metadata: authors: 0: author: João,Igor Custodio
1: author: Calice,Pietro
2: author: Lucas,Andre
3: author: Schaumburg,Julia
count: World
docna: 0: docna: Financial Development and Fragility : A
            Clustering Analysis
docty: Policy Research Working Paper
abstracts: cdata!: This paper explores the potential
            correlations between financial development and state
            fragility, using a sample of 137 countries observed over the
            period from 1998&#x2013;2019. The countries are grouped
            into clusters that capture the different joint states of
            financial development and fragility. The paper introduces a
            new switching methodology to further allow for a
            qualification of the evolution of countries in terms. of
            fragility scores with and without controlling for other
            variables. Irrespective of the precise methodology and state
            fragility measure as used in this paper, the findings
            indicate a negative correlation between financial
            development and state fragility, after controlling for
            several forms of observed and unobserved heterogeneity.
display_title: Financial Development and Fragility
            : A Clustering Analysis
listing_relative_url: /research/2024/07/34364723/financial-development-fragility-clustering-analysis
topicv3: Fragility, Conflict and Violence m1326586 2432,Financial Sector Development m1327681 1920
publishtoextweb_dt: 2024-07-15T00:00:00Z
dois: 10.1596/1813-9450-10850
origu: Off of Sr VP Dev Econ/Chief Econ (DECVP)
no_of_pages: 54
author_keys: 0: author_key: NA
1: author_key: 1297294
2: author_key: 2300357
3: author_key: 2298703
model_extractions: Policy Research Working Paper 10850 Financial Development and Fragility A Clustering Analysis _Igor Custodio João Pietro Calice Andre Lucas Julia Schaumburg_ Finance, Competitiveness and Innovation Global Practice July 2024
Policy Research Working Paper 10850

# **Abstract**

This paper explores the potential correlations between financial development and state fragility, using a sample of 137 countries observed over the period from 1998–2019. The countries are grouped into clusters that capture the different joint states of financial development and fragility. The paper introduces a new switching methodology to further allow for a qualification of the evolution of countries in terms.

of fragility scores with and without controlling for other variables. Irrespective of the precise methodology and state fragility measure as used in this paper, the findings indicate a negative correlation between financial development and state fragility, after controlling for several forms of observed and unobserved heterogeneity.

This paper is a product of the Finance, Competitiveness and Innovation Global Practice. It is part of a larger effort by the World Bank to provide open access to its research and make a contribution to development policy discussions around the world. Policy Research Working Papers are also posted on the Web at http://www.worldbank.org/prwp. The authors may be contacted at pcalice@worldbank.org.

_The Policy Research Working Paper Series disseminates the findings of work in progress to encourage the exchange of ideas about development issues. An objective of the series is to get the findings out quickly, even if the presentations are less than fully polished. The papers carry the names of the authors and should be cited accordingly. The findings, interpretations, and conclusions expressed in this paper are entirely those of the authors. They do not necessarily represent the views of the International Bank for Reconstruction and Development/World Bank and its affiliated organizations, or those of the Executive Directors of the World Bank or the governments they represent._ Produced by the Research Support Team
Financial Development and Fragility: A Clustering Analysis<sup>∗</sup> Igor Custodio Jo ̃ao<sup>_a_</sup> Pietro Calice<sup>_b_</sup> Andre Lucas<sup>_a_</sup> Julia Schaumburg<sup>_a_</sup>

> _a_ Vrije Universiteit Amsterdam and Tinbergen Institute

> _b_ The World Bank

> ~~∗~~ <u>This paper was funded by the</u> State and Peacebuilding 2.0 Umbrella Trust Fund (SPF2.0). SPF2.0 is a global multi-donor fund administered by the World Bank that works with partners to address the drivers and impacts of fragility, conflict, and violence (FCV) and strengthen the resilience of countries and affected p opulations, communities, and institutions. The SPF2.0 is kindly supported by : Denmark, Germany, Netherlands, Norway, Sweden, and Switzerland. The views expressed in this paper are those of the authors and do not necessarily represent the views of the World Bank and its affiliated organizations, or those of the Executive Directors of the World Bank or the governments they represent.
# **1 Introduction**

State fragility is one of the most pressing development challenges of our time, with an estimated 800 million people living in fragile states. In broadest terms, fragility refers to a situation where the state is unable or unwilling to fulfill the basic needs of its citizens. The definition centers on the potential for adverse outcomes and the absence of fundamental structures that ensure continuity and stability. Countries with higher levels of fragility more often struggle to provide basic services to their citizens, to maintain economic stability, or to implement effective policy (Stewart and Brown, 2009). Political instability in fragile countries is more likely to be linked to armed conflict, civil unrest, and violence (Chami et al., 2021).

In this paper, we examine the relationship between state fragility and financial development to determine which countries are more prone to extended periods of instability, other things being equal. Financial development is the process of development of institutions and markets to overcome the costs associated with the delivery of the key functions of the financial system: mobilizing savings, promoting greater information sharing, improving resource allocation, and facilitating diversification and management of risk (Levine, 2005). Financial development, especially when starting from low levels, plays a pivotal role in promoting efficient resource allocation and fostering entrepreneurship, thereby contributing to economic stability and potentially reducing state fragility. At the same time, however, state fragility can obstruct the process of financial development. Financial markets and, more broadly, financial activity rely critically on well-functioning institutions 2
that protect property rights and provide a minimum of certainty; on an environment of trust among economic agents; and on the ability of policy-makers to provide stable macroeconomic conditions. Fragile states, where these elements are in short supply, can be expected to have less developed financial institutions and markets—although the evidence on this is mixed (Barajas et al., 2021).

