.com]</figcaption></figure>

rule and excessive deficit procedures were launched against Portugal (in 2002), and France and Germany (in 2003), but sanctions were never implemented.

The SGP has been reformed several times (in 2005, the Six Pack<fnref n="19" /> in 2011, the Two Pack<fnref n="20" /> in 2013, the next reform is expected in 2024) to attempt to strengthen the preventive arm (to balance public finance in the medium term) and the dissuasive arm (defining the rules to respect and the sanctions in case of non-compliance<fnref n="21" />) of the SGP. The main idea is to foster balanced public budgets in the medium term to ensure compliance with the 3% rule should the economic

<footnote n="19">¹⁹The Six Pack was adopted in October 2011 by the European Parliament and the Council and came into force on 13 December 2011. It is a set of six new legislative texts: five regulations and one directive (hence the name six pack). See e.g. EU Regulation (2011a, 2011b, 2011c, 2011d, 2011e), EU Council Regulation (2011), EU Council Directive (2011).</footnote>

<footnote n="20">²⁰The Two Pack was adopted in May 2013 by the European Parliament and the Council and came into force immediately in all eurozone states. It consists of two regulations (hence the name Two Pack). See e.g. EU Regulation (2013a, 2013b).</footnote>

<footnote n="21">²¹Ranging from 0.5% to 2% of GDP.</footnote>
<figure><img src="image_9.png" /></figure>

situation deteriorate. The monitoring of national cyclically adjusted public balances and cyclical features has been reinforced in recent years. One of the key objectives of the MIP scoreboard introduced with the Six Pack (2011) is to identify risks of internal and external imbalances that could destabilise public finance in the long term.<fnref n="22" />

Unfortunately, these measures seem insufficient to assess the risk of a country exceeding the 3% threshold: many have continued to violate the 3% rule, as shown in Figure 1 and Table 1. We are interested in the preventive instrument and search for the best indicators to forecast compliance with the 3% rule. We are thus interested in the reasons why the SGP is still not satisfied after all these reforms. In this paper, we analyse data from the 28 EU member states over 2006–2018, a period that follows the first SGP reform and includes the Six-Pack (2011) and Two-Pack (2013) reforms. This allows us to evaluate government efforts in response to the SGP reforms.<fnref n="23" />

Figure 1 plots the SGP compliance of the 28 European countries between 2006 and 2018. It highlights substantial differences in government behaviour toward the SGP. As pointed out by the European Commission,<fnref n="24" /> the European fiscal framework and the SGP have become extremely complex for member countries to implement. Indeed, in addition to the target public finance indicators, many macroeconomic variables are monitored on which governments cannot intervene directly. Concurrently, since the early 1990s, the number of national fiscal rules self-imposed by EU member states has increased substantially, adding an extra layer of rules to be respected.

Table 1 provides an overview of public finance statistics and SGP compliance. This heterogeneity of national public finance reinforces the conclusions drawn from Figure 1, that member states do not adopt the same attitude toward the SGP. Countries such as Luxembourg, Estonia and Sweden complied with the SGP throughout the studied period while France complied in just 4 years (out of 12). Since member states react differently to the same symmetrical shocks (Bak and Maciejewski, 2017; Frenkel and Nickel, 2002; Velickovski and Stojkov, 2014), they also react differently to a single and general fiscal rule. The European Commission has already implemented the idea of providing country-specific recommendations for public finance plans as part of the European Semester.

Despite the SGP criterion not having been met in 36% of cases,<fnref n="25" /> sanctions have never been applied to avoid worsening the economic situation of non-compliant states. Ireland's public deficit reached 32% of GDP during the Sovereign Debt Crisis and no financial sanctions were applied. The major problem, if deficits slip away, is that the debt can become too large. For example,

<footnote n="22">²²This scoreboard covers primary and auxiliary indicators: (i) external imbalance and competitiveness indicators: current account balance (3 year average), net external investment (in % of GDP), real effective exchange rate (3 year % change), export market shares (5 year % change) and nominal unit labor cost (3 year % change); (ii) internal imbalance indicators: deflated house price index (1 year % change), consolidated private sector credit flow (% of GDP), consolidated private sector debt (% of GDP), general government sector debt (% of GDP), unemployment rate (3 year average), non-consolidated total financial sector liabilities (1 year % change).</footnote>

<footnote n="23">²³The data for many of the macroeconomic variables we use are incomplete until after 2005, and our algorithm is very sensitive to missing values.</footnote>

<footnote n="24">²⁴See the European Commission website and communication on EU governance review.</footnote>

<footnote n="25">²⁵This result is not surprising since similar compliance rates have been reported for national fiscal rules. Despite the 'magnet effect' of national fiscal rules, Eyraud et al. (2018) point out the 'poor track record of compliance' with fiscal rules. Similarly, Reuter (2015) showed that while governments do tend to move closer to national fiscal limits, in the end, fiscal rules are complied with in just 51% of the cases.</footnote>
<figure><img src="image_8.png" /></figure>

situation deteriorate. The monitoring of national cyclically adjusted public balances and cyclical features has been reinforced in recent years. One of the key objectives of the MIP scoreboard introduced with the Six Pack (2011) is to identify risks of internal and external imbalances that could destabilise public finance in the long term.<fnref n="22" />

Unfortunately, these measures seem insufficient to assess the risk of a country exceeding the 3% threshold: many have continued to violate the 3% rule, as shown in Figure 1 and Table 1. We are interested in the preventive instrument and search for the best indicators to forecast compliance with the 3% rule. We are thus interested in the reasons why the SGP is still not satisfied after all these reforms. In this paper, we analyse data from the 28 EU member states over 2006–2018, a period that follows the first SGP reform and includes the Six-Pack (2011) and Two-Pack (2013) reforms. This allows us to evaluate government efforts in response to the SGP reforms.<fnref n="23" />

Figure 1 plots the SGP compliance of the 28 European countries between 2006 and 2018. It highlights substantial differences in government behaviour toward the SGP. As pointed out by the European Commission,<fnref n="24" /> the European fiscal framework and the SGP have become extremely complex for member countries to implement. Indeed, in addition to the target public finance indicators, many macroeconomic variables are monitored on which governments cannot intervene directly. Concurrently, since the early 1990s, the number of national fiscal rules self-imposed by EU member states has increased substantially, adding an extra layer of rules to be respected.

Table 1 provides an overview of public finance statistics and SGP compliance. This heterogeneity of national public finance reinforces the conclusions drawn from Figure 1, that member states do not adopt the same attitude toward the SGP. Countries such as Luxembourg, Estonia and Sweden complied with the SGP throughout the studied period while France complied in just 4 years (out of 12). Since member states react differently to the same symmetrical shocks (Bak and Maciejewski, 2017; Frenkel and Nickel, 2002; Velickovski and Stojkov, 2014), they also react differently to a single and general fiscal rule. The European Commission has already implemented the idea of providing country-specific recommendations for public finance plans as part of the European Semester.

Despite the SGP criterion not having been met in 36% of cases,<fnref n="25" /> sanctions have never been applied to avoid worsening the economic situation of non-compliant states. Ireland's public deficit reached 32% of GDP during the Sovereign Debt Crisis and no financial sanctions were applied. The major problem, if deficits slip away, is that the debt can become too large. For example,

<footnote n="22">²²This scoreboard covers primary and auxiliary indicators: (i) external imbalance and competitiveness indicators: current account balance (3 year average), net external investment (in % of GDP), real effective exchange rate (3 year % change), export market shares (5 year % change) and nominal unit labor cost (3 year % change); (ii) internal imbalance indicators: deflated house price index (1 year % change), consolidated private sector credit flow (% of GDP), consolidated private sector debt (% of GDP), general government sector debt (% of GDP), unemployment rate (3 year average), non-consolidated total financial sector liabilities (1 year % change).</footnote>

<footnote n="23">²³The data for many of the macroeconomic variables we use are incomplete until after 2005, and our algorithm is very sensitive to missing values.</footnote>

<footnote n="24">²⁴See the European Commission website and communication on EU governance review.</footnote>

<footnote n="25">²⁵This result is not surprising since similar compliance rates have been reported for national fiscal rules. Despite the 'magnet effect' of national fiscal rules, Eyraud et al. (2018) point out the 'poor track record of compliance' with fiscal rules. Similarly, Reuter (2015) showed that while governments do tend to move closer to national fiscal limits, in the end, fiscal rules are complied with in just 51% of the cases.</footnote>
<table><caption>TABLE 1 Public finance statistics in European Union countries between 2004 and 2018.</caption><thead><tr><td>Key indicator</td><td>Mean</td><td>Country with best value</td><td>Country with worst value</td></tr></thead><tbody><tr><td>3% limit compliance (in %)</td><td>63.70</td><td>Estonia, Luxembourg, Sweden (complied with the rule every year)</td><td>France (complied only 4 times with the rule)</td></tr><tr><td>Public Budget Balance (in % GDP)</td><td>-2.556</td><td>Finland in 2008: 5.129 (highest public balance over the period)</td><td>Ireland in 2011: -32.028 (highest public deficit over the period)</td></tr><tr><td>General government gross public debt (in % GDP)</td><td>58.69</td><td>Estonia in 2009: 3.664 (lowest public debt over the period)</td><td>Greece in 2017: 183.45 (highest public debt over the period)</td></tr><tr><td>Gross fixed capital formation (in % GDP)</td><td>21.96</td><td>Slovak Republic in 2009: 37.4 (highest GFCF over the period)</td><td>Hungary in 2015: 11.5 (lowest GFCF over the period)</td></tr></tbody></table>

Greece's public debt got close to 200% in 2017, putting the EMU at risk of a domino effect (see e.g. Cooper et al. (2014)).

The persistent failure to comply with the SGP despite its reforms suggests that the way public finance are monitored should be modified. A simple solution to improve the monitoring process would be to strengthen the preventive arm of the SGP, focusing on the forecasting and surveillance process.

So, in our analysis we consider (i) that a simpler rule would be more easily maintained by the member states, (ii) that the focus should be on the preventive arm, and (iii) that the key drivers of non-compliance should be identified. To this end, we focus only on compliance with 'SGP 1.0' (as dubbed by Debrun and Jonung (2019)), the 3% limit on the public deficit. The proposed forecasting procedure involves two steps: (i) the identification of the key determinants of compliance using a feature selection procedure, (ii) the training of a machine learning model to forecast compliance with the rules one year in advance, giving member states enough time to alter predicted negative outcomes.

## 3.2 | Fiscal rule compliance: potential predictors

We tried to assemble a dataset covering all the relevant features for forecasting SGP compliance. These variables are listed in Table 2 (see also Appendix 2: Table A1 for descriptives statistics of the variables). We used data for the 28 EU members from 2006 to 2018. Our variables can be divided into three groups, namely economic integration indicators, MIP scoreboard indicators, and other macroeconomic variables. The MIP scoreboard is a good starting point because it contains variables intended to prevent external and internal imbalances<fnref n="26" /> and

<footnote n="26">²⁶See Eurostat's definition of the MIP scoreboard.</footnote>
<table><caption>TABLE 1 Public finance statistics in European Union countries between 2004 and 2018.</caption><thead><tr><td>Key indicator</td><td>Mean</td><td>Country with best value</td><td>Country with worst value</td></tr></thead><tbody><tr><td>3% limit compliance (in %)</td><td>63.70</td><td>Estonia, Luxembourg, Sweden (complied with the rule every year)</td><td>France (complied only 4 times with the rule)</td></tr><tr><td>Public Budget Balance (in % GDP)</td><td>-2.556</td><td>Finland in 2008: 5.129 (highest public balance over the period)</td><td>Ireland in 2011: -32.028 (highest public deficit over the period)</td></tr><tr><td>General government gross public debt (in % GDP)</td><td>58.69</td><td>Estonia in 2009: 3.664 (lowest public debt over the period)</td><td>Greece in 2017: 183.45 (highest public debt over the period)</td></tr><tr><td>Gross fixed capital formation (in % GDP)</td><td>21.96</td><td>Slovak Republic in 2009: 37.4 (highest GFCF over the period)</td><td>Hungary in 2015: 11.5 (lowest GFCF over the period)</td></tr></tbody></table>

Greece's public debt got close to 200% in 2017, putting the EMU at risk of a domino effect (see e.g. Cooper et al. (2014)).

The persistent failure to comply with the SGP despite its reforms suggests that the way public finance are monitored should be modified. A simple solution to improve the monitoring process would be to strengthen the preventive arm of the SGP, focusing on the forecasting and surveillance process.

So, in our analysis we consider (i) that a simpler rule would be more easily maintained by the member states, (ii) that the focus should be on the preventive arm, and (iii) that the key drivers of non-compliance should be identified. To this end, we focus only on compliance with 'SGP 1.0' (as dubbed by Debrun and Jonung (2019)), the 3% limit on the public deficit. The proposed forecasting procedure involves two steps: (i) the identification of the key determinants of compliance using a feature selection procedure, (ii) the training of a machine learning model to forecast compliance with the rules one year in advance, giving member states enough time to alter predicted negative outcomes.

## 3.2 | Fiscal rule compliance: potential predictors

We tried to assemble a dataset covering all the relevant features for forecasting SGP compliance. These variables are listed in Table 2 (see also Appendix 2: Table A1 for descriptives statistics of the variables). We used data for the 28 EU members from 2006 to 2018. Our variables can be divided into three groups, namely economic integration indicators, MIP scoreboard indicators, and other macroeconomic variables. The MIP scoreboard is a good starting point because it contains variables intended to prevent external and internal imbalances<fnref n="26" /> and

<footnote n="26">²⁶See Eurostat's definition of the MIP scoreboard.</footnote>
<figure><img src="image_10.png" /></figure>

<table><caption>TABLE 2 Overview of variables considered.</caption><thead><tr><th>Variables</th><th>Correspondence variables</th><th>Source/Database</th></tr></thead><tbody><tr><td>Y</td><td>Dummy variable = 1 if 3% limit was respected in period t</td><td>Authors' calculations</td></tr><tr><td>X1</td><td>Dummy variable reflecting whether the country's economy was rated advanced in period t-p</td><td>IMF Fiscal Rules Database</td></tr><tr><td>X2</td><td>Dummy variable reflecting whether the country's economy was rated emerging in period t-p</td><td>IMF Fiscal Rules Database</td></tr><tr><td>X3</td><td>Dummy variable reflecting EU membership in period t-p</td><td>IMF Fiscal Rules Database</td></tr><tr><td>X4</td><td>Dummy variable reflecting the country's federal nature in period t-p</td><td>IMF Fiscal Rules Database</td></tr><tr><td>X5</td><td>Dummy variable reflecting eurozone membership in period t-p</td><td>IMF Fiscal Rules Database</td></tr><tr><td>X6</td><td>Dummy variable reflecting whether the country was subject to a fiscal rule enforcement procedure in period t-p</td><td>IMF Fiscal Rules Database</td></tr><tr><td>X7</td><td>Gross domestic product, deflator, in period t-p</td><td>Eurostat</td></tr><tr><td>X8</td><td>Total investment in period t-p</td><td>Eurostat</td></tr><tr><td>X9</td><td>Gross national savings in period t-p</td><td>Eurostat</td></tr><tr><td>X10</td><td>Inflation, average consumer prices, in period t-p</td><td>Eurostat</td></tr><tr><td>X11</td><td>Population in period t-p</td><td>Eurostat</td></tr><tr><td>X12</td><td>General government revenue in period t-p</td><td>Eurostat</td></tr><tr><td>X13</td><td>General government total expenditure in period t-p</td><td>Eurostat</td></tr><tr><td>X14</td><td>General government net lending/borrowing in period t-p</td><td>Eurostat</td></tr><tr><td>X15</td><td>General government gross debt in period t-p</td><td>Eurostat</td></tr><tr><td>X16</td><td>Net External Positions in period t-p</td><td>Eurostat</td></tr><tr><td>X17</td><td>Current account balance in period t-p</td><td>Eurostat</td></tr><tr><td>X18</td><td>Current account balance in period t-p, consolidated over 3 years in period t-p</td><td>Eurostat</td></tr><tr><td>X19</td><td>Real Effective Exchange Rate in period t-p</td><td>Eurostat</td></tr><tr><td>X20</td><td>Global export market share -% change over 5 years-in period t-p</td><td>Eurostat</td></tr><tr><td>X21</td><td>Nominal unit wage cost -% change over 3 years-in period t-p</td><td>Eurostat</td></tr><tr><td>X22</td><td>Private sector debt in period t-p, consolidated -% of GDP</td><td>Eurostat</td></tr><tr><td>X23</td><td>Liabilities of the financial corporations sector, -% change over 1 year-in period t-p</td><td>Eurostat</td></tr><tr><td>X24</td><td>Unemployment rate-3-year average-in period t-p</td><td>Eurostat</td></tr><tr><td>X25</td><td>Unemployment rate in period t-p</td><td>Eurostat</td></tr><tr><td>X26</td><td>Gross domestic product (real GDP) -% change over 1 year-in period t-p</td><td>Eurostat</td></tr><tr><td>X27</td><td>Gross fixed capital formation in period t-p-% of GDP</td><td>Eurostat</td></tr><tr><td>X28</td><td>Gross domestic expenditure on R & D in period t-p-% of GDP</td><td>Eurostat</td></tr><tr><td>X29</td><td>Direct investment in the reporting economy (flow) in period t-p-% of GDP</td><td>Eurostat</td></tr><tr><td>X30</td><td>Direct investment in the reporting economy (stocks)-% of GDP</td><td>Eurostat</td></tr><tr><td>X31</td><td>Net trade balance of energy products in period t-p-% of GDP</td><td>Eurostat</td></tr><tr><td>X32</td><td>Real effective exchange rate, Euro area trading partners-% change over 3 years</td><td>Eurostat</td></tr><tr><td>X33</td><td>Terms of trade (goods and services) -% change over 5 years-in period t-p</td><td>Eurostat</td></tr><tr><td>X34</td><td>Market share of world exports, volumes-% change over 1 year-in period t-p</td><td>Eurostat</td></tr><tr><td>X35</td><td>Labor productivity -% change over 1 year-in period t-p</td><td>Eurostat</td></tr><tr><td>X36</td><td>Residential construction in period t-p-% of GDP</td><td>Eurostat</td></tr><tr><td>X37</td><td>Employment-% change over 1 year-in period t-p</td><td>Eurostat</td></tr><tr><td>X38</td><td>Dummy variable reflecting the occurrence of an economic crisis in period t-p</td><td>Author's research</td></tr><tr><td>X39</td><td>Output gap (production function approach) in period t-p</td><td>AMECO Database</td></tr><tr><td>X40</td><td>Oil prices in period t-p</td><td>FED</td></tr><tr><td>X41</td><td>Bond yields in period t-p</td><td>FED</td></tr><tr><td>X42</td><td>Foreign currency long-term sovereign debt ratings, index from 1-21, in period t-p</td><td>World Bank<sup>b</sup></td></tr><tr><td>X43</td><td>Fiscal space in period t-p</td><td>Author's calculations</td></tr><tr><td>X44</td><td>Fiscal Rule Index (European Commission) in period t-p</td><td>European Commission fiscal rules database</td></tr><tr><td>X45</td><td>Structural Breaks in period t-p</td><td>Author's calculations (using Bai and Perron test)</td></tr><tr><td>X46</td><td>Government fragmentation in period t-p</td><td>World Bank<sup>b</sup></td></tr></tbody></table>

Note: <sup>a</sup>A cross-country Database of Fiscal Space, 2019.

