# Detecting global financial crises with scarce data by
multivariate nonlinear filtering

Cécile Bastidon, Antoine Parent, Patrice Abry, Pierre Borgnat, Pablo Jensen,
Barbara Pascal

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Cécile Bastidon, Antoine Parent, Patrice Abry, Pierre Borgnat, Pablo Jensen, et al. Detecting global financial crises with scarce data by multivariate nonlinear filtering. Journal of Physics: Complexity, 2025, 6 (3), pp.035003. 10.1088/2632-072X/ade948. hal-05234050

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# Detecting global financial crises with scarce data by multivariate nonlinear filtering

To cite this article: Cécile Bastidon et al 2025 J. Phys. Complex. 6 035003

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PAPER

# Detecting global financial crises with scarce data by multivariate nonlinear filtering

Cécile Bastidon¹ ID, Antoine Parent² ID, Patrice Abry² ID, Pierre Borgnat³,* ID, Pablo Jensen³ ID and Barbara Pascal⁴ ID

¹ LEAD, Toulon University, Campus Porte d'Italie, 70, Avenue Roger Devoucoux, 83000 Toulon, France

² LED, Univ. Paris 8 & OFCE, 93526 Saint-Denis, France

³ CNRS, ENS de Lyon, LPENSL, UMR5672, 69342 Lyon Cedex 07, France

⁴ Nantes Université, École Centrale Nantes, CNRS, LS2N, UMR 6004, 44000 Nantes, France

* Author to whom any correspondence should be addressed.

E-mail: pierre.borgnat@ens-lyon.fr, bastidon@univ-tln.fr, antoine.parent@sciencespo.fr, patrice.abry@ens-lyon.fr, pablo.jensen@ens-lyon.fr and barbara.pascal@cnrs.fr

**Keywords:** multivariate time series analysis, nonlinear filtering, time reassignment, detection of financial crises, financial globalization, econophysics, cliometrics and complexity

## Abstract

An original procedure is devised for the automated detection of global financial crises from multivariate databases of share prices. It consists of: i) the construction of time series from the time-windowed estimations of crisis relevant information (cross-correlations or volatilities); ii) the piecewise-linear filtering of times series by nonlinear filtering, achieved by nonsmooth proximal minimization implemented by an efficient iterative algorithm; iii) the estimation of a reassigned time in each window; iv) the detection of crises and estimation of their intensities by exploiting the multivariate structure of denoised time series. Applied to a world dataset of 32 indices over 6 decades, this original model based procedure detects all major crises from the reference lists. It also permits to devise a typology in reference to an archetypal financial crisis. It is automated, data-driven and reproducible notably for the analysis of financial crises over history, or contemporary crises on worldwide databases, via a novel toolbox. Finally it is robust to scarce, incomplete and noisy data.

## 1. Introduction

A connected world requires indicators of major global financial disruptions. Dating global financial crises, defined as 'synchronized crises across many countries', by contrast to 'national isolated crises' (Jordà et al 2011) is a crucial issue of time series analysis, in economic history as well as in contemporary financial macroeconomics. In the present work, we propose an original indicator for the detection of global financial crises from scarce data. It is devised from a nonlinear filtering algorithm, based on nonsmooth optimization, and only needs share prices. There is a large body of literature in financial macroeconomics and cliometrics geared towards financial crisis detection, including expert and model based approaches, as summarized in table 1. The original indicator we propose belongs to the category of model-based approaches.