Numerous studies have empirically investigated the links between financial development and economic growth, often uncovering positive and significant correlations (Levine, 1997; Beck et al., 2000; Valickova et al., 2015), with causality running from finance to growth. Financial development has also been shown to reduce income inequality and is strongly associated with poverty alleviation (Beck et al., 2007). However, the relationship between financial development and state fragility (or its opposite, resilience), which may be heterogeneous across groups of countries and measures of financial development, has not been extensively addressed in the existing literature (Barajas et al., 2021). This leaves a gap that this paper aims to fill.

There are at least two major limitations of the current study that we want to mention upfront. First, measuring state fragility is not undisputed; see for instance Grimm et al. (2014). Similarly, proxying financial development is a difficult exercise given its multifaceted nature; see, for example, Svirydzenka (2016). In this paper we use World Bank and IMF measures for state fragility and financial development, respectively. For state fragility we use the Country Policy Institutional Assessment (CPIA) score, a widely used metric to classify fragile countries. This picks up aspects related to the level of institutional capacity and policy environment of a country. For financial development, we use the financial development (FD) index, introduced by Sahay et al. (2015), which is a relative ranking of countries on depth, access, and efficiency of financial institutions and financial markets.

Second, there is an obvious endogeneity issue with the current analysis. The relationship between financial development and state fragility may work both ways. The current data set does not allow us to solve this endogeneity 3
issue. Though we account for observed and unobserved static and dynamic heterogeneity, we cannot exclude further heterogeneity that might impact the results. Further corrections do not really seem possible given the patterns in the empirical data. Particularly the dynamics of the fragility proxies are quite stale, resulting only in a limited value added of the time-series dimension of the analysis and impeding hopes to seriously address the potential endogeneity biases. To accommodate this concern, we report a number of different robustness checks of the baseline analysis.

We explore the potential link of financial development with the degree of state fragility in a sample of 108 countries from 1998 to 2019 using a variety of methodologies. Our main research question is whether we can distinguish different groups in the data that show different dynamics of CPIA and CPIA-FD correlation over time. In particular, we ask whether countries remain rather stable within their CPIA-FD group for a longer period of time, or whether they can transition for instance from high-fragility to lowfragility groups (see also Safran and Sugiyarto, 2014). For this, we first group countries based on their aggregate CPIA and FD scores using a clustering approach to account for possible heterogeneity. Based on this preliminary analysis, we find two or three stable clusters in the data. Next, we use the approach of Bonhomme and Manresa (2015) to investigate the correlation between CPIA and FD, while controlling for cluster-specific time-varying heterogeneity as well as for a group of observed determinants of fragility, such as growth, inflation, military spending, and natural resource rents. Finally, we extend the methodology of Bonhomme and Manresa (2015) to account for switches of groups over time. We do so in a more generic setting than for instance Lumsdaine et al. (2023), who generalize Bonhomme and Manresa (2015) to a setting with structural breaks. In our approach, we do not enforce ex-ante that changes over time should be of a before-after structural break, but allow for more flexible changes, where patches of observations at different times may belong to the same regime, alternated by a different regime at other times. This is important in our current setting, where countries may 4
transit from high fragility to low fragility and back again several times over a longer period.

We find a negative association between state fragility as a left-hand side variable and financial development. This association between the two variables remains after controlling for other variables that can determine state fragility, such as growth, military spending and inflation. It also remains if we control for unobserved heterogeneity with or without clustering, and with or without allowing for cluster switches. Finally, the effect remains if we swap the CPIA scores by the Fund for Peace’s Fragile State Index (FSI) scores as a proxy for state fragility.

The remainder of this paper is set up as follows. In Section 2 we present the methodology used to divide the different countries into groups, both in a static and dynamic way. Section 3 describes the data and variables used. Section 4 presents the results, and Section 5 concludes. An appendix gathers additional empirical results and robustness checks not shown in the main paper.

# **2 Clustering methodologies**

## **2.1 The model**

Let _yi,t ∈_ R<sup>_G×_1</sup> for _i_ = 1 _, . . . , I_ and _t_ = 1 _, . . . , T_ be a vector of multivariate panel data observations. The panel can be unbalanced. In our context, the vector _yi,t_ may for instance contain different financial development proxies such as the FD index, state fragility proxies such as the CPIA score, and possibly other economic indicators, not all of which may be available at all times or for all countries. We assume that all elements of _yi,t_ are either observed or missing for a specific unit _i_ at time _t_ . We also define a matrix of observed covariates _Xi,t ∈_ R<sup>_G×M_</sup> .

At each time _t_ , unit _i_ belongs to one of _K_ groups or clusters. The corresponding group is indicated as _ki,t ∈K_ = _{_ 1 _, . . . , K}_ . For each group, we 5
specify the model for _yi,t_ as for some density _qε_ ( _·_ ; Σ _k, ψk_ ) with mean zero, covariance matrix Σ _k_ , and shape parameter(s) in the vector _ψk_ . The parameters **_α_** _k_ = ( _αk,_ 1 _, . . . , αk,T_ )<sup>_⊤_</sup> , _βk_ , Σ _k_ , and _ψk_ , for _k ∈K_ , are gathered into a parameter vector **_θ_** that needs to be estimated from the data together with the cluster assignments _ki,t_ .