<sup>b</sup>Database of Political Institutions.

contains complete series for many macroeconomic variables. Nevertheless, the MIP scoreboard was not designed to forecast SGP compliance. We therefore also include other variables.

Variables X1 to X6 are economic integration indicators: dummy variables for the country's status as an advanced or an emerging economy (X1 and X2); a dummy variable indicating whether the country was an EU Member (X3), and a dummy variable reflecting a country's federal nature (X4). A dummy variable indicating whether or not the country was a eurozone member (X5) is included to test whether eurozone members comply with the rule more often than non-members. We also included a variable to investigate whether formal procedures launched under the SGP affect compliance (X6).

The primary and auxiliary indicators of the MIP scoreboard are included as variables X7 to X37. These variables enable the assessment of trade balance,<sup>27</sup> foreign direct investment flows, labour markets, public debt, private debt, property prices, financial asset prices, etc.

Other macroeconomic parameters that are not monitored by the European Commission are included as variables X38 to X46. X38 is a binary variable indicating the presence of an

<footnote n="27"><sup>27</sup>See for instance Fichet de Clairfontaine et al. (2017) for a study of the link between fiscal and external balance.</footnote>
<figure><img src="image_9.png" /></figure>

<table><caption>TABLE 2 Overview of variables considered.</caption><thead><tr><th>Variables</th><th>Correspondence variables</th><th>Source/Database</th></tr></thead><tbody><tr><td>Y</td><td>Dummy variable = 1 if 3% limit was respected in period t</td><td>Authors' calculations</td></tr><tr><td>X1</td><td>Dummy variable reflecting whether the country's economy was rated advanced in period t-p</td><td>IMF Fiscal Rules Database</td></tr><tr><td>X2</td><td>Dummy variable reflecting whether the country's economy was rated emerging in period t-p</td><td>IMF Fiscal Rules Database</td></tr><tr><td>X3</td><td>Dummy variable reflecting EU membership in period t-p</td><td>IMF Fiscal Rules Database</td></tr><tr><td>X4</td><td>Dummy variable reflecting the country's federal nature in period t-p</td><td>IMF Fiscal Rules Database</td></tr><tr><td>X5</td><td>Dummy variable reflecting eurozone membership in period t-p</td><td>IMF Fiscal Rules Database</td></tr><tr><td>X6</td><td>Dummy variable reflecting whether the country was subject to a fiscal rule enforcement procedure in period t-p</td><td>IMF Fiscal Rules Database</td></tr><tr><td>X7</td><td>Gross domestic product, deflator, in period t-p</td><td>Eurostat</td></tr><tr><td>X8</td><td>Total investment in period t-p</td><td>Eurostat</td></tr><tr><td>X9</td><td>Gross national savings in period t-p</td><td>Eurostat</td></tr><tr><td>X10</td><td>Inflation, average consumer prices, in period t-p</td><td>Eurostat</td></tr><tr><td>X11</td><td>Population in period t-p</td><td>Eurostat</td></tr><tr><td>X12</td><td>General government revenue in period t-p</td><td>Eurostat</td></tr><tr><td>X13</td><td>General government total expenditure in period t-p</td><td>Eurostat</td></tr><tr><td>X14</td><td>General government net lending/borrowing in period t-p</td><td>Eurostat</td></tr><tr><td>X15</td><td>General government gross debt in period t-p</td><td>Eurostat</td></tr><tr><td>X16</td><td>Net External Positions in period t-p</td><td>Eurostat</td></tr><tr><td>X17</td><td>Current account balance in period t-p</td><td>Eurostat</td></tr><tr><td>X18</td><td>Current account balance in period t-p, consolidated over 3 years in period t-p</td><td>Eurostat</td></tr><tr><td>X19</td><td>Real Effective Exchange Rate in period t-p</td><td>Eurostat</td></tr><tr><td>X20</td><td>Global export market share -% change over 5 years-in period t-p</td><td>Eurostat</td></tr><tr><td>X21</td><td>Nominal unit wage cost -% change over 3 years-in period t-p</td><td>Eurostat</td></tr><tr><td>X22</td><td>Private sector debt in period t-p, consolidated -% of GDP</td><td>Eurostat</td></tr><tr><td>X23</td><td>Liabilities of the financial corporations sector, -% change over 1 year-in period t-p</td><td>Eurostat</td></tr><tr><td>X24</td><td>Unemployment rate-3-year average-in period t-p</td><td>Eurostat</td></tr><tr><td>X25</td><td>Unemployment rate in period t-p</td><td>Eurostat</td></tr><tr><td>X26</td><td>Gross domestic product (real GDP) -% change over 1 year-in period t-p</td><td>Eurostat</td></tr><tr><td>X27</td><td>Gross fixed capital formation in period t-p-% of GDP</td><td>Eurostat</td></tr><tr><td>X28</td><td>Gross domestic expenditure on R & D in period t-p-% of GDP</td><td>Eurostat</td></tr><tr><td>X29</td><td>Direct investment in the reporting economy (flow) in period t-p-% of GDP</td><td>Eurostat</td></tr><tr><td>X30</td><td>Direct investment in the reporting economy (stocks)-% of GDP</td><td>Eurostat</td></tr><tr><td>X31</td><td>Net trade balance of energy products in period t-p-% of GDP</td><td>Eurostat</td></tr><tr><td>X32</td><td>Real effective exchange rate, Euro area trading partners-% change over 3 years</td><td>Eurostat</td></tr><tr><td>X33</td><td>Terms of trade (goods and services) -% change over 5 years-in period t-p</td><td>Eurostat</td></tr><tr><td>X34</td><td>Market share of world exports, volumes-% change over 1 year-in period t-p</td><td>Eurostat</td></tr><tr><td>X35</td><td>Labor productivity -% change over 1 year-in period t-p</td><td>Eurostat</td></tr><tr><td>X36</td><td>Residential construction in period t-p-% of GDP</td><td>Eurostat</td></tr><tr><td>X37</td><td>Employment-% change over 1 year-in period t-p</td><td>Eurostat</td></tr><tr><td>X38</td><td>Dummy variable reflecting the occurrence of an economic crisis in period t-p</td><td>Author's research</td></tr><tr><td>X39</td><td>Output gap (production function approach) in period t-p</td><td>AMECO Database</td></tr><tr><td>X40</td><td>Oil prices in period t-p</td><td>FED</td></tr><tr><td>X41</td><td>Bond yields in period t-p</td><td>FED</td></tr><tr><td>X42</td><td>Foreign currency long-term sovereign debt ratings, index from 1-21, in period t-p</td><td>World Bank<sup>b</sup></td></tr><tr><td>X43</td><td>Fiscal space in period t-p</td><td>Author's calculations</td></tr><tr><td>X44</td><td>Fiscal Rule Index (European Commission) in period t-p</td><td>European Commission fiscal rules database</td></tr><tr><td>X45</td><td>Structural Breaks in period t-p</td><td>Author's calculations (using Bai and Perron test)</td></tr><tr><td>X46</td><td>Government fragmentation in period t-p</td><td>World Bank<sup>b</sup></td></tr></tbody></table>

Note: <sup>a</sup>A cross-country Database of Fiscal Space, 2019.

<sup>b</sup>Database of Political Institutions.

contains complete series for many macroeconomic variables. Nevertheless, the MIP scoreboard was not designed to forecast SGP compliance. We therefore also include other variables.

Variables X1 to X6 are economic integration indicators: dummy variables for the country's status as an advanced or an emerging economy (X1 and X2); a dummy variable indicating whether the country was an EU Member (X3), and a dummy variable reflecting a country's federal nature (X4). A dummy variable indicating whether or not the country was a eurozone member (X5) is included to test whether eurozone members comply with the rule more often than non-members. We also included a variable to investigate whether formal procedures launched under the SGP affect compliance (X6).

The primary and auxiliary indicators of the MIP scoreboard are included as variables X7 to X37. These variables enable the assessment of trade balance,<sup>27</sup> foreign direct investment flows, labour markets, public debt, private debt, property prices, financial asset prices, etc.

Other macroeconomic parameters that are not monitored by the European Commission are included as variables X38 to X46. X38 is a binary variable indicating the presence of an

<footnote n="27"><sup>27</sup>See for instance Fichet de Clairfontaine et al. (2017) for a study of the link between fiscal and external balance.</footnote>
economic crisis. It is assigned a value of one during the Global Financial crisis (2007–2008) and during the Sovereign Debt crisis (2010–2011). This is a simple but broad indicator that captures all potential changes in an economy. Nevertheless, according to Sancak et al. (2010) or Boschi and d’Addona (2019), this approach may be imperfect. We therefore use an alternative proxy of economic crises following Barbier-Gauchard et al. (2021), inspired by Wiese et al. (2018), who developed a measure of government fiscal volatility based on the Bai-Perron structural break filter. We thus expect crises and cyclical fluctuations to have an impact on the forecasting of fiscal rules compliance and test for the presence of structural breaks identified by the Bai-Perron test in the structural balance of each country (X46). We also include an indicator of the macroeconomic cycle (the output gap as measured by the production function approach) with variable X39. Reuter (2019) did not find the output gap to be an important determinant of national fiscal rule compliance, while Larch and Santacroce (2020) found a significant correlation between the output gap and SGP compliance. We also include variables X40 to X42 for oil prices, bond yields, foreign currency and long-term sovereign debt ratings.²⁸The fiscal space (X43) is simply measured as the difference between the level of public debt in a given country and the EU median (e.g. Cheng and Pitterle (2018) for an overview of fiscal space definitions). The fiscal space allows estimates of the flexibility of governments' public spending. The intuition is that the ability of governments to spend for as long as public debt seems manageable may affect fiscal rule compliance. The European Commission and the IMF rate the strictness of fiscal rules (their theoretical coercive power) in fiscal rule indices. The European Commission's national Fiscal Rules Strength Index (FRSI) considers the main features of fiscal rules: their legal basis (written as a law or just a government commitment?), the level of public finance coverage (does the rule apply to all public administrations or just to central government?), enforcement procedures (are sanctions imposed?), the presence of a monitoring institution (is there an independent fiscal watchdog in charge of fiscal rule compliance?), stabilisation power (are public investment and cyclical components excluded?). The European Commission's FRSI quantifies the strength of all national (and subnational) fiscal rules by applying a standardisation procedure to these scores. In short, while the FRSI measures the strength of each fiscal rule separately, the FRI (used in our study) represents the strength index of all fiscal rules at all levels of government in a given country. We thus include the FRI (X45) to see whether countries that also implement strong national fiscal rules are more likely to comply with the 3% rule of the SGP. Finally, following Annett (2000) and using the World Bank's Database of Political Institutions, variable X46 measures the level of government fragmentation in each country, as reflected by the dispersion of parties in parliament.²⁹We want to see if government fragmentation is related to SGP compliance. Y is our binary dependent variable describing a country's compliance with the SGP (3% limit) in year t. Our aim is to forecast Yt using lagged values of the 46 variables in the dataset Xij, i = 1, ..., 46, j = t-1, t-2, t-3. Y is the Dependent variable. The X variables are the

<footnote n="28">²⁸An index ranging from 1 to 21 from the World Bank's 'Cross-Country Database of Fiscal Space' (Ayhan Kose et al., 2017).</footnote>

<footnote n="29">²⁹We used the Annett (2000) definition of society fractionalisation applied to government fractionalisation:
$$Fractionalisation = 1 - \sum_{i=1}^{M} \left( \frac{n_i}{N} \right)^2 \quad i = 1, \dots, M$$</footnote>

<footnote>with N being the total number of seats in a country's parliament and $n_i$ is the number of seats belonging to the i-th party. Government fractionalisation is thus defined as the probability that two randomly chosen members of parliament come from two different parties (which also corresponds to the World Bank definition of government fragmentation).</footnote>
economic crisis. It is assigned a value of one during the Global Financial crisis (2007–2008) and during the Sovereign Debt crisis (2010–2011). This is a simple but broad indicator that captures all potential changes in an economy. Nevertheless, according to Sancak et al. (2010) or Boschi and d’Addona (2019), this approach may be imperfect. We therefore use an alternative proxy of economic crises following Barbier-Gauchard et al. (2021), inspired by Wiese et al. (2018), who developed a measure of government fiscal volatility based on the Bai-Perron structural break filter. We thus expect crises and cyclical fluctuations to have an impact on the forecasting of fiscal rules compliance and test for the presence of structural breaks identified by the Bai-Perron test in the structural balance of each country (X46). We also include an indicator of the macroeconomic cycle (the output gap as measured by the production function approach) with variable X39. Reuter (2019) did not find the output gap to be an important determinant of national fiscal rule compliance, while Larch and Santacroce (2020) found a significant correlation between the output gap and SGP compliance. We also include variables X40 to X42 for oil prices, bond yields, foreign currency and long-term sovereign debt ratings.²⁸The fiscal space (X43) is simply measured as the difference between the level of public debt in a given country and the EU median (e.g. Cheng and Pitterle (2018) for an overview of fiscal space definitions). The fiscal space allows estimates of the flexibility of governments' public spending. The intuition is that the ability of governments to spend for as long as public debt seems manageable may affect fiscal rule compliance. The European Commission and the IMF rate the strictness of fiscal rules (their theoretical coercive power) in fiscal rule indices. The European Commission's national Fiscal Rules Strength Index (FRSI) considers the main features of fiscal rules: their legal basis (written as a law or just a government commitment?), the level of public finance coverage (does the rule apply to all public administrations or just to central government?), enforcement procedures (are sanctions imposed?), the presence of a monitoring institution (is there an independent fiscal watchdog in charge of fiscal rule compliance?), stabilisation power (are public investment and cyclical components excluded?). The European Commission's FRSI quantifies the strength of all national (and subnational) fiscal rules by applying a standardisation procedure to these scores. In short, while the FRSI measures the strength of each fiscal rule separately, the FRI (used in our study) represents the strength index of all fiscal rules at all levels of government in a given country. We thus include the FRI (X45) to see whether countries that also implement strong national fiscal rules are more likely to comply with the 3% rule of the SGP. Finally, following Annett (2000) and using the World Bank's Database of Political Institutions, variable X46 measures the level of government fragmentation in each country, as reflected by the dispersion of parties in parliament.²⁹We want to see if government fragmentation is related to SGP compliance. Y is our binary dependent variable describing a country's compliance with the SGP (3% limit) in year t. Our aim is to forecast Yt using lagged values of the 46 variables in the dataset X_{i,j}, i = 1, ..., 46, j = t-1, t-2, t-3. Y is the Dependent variable. The X variables are the

$$Fractionalisation = 1 - \sum_{i=1}^{M} \left( \frac{n_i}{N} \right)^2 \quad i = 1, \dots, M$$

with N being the total number of seats in a country's parliament and $n_i$ is the number of seats belonging to the i-th party. Government fractionalisation is thus defined as the probability that two randomly chosen members of parliament come from two different parties (which also corresponds to the World Bank definition of government fragmentation).