Following the reference narrative of Goodhart and Delargy (1998), which highlights patterns of recurrence between the crises of the 'second globalization' (1970s and after) and those of the 'first globalization' (pre-1914), the expert-based dealings of Eichengreen and Bordo (2002), Reinhart and Rogoff (2009) and Laeven and Valencia (2012) remain the benchmark. Ongoing research seeks to provide updates performed by textual analysis and machine learning (e.g. Romer and Romer 2017, Chen et al 2020). The existing model-based procedures (e.g. Schularick and Taylor 2009, Danielsson et al 2018) make various assumptions on data and models. In particular, early warning indicators à la Kaminsky et al (1998), Kaminsky and Reinhart (1999) are based on detection thresholds selected by optimizing true/false positives from receiver operating characteristic curves, involving a benchmark crises database. Regime switching models (e.g. Baele et al 2010, Duprey et al 2017) or multinomial regressions (e.g. Berg and Pattillo 1999,
<table><caption>**Table 1.** Selective survey of crisis detection and dating methods organized chronologically and per categories, from the most expert-based to the most model-based. Berg and Pattillo (1999), Bussiere and Fratzscher (2006), Schularick and Taylor (2009) and Danielsson *et al* (2018) are located between the third and fourth columns because they involve both ‘early warning indicators’ and ‘regime models’ methods. In addition to differences in methods, expert-based and model-based strategies have different purposes. Expert-based works mainly aim at guiding the understanding of contemporary crises through the knowledge of past crises. Model-based approaches are designed to date crises by combining time series of relevant indicators of financial instability with models identifying breaks. The works from the late 1990s to about the mid-2010s address a broad set of macroeconomic indicators and are focused on emerging crises. More recent work addresses indicators of financial stress and focus on guiding macroprudential policies.</caption><thead><tr><th>Economic history<br>« expert based »</th><th>Textual data</th><th>Early Warning<br>Indicators<br>« model based »</th><th>Regime models &<br>multinomial regressions<br>« model based »</th><th>Multivariate<br>« model based »</th></tr></thead><tbody><tr><td>2000<br>Goodhart & Delargy 1998<br>Narrative economic history, 1st<br>era of globalization<br><br>Eichengreen & Bordo 2002<br>Crisis criteria (banking,<br>currency, twin)</td><td>Kaminsky, Lizondo & Reinhart 1997<br>"Signals approach": ROC on macro<br>indicators (currency crises,<br>emerging economies)<br><br>Kaminsky Reinhart 1999<br>"Signals approach": ROC on macro<br>indicators (banking and currency crises,<br>emerging economies)</td><td>Berg & Pattillo 1999<br>Comparison of EWT and prohibit<br>regressions (currency crises)</td><td>Forbes & Rigobon 2002<br>Bias in correlations<br>estimations of<br>heteroskedastic time series<br>of stock markets prices</td><td></td></tr><tr><td>2010<br>Reinhart & Rogoff 2009<br>Crisis criteria (defaults,<br>hyperinflation, banking,<br>currency, etc.)<br><br>Laeven & Valencia 2012<br>Crisis criteria (banking,<br>currency, systemic sovereign)</td><td>Bussière & Fratzscher 2006<br>EWT<br>+ Multinomial prohibit regressions<br>(emerging crisis)</td><td></td><td>Bacel 2005<br>MSM on stock markets<br>volatility</td><td></td></tr><tr><td>2020<br>Romer & Romer 2017<br>Expert based indicator of<br>financial crises built from<br>OECD textual data<br>+ Impulse response of output<br><br>Chen & al 2020<br>Machine learning indicator of<br>financial crises built from various<br>textual data</td><td>Drehmann Juselius 2014<br>Ranking of macro indicators<br>based on AUROC,<br>macroprudential purpose<br><br>Brave & Butter 2018<br>ROC on aggregated financial<br>stress indicators<br><br>Lee & al 2020<br>AUROC on financial stress<br>indicators à la Aikman et al 2017</td><td>Schularick & Taylor 2009<br>AUROC<br>+ Logit regressions (long<br>term relationship between<br>money and credit, advanced<br>economies)</td><td>Duprey et al 2017<br>MSM on financial<br>stress indicators</td><td>Aikman et al 2017<br>Heatmaps on time series of<br>aggregated financial indicators<br>and components<br><br>Bastidon et al 2020<br>Signal processing of network<br>indicators on stock markets prices</td></tr></tbody></table>

Bussiere and Fratzscher 2006) focus on switches between stable regimes. Most of time, this strand of literature does not rely on the analysis of multivariate signals. Aikman et al (2017) and Bastidon et al (2020) are examples of an emerging model-based literature focused on the use of long time series of multivariate financial information.