The set-up in Eq. (1) is quite general and nests a number of specifications from the literature. This includes specifications for _K_ -means clustering with time-varying cluster means (by setting _ki,t ≡ ki_ and _βki,t ≡_ 0) or timeinvariant cluster means (by in addition setting _αki,t,t ≡ αki_ ). Additionally, the specification covers clustering methods based on group-specific time-varying heterogeneity (by setting _ki,t ≡ ki_ and taking _yi,t_ as scalar, so _G_ = 1) as in Bonhomme and Manresa (2015). In addition, however, Eq. (1) also allows for group-specific time-varying heterogeneity specifications where units are allowed to switch cluster over time.

# **2.2 Static clustering membership**

Group membership is defined based on minimizing a membership loss function. For time-invariant or static group membership ( _ki,t ≡ ki_ ) as in Bonhomme and Manresa (2015), we define where _L_ ( _·_ ; **_θ_** _ki,t, Si,t_ ) is a loss-function, and **_θ_** _k,t_ contains the parameters _αk,t_ , _βk_ , _ψk_ , and Σ _k_ for group _k_ at time _t_ . For instance, _L_ ( _·_ ; **_θ_** _ki,t, Si,t_ ) can be a squared or absolute loss based on the prediction errors ( _yi,t − αk − Xi,tβk_ ). Alternatively, the loss might be based on the negative log-likelihood, for instance for a Gaussian likelihood _−_ log _qε_ ( _εi,t_ ; Σ _k, ψk_ ). Asymmetry in the loss-function is easily allowed for.

As in Bonhomme and Manresa (2015), estimation of **_θ_** and the group membership vector **_k_** = ( _k_ 1 _, . . . , kI_ ) works by alternating the following two 6
steps. From an initial (random) starting value for **_k_**<sup>ˆ</sup> ( _r_ ) for _r_ = 0, the parameters **_θ_** are estimated as<sup>1</sup> These optimizations can be carried out separately for each of the tuples ( _r_ ) ( _αk,_ 1 _, . . . , αk,T , βk, ψk,_ Σ _k_ ) . Next, the group membership vector **_k_**<sup>ˆ</sup> is updated to **_k_**<sup>ˆ</sup> ( _r_ +1) via Eq. (2) using the value **_θ_** ˆ ( _r_ )(ˆ **_k_** ( _r_ )). This procedure is repeated until convergence, which typically only uses a few iterations. For robustness, a large number of starting values **_k_**<sup>ˆ</sup> (0) is used, and the estimates ˆ ˆ **_k_** and **_θ_** are retained that give the best overall criterion value in Eq. (3).

# **2.3 Cluster switching**

If units are allowed to switch group membership over the period of the sample, different strategies are available. As in Lumsdaine et al. (2023), we can extend the approach of Bonhomme and Manresa (2015) to allow for a structural break. In our current context, however, that may not be desirable. Countries that have transited from a group of high fragility to one of low fragility might find themselves switching back to the former group if the country enters a stressed period due to war, political turmoil, economic distress, or other reasons. We therefore extend both Bonhomme and Manresa (2015) and Lumsdaine et al. (2023) in the following way, while retaining ease of estimation.

Time-varying group membership is determined as follows. First, we group the different time points _t ∈T_ = _{_ 1 _, . . . , T }_ into _H_ disjoint groups _Th_ for _h_ = 1 _, . . . , H_ , such that _∪_<sup>_H_</sup> _h_ =1<sup>_Th_=</sup><sup>_T_, and</sup><sup>_Th∩Th′_=</sup><sup>_∅_for all 1</sup><sup>_≤h < h′≤H_.</sup> We constrain _ki,t_ = _ki,t′_ = _k_<sup> ̃</sup> _i,h_ for all _t, t_<sup>_′_</sup> _∈Th_ , while no constraint is imposed if _t ∈Th_ and _t_<sup>_′_</sup> _∈Th′_ for _h_ = _h_<sup>_′_</sup> . This setting allows for much more flexibility of switches in group membership beyond the structural break setting, while

> 1 ˆ (0) ˆ(0) One can also start with an initial guess of **_θ_** rather than **_k_** .

7
still allowing for a structural break to emerge if for instance _H_ = 2 and _t < t_<sup>_′_</sup> for all _t ∈T_ 1 and _t_<sup>_′_</sup> _∈T_ 2.

To estimate the parameters **_θ_** , the time-partitions _{Th}_<sup>_H_</sup> _h_ =1<sup>, and the time-</sup> varying group memberships **_k_**<sup> ̃</sup> = ( _k_<sup> ̃</sup> 1 _,_ 1 _, . . . , k_<sup> ̃</sup> _I,H_ ) with _ki,t_ = _k_<sup> ̃</sup> _i,h_ for _t ∈Th_ ; compare Bonhomme and Manresa (2015).

1. Set _r_ = 0 and start from some some initial random time-partition _{T_<sup>ˆ</sup> _h_ ( _r_ ) _}Hh_ =1<sup>andtime-varyinggroupassignments</sup><sup>**_k_**ˆ ̃(</sup><sup>_r_),with</sup><sup>_k_ˆ</sup> _i,t_<sup>(</sup><sup>_r_)=</sup><sup>_k_ˆ ̃</sup> _i,h_<sup>(</sup><sup>_r_)</sup> ( _r_ )

for _t ∈ T_<sup>ˆ</sup> _h_ .

3. Given the partition _{T_<sup>ˆ</sup> _h_ ( _r_ ) _}Hh_ =1<sup>andtheparametervector</sup><sup>**_θ_**ˆ</sup> ( _r_ ), update the time-varying group assignments ( _r_ ) for every _i_ and _h_ and set _k_<sup>ˆ</sup> _i,t_<sup>(</sup><sup>_r_+1)</sup> = _k_<sup>ˆ ̃</sup> _i,h_<sup>(</sup><sup>_r_)for</sup><sup>_t ∈T_ˆ</sup><sup>_h_</sup> .