<footnote n="28">²⁸An index ranging from 1 to 21 from the World Bank's 'Cross-Country Database of Fiscal Space' (Ayhan Kose et al., 2017).</footnote>

<footnote n="29">²⁹We used the Annett (2000) definition of society fractionalisation applied to government fractionalisation:</footnote>
<figure><img src="image_11.png" /></figure>

potential predictors tested in the feature selection step. All variables used as predictors are lagged by p, with p = 1, 2, 3 for each feature, leading to 138 tested features (the 46 variables in the 3 lag periods).

# 4 | EMPIRICAL STRATEGY

We performed three sets of tests, comparing in each case the best SVM model with the best logit model on the same dataset. We first included just the main indicators of the MIP scoreboard, to check whether these variables, designed to prevent internal and external imbalances, might also be relevant to SGP compliance. We then tested our framework using the complete dataset presented in Table 2. The second model outperforming the first would therefore indicate that the MIP scoreboard is insufficient to forecast SGP compliance. In the third set of tests, we combined a well-established feature selection method with our forecasting scheme, to select only the necessary variables in the model.

## 4.1 | Forecasting algorithms

### 4.1.1 | The logistic function

We used a traditional econometric method (logit regression) and an emerging approach in economics from the field of machine learning (support vector machine), with the aim of creating the most accurate forecasting model by comparing the two.

The logistic function is commonly used for forecasting purposes, as for instance by Martin (1977) and Olson (1980), who used logistic regression to forecast bank and firm failures. Our forecasting problem can be viewed as one of binary classification: to predict whether the countries in the dataset will comply with the rule (class 1) or not (class 0) $Y \in \{0; 1\}$. The goal of our system is to use the input variables to find the linear or non-linear separator that correctly classifies the cases.

The collected data are represented by $x_{i,j}, i = 1, \dots, n$ and $j = 1, \dots, m$, describing $n = 364$ data points with $m = 138$ features, arranged in vectors $\mathbf{x}_i = [x_{i,1}, \dots, x_{i,138}]^T$. The logistic function constrains $Y$ to the range (0, 1) via the sigmoid function:

$$p(y_i = 1) = \pi_i = \frac{\exp \hat{\mathbf{x}}_i^T \beta}{1 + \exp \hat{\mathbf{x}}_i^T \beta} \quad (1)$$

where $\hat{\mathbf{x}}_i = [1, \mathbf{x}_i^T]^T$ corresponds to a feature-column (the 1 corresponds to the intercept of the regression) and $\beta$ is the column vector of the regression coefficients.

The classifier produces a probability score between 0 and 1, with values lower than 0.5 corresponding to class 0; and probabilities of 0.5 or above corresponding to class 1.

The goal is to find the $\beta$ according to $p(Y|X)$ that most accurately classifies the observed data points. The problem is equivalent to maximising the product of the likelihood probabilities:

$$l(\beta) = \sum_{i=1}^{n} [y_i \log(\pi_i) + (1 - y_i) \log(1 - \pi_i)] = \sum_{i=1}^{n} [y_i \log\left(\frac{\pi_i}{1 - \pi_i}\right) + \log(1 - \pi_i)] = \sum_{i=1}^{n} [y_i \hat{\mathbf{x}}_i^T \beta - \log(1 + \exp \hat{\mathbf{x}}_i^T \beta)] \quad (2)$$
<figure><img src="image_10.png" /></figure>

potential predictors tested in the feature selection step. All variables used as predictors are lagged by p, with p = 1, 2, 3 for each feature, leading to 138 tested features (the 46 variables in the 3 lag periods).

# 4 | EMPIRICAL STRATEGY

We performed three sets of tests, comparing in each case the best SVM model with the best logit model on the same dataset. We first included just the main indicators of the MIP scoreboard, to check whether these variables, designed to prevent internal and external imbalances, might also be relevant to SGP compliance. We then tested our framework using the complete dataset presented in Table 2. The second model outperforming the first would therefore indicate that the MIP scoreboard is insufficient to forecast SGP compliance. In the third set of tests, we combined a well-established feature selection method with our forecasting scheme, to select only the necessary variables in the model.

## 4.1 | Forecasting algorithms

### 4.1.1 | The logistic function

We used a traditional econometric method (logit regression) and an emerging approach in economics from the field of machine learning (support vector machine), with the aim of creating the most accurate forecasting model by comparing the two.

The logistic function is commonly used for forecasting purposes, as for instance by Martin (1977) and Olson (1980), who used logistic regression to forecast bank and firm failures. Our forecasting problem can be viewed as one of binary classification: to predict whether the countries in the dataset will comply with the rule (class 1) or not (class 0) $Y \in \{0; 1\}$. The goal of our system is to use the input variables to find the linear or non-linear separator that correctly classifies the cases.

The collected data are represented by $x_{i,j}, i = 1, \dots, n$ and $j = 1, \dots, m$, describing $n = 364$ data points with $m = 138$ features, arranged in vectors $\mathbf{x}_i = [x_{i,1}, \dots, x_{i,138}]^T$. The logistic function constrains $Y$ to the range (0, 1) via the sigmoid function:

$$p(y_i = 1) = \pi_i = \frac{\exp \hat{\mathbf{x}}_i^T \beta}{1 + \exp \hat{\mathbf{x}}_i^T \beta} \quad (1)$$

where $\hat{\mathbf{x}}_i = [1, \mathbf{x}_i^T]^T$ corresponds to a feature-column (the 1 corresponds to the intercept of the regression) and $\beta$ is the column vector of the regression coefficients.

The classifier produces a probability score between 0 and 1, with values lower than 0.5 corresponding to class 0; and probabilities of 0.5 or above corresponding to class 1.

The goal is to find the $\beta$ according to $p(Y|X)$ that most accurately classifies the observed data points. The problem is equivalent to maximising the product of the likelihood probabilities:

$$l(\beta) = \sum_{i=1}^{n} [y_i \log(\pi_i) + (1 - y_i) \log(1 - \pi_i)] = \sum_{i=1}^{n} [y_i \log\left(\frac{\pi_i}{1 - \pi_i}\right) + \log(1 - \pi_i)] = \sum_{i=1}^{n} [y_i \hat{\mathbf{x}}_i^T \beta - \log(1 + \exp \hat{\mathbf{x}}_i^T \beta)] \quad (2)$$
<figure><img src="image_12.png" /><figcaption>FIGURE 2 SVM classification according to public budget balance, output gap and oil prices. [Colour figure can be viewed at wileyonlinelibrary.com]</figcaption></figure>

## 4.1.2 | The support vector machine (SVM) algorithm

### *The SVM algorithm for linearly separable data*

SVM is a supervised machine learning method<fnref n="30" /> for the binary classification of a set of data points. SVM aims at identifying a small subset of data points from the initial dataset, called support vectors, that define the position of the linear separator between the two classes.

With $y_i$ as the binary outcome equal to either -1 or 1 (in the logistic model, $y_i$ is equal to 0 or 1). If the two classes are linearly separable, the separator is given by:

$$f(\mathbf{x}_i) = \mathbf{w}^T \mathbf{x}_i - b = 0 \qquad (3)$$

where $\mathbf{x}_i$ is the $i$-th m-sized data point (for our tests the data points are $i = 1, \dots, 364$ and the features are $m = 138$); $\mathbf{w}$ is the weight vector, $b$ is the bias. In that sense all data satisfy:

$$\mathbf{w}^T \mathbf{x}_i - b > 0 \quad \text{if} \quad y_i \in +1 \\ \mathbf{w}^T \mathbf{x}_i - b < 0 \quad \text{if} \quad y_i \in -1, \quad \text{thus} \quad y_i f(\mathbf{x}_i) > 0, \quad \forall i$$

Ideally, the optimal separator is defined as the decision boundary that classifies each data point in the correct subspace and has the maximum distance from each class. This distance is often called ‘margin’ and corresponds to the exact distance of the hyperplane with each class.

Figure 2 shows the representation of the classification for three-dimensional systems.<fnref n="31" /> The classification is given by the green separating plane. The two classes of points are represented in

<footnote n="30">³⁰Supervised learning involves providing a set of data and a set of observations to an algorithm, which creates a mapping function that describes the relationship between the data and the observations.</footnote>

<footnote n="31">³¹Our case involves more than three variables/features so it is impossible to show the cloud of data points in full. However, the 3D representation in Figure 2 was created using three variables from our dataset.</footnote>
<figure><img src="image_11.png" /><figcaption>FIGURE 2 SVM classification according to public budget balance, output gap and oil prices. [Colour figure can be viewed at wileyonlinelibrary.com]</figcaption></figure>

## 4.1.2 | The support vector machine (SVM) algorithm

### *The SVM algorithm for linearly separable data*

SVM is a supervised machine learning method<fnref n="30" /> for the binary classification of a set of data points. SVM aims at identifying a small subset of data points from the initial dataset, called support vectors, that define the position of the linear separator between the two classes.

With $y_i$ as the binary outcome equal to either -1 or 1 (in the logistic model, $y_i$ is equal to 0 or 1). If the two classes are linearly separable, the separator is given by:

$$f(\mathbf{x}_i) = \mathbf{w}^T \mathbf{x}_i - b = 0 \qquad (3)$$

where $\mathbf{x}_i$ is the $i$-th m-sized data point (for our tests the data points are $i = 1, \dots, 364$ and the features are $m = 138$); $\mathbf{w}$ is the weight vector, $b$ is the bias. In that sense all data satisfy:

$$\mathbf{w}^T \mathbf{x}_i - b > 0 \quad \text{if} \quad y_i \in +1 \\ \mathbf{w}^T \mathbf{x}_i - b < 0 \quad \text{if} \quad y_i \in -1, \quad \text{thus} \quad y_i f(\mathbf{x}_i) > 0, \quad \forall i$$

Ideally, the optimal separator is defined as the decision boundary that classifies each data point in the correct subspace and has the maximum distance from each class. This distance is often called ‘margin’ and corresponds to the exact distance of the hyperplane with each class.

Figure 2 shows the representation of the classification for three-dimensional systems.<fnref n="31" /> The classification is given by the green separating plane. The two classes of points are represented in

<footnote n="30">³⁰Supervised learning involves providing a set of data and a set of observations to an algorithm, which creates a mapping function that describes the relationship between the data and the observations.</footnote>

<footnote n="31">³¹Our case involves more than three variables/features so it is impossible to show the cloud of data points in full. However, the 3D representation in Figure 2 was created using three variables from our dataset.</footnote>
<figure><img src="image_13.png" /></figure>

different colours: compliant cases are shown as red points and non-compliant cases as blue points.

The separating hyperplane is identified by the Lagrangian relaxation of a quadratic equation:

$$ \min_{\mathbf{w},b} \max_{\mathbf{a}} \left( \frac{1}{2} \|\mathbf{w}\|^2 - \sum_{i=1}^{N} a_i [y_i(\mathbf{w}^T \mathbf{x}_i - b) - 1] \right) \quad (4) $$

In Equation (4), $\mathbf{a} = [a_1, \dots, a_n]^T$ are the non-negative Lagrange multipliers. (4) is not used to calculate the solution. Instead the simpler dual problem is solved, described by:

$$ \max_{\mathbf{a}} \left\{ \sum_{i=1}^{N} a_i - \sum_{j=1}^{N} \sum_{k=1}^{N} a_j a_k y_j y_k \mathbf{x}_j^T \mathbf{x}_k \right\} \quad (5) $$

with $\sum_{i=1}^{N} a_i y_i = 0$ and $0 \le a_i, \forall i$. The location of the hyperplane is obtained by solving (5), leading to:

$$ \hat{\mathbf{w}} = \sum_{i=1}^{N} a_i y_i \mathbf{x}_i \quad (6) $$

$$ \hat{b} = \hat{\mathbf{w}}^T \mathbf{x}_i - y_i, \quad i \in V, \quad (7) $$

where $V = \{i: 0 < a_i\}$ is the set of support vector indices.

To account for noise and outliers in the dataset, Cortes and Vapnik (1995) introduced non-negative slack variables $\xi_i \ge 0$, $\forall i$ that are tolerant of a certain level of misclassification. To keep the misclassification rate as low as possible, each misclassification is counted as an additional cost in the objective function to be minimised.

$$ \min_{\mathbf{w},b,\xi} \max_{\mathbf{a},\mu} \left\{ \frac{1}{2} \|\mathbf{w}\|^2 + C \sum_{i=1}^{N} \xi_i - \sum_{j=1}^{N} a_j [y_j(\mathbf{w}^T \mathbf{x}_j - b) - 1 + \xi_j] - \sum_{k=1}^{N} \mu_k^T \xi_k \right\} \quad (8) $$

where the non-negative slack $\xi_i$ correspond to the distance of vector $\mathbf{x}_i$ from the hyperplane when classified erroneously. $\mu_k = [\mu_1, \dots, \mu_n]$ are Lagrange multipliers. The optimal hyperplane is finally given by:

$$ \hat{\mathbf{w}} = \sum_{i=1}^{N} a_i y_i \mathbf{x}_i \quad (9) $$

$$ \hat{b} = \hat{\mathbf{w}}^T \mathbf{x}_i - y_i, \quad i \in V, \quad (10) $$

where $V = \{i: 0 < a_i < C\}$ is the set of support vector indices. Parameter C is found using a power-of-2 grid search and $2^{-7} \le C \le 2^7$.

### The SVM algorithm for non-linearly separable data

Real-world phenomena are often nonlinear. Linear models such as SVM are unable to correctly reproduce these systems (see e.g. Cortes and Vapnik (1995)), but the problem of non-linearity can be overcome by combining SVM with the kernel trick. Kernels project the initial data
<figure><img src="image_12.png" /></figure>

different colours: compliant cases are shown as red points and non-compliant cases as blue points.

The separating hyperplane is identified by the Lagrangian relaxation of a quadratic equation:

$$ \min_{\mathbf{w},b} \max_{\mathbf{a}} \left( \frac{1}{2} \|\mathbf{w}\|^2 - \sum_{i=1}^{N} a_i [y_i(\mathbf{w}^T \mathbf{x}_i - b) - 1] \right) \quad (4) $$

In Equation (4), $\mathbf{a} = [a_1, \dots, a_n]^T$ are the non-negative Lagrange multipliers. (4) is not used to calculate the solution. Instead the simpler dual problem is solved, described by:

$$ \max_{\mathbf{a}} \left\{ \sum_{i=1}^{N} a_i - \sum_{j=1}^{N} \sum_{k=1}^{N} a_j a_k y_j y_k \mathbf{x}_j^T \mathbf{x}_k \right\} \quad (5) $$

with $\sum_{i=1}^{N} a_i y_i = 0$ and $0 \le a_i, \forall i$. The location of the hyperplane is obtained by solving (5), leading to:

$$ \hat{\mathbf{w}} = \sum_{i=1}^{N} a_i y_i \mathbf{x}_i \quad (6) $$

$$ \hat{b} = \hat{\mathbf{w}}^T \mathbf{x}_i - y_i, \quad i \in V, \quad (7) $$

where $V = \{i: 0 < a_i\}$ is the set of support vector indices.

To account for noise and outliers in the dataset, Cortes and Vapnik (1995) introduced non-negative slack variables $\xi_i \ge 0$, $\forall i$ that are tolerant of a certain level of misclassification. To keep the misclassification rate as low as possible, each misclassification is counted as an additional cost in the objective function to be minimised.

$$ \min_{\mathbf{w},b,\xi} \max_{\mathbf{a},\mu} \left\{ \frac{1}{2} \|\mathbf{w}\|^2 + C \sum_{i=1}^{N} \xi_i - \sum_{j=1}^{N} a_j [y_j(\mathbf{w}^T \mathbf{x}_j - b) - 1 + \xi_j] - \sum_{k=1}^{N} \mu_k^T \xi_k \right\} \quad (8) $$

where the non-negative slack $\xi_i$ correspond to the distance of vector $\mathbf{x}_i$ from the hyperplane when classified erroneously. $\mu_k = [\mu_1, \dots, \mu_n]$ are Lagrange multipliers. The optimal hyperplane is finally given by:

$$ \hat{\mathbf{w}} = \sum_{i=1}^{N} a_i y_i \mathbf{x}_i \quad (9) $$

$$ \hat{b} = \hat{\mathbf{w}}^T \mathbf{x}_i - y_i, \quad i \in V, \quad (10) $$

where $V = \{i: 0 < a_i < C\}$ is the set of support vector indices. Parameter C is found using a power-of-2 grid search and $2^{-7} \le C \le 2^7$.