With regard to the data used to detect crises, a first observation is that available data is scarce in the fields of economic history, development economics, and real-time crisis analysis. In particular, the availability of non-financial data with relevant frequencies for large samples of countries is often an issue. In this context, a specific branch of literature in economic history uses banking sector share prices to study banking crises over history (Jalil 2015, Campbell et al 2016, Baron et al 2021, Kenny et al 2021, not shown in table 1 as they specifically relate to banking crises and not global financial crises). We also observe that one of the difficulties of model-based detections, when implemented over long periods, is that, depending on the nature of the indicators, their relevance can vary sharply over time (Drehmann and Juselius 2014). This leads some authors to limit their periods of study (Aikman et al 2017). We use equity market prices, which are appropriate for detecting crises caused at different times by deteriorated fundamentals of different nature, as they are permanently regarded as a benchmark of global financial markets.

With regard to data pre-processing, in the existing literature crisis detection exploits correlations, often combined with volatilities which does not permit to disentangle their respective contributions to crises. The use of correlation is justified, from the point of view of economic history, by an extensive reference literature (e.g. O'Rourke and Williamson 1999, Obstfeld and Taylor 2004, Bekaert et al 2011) which shows that the convergence of asset prices dynamics, i.e. increasing overall correlations, is a major indicator of financial globalization. The correlation-based approach has also been dominant in the study of the contagion of financial crises for the last two decades, in particular since Forbes and Rigobon (2002). In the most recent literature, correlations of asset returns are often taken into account as weights in the construction of aggregate indicators (e.g. Brave and Butters 2012, Danielsson et al 2018). We separately use correlations and volatilities, usually combined in aggregate indicators, to disentangle their contributions to crisis detection.

With regard to the model itself, the overarching contribution of the present work is to propose a pluridisciplinary strategy to renew existing methods for detecting and characterizing financial crises, combining tools borrowed from statistical signal processing with classical macroeconomic analyses. The proposed strategy, grounded in multivariate data analysis, is fully automated and thus does not rely on any expert knowledge. It does not assume regimes with stable characteristics over time, which is particularly relevant in a historical approach on long time series.
This yields an original indicator of global financial crises that permits us not only to identify benchmark
crises in expert-based databases, but also to provide a crisis typology. This indicator enables us to disentangle
contributions of correlations from those of volatilities to financial crises detection, in contrast to existing
literature. This possibility of performing a specific volatility-based detection allows for the measurement and
hierarchization of per-country contributions to crises.

To that end, the method is organized into an original four-step algorithm applied, in the present work, to a database of 32 countries' equity markets, from January 1960 to present, comprising 18 300 sample points for an average of 47 years of data per country. This period includes financial crises of diverse causes and geographical origins, identified in the reference expert-based literature. It has to be noted that the proposed method deals with scarce datasets, whereas statistical physics applications in Sociophysics and in particular Economophysics generally concern fields where huge datasets are available Jusup et al (2022). Recent works on stock markets prices, currencies, or cryptoassets currently deal with portfolios of hundreds or thousands of assets at daily or intraday frequency (see, e.g. the study of the efficiency of cryptoassets markets of Sigaki et al 2019 which deals with 400 cryptoassets out of over a thousand available on reference platform, in daily frequency). By contrast, our work applies to historical data sets, for which information is scarce, sometimes incomplete and highly noisy. We show that it is possible to mobilize robust statistical physics applications for the dating and categorization of global financial crises from such data.

In the first step, we design multivariate time series by performing a sliding window-based estimation of
informational content from the available data, which can be based independently on correlation structures
or per-country volatilities. In the second step, we denoise the obtained time series of information using a
multivariate piecewise nonlinear filtering procedure, framed as an inverse problem and solved by nonsmooth
convex optimization. It has to be noted that contrary to common practice in financial macroeconomics or
empirical finance, the nonlinear filtering is not used as a data pre-processing and it constitutes the core of the
proposed indicator. The general method elaborates on the nonlinear filtering literature initiated by Kim *et al* (2009), which itself generalizes the (linear) Hodrick–Prescott (HP) filtering, a widely used method for trend
estimation (Hodrick and Prescott 1997). More specifically, we extend to global financial crises detection, the
procedure developed in Pascal *et al* (2021) that seeks a piecewise linear approximation of the multivariate
data, having a small number of simultaneous changepoints. Contrary to HP filters or other linear filtering
methods, the nonlinear approach allows us to detect abrupt changes. The proposed crisis detection
procedure extends the existing algorithm to ensure robustness against missing samples. The third step
consists of an improved localization of the changepoints within each window by a time-reassignment
procedure. In the fourth step, we select the changepoints with an automated criterion actually identifying
relevant global crisis events.