4. Re-estimate the parameter **_θ_**<sup>ˆˆ(</sup><sup>_r_)</sup> as in step 2 using the new group assignments.

5. Update the partition as i.e., allocate each time point to that subset in the partition that minimizes its total (across units _i_ ) loss function, given the cluster memberships _k_<sup> ̃</sup> _i,h_ as estimated in the previous step.

6. Increment _r_ by 1 and repeat steps 2–5 until convergence.

7. Use different random initial starting values and repeat steps 1–6, retaining the estimates that yield the best criterion function value.

8
## **2.4 Identification and the number of clusters**

So far, the analysis has been based on an assumed number of clusters _K_ and number of subsets _H_ in the partition of time points _T_ . To determine these numbers in a data-driven way, different strategies can be followed. First, if the loss function is taken as the negative log-likelihood value, one can use standard model selection criteria, where the number of parameters is given by the length of **_θ_** , which increases in _K_ .

Note that the model specification with dynamic group membership as introduced in Section 2.3 is prone to a standard clustering identification challenge. To illustrate this, consider a setting with _H_ = 2 partitions, _K_ = 2 groups, and _I_ = 2 units, while abstracting from the presence of control variables ( _βk ≡_ 0). Let unit 1 and 2 be permanently in cluster 1 and 2, respectively, i.e., _α_ 1 _,t_ = _α_ ̄1, _α_ 2 _,t_ = _α_ ̄2, and _ki,t_ = _i_ for all _t_ and _i_ = 1 _,_ 2. Then this setting has the same loss value as one where (i) both units switch cluster halfway the sample (i.e., _k_ 1 _,t_ = 1 for _t_ = 1 _, . . . , ⌊T/_ 2 _⌋_ , and _k_ 1 _,t_ = 2 for _t_ = _⌊T/_ 2 _⌋_ + 1 _, . . . , T_ , and similarly for _k_ 2 _,t_ ); and (ii) the cluster means swith at the same time (i.e., _α_ 1 _,t_ = _α_ ̄1 and _α_ 2 _,t_ = _α_ ̄2 for _t_ = 1 _, . . . , ⌊T/_ 2 _⌋_ , and _α_ 1 _,t_ = _α_ ̄2 and _α_ 2 _,t_ = _α_ ̄1 for _t_ = _⌊T/_ 2 _⌋_ + 1 _, . . . , T_ ). There are different solutions to this identification problem. One of these maximize the overlap of the units in cluster _k_ between times _t_ and _t_ +1 to identify the cluster identities (see, for instance, Custodio Jo ̃ao et al., 2022). Given the limited number of clusters we find in our empirical applcation and the clear-cut stability over time, this identification issue does not seem to be a particular challenge in the current application.

# **3 The data**

As our state fragility variable in the baseline analysis, we use the CPIA score, which is a widely accepted proxy for state fragility. The CPIA is a composite index of 16 policies grouped into four different clusters. These clusters relate to economic management; structural policies; policies for social inclusion; 9
and public sector management and institutions. Individual policy scores are collected from the World Bank through surveys of its staff. For each policy, a score between 1 and 6 is provided, with higher scores indicating higher degrees of policy and institutional effectiveness, making the CPIA actually more a matter of resilience than of fragility. These scores are averaged at both the cluster and country level to provide an overall CPIA score falling between 1 and 6, with an unweighted average score of 3.2 or less generally seen as the threshold of state fragility. However, in our analysis, we treat state fragility as a continuum rather than a binary concept. We, therefore, refer to state fragility more broadly as countries with lower CPIA score without using the 3.2 cutoff point.

CPIA scores are available since 1977 for World Bank client countries though only those for low-income IDA-eligible countries are published. However, in our sample we focus on the period 1998-2019 to ensure comparability of CPIA scores to account for several changes occurred to the methodology over the years (see Gonzalez and Nishiuchi, 2018, for an overview). For the sample, we concentrate on countries without too many missing values for the CPIA. This means we leave out from the sample a number of World Bank client countries that have high-income status. As these countries would typically have high CPIA values, there would be little to explain from the covariates.

As a robustness check later in the paper, we also perform part of the analyses with another measure of fragility, namely the Fund for Peace’s Fragile States Index (FSI). This index consists of five groups of indicators measuring state cohesion, economic conditions, political legitimacy and public service provision, social conditions, and external intervention. The FSI is only available from 2006 onwards. We therefore do not use it in our baseline analysis, which spans a longer timeframe. The FSI also covers more countries than the CPIA in the sense that the latter has missing values for a fair number of developed OECD countries. In our robustness analysis, we therefore discard these countries as well in order to have similar samples for the CPIA and 10
FSI analyses.

As a proxy for financial development, we use the FD index, a summary measure of how developed financial institutions (banks, insurance companies, mutual funds, and pension funds) and financial markets (stock and bond markets) are in terms of their depth (size and liquidity of markets), access (ability of individuals and companies to access financial services), and efficiency (ability of institutions to provide financial services at low cost and with sustainable revenues, and the level of activity of capital markets). The FD index was developed by Sahay et al. (2015) (see also Svirydzenka, 2016) to overcome the shortcomings of single indicators (such as ratio of private credit to GDP) as proxies of financial development, which is essentially a multidimensional process. To that end, a number of sub-indices are created for financial institutions and markets across depth, access, and efficiency, culminating in the final index of financial development.