### The SVM algorithm for non-linearly separable data

Real-world phenomena are often nonlinear. Linear models such as SVM are unable to correctly reproduce these systems (see e.g. Cortes and Vapnik (1995)), but the problem of non-linearity can be overcome by combining SVM with the kernel trick. Kernels project the initial data
space into a higher-dimension feature space. Instead of searching for the optimal separator in the data space, this is done in the feature space and the solution is then transformed back into the original space. Although the separator in the feature space is linear (SVM separators are always linear), the nonlinear kernel yields a nonlinear separator in the data space. The kernel trick reduces the computational cost: the projection is performed in the inner product space instead of projecting each point separately into the feature space. Adding the kernel projection into the objective function leads to the following minimisation equation:

$$ \max_{\mathbf{a}} = \sum_{i=1}^{N} a_i - \frac{1}{2} \sum_{j=1}^{N} \sum_{k=1}^{N} a_j a_k y_j y_k K(\mathbf{x}_j, \mathbf{x}_k). \quad (11) $$

with $\sum_{i=1}^{N} a_i y_i = 0$ and $0 \le a_i \le C$, $\forall i$. We tested the three most commonly used kernels:

$$ \text{Linear} \qquad K_1(\mathbf{x}_i, \mathbf{x}_j) = \mathbf{x}_i^T \mathbf{x}_j + r, \qquad (12) $$

$$ \text{Radial Basis Function (RBF)} \qquad K_2(\mathbf{x}_i, \mathbf{x}_j) = e^{-r \|\mathbf{x}_i - \mathbf{x}_j\|^2}, \qquad (13) $$

$$ \text{Polynomial} \qquad K_3(\mathbf{x}_i, \mathbf{x}_j) = (\mathbf{x}_i^T \mathbf{x}_j + r)^d, \qquad (14) $$

Now, the rule for classifying a data point $\mathbf{x}$ is:

$$ f(\mathbf{x}) = \text{sign} \left\{ \sum_{i=1}^{N} a_i y_i K(\mathbf{x}_i, \mathbf{x}) + b \right\} \qquad (15) $$

Indeed, if $f(\mathbf{x}) > 0$ the point is classified in class + 1; otherwise, it belongs to class - 1.

For further details of the SVM method, see Gogas and Papadimitriou (2021).

## 4.2 | Feature selection: logistic LASSO regression

The goal of feature selection is to reduce the input set by removing irrelevant or redundant features in the model. Reducing the input set decreases the computational cost of training and limits the risk of overfitting.

Friedman et al. (2009) proposed LASSO as an alternative regularisation method in cases where ridge regression fails to reduce the number of predictors in the model. LASSO involves penalising the coefficients of some of the input variables and shrinking them to zero. The main goal of the method is to minimise the prediction error, yielding selected features among the variables as a by-product.

The shrinkage operation identifies the key features in the dataset, avoiding the problems associated with transformation-based dimension reduction approaches such as factor analysis, principal component analysis and independent component analysis, whose factors are uninterpretable.

The LASSO estimator in logistic regression is:

$$ \hat{\beta}(\lambda) = \underset{\beta}{\operatorname{argmin}} (n^{-1} \sum_{i=1}^{n} \rho_{(\beta)}(X_i, Y_i) + \lambda ||\beta||_1) \qquad (16) $$
space into a higher-dimension feature space. Instead of searching for the optimal separator in the data space, this is done in the feature space and the solution is then transformed back into the original space. Although the separator in the feature space is linear (SVM separators are always linear), the nonlinear kernel yields a nonlinear separator in the data space. The kernel trick reduces the computational cost: the projection is performed in the inner product space instead of projecting each point separately into the feature space. Adding the kernel projection into the objective function leads to the following minimisation equation:

$$ \max_{\mathbf{a}} = \sum_{i=1}^{N} a_i - \frac{1}{2} \sum_{j=1}^{N} \sum_{k=1}^{N} a_j a_k y_j y_k K(\mathbf{x}_j, \mathbf{x}_k). \quad (11) $$

with $\sum_{i=1}^{N} a_i y_i = 0$ and $0 \le a_i \le C, \forall i$. We tested the three most commonly used kernels:

$$ \text{Linear} \qquad K_1(\mathbf{x}_i, \mathbf{x}_j) = \mathbf{x}_i^T \mathbf{x}_j + r, \qquad (12) $$

$$ \text{Radial Basis Function (RBF)} \qquad K_2(\mathbf{x}_i, \mathbf{x}_j) = e^{-r \|\mathbf{x}_i - \mathbf{x}_j\|^2}, \qquad (13) $$

$$ \text{Polynomial} \qquad K_3(\mathbf{x}_i, \mathbf{x}_j) = (\mathbf{x}_i^T \mathbf{x}_j + r)^d, \qquad (14) $$

Now, the rule for classifying a data point $\mathbf{x}$ is:

$$ f(\mathbf{x}) = \text{sign} \left\{ \sum_{i=1}^{N} a_i y_i K(\mathbf{x}_i, \mathbf{x}) + b \right\} \qquad (15) $$

Indeed, if $f(\mathbf{x}) > 0$ the point is classified in class + 1; otherwise, it belongs to class - 1.

For further details of the SVM method, see Gogas and Papadimitriou (2021).

## 4.2 | Feature selection: logistic LASSO regression

The goal of feature selection is to reduce the input set by removing irrelevant or redundant features in the model. Reducing the input set decreases the computational cost of training and limits the risk of overfitting.

Friedman et al. (2009) proposed LASSO as an alternative regularisation method in cases where ridge regression fails to reduce the number of predictors in the model. LASSO involves penalising the coefficients of some of the input variables and shrinking them to zero. The main goal of the method is to minimise the prediction error, yielding selected features among the variables as a by-product.

The shrinkage operation identifies the key features in the dataset, avoiding the problems associated with transformation-based dimension reduction approaches such as factor analysis, principal component analysis and independent component analysis, whose factors are uninterpretable.

The LASSO estimator in logistic regression is:

$$ \hat{\beta}(\lambda) = \underset{\beta}{\operatorname{argmin}} (n^{-1} \sum_{i=1}^{n} \rho_{(\beta)}(X_i, Y_i) + \lambda ||\beta||_1) \qquad (16) $$
<figure><img src="image_14.png" /></figure>

<figure><img src="image_15.png" /><figcaption>FIGURE 3 A graphical representation of the 5-fold cross validation. [Colour figure can be viewed at wileyonlinelibrary.com]</figcaption></figure>

Parameter $\lambda$ is optimised by grid search (view Appendix 1: Figure A1) using the one-standard error rule. Among models with similar performance, the most parsimonious one was retained.

## 4.3 | Assessment of forecasting performance

The performance of our models was calculated as the proportion of correctly forecasted observations:

$$ accuracy = \frac{TP + TN}{TP + FP + FN + TN} \qquad (17) $$

where TP is the number of true positives (correctly forecasted positive instances), TN is the number of true negatives (correctly forecasted negative instances), FP is the number of false positives (incorrectly forecasted positive instances), and FN is the number of false negative instances (incorrectly forecasted negative instances). Note that in our setup, a positive (negative) instance is a member state that respected (broke) the rule in year t.

The forecasting accuracy is a simple and easy-to-use metric but is somewhat coarse and superficial. Consider, for example, a dataset of 90 positive and 10 negative cases, analysed using a naive model yielding only positive forecasts. The accuracy of the model is 90%, which is rather misleading since it missed all the negative cases. For a deeper and richer representation of the model's performance, we used the confusion matrix, which assesses the performance of the model for two potential outcomes separately (see e.g. Kulkarni et al. (2020)).

For our model indeed, false positives would be damaging to the EU economy: countries incorrectly forecasted as compliant with SGP would not receive any recommendations to improve their public finance management. Too many false positives could jeopardize the sustainability of the entire currency area. Limiting false positives is therefore important, but not at the cost of too many false negatives, as illustrated by the naive example above. False negatives (i.e. countries incorrectly forecasted as non-compliant), would lead the commission to unnecessarily recommend strict measures that could harm the countries' economy by reducing its fiscal capacity. It is therefore important to assess the performance of the model for both positive and negative cases.
<figure><img src="image_13.png" /></figure>

<figure><img src="image_14.png" /><figcaption>FIGURE 3 A graphical representation of the 5-fold cross validation. [Colour figure can be viewed at wileyonlinelibrary.com]</figcaption></figure>

Parameter $\lambda$ is optimised by grid search (view Appendix 1: Figure A1) using the one-standard error rule. Among models with similar performance, the most parsimonious one was retained.

## 4.3 | Assessment of forecasting performance

The performance of our models was calculated as the proportion of correctly forecasted observations:

$$ accuracy = \frac{TP + TN}{TP + FP + FN + TN} \qquad (17) $$

where TP is the number of true positives (correctly forecasted positive instances), TN is the number of true negatives (correctly forecasted negative instances), FP is the number of false positives (incorrectly forecasted positive instances), and FN is the number of false negative instances (incorrectly forecasted negative instances). Note that in our setup, a positive (negative) instance is a member state that respected (broke) the rule in year t.

The forecasting accuracy is a simple and easy-to-use metric but is somewhat coarse and superficial. Consider, for example, a dataset of 90 positive and 10 negative cases, analysed using a naive model yielding only positive forecasts. The accuracy of the model is 90%, which is rather misleading since it missed all the negative cases. For a deeper and richer representation of the model's performance, we used the confusion matrix, which assesses the performance of the model for two potential outcomes separately (see e.g. Kulkarni et al. (2020)).

For our model indeed, false positives would be damaging to the EU economy: countries incorrectly forecasted as compliant with SGP would not receive any recommendations to improve their public finance management. Too many false positives could jeopardize the sustainability of the entire currency area. Limiting false positives is therefore important, but not at the cost of too many false negatives, as illustrated by the naive example above. False negatives (i.e. countries incorrectly forecasted as non-compliant), would lead the commission to unnecessarily recommend strict measures that could harm the countries' economy by reducing its fiscal capacity. It is therefore important to assess the performance of the model for both positive and negative cases.
<table><caption>TABLE 3 ‘Compliance with 3% limit’ forecasting accuracy: comparison of models (%). [Colour table can be viewed at wileyonlinelibrary.com]</caption><thead><tr><th>Model</th><th>Primary MIP<br>scoreboard<br>variables only</th><th>All features<br>(138)</th><th>Primary MIP<br>scoreboard<br>variables only</th><th>All features<br>(138)</th></tr></thead><tbody><tr><td>Linear SVM model</td><td>64.6</td><td>83.5</td><td>85.1</td><td>87.0</td></tr><tr><td>Quadratic SVM model</td><td>69.5</td><td>83.0</td><td>79.6</td><td>88.9</td></tr><tr><td>RBF SVM model</td><td>62.9</td><td>80.2</td><td>77.8</td><td>88.9</td></tr><tr><td>Logistic model</td><td>63.7</td><td>75.5</td><td>75.9</td><td>73.3</td></tr><tr><td>Cross-validation method</td><td>k-Fold</td><td>k-fold</td><td>Hold-out</td><td>Hold-out</td></tr></tbody></table>

Note: Hold-out splits the dataset into a ‘training set’ (85%) and a ‘test set’ (15%). The results are from the test set. For k-fold cross-validation the process was repeated 5 times; the values shown are the mean results. Parameter C in the SVM algorithm is equal to 2¹ and was obtained using a power-of-2 grid search. Best accuracy results are highlighted in yellow.

## 4.4 Validation

Machine learning methods are, in general, unaffected by reverse causality, which is a common problem in classic econometrics. They are liable to overfitting, however, a common error whereby the model learns to describe the training data instead of the underlying phenomenon. Overfitting can be avoided by performing hold-out validation or the more powerful k-fold cross-validation (see Figure 3 for illustration). In hold-out validation, the dataset is split into a ‘training’ set and ‘test’ set. The model is trained on the training set and the test set is used to evaluate the performance of the model on unknown data. If the training accuracy is much higher than the testing accuracy, this is a strong indication that the model overfitted the training dataset. Typically, around 80% of the data is used for training and 20% for testing. The k-fold cross-validation involves repeating the process k times on k equally sized subsets.<fnref n="32" /> A different subset is used as the ‘test’ set in each run, and the remaining k-1 subsets constitute the ‘training’ set. The average performance over the k runs is used to select the optimal model.

Figure 3 provides an illustration of iterations of a 5-fold cross validation process. In each iteration, a different chunk of the dataset is used for testing (show in red), while the remaining four parts are used for training (green). In each iteration, the position of the separator is fixed using the training subset, and the model is evaluated using the testing subset. The overall performance of the model is quantified by the mean classification error over the five iterations.

# 5 RESULTS

## 5.1 Forecasting SGP compliance

We tested the SVM classification algorithm with three kernels (linear, RBF and polynomial) and used the performance of the logistic regression model on the same datasets as a benchmark. In a first stage, the models were trained using the primary indicators of the MIP scoreboard to

<footnote n="32">³²The appropriate value of *k* is somewhat arbitrary. Yadav and Shukla (2016) found that *k* = 10 was suitable for a large dataset and that *k* = 2 produced and underfitted model, whereas Chernozhukov et al. (2018) found that *k* = 2 and *k* = 5 led to similar results. Given the limited size of our dataset, *k* = 5 seems a reasonable choice.</footnote>
<table><caption>TABLE 3 ‘Compliance with 3% limit’ forecasting accuracy: comparison of models (%). [Colour table can be viewed at wileyonlinelibrary.com]</caption><thead><tr><th>Model</th><th>Primary MIP<br>scoreboard<br>variables only</th><th>All features<br>(138)</th><th>Primary MIP<br>scoreboard<br>variables only</th><th>All features<br>(138)</th></tr></thead><tbody><tr><td>Linear SVM model</td><td>64.6</td><td>83.5</td><td>85.1</td><td>87.0</td></tr><tr><td>Quadratic SVM model</td><td>69.5</td><td>83.0</td><td>79.6</td><td>88.9</td></tr><tr><td>RBF SVM model</td><td>62.9</td><td>80.2</td><td>77.8</td><td>88.9</td></tr><tr><td>Logistic model</td><td>63.7</td><td>75.5</td><td>75.9</td><td>73.3</td></tr><tr><td>Cross-validation method</td><td>k-Fold</td><td>k-fold</td><td>Hold-out</td><td>Hold-out</td></tr></tbody></table>

*Note:* Hold-out splits the dataset into a ‘training set’ (85%) and a ‘test set’ (15%). The results are from the test set. For *k*-fold cross-validation the process was repeated 5 times; the values shown are the mean results. Parameter *C* in the SVM algorithm is equal to 2¹ and was obtained using a power-of-2 grid search. Best accuracy results are highlighted in yellow.

## 4.4 Validation

Machine learning methods are, in general, unaffected by reverse causality, which is a common problem in classic econometrics. They are liable to overfitting, however, a common error whereby the model learns to describe the training data instead of the underlying phenomenon. Overfitting can be avoided by performing hold-out validation or the more powerful *k*-fold cross-validation (see Figure 3 for illustration). In hold-out validation, the dataset is split into a ‘training’ set and ‘test’ set. The model is trained on the training set and the test set is used to evaluate the performance of the model on unknown data. If the training accuracy is much higher than the testing accuracy, this is a strong indication that the model overfitted the training dataset. Typically, around 80% of the data is used for training and 20% for testing. The *k*-fold cross-validation involves repeating the process *k* times on *k* equally sized subsets.<fnref n="32" /> A different subset is used as the ‘test’ set in each run, and the remaining *k*-1 subsets constitute the ‘training’ set. The average performance over the *k* runs is used to select the optimal model.

Figure 3 provides an illustration of iterations of a 5-fold cross validation process. In each iteration, a different chunk of the dataset is used for testing (show in red), while the remaining four parts are used for training (green). In each iteration, the position of the separator is fixed using the training subset, and the model is evaluated using the testing subset. The overall performance of the model is quantified by the mean classification error over the five iterations.

# 5 RESULTS

## 5.1 Forecasting SGP compliance

We tested the SVM classification algorithm with three kernels (linear, RBF and polynomial) and used the performance of the logistic regression model on the same datasets as a benchmark. In a first stage, the models were trained using the primary indicators of the MIP scoreboard to

<footnote n="32">³²The appropriate value of *k* is somewhat arbitrary. Yadav and Shukla (2016) found that *k*=10 was suitable for a large dataset and that *k*=2 produced and underfitted model, whereas Chernozhukov et al. (2018) found that *k*=2 and *k*=5 led to similar results. Given the limited size of our dataset, *k*=5 seems a reasonable choice.</footnote>
<figure><img src="image_16.png" /></figure>

evaluate the potential of the MIP scoreboard variables to forecast compliance with the 3% rule. The training process was then repeated using the entire 138-variable dataset. The results of these two analyses are reported in Table 3.

The results obtained with the two datasets show that in almost every case, the models trained with the full dataset outperformed those trained on the primary MIP scoreboard only. The highest prediction accuracy with the MIP scoreboard variables (85.1%) was achieved with the linear kernel SVM model and hold-out validation (the corresponding accuracy with the full dataset was 87%). With the full dataset, the SVM models with non-linear kernels (the quadratic and the RBF kernel) had a forecasting accuracy of 88.9% in hold-out cross-validation, the best performance achieved with any of the models in hold-out validation. The gains in performance of the models trained on the full dataset were more substantial in *k*-fold cross-validation: the accuracy of the models trained on the restricted set ranged from 62.9% with the RBF-SVM model, to 69.5% for the quadratic kernel SVM model, whereas for the models trained on the full dataset ranged forecasting accuracies ranged from 75.5% for the logit model to 83.5% for the linear SVM model. These results suggest that the full dataset has a greater forecasting power than the primary MIP scoreboard indicators alone. The next step of our study was to identify the optimal input dataset using LASSO-based feature selection.