The rest of the article is organized as follows. Section 2 describes the four steps of the procedure.
Section 3 presents and discusses achieved results, and proposes a new typology of global crises. For
reproducibility, the method is encapsulated in a novel ClioCrises Toolbox.

## 2. Crisis indicator methodology construction

The detection procedure developed for the present work consists of steps related to the pipeline used in the model-based literature. The first step consists of extracting, from the database, relevant information from the asset prices (2.1). We then compute multivariate signals from these series: on one hand, the moments of the distribution of pairwise correlations, and on the other hand the standard deviations of equity markets returns. The second step consists in filtering the multivariate signal to extract multivariate piecewise linear series (2.2). We employ for that a multivariate nonlinear filtering method. The next step is to correctly attribute the dates of the candidate change points by a time reassignment procedure (2.3). Finally, the last step consists in selecting these candidate changepoints in order to keep only the relevant dates of crises (2.4).

2.1. From asset prices to information structured in multivariate time series

2.1.1. *Asset price database*

The database is a proxy of world stock markets as it contains 32 advanced, emerging and developing
economies<fnref n="5" />. It consists of national stock market benchmark indices, for the period 1960–2020, in monthly
frequency. The data are extracted from the FRED database (Federal Reserve Economic Data). In 1960, 15
countries were available. This number gradually increases to 32 from 1999 onwards (see figure 1(a)).

<footnote n="5">⁵ Australia, Austria, Belgium, Brazil, Canada, Chile, China, Czech Rep., Denmark, Finland, France, Germany, Greece, India, Italy, Ireland, Israel, Japan, Korea, Luxembourg, Mexico, Netherlands, New Zealand, Norway, Poland, Russian Fed., South Africa, Spain, Sweden, Switzerland, United Kingdom, United States of America.</footnote>
<figure><img src="image_4.png" /><figcaption>**Figure 1.** Data. Time series reporting, from top to bottom, the number of countries available for analysis, log-returns per country, pairwise correlations, summary moments of the distribution of pairwise correlations, and volatilities per country. The dataset covers returns from 1960 to 2020. Pairwise correlations and volatilities are estimated in sliding windows. Corresponding estimates are plotted as time series attached to the central time of each window (hence represented on a reduced period of times). Each color corresponds to a different times series, hence 32 countries for returns and volatilities and 496 pairs of countries for crosscorrelations.</figcaption></figure>

The data are converted to a single currency, so as to be used to estimate series of cross-correlations and standard deviations that will be employed as a multivariate signal. All indices, expressed in national currency, are converted into U.S. dollars using bilateral rates. For the fixed exchange rate period, the FRED database is incomplete for exchange rates. We use the exchange rates and realignment dates from Bordo (1993). All index time series converted to US dollars are then converted back to constant dollars. Finally, returns are calculated by log-differentiation of prices (figure 1(b)).
### 2.1.2. *Crisis related information time series*

In the relevant literature, crisis detections are based on aggregated indicators, which hence do not enable us to disentangle the contributions of cross-correlations and volatilities to crises. To address this shortcoming, we perform two crisis detections, independently based on cross-correlations and on volatilities; then we compare them. To that end, correlations are first estimated within centered 100-sample long sliding windows. This yields two independent sets of time series of length 727 datapoints (one datapoint per month): the Volatility set comprises the $M = 32$ time series of volatility, one per country, estimated as the empirical standard deviations of the returns within the sliding windows; the CrossCorrelation set consists of four time series obtained as the first four moments (mean, standard deviation, skewness and kurtosis) of the per-window (hence local) distributions of the $(32*31)/2 = 496$ pairwise Spearman correlation coefficients, computed for each of the sliding windows (hence $M = 4$). For both sets, missing values exist for countries not yet entered in the database.