We consider four covariates for the analysis where we cluster countries based on unobserved time-varying heterogeneity while controlling for linear effects of other observables. In particular, drawing from the empirical literature on the determinants of state fragility (see, for example, Akanbi et al., 2021; Simon Feeny and Regan-Beasley, 2015; Blomberg et al., 2011; Carment et al., 2008, 2011), we consider GDP growth ( `gdpgrowth` ), inflation ( `inflation w` , winsorized at 0.5%), military expenditure as a share of GDP ( `militarygdp` ), and total natural resources rents as a share of GDP ( `natresrentgdp` ). Other controls were also investigated in earlier preliminary analyses (e.g. gross capital formation as a share of GDP, life expectancy, changes in effective executive of the country), but none of these led to stable results and their coefficients were typically estimated very close to zero.

There are quite some missing values in the data. Depending on what selection mechanism is used, we therefore end up with a different number of countries and time-series observations in the sample. Table 1 summarizes the main differences. Particularly the requirement that the additional controls exist for all time points can cause substantial sample attrition. Therefore, 11
Table 1: Sample selection and observation numbers This table summarizes the different numbers of observations used in the different analyses based on the sample selection criteria. In each case, for a particular combination ( _i, t_ ) the CPIA, FD (or all its constituents), and the controls need to be available. The sample selection relates to the number of countries _i_ and the number of observations <u>(</u> _i, t_ <u>).</u>

|required|linear|clustering|clustering|clustering|clustering|
|---|---|---|---|---|---|
|full|panel|(2-dim)|with|(7-dim)|with|
|1998–2019|model||controls||controls|
|CPIA||yes||yes||
|FD||yes||||
|FD consituents||||yes||
|controls||||||
|nr. countries|137|125|118|49|54|
|nr. observations|2896|2750|2236|1078|1068|

for our clustering analyses without controls, we only require the CPIA and financial development proxies to exist. Given that clustering is performed with time-varying cluster means, we require the variables to be present for all times. For the panel data analysis, we take all observations into account for which we observe the CPIA, financial development proxies, and the other controls.

Averages of CPIA and FD are presented in Figure 1. The time-series averages per country in Figure 1a show some expected differences, such as higher financial development scores in economies such as for instance the Republic of Korea, Brazil, China, India, and South Africa, and lower scores in most of Sub-Saharan Africa.

The CPIA scores are only partially correlated with the FD scores, at best. Some countries that have relatively low FD scores can at the same time, have relatively high CPIA scores, and the other way around. This result is in line with Barajas et al. (2021). Notably the largest part of Sub-Saharan Africa has low FD scores, but at the same time shows much more variety in the 12
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clear and is investigated in more detail in the empirical analysis in the next section.

# **4 Results**

Our results section is split into three parts. In Section 4.1 we provide a standard linear panel data analysis with fixed effects as a benchmark. In that setting, there is a significantly negative relation between financial development and state fragility. To account for omitted heterogeneity beyond the fixed effects, we provide two first clustering analyses in Section 4.2. The cluster assignments in this analysis are static. In Section 4.3 we relax this assumption and allow countries to (repeatedly) switch cluster membership over time. Finally, Section 4.4 presents a number of robustness checks, including disaggregating the FD index into all 6 (institutions and markets-related) constituents or into 3 (institutions-related only), and redoing the core analyses, and a robustness check where we replace the CPIA index by the FSI index as an alternative measure for state fragility.

## **4.1 Linear model with two-way fixed effects**

As a first benchmark, we estimate a two-way panel data model with countryfixed effects and time-fixed effects, This first model has no clustering effects, i.e., there are no separate groups. The dependent variable is country _i_ ’s CPIA score at time _t_ , and the regressors include financial development (FD) and our four control variables _Xi,t_ from Section 3.

A graphical representation of the coefficient estimates can be found in Figure 2a. We find that increases in FD and GDP growth are associated with lower fragility (higher CPIA scores), whereas higher inflation is associated 14
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might, however, be further heterogeneity. As in Bonhomme and Manresa (2015) we address this by allowing for group or cluster-specific time-varying effects in the model. As a first step in this analysis, we investigate the number of clusters directly using a bivariate location-scale model without regressors and with static cluster compositions, corresponding to equation (1) with _yi,t_ = ( _CPIAi,t, FDi,t_ )<sup>_′_</sup> and _βki,t_ = 0.

Using the algorithm described in Section 2.2, we estimate the model for different numbers of clusters, _K ∈{_ 2 _, ...,_ 6 _}_ . We note that is quite different from doing bivariate clustering on ( _CPIAi,t, FDi,t_ ) only, as we allow for time-variation in the means. In fact, the clustering takes place in a 2 _T_ -dimensional space ( _CPIAi,_ 1 _, . . . , CPIAi,T , FDi,_ 1 _, . . . , FDi,T_ )<sup>_′_</sup> . The numbers of observations for the different analyses was presented in Table 1. We account for scale differences between CPIA and FD by using standardized versions of the variables, _CPIA_<sup>�</sup> _i,t_ = _m_<sup>_CPIA_</sup> _t_ +( _CPIAi,t − m_<sup>_CPIA_</sup> _t_ ) _/s_<sup>_CPIA_</sup> _t_ and � _FDi,t_ = _m_<sup>_FD_</sup> _t_ + ( _FDi,t − m_<sup>_FD_</sup> _t_ ) _/s_<sup>_FD_</sup> _t_ , where _mt_ and _st_ are the cross-sectional means and standard deviations of the different variables.