LASSO feature selection³³highlighted the 12 key variables for forecasting SGP compliance. Following Gogas et al. (2018), these variables were introduced into the SVM forecasting model and the input dataset was further shrunk by comparing the results with all 12 selected variables with those obtained when each of these variables was removed in turn. The optimal set was retained and the procedure was repeated until no improvement was achieved.

The final dataset consisted of eight features: (a) the general government fiscal balance in period $t-1$ (as expected since a weak fiscal balance in one year will tend to affect the next), (b) financial sector liabilities in period $t-1$ (the global financial crisis highlighted the interdependence of financial institutions, their liabilities and public finance sustainability);³⁴the dummy variables reflecting the occurrence of a crisis in periods $t-1$ and $t-2$; economic crises indeed have a dual impact on public deficits: they induce economic recessions and increase investment needs (furthermore, the fact that these two features are identified as crucial indicates that the SGP escape clause should account for the length of crises and not just focus on drops in GDP during recessions), (d) the output gap in period $t-1$ (the output gap is an indicator of position in the economic cycle—increased GDP volatility under poor economic conditions impacts public deficit and thus SGP compliance), (e) oil prices in period $t-1$ (increases in oil prices have led to crises directly, as in 1973 and 1979, or by proxy, as in 2008—the ‘yellow vests’ movement, triggered in 2018 by oil prices in France, is a further example of the consequences on public deficits that these situations can have), (f) the bond yield in period $t-1$ and (g) the fiscal space in period $t-1$ (both variables are measures of government fiscal flexibility, especially in periods of crises—this is in line with Romer and Romer (2018), who highlight the importance of fiscal space during financial crises and normal recessions). Table 4 presents the performance of the different models trained with the selected features.

The linear SVM model performed best in both hold-out and *k*-fold cross-validation (98.1% and 90.4% accuracy, respectively). Note that for *k*-fold cross-validation, the result is the mean accuracy of five iterations. The forecasting accuracy of the best performing model using the complete dataset was 91.7%. The corresponding confusion matrix is shown in Figure 4.

<footnote n="33">³³The LASSO procedure results are reported in Appendix 1: Figure B1. Twelve features were identified as important according to the one-standard error rule.</footnote>

<footnote n="34">³⁴For instance, commercial banks hold large quantities of treasury bills (see e.g. Cook (1993)).</footnote>
<figure><img src="image_15.png" /></figure>

evaluate the potential of the MIP scoreboard variables to forecast compliance with the 3% rule. The training process was then repeated using the entire 138-variable dataset. The results of these two analyses are reported in Table 3.

The results obtained with the two datasets show that in almost every case, the models trained with the full dataset outperformed those trained on the primary MIP scoreboard only. The highest prediction accuracy with the MIP scoreboard variables (85.1%) was achieved with the linear kernel SVM model and hold-out validation (the corresponding accuracy with the full dataset was 87%). With the full dataset, the SVM models with non-linear kernels (the quadratic and the RBF kernel) had a forecasting accuracy of 88.9% in hold-out cross-validation, the best performance achieved with any of the models in hold-out validation. The gains in performance of the models trained on the full dataset were more substantial in *k*-fold cross-validation: the accuracy of the models trained on the restricted set ranged from 62.9% with the RBF-SVM model, to 69.5% for the quadratic kernel SVM model, whereas for the models trained on the full dataset ranged forecasting accuracies ranged from 75.5% for the logit model to 83.5% for the linear SVM model. These results suggest that the full dataset has a greater forecasting power than the primary MIP scoreboard indicators alone. The next step of our study was to identify the optimal input dataset using LASSO-based feature selection.

LASSO feature selection³³highlighted the 12 key variables for forecasting SGP compliance. Following Gogas et al. (2018), these variables were introduced into the SVM forecasting model and the input dataset was further shrunk by comparing the results with all 12 selected variables with those obtained when each of these variables was removed in turn. The optimal set was retained and the procedure was repeated until no improvement was achieved.

The final dataset consisted of eight features: (a) the general government fiscal balance in period $t-1$ (as expected since a weak fiscal balance in one year will tend to affect the next), (b) financial sector liabilities in period $t-1$ (the global financial crisis highlighted the interdependence of financial institutions, their liabilities and public finance sustainability);³⁴the dummy variables reflecting the occurrence of a crisis in periods $t-1$ and $t-2$; economic crises indeed have a dual impact on public deficits: they induce economic recessions and increase investment needs (furthermore, the fact that these two features are identified as crucial indicates that the SGP escape clause should account for the length of crises and not just focus on drops in GDP during recessions), (d) the output gap in period $t-1$ (the output gap is an indicator of position in the economic cycle—increased GDP volatility under poor economic conditions impacts public deficit and thus SGP compliance), (e) oil prices in period $t-1$ (increases in oil prices have led to crises directly, as in 1973 and 1979, or by proxy, as in 2008—the ‘yellow vests’ movement, triggered in 2018 by oil prices in France, is a further example of the consequences on public deficits that these situations can have), (f) the bond yield in period $t-1$ and (g) the fiscal space in period $t-1$ (both variables are measures of government fiscal flexibility, especially in periods of crises—this is in line with Romer and Romer (2018), who highlight the importance of fiscal space during financial crises and normal recessions). Table 4 presents the performance of the different models trained with the selected features.

The linear SVM model performed best in both hold-out and *k*-fold cross-validation (98.1% and 90.4% accuracy, respectively). Note that for *k*-fold cross-validation, the result is the mean accuracy of five iterations. The forecasting accuracy of the best performing model using the complete dataset was 91.7%. The corresponding confusion matrix is shown in Figure 4.

<footnote n="33">³³The LASSO procedure results are reported in Appendix 1: Figure B1. Twelve features were identified as important according to the one-standard error rule.</footnote>

<footnote n="34">³⁴For instance, commercial banks hold large quantities of treasury bills (see e.g. Cook (1993)).</footnote>
<table><caption>TABLE 4 ‘Compliance with 3% limit’ forecasting accuracy with only Best Predictors(%). [Colour table can be viewed at wileyonlinelibrary.com]</caption><thead><tr><th>Model</th><th>Features selected by LASSO</th><th>Features selected by LASSO</th></tr></thead><tbody><tr><td>Linear SVM model</td><td>90.4</td><td>98.1</td></tr><tr><td>Quadratic SVM model</td><td>84.6</td><td>87.0</td></tr><tr><td>RBF SVM model (γ = 12)</td><td>86.5</td><td>88.9</td></tr><tr><td>Logistic model</td><td>78.5</td><td>76.3</td></tr><tr><td>Cross-validation method</td><td>k-fold</td><td>Hold-out</td></tr></tbody></table>

Note: Hold-out splits the dataset into a ‘training set’ (85%) and a ‘test set’ (15%). The results are from the test set. For *k*-fold cross-validation the process was repeated 5 times; the values shown are the mean results. Parameter C in the SVM algorithm is equal to 2¹ and was obtained using a power of 2 grid search. Best accuracy results are highlighted in yellow.

The confusion matrix shows that the model correctly forecasted 112 of 135 cases of non-compliance with the 3% rule, with a low rate of false alarms (7 false alerts among 229 cases of compliance; a false alarm involves the model inaccurately forecasting non-compliance with the 3% rule). The model is therefore highly accurate in identifying non-compliance, with relatively few instances of countries incorrectly being given recommendations to avoid non-compliance (false alarms).

Although assessments of the input data are limited because the models may be slightly sub-optimal due to the use of a grid search to identify the best model hyperparameters, these results strongly suggest that the main MIP scoreboard indicators are less appropriate than the LASSO-selected features set to monitor internal imbalances. This is not surprising since the MIP scoreboard indicators were not developed to predict SGP compliance. Moreover, some of these indicators are defined over several years and would increase the complexity of the monitoring system. These results indicate that the variables in our optimal input set would be a better choice to reinforce European fiscal framework surveillance.

In terms of countries, the following cases of non-compliance were missed<fnref n="35" />: Belgium (2009 and 2011), Bulgaria (2014), Croatia (2006, 2013 and 2015), Czech Republic (2012), Denmark (2012), Finland (2014), France (2016), Germany (2009 and 2010), Ireland (2008), Italy (2006 and 2011), Lithuania (2008), Malta (2012), Poland (2014), Slovak Republic (2006), Romania (2007 and 2018), Slovenia (2013), Spain (2008). Twelve of the 23 cases involved either the Global Financial Crisis (2007-2009) or the Sovereign Debt Crisis (2010–2012). Clearly, during these periods, the member states were more concerned with the effects of the crisis than complying with the SGP. This change in macroeconomic priorities is usually unexpected and cannot be forecast in the lagged instances of the feature variables. Remember that these crises led to the introduction of the escape clause as part of the Six-Pack reform in 2011.

## 5.2 | Distance from separator hyperplane and policy implications

Although the vast majority of machine learning algorithms work as ‘black boxes’ in that they do not yield an analytical expression of the forecasting model (as econometric models typically do) the proposed SVM model can be expressed analytically. Indeed, the optimal

<footnote n="35">³⁵The missed instances of compliance were Bulgaria (2009), Croatia (2008), Hungary (2012), Italy (2014), Malta (2013), Slovak Republic (2013), United Kingdom (2016).</footnote>
<table><caption>TABLE 4 ‘Compliance with 3% limit’ forecasting accuracy with only Best Predictors(%). [Colour table can be viewed at wileyonlinelibrary.com]</caption><thead><tr><th>Model</th><th>Features selected by LASSO</th><th>Features selected by LASSO</th></tr></thead><tbody><tr><td>Linear SVM model</td><td>90.4</td><td>98.1</td></tr><tr><td>Quadratic SVM model</td><td>84.6</td><td>87.0</td></tr><tr><td>RBF SVM model (γ = 12)</td><td>86.5</td><td>88.9</td></tr><tr><td>Logistic model</td><td>78.5</td><td>76.3</td></tr><tr><td>Cross-validation method</td><td>k-fold</td><td>Hold-out</td></tr></tbody></table>

*Note:* Hold-out splits the dataset into a ‘training set’ (85%) and a ‘test set’ (15%). The results are from the test set. For *k*-fold cross-validation the process was repeated 5 times; the values shown are the mean results. Parameter C in the SVM algorithm is equal to 2¹ and was obtained using a power of 2 grid search. Best accuracy results are highlighted in yellow.

The confusion matrix shows that the model correctly forecasted 112 of 135 cases of non-compliance with the 3% rule, with a low rate of false alarms (7 false alerts among 229 cases of compliance; a false alarm involves the model inaccurately forecasting non-compliance with the 3% rule). The model is therefore highly accurate in identifying non-compliance, with relatively few instances of countries incorrectly being given recommendations to avoid non-compliance (false alarms).

Although assessments of the input data are limited because the models may be slightly sub-optimal due to the use of a grid search to identify the best model hyperparameters, these results strongly suggest that the main MIP scoreboard indicators are less appropriate than the LASSO-selected features set to monitor internal imbalances. This is not surprising since the MIP scoreboard indicators were not developed to predict SGP compliance. Moreover, some of these indicators are defined over several years and would increase the complexity of the monitoring system. These results indicate that the variables in our optimal input set would be a better choice to reinforce European fiscal framework surveillance.

In terms of countries, the following cases of non-compliance were missed<fnref n="35" />: Belgium (2009 and 2011), Bulgaria (2014), Croatia (2006, 2013 and 2015), Czech Republic (2012), Denmark (2012), Finland (2014), France (2016), Germany (2009 and 2010), Ireland (2008), Italy (2006 and 2011), Lithuania (2008), Malta (2012), Poland (2014), Slovak Republic (2006), Romania (2007 and 2018), Slovenia (2013), Spain (2008). Twelve of the 23 cases involved either the Global Financial Crisis (2007-2009) or the Sovereign Debt Crisis (2010–2012). Clearly, during these periods, the member states were more concerned with the effects of the crisis than complying with the SGP. This change in macroeconomic priorities is usually unexpected and cannot be forecast in the lagged instances of the feature variables. Remember that these crises led to the introduction of the escape clause as part of the Six-Pack reform in 2011.

## 5.2 | Distance from separator hyperplane and policy implications

Although the vast majority of machine learning algorithms work as ‘black boxes’ in that they do not yield an analytical expression of the forecasting model (as econometric models typically do) the proposed SVM model can be expressed analytically. Indeed, the optimal

<footnote n="35">³⁵The missed instances of compliance were Bulgaria (2009), Croatia (2008), Hungary (2012), Italy (2014), Malta (2013), Slovak Republic (2013), United Kingdom (2016).</footnote>
<figure><img src="image_17.png" /></figure>

<figure><img src="image_18.png" /><figcaption>FIGURE 4 Confusion matrix for the linear SVM model (hold-out cross-validation).</figcaption></figure>

forecasting model derived from the results in section 5.1 corresponds to a linear decision boundary (also called separator hyperplane) that separates cases of SGP compliance and non-compliance with 91.7% accuracy using 8 explanatory variables. The analytical form of the separator hyperplane is:

$$H: 3, 546x_1 - 0, 285x_2 - 0, 338x_3 - 0, 458x_4 - 1, 014x_5 - 0, 416x_6 - 0, 581x_7 - 0, 175x_8 + 0, 793 = 0 \quad (18)$$

where $x_1$ corresponds to the general government fiscal balance in period $t-1$, $x_2$ to financial sector liabilities (% change over 1 year) in period $t-1$, $x_3$ and $x_4$ to the crisis dummies in periods $t-1$ and $t-2$, respectively, $x_5$ to the output gap in period $t-1$, $x_6$ to oil prices in period $t-1$, $x_7$ to the bond yield in period $t-1$ and $x_8$ to the fiscal space in period $t-1$.

These results have several implications: First, the analytical form of equation (18) can be used to calculate the distance between any point and the hyperplane, which corresponds to the distance a country needs to be moved by to pass from one subspace to the other. If a country is forecast not to comply with the SGP, the European Commission's recommendations should produce a displacement large enough to pass onto the other side. Obviously, it is easier to rectify the 'fate' of 'non-compliers' closer to the separation hyperplane. Similarly, the distance from the boundary can be used as a confidence parameter for countries forecast to comply. Compliant countries close to the separator should be monitored closely, since small economic perturbations or budget failures may push it into the non-compliant subspace. The same level of alertness is not required for countries forecast to be further from the separator in the compliant subspace.

Considering an observation A with coordinates $(x_A, y_A, z_A, r_A, s_A, t_A, v_A, w_A)$, its distance from the separator hyperplane is defined as follows:

$$d(A) = \frac{|3, 546x_A - 0, 285y_A - 0, 338z_A - 0, 458r_A - 1, 014s_A - 0, 416t_A - 0, 581v_A - 0, 175w_A + 0, 793|}{\sqrt{3, 546^2 + (-0, 285)^2 + (-0, 338)^2 + (-0, 458)^2 + (-1, 014)^2 + (-0, 416)^2 + (-0, 581)^2 + (-0, 175)^2}} \quad (19)$$

According to this definition, the points in the compliant subspace are 0.00076 to 1.7305 from the decision boundary, whereas the instances of non-compliance are 0.0027 to 5.0717 from the boundary. There are therefore some extreme cases of non-compliance, such as Ireland and Portugal in 2011, Greece from 2008 to 2013 and Slovenia in 2014. It is very hard in these cases for the countries to move back into the compliant subspace, Greece for example
<figure><img src="image_16.png" /></figure>

<figure><img src="image_17.png" /><figcaption>FIGURE 4 Confusion matrix for the linear SVM model (hold-out cross-validation).</figcaption></figure>

forecasting model derived from the results in section 5.1 corresponds to a linear decision boundary (also called separator hyperplane) that separates cases of SGP compliance and non-compliance with 91.7% accuracy using 8 explanatory variables. The analytical form of the separator hyperplane is:

$$H: 3, 546x_1 - 0, 285x_2 - 0, 338x_3 - 0, 458x_4 - 1, 014x_5 - 0, 416x_6 - 0, 581x_7 - 0, 175x_8 + 0, 793 = 0 \quad (18)$$

where $x_1$ corresponds to the general government fiscal balance in period $t-1$, $x_2$ to financial sector liabilities (% change over 1 year) in period $t-1$, $x_3$ and $x_4$ to the crisis dummies in periods $t-1$ and $t-2$, respectively, $x_5$ to the output gap in period $t-1$, $x_6$ to oil prices in period $t-1$, $x_7$ to the bond yield in period $t-1$ and $x_8$ to the fiscal space in period $t-1$.