Figure 1 illustrates the time series obtained from the data, and used in the present study. It shows, from top to bottom, the number of actually available countries included in the analysis, the 32 return time series, the 496 pairwise cross-correlation time series, the four first moments of the distributions of cross-correlations, used as summary for the 496 estimated pairwise correlation time series and the 32 estimated volatility time series. The time series of correlations show strong nonstationary patterns over time. In the 1960s, they increase slightly (increasing mean); in the 1970s, with the increase of the panel of countries including developing and emerging economies, they seem to diverge (increasing standard deviation), but in reality this dispersion is due to a small number of pairwise correlations with atypical values (increasing kurtosis). For the 1980s, 1990s and 2000s, most correlations are stable, then increasing and converging (increasing mean, decreasing standard deviation). In the second half of the 2000s, they are clustered into two groups of high values (around 0.7–0.9) and lower values (around 0.3–0.5). Finally, in the later period, the correlations decrease (decreasing mean) and the clusters no longer appear.

Regarding volatilities, i.e. standard deviations of returns (figure 1(e)), the lowest values are stable over time. The largest values increase over the period of study with the exception of the last decade. Countries entering the dataset are always characterized by high atypical values that converge rapidly. Finally, the end of the study period shows the most homogeneous evolutions: all the series show an upward jump around years 2007–2009 and almost all of them then decrease in two stages, slowly and then rapidly.

## 2.2. Non linear multivariate filtering

### 2.2.1. *Piecewise linear smooth multivariate filtering: principles*

The nonlinear filtering constitutes the core of the proposed indicator, as it allows to detect abrupt changes. In time series statistical analysis, classical approaches were proposed to detect zones in financial data that have often relied on temporal segmentation of covariance functions (or matrices). Such methods are usually grounded on the demanding assumption of temporal piecewise stationarity in correlations. They also rely on the availability of all time series across the whole temporal period of interest. Both of these a priori assumptions are likely to be questioned for crisis detection in economic history: indeed, crises are likely to occur as the end of periods characterized by homogeneous evolutions (growth, trends, ...) rather than by stationary statistics; further, as is often the case in economic history, the data considered here are left-censored for a number of countries and data availability thus does start at different dates for different countries.

These issues preclude or significantly impair the use of standard parametric segmentation methods based on covariance estimation, e.g. those in Gibberd and Nelson (2016) or Hallac et al (2019)) that segment the time-dependent estimated correlation matrix, or the precision matrix (Hallac et al 2017); or using Bayesian frameworks (e.g. Xuan and Murphy 2007); or with specific optimization procedures for penalties on changes in covariance structure (Angelosante and Giannakis 2011), which can be used for financial data (Xuan and Murphy 2007).

To overcome these issues, two key ideas are proposed here. First, we propose to avoid exploiting the structure of the estimated covariance matrix $R(t)$, and instead make use of pairwise correlations for available pairs only. This approach shows the crucial ability of being applicable to multivariate dataset of any sizes and of time dependent sizes. Notably, it can be applied to the two sets of multivariate series described above, either the 4-variate one based on the first four moments of the distribution of available estimated correlations, or the 32-variate signal of univariate return volatility time series. Second, as piecewise stationarity is a feature which is not expected in the present nonstationary setting with punctual crisis events, it is proposed to avoid relying on it and to devise an approach grounded on the less restrictive assumption of piecewise linear behaviors of the time series to be denoised.

The rationale of the multivariate nonlinear filtering procedure is that denoised time series should consist of piecewise linear segments continuously connected at a small number of discontinuities (thus consisting of
non-zero second derivatives, or, equivalently, changes in the first-derivative), and whose sparse location estimations can be data driven and shared by all components. Time series filtering by linear filtering would end up in smoothing out change points, together with the noise, and hence in impairing the detection of such change points. On the contrary, nonlinear filtering smooths the data when fluctuations are due to noise only and avoids smoothing when actual changes are detected from time series. Further, it is worth mentioning that even if volatilities, unlike correlations, do not directly account for dependencies across components, the multivariate nature of the proposed nonlinear filtering relies on joint cross-component information, by favoring that estimated change points are colocated across components.