To choose the optimal number of clusters, we compute a variety of clustervalidation indices. The outcome is presented in Figure 3, together with the mean squared error criterion which is used to find the parameters.<sup>2</sup> We find that most indices point towards two clusters. The mean squared error, which is used as the objective function for estimating the parameters, decreases monotonically in the number of clusters by construction and is therefore not useful to select the number of clusters. Given most criteria point to _K_ = 2 clusters, we use this number as our benchmark in the remainder of our analysis.

The estimated cluster assignments based on the above analysis are shown in Figure 4. Besides most countries in Sub-Saharan Africa and the Middle East and North Africa, Cluster 2 comprises several countries in Central America (El Salvador, Guatemala, Honduras, Nicaragua), South America

> 2For a more detailed overview of the different cluster indices, see, for instance, De Amorim and Hennig (2015).

16
-_—TM—— ——_

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the correction for control variables during clustering can make a substantial difference in the cluster compositions.

In Figure 6b we show the patterns of the estimated time-varying intercepts. These can be compared to the time-varying cluster means without regression controls as shown in Figure 5. Observed and unobserved heterogeneity may thus have substantial effects on the clustering results. As mentioned earlier, however, the association between CPIA and FD remains largely materially unaffected by the cluster the country belongs to; see Figure 6a.

# **4.3 Allowing for cluster membership switches**

In this section, we extend the previous analysis by allowing countries to switch cluster membership over time. In this way, countries are allowed to switch back and forth between for instance a high and low fragility group, depending on their historical development. We stress again that our analysis departs from that of Lumsdaine et al. (2023), who extend Bonhomme and Manresa (2015) to accommodate a structural break. In our setting, we divide up time into _H_ time buckets. The time points in these buckets may be consecutive, such as in the case of a structural break, or not. By allowing for this additional flexibility, a structural break is still possible, but will come out of the analysis endogenously through the estimation procedure laid out in Section 2.3.

We experiment with _K_ = 2 or _K_ = 3 clusters, and _H_ = 2 or _H_ = 3 time buckets, based on the preliminary analysis in Section 4.2. Given that we allow for different time buckets, it is also important to be somewhat more lenient on the number of clusters, as the possible cluster transitions of countries may land them in intermediate groups that do not have the properties of either of the former two groups in the clustering results with static cluster membership; see Section 4.2. Figure 7 presents an overview of the number of cluster switches in these four different settings.

There are two main features to notice in Figure 7. First, the time21
<!-- Start of picture text --> 120-<br>58 —- Cluster<br>a) 60- a gf | ;<br>3 ———— 2<br>Zz<br>30-<br>o-<br>1: 1998to 2002 2: 2003't0 2019<br><!-- End of picture text -->

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<!-- Start of picture text --> = ST Cluster<br>8 oo. U | a | ‘<br>33 <a —— 2<br>Zz<br>30-<br>o-<br>1: 1998to 2002 2: 2003'to 2009 3: 2010'to 2019<br><!-- End of picture text -->

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|<br>3 — —<br>; 60- as Cluster1<br>204<br>— — j=—s—<br>o-<br>1 1998 t02002 2:2003 'to2009, 3:2010'to 2019<br><!-- End of picture text -->
countries that switch from cluster 1 to 2, or reversely, as indicated by the smaller bands in the figure. This holds for both _H_ = 2 and _H_ = 3 time buckets. We also see that the number of countries can change over time, as the regression methodology can easily deal with unbalanced panels. For _K_ = 3 clusters, we see that the results become somewhat less stable. Particularly the countries in the middle CPIA group (cluster 2) have sizable transitions upward and downward. This might signal that this number of clusters is over-fitting the data. It is interesting to see that for _K_ = 3 transitions over two clusters are very rare: only for the setting of _H_ = 3 time buckets we see an incidental transition from the lowest CPIA cluster (number 3) to the highest one (number 1).

In Figure 8 we provide the results for the regression parameters for a variety of numbers of clusters _K_ = 2 _,_ 3 _,_ 4 and time partitions _H_ = 1 _,_ 2 _,_ 3 _,_ 4. The regression coefficients show quite a consistent picture. In all cases, we see that high financial development (FD) is negatively associated with fragility (low CPIA score). Also GDP growth is associated with higher CPIA scores for all settings considered. Conversely, inflation and natural resource rents have a negative association with CPIA scores. Only military spending seems not robust, the coefficient being sometimes positive, sometimes negative, and sometimes insignificant.