These results have several implications: First, the analytical form of equation (18) can be used to calculate the distance between any point and the hyperplane, which corresponds to the distance a country needs to be moved by to pass from one subspace to the other. If a country is forecast not to comply with the SGP, the European Commission's recommendations should produce a displacement large enough to pass onto the other side. Obviously, it is easier to rectify the 'fate' of 'non-compliers' closer to the separation hyperplane. Similarly, the distance from the boundary can be used as a confidence parameter for countries forecast to comply. Compliant countries close to the separator should be monitored closely, since small economic perturbations or budget failures may push it into the non-compliant subspace. The same level of alertness is not required for countries forecast to be further from the separator in the compliant subspace.

Considering an observation A with coordinates $(x_A, y_A, z_A, r_A, s_A, t_A, v_A, w_A)$, its distance from the separator hyperplane is defined as follows:

$$d(A) = \frac{|3, 546x_A - 0, 285y_A - 0, 338z_A - 0, 458r_A - 1, 014s_A - 0, 416t_A - 0, 581v_A - 0, 175w_A + 0, 793|}{\sqrt{3, 546^2 + (-0, 285)^2 + (-0, 338)^2 + (-0, 458)^2 + (-1, 014)^2 + (-0, 416)^2 + (-0, 581)^2 + (-0, 175)^2}} \quad (19)$$

According to this definition, the points in the compliant subspace are 0.00076 to 1.7305 from the decision boundary, whereas the instances of non-compliance are 0.0027 to 5.0717 from the boundary. There are therefore some extreme cases of non-compliance, such as Ireland and Portugal in 2011, Greece from 2008 to 2013 and Slovenia in 2014. It is very hard in these cases for the countries to move back into the compliant subspace, Greece for example
<figure><img src="image_19.png" /><figcaption>**FIGURE 5** Linear decision boundary in three dimensions. [Colour figure can be viewed at wileyonlinelibrary.com]</figcaption></figure>

having been placed under strict European Commission monitoring for 10 years following the Sovereign Debt Crisis.

Figure 5 shows 1/5 of the dataset<fnref n="36" /> and the linear separator hyperplane.<fnref n="37" /> The red dots circled in purple are the instances of non-compliance that are closest to the hyperplane (e.g. Belgium in 2014 and Croatia in 2012) and therefore would have been the easiest to rectify. The black dots circled in purple would also have triggered increased attention because they are close to the ‘cliff edge’ (the Slovak Republic in 2007 for example).

In summary, the feature selection step revealed the key variables that governments and institutions in charge of fiscal surveillance should monitor, i.e. the general government fiscal balance in the preceding year, the fiscal space, output gap, oil prices, bond yields, corporate sector financial liabilities, and crisis dummies. Second, the optimal model can be used to forecast SGP (non-)compliance with better than 90% accuracy. By including the key determinants of SGP compliance into the model, predicted deviations from the rule can be prevented. Moreover, the distance of each country from the decision boundary of the model, can be used to adapt recommendations toward SGP compliance.

This model could therefore be implemented to promote future public finance sustainability. This original method could indeed be used to monitor countries within the European Semester framework, provide precise forecasts of SGP compliance, and adapt recommendations on a country-by-country basis with greater distances from the boundary triggering more urgent and stricter measures. More accurate diagnoses of SGP violation risks could be used by member states when preparing their draft budgets. Strengthening the preventive instrument of the SGP in this way may contribute to overcoming one of the main limitations of the dissuasive instrument, namely that once the 3% threshold has been violated and no escape clauses apply, it is often too late to rectify the ongoing public finance trend.

<footnote n="36">³⁶These observations were selected at random to make the figure legible.</footnote>

<footnote n="37">³⁷The separator in Figure 5 only represents three dimensions of the 8-dimensional separator hyperplane summarised as $H: \alpha x + \beta y + \gamma z + b = 0$.</footnote>
<figure><img src="image_18.png" /><figcaption>**FIGURE 5** Linear decision boundary in three dimensions. [Colour figure can be viewed at wileyonlinelibrary.com]</figcaption></figure>

having been placed under strict European Commission monitoring for 10 years following the Sovereign Debt Crisis.

Figure 5 shows 1/5 of the dataset<fnref n="36" /> and the linear separator hyperplane.<fnref n="37" /> The red dots circled in purple are the instances of non-compliance that are closest to the hyperplane (e.g. Belgium in 2014 and Croatia in 2012) and therefore would have been the easiest to rectify. The black dots circled in purple would also have triggered increased attention because they are close to the ‘cliff edge’ (the Slovak Republic in 2007 for example).

In summary, the feature selection step revealed the key variables that governments and institutions in charge of fiscal surveillance should monitor, i.e. the general government fiscal balance in the preceding year, the fiscal space, output gap, oil prices, bond yields, corporate sector financial liabilities, and crisis dummies. Second, the optimal model can be used to forecast SGP (non-)compliance with better than 90% accuracy. By including the key determinants of SGP compliance into the model, predicted deviations from the rule can be prevented. Moreover, the distance of each country from the decision boundary of the model, can be used to adapt recommendations toward SGP compliance.

This model could therefore be implemented to promote future public finance sustainability. This original method could indeed be used to monitor countries within the European Semester framework, provide precise forecasts of SGP compliance, and adapt recommendations on a country-by-country basis with greater distances from the boundary triggering more urgent and stricter measures. More accurate diagnoses of SGP violation risks could be used by member states when preparing their draft budgets. Strengthening the preventive instrument of the SGP in this way may contribute to overcoming one of the main limitations of the dissuasive instrument, namely that once the 3% threshold has been violated and no escape clauses apply, it is often too late to rectify the ongoing public finance trend.

<footnote n="36">³⁶These observations were selected at random to make the figure legible.</footnote>

<footnote n="37">³⁷The separator in Figure 5 only represents three dimensions of the 8-dimensional separator hyperplane summarised as $H: \alpha x + \beta y + \gamma z + b = 0$.</footnote>
<figure><img src="image_20.png" /></figure>

# 6 | CONCLUSION

Our analysis focuses on the 28 EU member states over the period 2006–2018 and offers a new perspective in the debate on the dissuasive vs the preventive arm of the SGP: we propose a simpler but more powerful preventive arm, to minimise the number of cases in which the dissuasive arm is required. Indeed, the proposed method can be used to tailor recommendations from the central authority to non-compliant countries. Instead of sanctions, the central authority could preemptively propose a set of well-targeted interventions, one year in advance, to move a country from the non-compliant to the compliant subspace of the model.

This paper describes a new machine learning-based forecasting model of SGP compliance among EU member states. We focused on the public deficit rule, since a promptly forecasted breach of the 3% deficit limit can be corrected in the medium term by multiannual programming. This is not the case for public debt, which is at the heart of the third reform of the SGP and takes much longer to rectify.

We identified a set of 8 features from a dataset of 138 variables using LASSO feature selection, and compared results from SVM models with three different kernels, using a logit model as a benchmark. The best results were obtained with the linear kernel SVM model, which when trained with a k-fold cross-validation setup had a forecasting accuracy of 91.7% (accurately forecasting 112 out of the 135 cases of non-compliance and 222 out of the 229 cases of compliance).

Our findings contribute to several areas of research:

First, they feed the discussion on ‘the impact of Machine Learning on economics’ (Athey, 2018). Indeed, machine learning models provide high forecasting power, and should be considered in fiscal policy outcome forecasting and risk prevention. In our case, the machine learning models outperformed the standard econometric benchmark in all but one case. Our study may open the way to a wider use of this type of model in macroeconomics.

Second, our paper can be viewed as taking a ‘risk-management approach’ to fiscal surveillance offering a solution to simplify fiscal frameworks. Simplifications appear necessary for forecasts to be endorsed by independent fiscal councils (Darvas et al., 2018; Debrun et al., 2019). Our findings could constitute a first step in reforming the European fiscal framework: (i) MIP scoreboard indicators could be used in European Commission recommendations to help countries overcome fiscal difficulties rather than to implement excessive imbalance/deficit procedures; (ii) simple advanced indicators could be incorporated into an alert mechanism to prevent SGP deviations.

Third, there are several possibilities for future research: (i) this approach could be used with the Compliance Tracker Database (Larch and Santacroce, 2020) to analyse compliance with the other fiscal rules in the SGP, such as the structural balance rule and the expenditure rule; (ii) the model could also be applied to forecast compliance with national fiscal policy outcomes using national-level datasets; (iii) these models could also be applied to other macroeconomic outcomes and forecast compliance with other objectives, in monetary and macroprudential policy for instance.

# ACKNOWLEDGEMENTS

This paper was supported by the European Union. However, the views and opinions expressed are those of the authors alone and do not necessarily reflect those of the European Union. The European Union cannot be held responsible for them. The authors want to thank Xavier Debrun for all his support and relevant comments during the UCA-CNRS-IRD-CERDI International Workshop on the Policy Mix, January 30–31, 2020. The paper also benefited from valuable discussions and comments from Alexandru Minea, Piotr Stanek, António
<figure><img src="image_19.png" /></figure>

6 | CONCLUSION

Our analysis focuses on the 28 EU member states over the period 2006–2018 and offers a new perspective in the debate on the dissuasive vs the preventive arm of the SGP: we propose a simpler but more powerful preventive arm, to minimise the number of cases in which the dissuasive arm is required. Indeed, the proposed method can be used to tailor recommendations from the central authority to non-compliant countries. Instead of sanctions, the central authority could preemptively propose a set of well-targeted interventions, one year in advance, to move a country from the non-compliant to the compliant subspace of the model.

This paper describes a new machine learning-based forecasting model of SGP compliance among EU member states. We focused on the public deficit rule, since a promptly forecasted breach of the 3% deficit limit can be corrected in the medium term by multiannual programming. This is not the case for public debt, which is at the heart of the third reform of the SGP and takes much longer to rectify.

We identified a set of 8 features from a dataset of 138 variables using LASSO feature selection, and compared results from SVM models with three different kernels, using a logit model as a benchmark. The best results were obtained with the linear kernel SVM model, which when trained with a k-fold cross-validation setup had a forecasting accuracy of 91.7% (accurately forecasting 112 out of the 135 cases of non-compliance and 222 out of the 229 cases of compliance).

Our findings contribute to several areas of research:

First, they feed the discussion on ‘the impact of Machine Learning on economics’ (Athey, 2018). Indeed, machine learning models provide high forecasting power, and should be considered in fiscal policy outcome forecasting and risk prevention. In our case, the machine learning models outperformed the standard econometric benchmark in all but one case. Our study may open the way to a wider use of this type of model in macroeconomics.

Second, our paper can be viewed as taking a ‘risk-management approach’ to fiscal surveillance offering a solution to simplify fiscal frameworks. Simplifications appear necessary for forecasts to be endorsed by independent fiscal councils (Darvas et al., 2018; Debrun et al., 2019). Our findings could constitute a first step in reforming the European fiscal framework: (i) MIP scoreboard indicators could be used in European Commission recommendations to help countries overcome fiscal difficulties rather than to implement excessive imbalance/deficit procedures; (ii) simple advanced indicators could be incorporated into an alert mechanism to prevent SGP deviations.

Third, there are several possibilities for future research: (i) this approach could be used with the Compliance Tracker Database (Larch and Santacroce, 2020) to analyse compliance with the other fiscal rules in the SGP, such as the structural balance rule and the expenditure rule; (ii) the model could also be applied to forecast compliance with national fiscal policy outcomes using national-level datasets; (iii) these models could also be applied to other macroeconomic outcomes and forecast compliance with other objectives, in monetary and macroprudential policy for instance.

ACKNOWLEDGEMENTS

This paper was supported by the European Union. However, the views and opinions expressed are those of the authors alone and do not necessarily reflect those of the European Union. The European Union cannot be held responsible for them. The authors want to thank Xavier Debrun for all his support and relevant comments during the UCA-CNRS-IRD-CERDI International Workshop on the Policy Mix, January 30–31, 2020. The paper also benefited from valuable discussions and comments from Alexandru Minea, Piotr Stanek, António
<figure><img src="image_21.png" /></figure>

Afonso, Stefano Bianchini and participants in the GEBA 2019, ERMEES international workshop 2019 and UCA-CNRS-IRD-CERDI International Workshop on the Policy Mix 2020. All remaining errors are our own.

## FUNDING INFORMATION

The work (English proof reading costs) was funded by European Union through Amélie Barbier-Gauchard Jean Monnet Chair.

## DATA AVAILABILITY STATEMENT

I confirm that my article contains a Data Availability Statement even if no data is available (list of sample statements) unless my article type does not require one (e.g. Editorials, Corrections, Book Reviews, etc.).

## ORCID

Kéa Baret https://orcid.org/0000-0002-4696-9592

Amélie Barbier-Gauchard https://orcid.org/0000-0003-4935-227X

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<figure><img src="image_20.png" /></figure>

Afonso, Stefano Bianchini and participants in the GEBA 2019, ERMEES international workshop 2019 and UCA-CNRS-IRD-CERDI International Workshop on the Policy Mix 2020. All remaining errors are our own.

## FUNDING INFORMATION

The work (English proof reading costs) was funded by European Union through Amélie Barbier-Gauchard Jean Monnet Chair.

## DATA AVAILABILITY STATEMENT

I confirm that my article contains a Data Availability Statement even if no data is available (list of sample statements) unless my article type does not require one (e.g. Editorials, Corrections, Book Reviews, etc.).

## ORCID

Kéa Baret https://orcid.org/0000-0002-4696-9592

Amélie Barbier-Gauchard https://orcid.org/0000-0003-4935-227X

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<figure><img src="image_22.png" /></figure>

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Eyraud, L., Debrun, X., Hodge, A., Lledó, V., & Pattillo, C. A. (2018). Second-generation fiscal rules: Balancing simplicity, flexibility, and enforceability. *Staff Discussion Notes No. 18/04*.

Fichet de Clairfontaine, A., Badinger, H., & Reuter, W. H. (2017). Fiscal rules and twin deficits: the link between fiscal and external balances. *World Economy*, *40*, 21–35.

Foremny, D. (2014). Sub-national deficits in European Countries: The impact of fiscal rules and tax autonomy. *European Journal of Political Economy*, *34*, 86–110.

Frenkel, M., & Nickel, C. (2002). How symmetric are the shocks and the shock adjustment dynamics between the euro area and central and eastern European countries? *IMF Working Papers*.

Friedman, J., Hastie, T., & Tibshirani, R. (2009). *Regularization for generalized linear models via coordinate descent*. *Department of Statistics Stanford University*.

Gogas, P., & Papadimitriou, T. (2021). Machine learning in economics and finance. *Computational Economics*, *57*, 1–4.

Gogas, P., Papadimitriou, T., & Agrapetidou, A. (2018). Forecasting bank failures and stress testing: A machine learning approach. *International Journal of Forecasting*, *34*, 440–455.
<figure><img src="image_21.png" /></figure>

Cordes, T., Kinda, T., Muthoora, P., & Weber, A. (2015). Expenditure rules: Effective tools for sound fiscal policy? *IMF Working Paper 15/29*.

Cortes, C., & Vapnik, V. (1995). Support Vector Network. *Machine Learning*, 20, 273–297.

Creel, J. (2003). Ranking fiscal policy rules: The golden rule of public finance vs. the stability and growth pact. *Documents de Travail de l'OFCE 2003-04*.

Dabán, T. (2011). *A second-generation of fiscal rules for Latin America*. IMF Public Financial Management Blog.

Darvas, Z., Martin, P., & Ragot, X. (2018). European fiscal rules require a major overhaul. *Notes du Conseil D'analyse Économique*, 47, 1–12.

Debrun, X., Gerard, M., & Harris, J. (2019). Fiscal watchdogs and sound fiscal policy: Is the barking loud enough to tame politicians? *MPRA Paper No. 96683*.

Debrun, X., & Jonung, L. (2019). Under threat: Rules-based fiscal policy and how to preserve it. *European Journal of Political Economy*, 57, 142–157.

Debrun, X., Moulin, L., Turrini, A., Ayuso i Casals, J., & Kumar, M. (2008). Tied to the mast? National fiscal rules in the European Union. *Economic Policy*, 23, 297–362.

Delgado-Téllez, M., Lledó, V. D., & Pérez, J. J. (2017). On the determinants of fiscal non-compliance: an empirical analysis of Spain's regions. *IMF Working Paper 17/5*.

EU Council Directive. (2011). 2011/85/EU of 8 November 2011 on the requirements for member states' fiscal frameworks.

EU Council Regulation. (1997a). No 1466/97 of the council of 7 July 1997 on the strengthening of the surveillance of budgetary positions and the surveillance and coordination of economic policies.

EU Council Regulation. (1997b). No 1467/97 of the council of 7 July 1997 on speeding up and clarifying the implementation of the excessive deficit procedure.

EU Council Regulation. (2011). No 1177/2011 of 8 November 2011 amending regulation (EC) no 1467/97 on speeding up and clarifying the implementation of the excessive deficit procedure.

EU Regulation. (2011c). No 1175/2011 of the European parliament and of the council of 16 November 2011 amending council regulation (EC) no 1466/97 on the strengthening of the surveillance of budgetary positions and the surveillance and coordination of economic policies.

EU Regulation. (2011d). No 1176/2011 of the European parliament and of the council of 16 November 2011 on the prevention and correction of macroeconomic imbalances.