Nonlinear filtering has initially been proposed as trend extractor in Kim et al (2009), and it compares well to data-driven methods (Moghtaderi et al 2011, 2013). It was then further elaborated for multivariate series in Pascal et al (2020) to include improved large-scale optimization algorithm, and an automated selection of hyperparameter. The core of the method is to obtain the M denoised time series $Y(t) = \{Y_1(t), \dots, Y_M(t)\}, t = 1, \dots, T$ from the original multivariate times series $X(t) = \{X_1(t), \dots, X_M(t)\}, t = 1, \dots, T$ and of size M, conveying crisis related information, by minimizing a functional designed from combining a data fidelity term with a regularization (also called penalization or constraint) term:

$$Y(t) = \arg\min_{U} \|X - U\|^2 + \lambda \sum_{t} \sqrt{\sum_{k} (U_k(t+1) - 2U_k(t) + U_k(t-1))^2}, \quad (1)$$

where $\lambda \in \mathbb{R}_+$ weights the trade-off between the data fidelity term and the penalization term. The data fidelity term is classically quantified by an $L^2$-norm: $\|X - U\|^2$. The penalization term is designed with two objectives in mind: i) perform joint (or multivariate) filtering of the times series, with colocated change points across components, and ii) enforce piecewise linearity in each denoised time series<fnref n="6" />. Indeed, the second order derivative (also called Laplacian operator) $U_k(t+1) - 2U_k(t) + U_k(t-1)$ that is applied to each time series will penalize any change of slope. The use of a $L^{1,2}$-norm ($L^2$-norm on the output of the Laplacian operator applied to each component, followed by a $L^1$-norm across components), favors the colocation of changes in the second derivatives, and thus ensures that the nonlinear filtering is truly multivariate in nature. While the data fidelity term relies on a classical $L^2$-norm, the originality of this approach lies in the use of a mixed $L^{1,2}$-norm in the penalization term to produce both multivariate (across components) and nonlinear (piecewise smooth) filtering. In order to be used on a large scale dataset, a specific optimization procedure has been developed.

The data fidelity term and the penalization term are balanced by the hyperparameter $\lambda$. Intuitively, $\lambda$ is reminiscent of the bandwidth in linear low-pass filtering: the larger $\lambda$, the more dominant the penalization term, thus the stronger the smoothing, hence decreasing the actual number of discontinuities (changes in the first-derivative) of the denoised times series. Conversely, the smaller $\lambda$, the more dominant the data fidelity term, thus decreasing the impact of filtering. The hyperparameter $\lambda$ hence controls the sparsity level of the (Laplacian of the) denoised time series and hence indirectly the number of discontinuities. It is however crucial to note that neither the locations nor the intensities of the first-derivative changes are decided a priori from an expert-knowledge or from a human decision.

The hyperparameter $\lambda$ remains the only tunable parameter of the proposed filtering procedure, yet with considerable impact on achieved denoised times series. To avoid the arbitrariness stemming from using expert-knowledge, or the variability between different expert inputs, or even the lack of reproducibility in one same expert input, or the recourse to a learning procedure that would require the prior expert-knowledge based labeled ground truth, use is made here of a recently devised data driven automated procedure (Pascal et al 2021). This procedure is based on a theoretical extension to multivariate dependent time series of the Stein Unbiased Risk Estimation (SURE) criterion, originally designed for univariate uncorrelated data (Stein 1981, Tibshirani 2015). The use of this automated hyperparameter selection is further enriched here to permit to assess the robustness of the achieved crisis detections, by quantifying the sensitivity of changepoint detection with respect to the tuning of $\lambda$ (cf section 3.1).

In a nutshell, the proposed procedure adaptively filters out noise fluctuations in multivariate data, except over a finite, sparse and data-selected subsets of locations where jumps in data are actual and relevant, without recourse to prior and/or expert knowledge.

### 2.2.2. Multivariate non linear denoising: implementation and toolbox

The inverse problem formulation of the nonlinear filtering problem implies the minimization of the functional of the data defined in equation (1). To ensure that all series equally contribute, they are centered

<footnote n="6">⁶ On the contrary, a linear method as derived from the HP filter (Hodrick and Prescott 1997) would favor to extract smooth series, but not piecewise linear series, hence the change points would not be well defined.</footnote>
(zero mean) and reduced (unit variance) prior to applying the filtering procedure. Because of the use of L¹,²-norm in the penalization term, the functional is by construction non differentiable (nonsmooth), thus excluding gradient descent procedures for minimization. However, as the functional has been designed carefully to remain convex, minimization can be achieved by accelerated iterative algorithms based on proximal operators. The devised iterative algorithm has been shown to yield fast, time and memory efficient and accurate optimisation and is not detailed here.