Some of the estimated group-specific time-effects are shown in Figure 9 and in line with those in Figure 9a for static cluster membership, though slightly smoother for all settings considered. Note that the group-specific effects in Figure 9a are based on different sets of countries at different times due to the possible cluster membership switches. This partly explains why the pattern of _αk,t_ is smoother in the latter case. Note that for _K_ = 2 the high and low CPIA groups show an increasing trend in fragility (or a negative trend in CPIA scores) in the latter part of the sample after controlling for financial development and the other regressors in the model, irrespective of whether we allow for cluster switches or not. This decline appears much more pronounced after controlling for all these effects than it does in the raw data 23
<!-- Start of picture text --> 3+<br>a) O ee ee<br>2-<br>a<br>1-<br>0 eS<br>1.25-<br>1.00-<br>0.75- 2<br>Py<br>0.50- g<br>0.25-<br>000— === —-—-=-—----------------------=-:<br>00m = — eee eee ee ee ee ee eH RH RH HHH HK KKH<br>- , t }<br>04+<br>25° ¢ +<br>me eg<br>-5.0- +<br>715-<br>00m = eee eee ee ee ee eee eRe KH Ke Ke KH Ke Ke KH KKH<br>05:<br>; ty *, ¢ :<br>AS- ¢<br>i 2 3 4<br>N. of Partitions<br>N.of Clusters 20304<br><!-- End of picture text -->
<!-- Start of picture text --> 3.75-<br>3.50-<br>=<br>8 3.25-<br>3.00-<br>2.75-<br>2000 2005 2010 2015 2021<br>Year<br>Cluster 1 2<br><!-- End of picture text --> <!-- Start of picture text --> 3.50-<br>a...<br>gees<br>=<br>3.00-<br>275-<br>2000 2005 2010 2015 2021<br>Year<br>Cluster 1 2<br><!-- End of picture text --> <!-- Start of picture text --> 3.75-<br>3.50-<br>3.50-<br>= a...<br>8 3.25- gees<br>22<br>==<br>3.00- 3.00-<br>2.75- 275-<br>2000 2005 2010 2015 2021 2000 2005 2010 2015 2021<br>Year Year<br>Cluster 1 2 Cluster 1 2<br>35- 3.5-<br>$ $& 30-<br>@ 30- ey<br>= z<br>25- ee ee ee eee 25-<br>2000 2005 2010 2015 2021 2000 2005 2010 2015 2021<br>Year Year<br>Cluster 1 2—3 Cluster 1 2—3<br><!-- End of picture text --> <!-- Start of picture text --> 35-<br>@ 30-<br>=<br>25- ee ee ee eee<br>2000 2005 2010 2015 2021<br>Year<br>Cluster 1 2—3<br><!-- End of picture text --> <!-- Start of picture text --> 3.5-<br>& 30-<br>ey<br>z<br>25-<br>2000 2005 2010 2015 2021<br>Year<br>Cluster 1 2—3<br><!-- End of picture text -->
figures are clearly separated throughout the sample, irrespective of the cluster reshuffling from one time-partition to the next. Again, this underlines our earlier finding of a positive association between state fragility on the one hand, and financial development and inflation on the other. The association of the averages is somewhat less clear for the other three variables, with periods of positive and negative association alternating over time.

In Figure 10, we show the cluster membership for _K_ = 2 and _H_ = 2. Results for the other settings are provided in the appendix. Given that countries may switch cluster, we provide two maps: one for time bucket 1 (top panel, 1998–2002), and one for bucket 2 (bottom panel, 2003–2019). There are some interesting differences with the static clustering results in Figure 6c. For instance, countries like Kazakhstan and Madagascar were allocated to cluster 1 under static cluster membership after controlling for all effects in the analysis. In Figure 10, however, they are allocated to this same group only for the first years in the sample. After that, they switch to cluster 2. The converse also occurs. For instance, countries like Mali and Kenya belong to cluster 1 in the static analysis. In the dynamic analysis, however, they switch from cluster 2 in the first few years of the sample to cluster 1 in the more recent years. The dynamic patterns above thus allow for a much richer characterization of countries in terms of the evolution of their relative degree of fragility and the status of their financial development. A full list of transitions for the case of _K_ = 2 clusters and _H_ = 2 time buckets is given in Table 2.

# **4.4 Robustness**

Thus far, we have concentrated the analysis on the aggregate FD index. This index, however, is composed out of 2 _×_ 3 (or 3 _×_ 2) sub-indices. In particular, there are sub-indices on financial markets (FM) and financial institutions (FI), each sub-divided into Access, Depth, and Efficiency as FMA, FMD, FME, and FIA, FID, FIE. In this robustness section we investigate three additional analyses. First, in Section 4.4.1 we use 7 variables: all FD 26
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<!-- Start of picture text --> "y ane<br>Cluster: [7]1 Mil2 LIN/A iy ;<br><!-- End of picture text -->
Table 2: Country transitions for _K_ = 2 clusters and _H_ = 2 time partitions

|From||To cl|uster|||
|---|---|---|---|---|---|
|cluster|1|||2||
|1|||Bolivia,|Dominican|Repub-|
||||lic, Hun|gary, Kaz|akhstan,|
||||Sri<br>La|nka,<br>Mad|agascar,|
||||Malawi,|T ̈urkiye||
|2|Benin, Georgia,|Kenya, Ko-||||
||rea, Rep., Mali|||||

components and CPIA. Given that there are many missing values for the market-related FD components (FM), we perform a second robustness check in Section 4.4.2 using CPIA and the institutions related (FI) components of FD, only. Finally, in Section 4.4.3 we replace the CPIA score by the FSI index as an alternative proxy for state fragility and redo the core analysis.

# **4.4.1 Clustering CPIA and all FD components (7 variables)**

In this section, we re-trace our steps using these 6 FD constituents in the analysis, rather than FD itself. For instance, in the clustering analysis, this means that we perform clustering with all 6 FD constituents together with CPIA (7 variables in total) over all time periods, i.e., clustering 7 _T_ -dimensional observation vectors. The FD constituents are much less complete in the database, substantially limiting the sample size; see also the observation numbers in Table 1.

Figure 11 shows the cluster validation criteria for the raw clustering analysis. As for the case with aggregate FD, the disaggregated analysis points to at most two clusters. One might even question whether two clusters is appropriate given the much reduced number of observations. For comparability, however, we proceed with allocating countries into two clusters. The result is provided in Figure 12.