EU Regulation. (2011e). No 1177/2011 of 8 November 2011 amending regulation (EC) no 1467/97 on speeding up and clarifying the implementation of the excessive deficit procedure.

EU Regulation. (2013a). No 472/2013 of the European parliament and of the council of 21 may 2013 on the strengthening of economic and budgetary surveillance of euro area member states that are experiencing, or are at risk of experiencing, serious financial stability problems.

EU Regulation. (2013b). No 473/2013 of the European parliament and of the council of 21 may 2013 laying down common provisions for the monitoring and evaluation of draft budgetary plans and for the correction of excessive deficits in euro area member states.

European Commission. (2017). *European Commission fiscal rules database*.

European Commission. (2020). *Report on the application of the economic governance framework*.

European Fiscal Board. (2020). Assessment of the fiscal stance appropriate for the euro area in 2021. *Report*.

Eyraud, L., Debrun, X., Hodge, A., Lledó, V., & Pattillo, C. A. (2018). Second-generation fiscal rules: Balancing simplicity, flexibility, and enforceability. *Staff Discussion Notes No. 18/04*.

Fichet de Clairfontaine, A., Badinger, H., & Reuter, W. H. (2017). Fiscal rules and twin deficits: the link between fiscal and external balances. *World Economy*, 40, 21–35.

Foremny, D. (2014). Sub-national deficits in European Countries: The impact of fiscal rules and tax autonomy. *European Journal of Political Economy*, 34, 86–110.

Frenkel, M., & Nickel, C. (2002). How symmetric are the shocks and the shock adjustment dynamics between the euro area and central and eastern European countries? *IMF Working Papers*.

Friedman, J., Hastie, T., & Tibshirani, R. (2009). *Regularization for generalized linear models via coordinate descent*. *Department of Statistics Stanford University*.

Gogas, P., & Papadimitriou, T. (2021). Machine learning in economics and finance. *Computational Economics*, 57, 1–4.

Gogas, P., Papadimitriou, T., & Agrapetidou, A. (2018). Forecasting bank failures and stress testing: A machine learning approach. *International Journal of Forecasting*, 34, 440–455.
Gogas, P., Papadimitriou, T., & Chrysanthidou, E. (2015). Yield curve point triplets in recession forecasting. *International Finance, 18*, 207–226.

Gogas, P., Papadimitriou, T., & Sofianos, E. (2022). Forecasting unemployment in the euro area with machine learning. *Journal of Forecasting, 41*(3), 551–566.

Guerguil, M., Mandon, P., & Tapsoba, R. (2017). Flexible fiscal rules and countercyclical fiscal policy. *Journal of Macroeconomics, 52*, 189–220.

Härdle, W., Lee, Y.-J., Schäfer, D., & Yeh, Y.-R. (2009). Variable selection and oversampling in the use of smooth support vector machines for predicting the default risk of companies. *Journal of Forecasting, 28*, 512–534.

Hauptmeier, S., & Kamps, C. (2020). Debt rule design in theory and practice – the SGP’s debt benchmark revisited. *Presentation at ECFIN workshop “Fiscal rules in Europe”*.

Huang, Z., Chen, H., Hsu, C.-J., Chen, W.-H., & Wuc, S. (2004). Credit rating analysis with support vector machines and neural networks: A market comparative study. *Decision Support Systems, 37*, 543–558.

IMF. (2009). Fiscal rules-anchoring expectations for sustainable public finances. *Fiscal Affairs Department, IMF Policy Paper*.

Ince, H., & Trafalis, T. (2006). A hybrid model for exchange rate prediction. *Decision Support Systems, 42*, 1054–1062.

Kopits, G., & Symansky, S. A. (1998). Fiscal policy rules. *Occasional Paper No. 162*.

Kulkarni, A., Chong, D., & Batarseh, F. A. (2020). 5 – Foundations of data imbalance and solutions for a data democracy. In F. A. Batarseh & R. Yang (Eds.), *Data democracy* (pp. 83–106). Academic Press.

Larch, M., Orseau, E., & van der Wielen, W. (2021). Do EU fiscal rules support or hinder counter-cyclical fiscal policy? *Journal of International Money and Finance, 112*, 102328.

Larch, M., & Santacroce, S. (2020). *Numerical compliance with EU fiscal rules: The compliance database of the Secretariat of the European Fiscal Board (Database)*.

Martin, D. (1977). Early warning of bank failure: A logit regression approach. *Journal of Banking and Finance, 1*(3), 249–276.

Medeiros, M. C., Vasconcelos, G. F. R., Veiga, A., & Zilberman, E. (2021). Forecasting inflation in a data-rich environment: The benefits of machine learning methods. *Journal of Business and Economic Statistics, 39*(1), 98–119.

Nandelenga, M. W., & Ellyne, M. J. (2020). Fiscal rules and the compliance debate: Why do countries adopt rules and fail to comply. *ERSA Working Paper No 815*.

Olson, J. A. (1980). Financial rations and the probabilistic prediction of bankruptcy. *Journal of Accounting Research, 18*(1), 109–131.

Plakandaras, V., Papadimitriou, T., & Gogas, P. (2013). Directional forecasting in financial time series using support vector machines: The US/Euro exchange rate. *Journal of Computational Optimization in Economics and Finance, 5*, 125–139.

Regulation, E. U. (2011a). *No 1173/2011 of the European parliament and of the council of 16 November 2011 on the effective implementation of fiscal surveillance in the euro area*.

Regulation, E. U. (2011b). *No 1174/2011 of the European parliament and of the council of 16 November 2011 laying down implementing measures to correct excessive macroeconomic imbalances in the euro area*.

Reuter, W. H. (2015). National numerical fiscal rules: not complied but still effective? *European Journal of Economy, 39*, 67–81.

Reuter, W. H. (2019). When and why do countries break their national fiscal rules? *European Journal of Political Economy, 57*, 125–141.

Romer, C. D., & Romer, D. H. (2018). Phillips lecture – why some times are different: macroeconomic policy and the aftermath of financial crises. *Economica, 85*, 1–40.

Sacchi, A., & Salotti, S. (2015). The impact of national fiscal rules on the stabilisation function of fiscal policy. *European Journal of Political Economy, 37*, 1–20.

Sancak, C., Xing, J., & Velloso, R. (2010). Tax revenue response to the business cycle. *IMF Working Paper No. WP/10/71*.

Schaechter, A., Kinda, T., Budinaand, N., & Weber, A. (2016). *Fiscal rules in response to the crisis–towards the ‘Next-Generation’ rules. A new dataset*. IMF Fiscal Affairs Department.

Schick, A. (2010). Post-crisis fiscal rules: Stabilising public finance while responding to economic aftershocks. *OECD Journal on Budgeting, 10*(2), 1–18.

Tapsoba, R. (2012). Do national numerical fiscal rules really shape fiscal behaviours in developing countries? A treatment effect evaluation. *Economic Modelling, 29*, 1356–1369.
<figure><img src="image_22.png" /></figure>

Gogas, P., Papadimitriou, T., & Chrysanthidou, E. (2015). Yield curve point triplets in recession forecasting. *International Finance, 18*, 207–226.

Gogas, P., Papadimitriou, T., & Sofianos, E. (2022). Forecasting unemployment in the euro area with machine learning. *Journal of Forecasting, 41*(3), 551–566.

Guerguil, M., Mandon, P., & Tapsoba, R. (2017). Flexible fiscal rules and countercyclical fiscal policy. *Journal of Macroeconomics, 52*, 189–220.

Härdle, W., Lee, Y.-J., Schäfer, D., & Yeh, Y.-R. (2009). Variable selection and oversampling in the use of smooth support vector machines for predicting the default risk of companies. *Journal of Forecasting, 28*, 512–534.

Hauptmeier, S., & Kamps, C. (2020). Debt rule design in theory and practice – the SGP’s debt benchmark revisited. *Presentation at ECFIN workshop “Fiscal rules in Europe”*.

Huang, Z., Chen, H., Hsu, C.-J., Chen, W.-H., & Wuc, S. (2004). Credit rating analysis with support vector machines and neural networks: A market comparative study. *Decision Support Systems, 37*, 543–558.

IMF. (2009). Fiscal rules-anchoring expectations for sustainable public finances. *Fiscal Affairs Department, IMF Policy Paper*.

Ince, H., & Trafalis, T. (2006). A hybrid model for exchange rate prediction. *Decision Support Systems, 42*, 1054–1062.

Kopits, G., & Symansky, S. A. (1998). Fiscal policy rules. *Occasional Paper No. 162*.

Kulkarni, A., Chong, D., & Batarseh, F. A. (2020). 5 – Foundations of data imbalance and solutions for a data democracy. In F. A. Batarseh & R. Yang (Eds.), *Data democracy* (pp. 83–106). Academic Press.

Larch, M., Orseau, E., & van der Wielen, W. (2021). Do EU fiscal rules support or hinder counter-cyclical fiscal policy? *Journal of International Money and Finance, 112*, 102328.

Larch, M., & Santacroce, S. (2020). *Numerical compliance with EU fiscal rules: The compliance database of the Secretariat of the European Fiscal Board (Database)*.

Martin, D. (1977). Early warning of bank failure: A logit regression approach. *Journal of Banking and Finance, 1*(3), 249–276.

Medeiros, M. C., Vasconcelos, G. F. R., Veiga, A., & Zilberman, E. (2021). Forecasting inflation in a data-rich environment: The benefits of machine learning methods. *Journal of Business and Economic Statistics, 39*(1), 98–119.

Nandelenga, M. W., & Ellyne, M. J. (2020). Fiscal rules and the compliance debate: Why do countries adopt rules and fail to comply. *ERSA Working Paper No 815*.

Olson, J. A. (1980). Financial rations and the probabilistic prediction of bankruptcy. *Journal of Accounting Research, 18*(1), 109–131.

Plakandaras, V., Papadimitriou, T., & Gogas, P. (2013). Directional forecasting in financial time series using support vector machines: The US/Euro exchange rate. *Journal of Computational Optimization in Economics and Finance, 5*, 125–139.

Regulation, E. U. (2011a). *No 1173/2011 of the European parliament and of the council of 16 November 2011 on the effective implementation of fiscal surveillance in the euro area*.

Regulation, E. U. (2011b). *No 1174/2011 of the European parliament and of the council of 16 November 2011 laying down implementing measures to correct excessive macroeconomic imbalances in the euro area*.

Reuter, W. H. (2015). National numerical fiscal rules: not complied but still effective? *European Journal of Economy, 39*, 67–81.

Reuter, W. H. (2019). When and why do countries break their national fiscal rules? *European Journal of Political Economy, 57*, 125–141.

Romer, C. D., & Romer, D. H. (2018). Phillips lecture – why some times are different: macroeconomic policy and the aftermath of financial crises. *Economica, 85*, 1–40.

Sacchi, A., & Salotti, S. (2015). The impact of national fiscal rules on the stabilisation function of fiscal policy. *European Journal of Political Economy, 37*, 1–20.

Sancak, C., Xing, J., & Velloso, R. (2010). Tax revenue response to the business cycle. *IMF Working Paper No. WP/10/71*.

Schaechter, A., Kinda, T., Budinaand, N., & Weber, A. (2016). *Fiscal rules in response to the crisis–towards the ‘Next-Generation’ rules. A new dataset*. IMF Fiscal Affairs Department.

Schick, A. (2010). Post-crisis fiscal rules: Stabilising public finance while responding to economic aftershocks. *OECD Journal on Budgeting, 10*(2), 1–18.

Tapsoba, R. (2012). Do national numerical fiscal rules really shape fiscal behaviours in developing countries? A treatment effect evaluation. *Economic Modelling, 29*, 1356–1369.
<figure><img src="image_23.png" /></figure>

Velickovski, I., & Stojkov, A. (2014). Is the European integration speeding up the economic convergence process of the Central and South-Eastern European countries? A shock perspective. *Empirica*, *41*, 287–321.

Wiese, R., Jong-A-Pin, R., & de Haan, J. (2018). Can successful fiscal adjustments only be achieved by spending cuts? *Economic Policy*, *54*, 145–166.

Wyplosz, C. (2012). Fiscal rules: Theoretical issues and historical experience. *NBER Working Paper No. 17884*.

Yadav, S., & Shukla, S. (2016). Analysis of k-fold cross-validation over hold-out validation on colossal datasets for quality classification. *2016 IEEE 6th international conference on advanced computing (IACC)*.

Yoon, J. (2021). Forecasting of real GDP growth using machine learning models: Gradient boosting and random forest approach. *Computational Economics*, *57*, 247–265.

**How to cite this article:** Baret, K., Barbier-Gauchard, A., & Papadimitriou, T. (2024). Forecasting stability and growth pact compliance using machine learning. *The World Economy*, *47*, 188–216. https://doi.org/10.1111/twec.13518

## APPENDIX 1

<table><caption>TABLE A1 Descriptive statistics.</caption><thead><tr><th>Variables</th><th>Correspondance variables</th><th>N</th><th>Mean</th><th>Min</th><th>Max</th><th>SD</th></tr></thead><tbody><tr><td>Y</td><td>Dummy variable = 1 if 3% limit was complied in t</td><td>364</td><td>0.629</td><td>0.00</td><td>1.00</td><td>0.483</td></tr><tr><td>X1</td><td>Dummy variable reflecting if the country was an advanced country in period t-p</td><td>364</td><td>0.785</td><td>0.00</td><td>1.00</td><td>0.410</td></tr><tr><td>X2</td><td>Dummy variable reflecting if the country was an Emerging country in period t-p</td><td>364</td><td>0.214</td><td>0.00</td><td>1.00</td><td>0.410</td></tr><tr><td>X3</td><td>Dummy variable reflecting if the country was a Resource-rich country in period t-p</td><td>364</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td></tr><tr><td>X4</td><td>Dummy variable reflecting if the country was an EU membership in period t-p</td><td>364</td><td>0.967</td><td>0.00</td><td>1.00</td><td>0.178</td></tr><tr><td>X5</td><td>Dummy variable reflecting if the country was a Federal Country in period t-p</td><td>364</td><td>0.107</td><td>0.00</td><td>1.00</td><td>0.309</td></tr><tr><td>X6</td><td>Dummy variable reflecting if the country was a eurozone member in period t-p</td><td>364</td><td>0.642</td><td>0.00</td><td>1.00</td><td>0.479</td></tr><tr><td>X7</td><td>Dummy variable reflecting if the country was submitted to enforcement procedure related to the supranational fiscal rules in period t-p</td><td>364</td><td>0.967</td><td>0.00</td><td>1.00</td><td>0.178</td></tr></tbody></table>
<figure><img src="image_23.png" /></figure>

Velickovski, I., & Stojkov, A. (2014). Is the European integration speeding up the economic convergence process of the Central and South-Eastern European countries? A shock perspective. *Empirica, 41*, 287–321.

Wiese, R., Jong-A-Pin, R., & de Haan, J. (2018). Can successful fiscal adjustments only be achieved by spending cuts? *Economic Policy, 54*, 145–166.

Wyplosz, C. (2012). Fiscal rules: Theoretical issues and historical experience. *NBER Working Paper No. 17884*.

Yadav, S., & Shukla, S. (2016). Analysis of k-fold cross-validation over hold-out validation on colossal datasets for quality classification. *2016 IEEE 6th international conference on advanced computing (IACC)*.

Yoon, J. (2021). Forecasting of real GDP growth using machine learning models: Gradient boosting and random forest approach. *Computational Economics, 57*, 247–265.