In addition to solving the optimization problem, the algorithm includes the automated and data driven selection of the hyperparameter λ. The devised procedure performs jointly the selection of the hyperparameter λ and the estimation of the denoised time series, via the minimization of equation (1). This joint procedure is implemented as a bi-level optimization scheme, thus resulting in limited additional computational or memory costs.

As a final feature of significant practical interest, the toolbox is designed to handle missing samples in time series. This is of particular interest in the analysis of cliometric, economic or financial data, as it permits to work with left-censored data (with data availability not starting at same date for all countries).

## 2.3. Reassigned changepoints

### 2.3.1. Identification of the changepoints

One output of the previous step is the denoised time series $Y(t)$ that are piecewise linear. Henceforth, the changepoints are naturally the times where the second order differences (i.e. $Y_m(t+1) - 2Y_m(t) + Y_m(t-1)$) of the denoised time series $Y_m(t)$ are non-zero. Additionally, the values of this difference at changepoints quantify the intensities of the changes, corresponding to differences in the first derivatives prior and after the changepoints. This will be used to define the crisis indicators (see 2.4).

### 2.3.2. Reassignment of the changepoints

Before using these changepoints to detect crises, let us point that the time series extracted according to 2.1.2 are computed over sliding windows, whose durations are long enough so that the estimation of pairwise correlations (or of volatilities) makes sense. This leaves some indeterminacy because there are usually some nonstationary behaviors inside each window. Especially, we would like to emphasize that if a changepoint is obtained for a window, it can be in fact at any time in it; e.g. it is not necessarily at the middle of the window, nor at its end or its beginning. To solve this issue, we introduce a reassignment procedure that captures the time where most of the correlation (or more generally, of the information) occurs in each considered sliding window.

This operation is reminiscent of the so-called technique of time-frequency reassignment (Auger and Flandrin 1995), or of synchrosqueezing (Daubechies and Maes 1996) in time-frequency / time-scale analysis of nonstationary series, and of the idea of ‘change of clock’ proposed by Berthon (1994) for nonstationary time series analysis. More precisely, for the present study, let us introduce reassigned times in each window $W_t = [t - L/2; t + L/2]$, indexed by its mean time $t$ and having duration $L$, for each considered multivariate series.

For the correlation series $R_{ij}(t)$ between countries $i$ and $j$ the reassigned times are defined as:

$$ \hat{u}_{ij}(t) = t + \frac{\sum_{u \in W_t} (u-t) x_i^c(u) x_j^c(u)}{|\sum_{u \in W_t} x_i^c(u) x_j^c(u)|}, \quad (2) $$

where the $x^c(u)$ and the centered are normalized times series in the window, hence $u \in W_t$. This estimator, built on point-wise correlations $x_i^c(u)x_j^c(u)$, has large fluctuations, but we have as many estimates as the number of pairs of countries in the dataset. As we are interested in a global view of these correlations, and more precisely of their parametric descriptors in terms of mean, variance, skewness and kurtosis, we keep as an estimate of the average reassigned time the median of these times (as the median is known to be robust and to mitigate fluctuations):

$$ \bar{t}_R(t) = \text{median}_{(i,j)} \{ \hat{u}_{ij}(t) \}. \quad (3) $$

We cannot expect the estimator to point to the exact time of interest, yet it helps in assigning relevant times to windows despite the heterogeneities inside a window. Without reassignment, the initial indeterminacy in dating of the changepoints (with $t$) is the length of the sliding window, here $L = 100$ months, i.e. more than 8 years. With reassignment, we cannot claim to have a precision for $\bar{t}_R$ of one month in position. Yet we have checked in preliminary simulations that, given the level of fluctuations existing in the series, the uncertainty in time position is reduced to be around 3 months.

When studying the series of volatilities, we introduce accordingly a reassigned median time, now based on the multivariate volatilities. Let us call $V_i(t)$ the volatility of data of country of index $i$ (going from 1 to up
to 32 at the end of the period of the study). The estimator of this reassigned time for volatilities is:

$$ \hat{u}_i(t) = t + \frac{\sum_{u \in W_i} (u-t)(V_i(u))^2}{\sum_{u \in W_i} V_i(u)^2}, \quad \text{and} \quad \bar{t}_V(t) = \text{median}_{(i)} \{\hat{u}_i(t)\}. \qquad (4) $$

As for $\bar{t}_R$, the expected precision in dating for $\bar{t}_V$ has been controlled to be reduced from the initial $L = 100$ months to a trimester.