28
<!-- Start of picture text --> Calinski.Harabasz Davies.Bouldin Hartigan<br>10-<br>1.70-<br>16- 1.65- 8-<br>1.60 -<br>14 - 1.55- 6 -<br>1.50- 4-<br>12-<br>1.45-<br>Krzanowski.Lai MSE Silhouette<br>0.23 -<br>2500 -<br>2.0-<br>0.21 -<br>2000 -<br>1.6- 1500- 0.18 4<br>1.2- 1000 - 0.17-<br>1 ' ' 1 ' 500 7 ' ' ' ' ' 0. 1 5 7 ' ' 1 ' '<br>234562345623456<br>K<br><!-- End of picture text -->
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components.

Directions for future research include an extension of the multivariate model with grouped heterogeneity, with or without cluster switching, towards the inclusion of control variables and/or lags of the dependent variables. Furthermore, as mentioned above, our current methodology detects associations in the data, but does not permit a causal interpretation of the coefficients due to the endogeneity among country fragility, financial development, and, potentially, other country characteristics. An investigation of the causal relationships between these variables could, for instance, be conducted using a structural model. Even then, however, we would still face the challenge of either finding suitable instrumental variables, or developing an alternative identification strategy. From a methodological point of view, combining structural simultaneous equations modeling for panel data with unobserved, grouped heterogeneity could be a promising avenue for future research.

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<!-- Start of picture text --> 3: 2010-2019<br>a - ae = 5<br>we = ar : sy<br>A.<br>o<br>Cluster: [1 [2 M3 INA<br>marae|<br><!-- End of picture text -->
<!-- Start of picture text --> a eeee |<br>5oree ee eee ||<br>5aee esa a es<br>5aoo a ee ;<br>5aa a a ||<br>ee Poa<br>ia<br>a<br>“tT — | ||<br>eee<br>“ti | |<br>me |_| __| _{__j|_|__j|_|{_|__|<br>Hl Po<br>an [| — {| ||<br>ae |<br>- ne<br>2000 2005 2010 2015 2020<br>Year<br>Cluster = 1 -© 2<br><!-- End of picture text -->
<!-- Start of picture text --> 0.275-<br>0.250-<br>0.225 - ar}<br>0.200-<br>0.175-<br>0.06-<br>fo}<br>0.04- ©33io}<br>=<br>0.02-<br>0.00-<br>03-<br>0.2-<br>a°g<br>O1-<br>0.023-<br>0.021 -<br>=<br>0.019- &Fi<br>0.017-<br>0.10-<br>0.08 - 2€a3Ffz<br>0.06-<br>2000 2005 2010 2015 2020<br>Year<br>Cluster 1 2<br><!-- End of picture text -->
<!-- Start of picture text --> 1 2<br>1 Dominican Republic, Kaza-<br>khstan, Sri Lanka, Malawi,<br>T ̈urkiye<br>2 Azerbaijan, Algeria, Georgia, Ko-<br>rea, Rep., Madagascar, Mali, Ro-<br>mania, South Africa<br><!-- End of picture text --> Table A.1: Transitions from partition 1 (rows) to 2 (columns), 2 clusters, 3 partition total, 4 controls

||1|2|
|---|---|---|
|1||Bolivia, Honduras, Madagascar,|
|||Trinidad & Tobago, Yemen, Rep.,|
|||South Africa|
|2|Kazakhstan, Kenya||

Table A.2: Transitions from partition 2 (rows) to 3 (columns), 2 clusters, 3 partition total, 4 controls 49
||1|2|3|
|---|---|---|---|
|1||Hungary, Mauri-<br>tania, Tunisia||
|2|Armenia,<br>Azer-||Bangladesh,|
||baijan, Burkina||Belarus,<br>Brazil,|
||Faso,<br>Bulgaria,||Congo<br>-<br>Kin-|
||Georgia, Ghana,||shasa, Ecuador,|
||Kyrgyzstan,||Iran,<br>Islamic|
||Mexico,<br>North||Rep., Sri Lanka,|
||Macedonia,||Mongolia,|
||Nicaragua,||Chad,<br>T ̈urkiye,|
||Peru, Romania,||Ukraine|
||Rwanda|||
|3||Burundi,<br>In-||
|||donesia,<br>Korea,||
|||Rep., Paraguay,||
|||Tajikistan||

Table A.3: Transitions from partition 1 (rows) to 2 (columns), 3 clusters, 2 partition total, 4 controls 50
||1|2|3||
|---|---|---|---|---|
|1||Albania,<br>Bo-|||
|||livia, Colombia,|||
|||Dominican<br>Re-|||
|||public, Ethiopia,|||
|||Kyrgyzstan,<br>Malawi|||
|2|Azerbaijan,||Banglade|sh,|
||Burkina<br>Faso,||Belarus,|Egypt,|
||Bulgaria,<br>Mex-||Arab|rep.,|
||ico,<br>North||Guinea-B|issau,|
||Macedonia||Cambodi|a,|
||||Lao|PDR,|
||||Chad, T|hailand,|
||||T ̈urkiye||
|3|Romania|Ecuador,|||
|||Paraguay,|||
|||Ukraine|||

Table A.4: Transitions from partition 1 (rows) to 2 (columns), 3 clusters, 3 partition total, 4 controls 51
||1|2|3|
|---|---|---|---|
|1||Tunisia||
|2|Georgia, Kenya,||Brazil, Ecuador,|
||Kyrgyzstan,||Sri Lanka, Mon-|
||Nicaragua,||golia, Ukraine|
||Rwanda, Serbia|||
|3||Belarus, Indone-||
|||sia,<br>Cambodia,||
|||Korea,<br>Rep.,||
|||Lao<br>PDR,||
|||Tajikistan||

Table A.5: Transitions from partition 2 (rows) to 3 (columns), 3 clusters, 3 partition total, 4 controls 52