**How to cite this article:** Baret, K., Barbier-Gauchard, A., & Papadimitriou, T. (2024). Forecasting stability and growth pact compliance using machine learning. *The World Economy, 47*, 188–216. https://doi.org/10.1111/twec.13518

## APPENDIX 1

<table><caption>TABLE A1 Descriptive statistics.</caption><thead><tr><th>Variables</th><th>Correspondance variables</th><th>N</th><th>Mean</th><th>Min</th><th>Max</th><th>SD</th></tr></thead><tbody><tr><td>Y</td><td>Dummy variable = 1 if 3% limit was complied in t</td><td>364</td><td>0.629</td><td>0.00</td><td>1.00</td><td>0.483</td></tr><tr><td>X1</td><td>Dummy variable reflecting if the country was an advanced country in period t-p</td><td>364</td><td>0.785</td><td>0.00</td><td>1.00</td><td>0.410</td></tr><tr><td>X2</td><td>Dummy variable reflecting if the country was an Emerging country in period t-p</td><td>364</td><td>0.214</td><td>0.00</td><td>1.00</td><td>0.410</td></tr><tr><td>X3</td><td>Dummy variable reflecting if the country was a Resource-rich country in period t-p</td><td>364</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td></tr><tr><td>X4</td><td>Dummy variable reflecting if the country was an EU membership in period t-p</td><td>364</td><td>0.967</td><td>0.00</td><td>1.00</td><td>0.178</td></tr><tr><td>X5</td><td>Dummy variable reflecting if the country was a Federal Country in period t-p</td><td>364</td><td>0.107</td><td>0.00</td><td>1.00</td><td>0.309</td></tr><tr><td>X6</td><td>Dummy variable reflecting if the country was a eurozone member in period t-p</td><td>364</td><td>0.642</td><td>0.00</td><td>1.00</td><td>0.479</td></tr><tr><td>X7</td><td>Dummy variable reflecting if the country was submitted to enforcement procedure related to the supranational fiscal rules in period t-p</td><td>364</td><td>0.967</td><td>0.00</td><td>1.00</td><td>0.178</td></tr></tbody></table>
<table><caption>TABLE A1 (Continued)</caption><thead><tr><th>Variables</th><th>Correspondance variables</th><th>N</th><th>Mean</th><th>Min</th><th>Max</th><th>SD</th></tr></thead><tbody><tr><td>X8</td><td>Gross domestic product, deflator, in period t-p</td><td>364</td><td>100.84</td><td>62.69</td><td>146.3</td><td>10.31</td></tr><tr><td>X9</td><td>Total investment in period t-p</td><td>364</td><td>22.47</td><td>9.819</td><td>41.53</td><td>4.756</td></tr><tr><td>X10</td><td>Gross national savings t-p</td><td>364</td><td>21.61</td><td>5.099</td><td>33.70</td><td>5.525</td></tr><tr><td>X11</td><td>Inflation, average consumer prices index, in period t-p</td><td>364</td><td>98.10</td><td>67.04</td><td>169.8</td><td>14.89</td></tr><tr><td>X12</td><td>Population in period t-p</td><td>364</td><td>17.92</td><td>0.403</td><td>82.66</td><td>22.63</td></tr><tr><td>X13</td><td>General government revenue in period t-p</td><td>364</td><td>41.96</td><td>25.94</td><td>56.36</td><td>6.546</td></tr><tr><td>X14</td><td>General government total expenditure in period t-p</td><td>364</td><td>42.63</td><td>-7.824</td><td>65.047</td><td>11.19</td></tr><tr><td>X15</td><td>General government net lending/ borrowing in period t-p</td><td>364</td><td>-2.601</td><td>-32.02</td><td>5.129</td><td>3.590</td></tr><tr><td>X16</td><td>General government gross debt in period t-p</td><td>364</td><td>60.41</td><td>3.664</td><td>183.4</td><td>34.96</td></tr><tr><td>X17</td><td>Net External Positions in period t-p</td><td>364</td><td>-37.31</td><td>-198.7</td><td>65.2</td><td>50.93</td></tr><tr><td>X18</td><td>Current account balance in period t-p</td><td>364</td><td>-0.869</td><td>-23.90</td><td>13.80</td><td>6.021</td></tr><tr><td>X19</td><td>Current account balance variations over 3 years in period t-p</td><td>364</td><td>-1.214</td><td>-21.00</td><td>11.00</td><td>5.722</td></tr><tr><td>X20</td><td>Real Effective Exchange Rate in period t-p</td><td>364</td><td>0.770</td><td>-20.40</td><td>36.00</td><td>6.681</td></tr><tr><td>X21</td><td>Global export market share—% change over 5 years—in period t-p</td><td>364</td><td>4.547</td><td>-31.68</td><td>95.57</td><td>23.88</td></tr><tr><td>X22</td><td>Nominal unit wage cost—% change over 3 years—in period t-p</td><td>364</td><td>6.948</td><td>-21.00</td><td>78.30</td><td>10.35</td></tr><tr><td>X23</td><td>Debt of private sector in period t-p, consolidated—% of GDP</td><td>364</td><td>144.9</td><td>39.10</td><td>379.4</td><td>70.46</td></tr><tr><td>X24</td><td>Liabilities of the financial corporations sector—% change over 1 year—in period t-p</td><td>364</td><td>8.145</td><td>-17.60</td><td>115.6</td><td>12.85</td></tr><tr><td>X25</td><td>Unemployment rate—3-year average—in period t-p</td><td>364</td><td>9.047</td><td>3.700</td><td>26.30</td><td>4.101</td></tr><tr><td>X26</td><td>Unemployment rate in period t-p</td><td>364</td><td>8.976</td><td>2.900</td><td>27.50</td><td>4.324</td></tr><tr><td>X27</td><td>Gross domestic product (real GDP)—% change over 1 year—in period t-p</td><td>364</td><td>1.966</td><td>-14.80</td><td>25.10</td><td>3.837</td></tr><tr><td>X28</td><td>Gross fixed capital formation in period t-p—% of GDP—</td><td>364</td><td>21.93</td><td>11.50</td><td>37.40</td><td>4.196</td></tr><tr><td>X29</td><td>Gross domestic expenditure on R &amp; D in period t-p—% of GDP—</td><td>352</td><td>1.504</td><td>0.370</td><td>3.750</td><td>0.877</td></tr><tr><td>X30</td><td>Direct investment in the reporting economy (flow) in period t-p—% of GDP—</td><td>364</td><td>25.01</td><td>-264.1</td><td>1336.6</td><td>118.6</td></tr></tbody></table>
<table><caption>TABLE A1 (Continued)</caption><thead><tr><th>Variables</th><th>Correspondance variables</th><th>N</th><th>Mean</th><th>Min</th><th>Max</th><th>SD</th></tr></thead><tbody><tr><td>X8</td><td>Gross domestic product, deflator, in period t-p</td><td>364</td><td>100.84</td><td>62.69</td><td>146.3</td><td>10.31</td></tr><tr><td>X9</td><td>Total investment in period t-p</td><td>364</td><td>22.47</td><td>9.819</td><td>41.53</td><td>4.756</td></tr><tr><td>X10</td><td>Gross national savings t-p</td><td>364</td><td>21.61</td><td>5.099</td><td>33.70</td><td>5.525</td></tr><tr><td>X11</td><td>Inflation, average consumer prices index, in period t-p</td><td>364</td><td>98.10</td><td>67.04</td><td>169.8</td><td>14.89</td></tr><tr><td>X12</td><td>Population in period t-p</td><td>364</td><td>17.92</td><td>0.403</td><td>82.66</td><td>22.63</td></tr><tr><td>X13</td><td>General government revenue in period t-p</td><td>364</td><td>41.96</td><td>25.94</td><td>56.36</td><td>6.546</td></tr><tr><td>X14</td><td>General government total expenditure in period t-p</td><td>364</td><td>42.63</td><td>-7.824</td><td>65.047</td><td>11.19</td></tr><tr><td>X15</td><td>General government net lending/ borrowing in period t-p</td><td>364</td><td>-2.601</td><td>-32.02</td><td>5.129</td><td>3.590</td></tr><tr><td>X16</td><td>General government gross debt in period t-p</td><td>364</td><td>60.41</td><td>3.664</td><td>183.4</td><td>34.96</td></tr><tr><td>X17</td><td>Net External Positions in period t-p</td><td>364</td><td>-37.31</td><td>-198.7</td><td>65.2</td><td>50.93</td></tr><tr><td>X18</td><td>Current account balance in period t-p</td><td>364</td><td>-0.869</td><td>-23.90</td><td>13.80</td><td>6.021</td></tr><tr><td>X19</td><td>Current account balance variations over 3 years in period t-p</td><td>364</td><td>-1.214</td><td>-21.00</td><td>11.00</td><td>5.722</td></tr><tr><td>X20</td><td>Real Effective Exchange Rate in period t-p</td><td>364</td><td>0.770</td><td>-20.40</td><td>36.00</td><td>6.681</td></tr><tr><td>X21</td><td>Global export market share—% change over 5 years—in period t-p</td><td>364</td><td>4.547</td><td>-31.68</td><td>95.57</td><td>23.88</td></tr><tr><td>X22</td><td>Nominal unit wage cost—% change over 3 years—in period t-p</td><td>364</td><td>6.948</td><td>-21.00</td><td>78.30</td><td>10.35</td></tr><tr><td>X23</td><td>Debt of private sector in period t-p, consolidated—% of GDP</td><td>364</td><td>144.9</td><td>39.10</td><td>379.4</td><td>70.46</td></tr><tr><td>X24</td><td>Liabilities of the financial corporations sector—% change over 1 year—in period t-p</td><td>364</td><td>8.145</td><td>-17.60</td><td>115.6</td><td>12.85</td></tr><tr><td>X25</td><td>Unemployment rate—3-year average—in period t-p</td><td>364</td><td>9.047</td><td>3.700</td><td>26.30</td><td>4.101</td></tr><tr><td>X26</td><td>Unemployment rate in period t-p</td><td>364</td><td>8.976</td><td>2.900</td><td>27.50</td><td>4.324</td></tr><tr><td>X27</td><td>Gross domestic product (real GDP)—% change over 1 year—in period t-p</td><td>364</td><td>1.966</td><td>-14.80</td><td>25.10</td><td>3.837</td></tr><tr><td>X28</td><td>Gross fixed capital formation in period t-p—% of GDP—</td><td>364</td><td>21.93</td><td>11.50</td><td>37.40</td><td>4.196</td></tr><tr><td>X29</td><td>Gross domestic expenditure on R &amp; D in period t-p—% of GDP—</td><td>352</td><td>1.504</td><td>0.370</td><td>3.750</td><td>0.877</td></tr><tr><td>X30</td><td>Direct investment in the reporting economy (flow) in period t-p—% of GDP—</td><td>364</td><td>25.01</td><td>-264.1</td><td>1336.6</td><td>118.6</td></tr></tbody></table>
<figure><img src="image_24.png" /></figure>

<table><caption>TABLE A1 (Continued)</caption><thead><tr><td>Variables</td><td>Correspondance variables</td><td>N</td><td>Mean</td><td>Min</td><td>Max</td><td>SD</td></tr></thead><tbody><tr><td>X31</td><td>Direct investment in the reporting economy (stocks)—% of GDP</td><td>364</td><td>350.2</td><td>4.200</td><td>9479.1</td><td>1135.5</td></tr><tr><td>X32</td><td>Net trade balance of energy products in period t-p—% of GDP—</td><td>364</td><td>-3.212</td><td>-14.90</td><td>2.300</td><td>2.062</td></tr><tr><td>X33</td><td>Real effective exchange rate, Euro area trading partners—% change over 3 years</td><td>364</td><td>1.487</td><td>-21.70</td><td>38.90</td><td>6.335</td></tr><tr><td>X34</td><td>Terms of trade (goods and services)—% change over 5 years—in period t-p</td><td>364</td><td>1.102</td><td>-10.20</td><td>28.30</td><td>4.870</td></tr><tr><td>X35</td><td>Market share of world exports, volumes—% change over 1 year—in period t-p</td><td>364</td><td>0.495</td><td>-10.30</td><td>36.40</td><td>4.816</td></tr><tr><td>X36</td><td>Labor productivity—% change over 1 year—in period t-p</td><td>364</td><td>1.294</td><td>-7.700</td><td>20.90</td><td>2.784</td></tr><tr><td>X37</td><td>Residential construction in period t-p—% of GDP</td><td>350</td><td>4.347</td><td>0.600</td><td>13.50</td><td>2.121</td></tr><tr><td>X38</td><td>Employment—% change over 1 year—in period t-p</td><td>364</td><td>0.658</td><td>-14.30</td><td>6.5</td><td>2.377</td></tr><tr><td>X39</td><td>Dummy variable reflecting if there is a Crisis in period t-p</td><td>364</td><td>0.307</td><td>0.000</td><td>1.000</td><td>0.462</td></tr><tr><td>X40</td><td>Output gap (production function approach) in period t-p</td><td>364</td><td>0.167</td><td>-12.89</td><td>20.29</td><td>4.345</td></tr><tr><td>X41</td><td>Oil Prices in period t-p</td><td>364</td><td>73.76</td><td>43.29</td><td>99.67</td><td>19.84</td></tr><tr><td>X42</td><td>Bonds yield in period t-p</td><td>351</td><td>3.849</td><td>0.090</td><td>22.50</td><td>2.460</td></tr><tr><td>X43</td><td>Foreign currency long-term sovereign debt ratings, index from 1 to 21, in period t-p</td><td>364</td><td>16.71</td><td>2.842</td><td>21.00</td><td>4.019</td></tr><tr><td>X44</td><td>Fiscal Space in period t-p</td><td>364</td><td>3.966</td><td>-60.18</td><td>119.1</td><td>33.11</td></tr><tr><td>X45</td><td>Fiscal Rules Index (by European Commission) in period t-p</td><td>364</td><td>0.542</td><td>-0.948</td><td>3.404</td><td>1.068</td></tr><tr><td>X46</td><td>Structural Breaks in period t-p</td><td>364</td><td>0.225</td><td>0.000</td><td>1.000</td><td>0.418</td></tr><tr><td>X47</td><td>Government fragmentation in period t-p</td><td>364</td><td>0.707</td><td>0.491</td><td>0.861</td><td>0.097</td></tr></tbody></table>

Note: Y is the dependent variable. X are the potential predictors tested in the feature selection step. All variables used as predictors are lagged by p periods. We only show descriptive statistics for the variables lagged by one period to save space and because these contain information about the variables lagged by 2 and 3 periods, which were also tested in the paper. Fiscal space is measured as the difference between a country's public debt and EU median debt in each year.
<figure><img src="image_24.png" /></figure>

<table><caption>TABLE A1 (Continued)</caption><thead><tr><th>Variables</th><th>Correspondance variables</th><th>N</th><th>Mean</th><th>Min</th><th>Max</th><th>SD</th></tr></thead><tbody><tr><td>X31</td><td>Direct investment in the reporting economy (stocks)—% of GDP</td><td>364</td><td>350.2</td><td>4.200</td><td>9479.1</td><td>1135.5</td></tr><tr><td>X32</td><td>Net trade balance of energy products in period t-p—% of GDP—</td><td>364</td><td>-3.212</td><td>-14.90</td><td>2.300</td><td>2.062</td></tr><tr><td>X33</td><td>Real effective exchange rate, Euro area trading partners—% change over 3 years</td><td>364</td><td>1.487</td><td>-21.70</td><td>38.90</td><td>6.335</td></tr><tr><td>X34</td><td>Terms of trade (goods and services)—% change over 5 years—in period t-p</td><td>364</td><td>1.102</td><td>-10.20</td><td>28.30</td><td>4.870</td></tr><tr><td>X35</td><td>Market share of world exports, volumes—% change over 1 year—in period t-p</td><td>364</td><td>0.495</td><td>-10.30</td><td>36.40</td><td>4.816</td></tr><tr><td>X36</td><td>Labor productivity—% change over 1 year—in period t-p</td><td>364</td><td>1.294</td><td>-7.700</td><td>20.90</td><td>2.784</td></tr><tr><td>X37</td><td>Residential construction in period t-p—% of GDP</td><td>350</td><td>4.347</td><td>0.600</td><td>13.50</td><td>2.121</td></tr><tr><td>X38</td><td>Employment—% change over 1 year—in period t-p</td><td>364</td><td>0.658</td><td>-14.30</td><td>6.5</td><td>2.377</td></tr><tr><td>X39</td><td>Dummy variable reflecting if there is a Crisis in period t-p</td><td>364</td><td>0.307</td><td>0.000</td><td>1.000</td><td>0.462</td></tr><tr><td>X40</td><td>Output gap (production function approach) in period t-p</td><td>364</td><td>0.167</td><td>-12.89</td><td>20.29</td><td>4.345</td></tr><tr><td>X41</td><td>Oil Prices in period t-p</td><td>364</td><td>73.76</td><td>43.29</td><td>99.67</td><td>19.84</td></tr><tr><td>X42</td><td>Bonds yield in period t-p</td><td>351</td><td>3.849</td><td>0.090</td><td>22.50</td><td>2.460</td></tr><tr><td>X43</td><td>Foreign currency long-term sovereign debt ratings, index from 1 to 21, in period t-p</td><td>364</td><td>16.71</td><td>2.842</td><td>21.00</td><td>4.019</td></tr><tr><td>X44</td><td>Fiscal Space in period t-p</td><td>364</td><td>3.966</td><td>-60.18</td><td>119.1</td><td>33.11</td></tr><tr><td>X45</td><td>Fiscal Rules Index (by European Commission) in period t-p</td><td>364</td><td>0.542</td><td>-0.948</td><td>3.404</td><td>1.068</td></tr><tr><td>X46</td><td>Structural Breaks in period t-p</td><td>364</td><td>0.225</td><td>0.000</td><td>1.000</td><td>0.418</td></tr><tr><td>X47</td><td>Government fragmentation in period t-p</td><td>364</td><td>0.707</td><td>0.491</td><td>0.861</td><td>0.097</td></tr></tbody></table>

Note: Y is the dependent variable. X are the potential predictors tested in the feature selection step. All variables used as predictors are lagged by p periods. We only show descriptive statistics for the variables lagged by one period to save space and because these contain information about the variables lagged by 2 and 3 periods, which were also tested in the paper. Fiscal space is measured as the difference between a country's public debt and EU median debt in each year.
APPENDIX 2

<figure><img src="image_25.png" /><figcaption>FIGURE A1 LASSO results. Source: Authors. [Colour figure can be viewed at wileyonlinelibrary.com]</figcaption></figure>
APPENDIX 2

<figure><img src="image_25.png" /><figcaption>FIGURE A1 LASSO results. Source: Authors. [Colour figure can be viewed at wileyonlinelibrary.com]</figcaption></figure>