## 2.4. Crisis index design

The proposed changepoints of 2.3 for correlations series (resp. volatilities) considered at the reassigned times $\bar{t}_R$ (resp. $\bar{t}_V$) can now be used to detect crises. Classically, in the financial macroeconomics literature, the labelling of changepoints as crises relies on the use of additional non financial data and involves associating candidate changepoints with a degradation of non-financial indicators (e.g. Gross Domestic Product in Duprey et al 2017). In the same spirit, the most recent chronologies of banking crises are often complemented by a study of their consequences (for example, Jalil 2015, Baron et al 2021, Kenny et al 2021). However, longitudinal data, especially non-financial data, are not available with monthly frequencies in a historical approach.

We avoid the recourse to any external information by designing a selection procedure that does not involve non financial data. A main assumption underlying our analysis and shown by ample evidence in the finance literature is that major crises are associated with local maxima in correlations. In more detail, in the long term, equity market correlations tend to rise with globalization (Goetzmann et al 2001), sometimes heterogeneously (Bekaert et al 2009). In the shorter term, the identification of downward breaks associated to major crises is consensual and robust to biases in the measurement of correlations (in particular, Longin and Solnik 1995, 2001, Campbell et al 2002). Indicators of systemic risk based on these maxima have even been proposed by, e.g. Zheng et al (2012) or Nobi et al (2014). These local maxima are mainly related to variations in the holding of international assets (Vermeulen 2013) linked to the persistence of the opportunity for diversification behaviour (Ang and Bekaert 2002); and variations in the risk aversion of international investors (Demirer et al 2018). We thus take as selection criterion the coincidence of candidate changepoints with a negative slope variation in the mean correlation.

For the selection applied to the list of candidate crises from the volatility dataset, we rely on the finding, which is ubiquitous in the finance literature (see Guidolin 2011 for a detailed survey), that specific high volatility regimes are associated with financial crises. We thus take as selection criterion the coincidence of candidate changepoints with a negative slope variation in the mean volatilities. The criterion of a negative slope variation in the mean is thus valid for both correlations and volatilities.

Let us define the crisis indicators by retaining the changepoints where the condition of negative slope variation is met. The contributions of each component to the changepoints permit to characterize the nature of the crises. Then, the sum across components of the absolute values of second order differences quantifies their magnitude. Note that in the case of incomplete series, the indicator is corrected by weighting according to the number of times entered in the dataset at each window.

## 2.5. ClioCrises toolbox

These four steps are implemented by an original ClioCrises Toolbox allowing both to solve the optimization problem and to automatically select the range of values for the penalty parameter. Within this interval the outcome of the procedure is robust. This toolbox is available online https://src.koda.cnrs.fr/pierre.borgnat/cliocrises.

# 3. Results and discussion

This section is devoted to the presentation and discussion of the results. The robustness of the method to the choice of the hyperparameter $\lambda$ is discussed in section 3.1. In sections 3.2 and 3.3, as is standard practice in the model-based literature on global financial crises, we compare the results with expert-based benchmark datings, both at the aggregate level and at the level of country contributions. Section 3.2 is devoted to the correlations dataset. At the aggregate level, we identify all benchmark global crises of the study period. Furthermore, we define the features of an ideal-type crisis in connexion with the established characteristics of global financial crises in the broad financial macroeconomics literature. Section 3.3 is devoted to the volatilities dataset. The decomposition of country contributions is compared with the reference datings of the expert-based literature at the country level. We disentangle the contribution of correlations and volatilities to crises detection. We show that volatilities allow us to identify specific crises, causing volatility spillovers.
<figure><img src="image_5.png" /><figcaption>**Figure 2.** Robustness of crises detection to changes in the hyperparameter λ. Left: distribution of estimated λ values in equation (1) produced by the automated selection procedure for 200 independent trials. Right plots assess the consistency amongst the crisis indicator lists obtained with three different choices of parameter λ: λ<sub>opt</sub> = 185, λ<sub>min</sub> ≈ 100 and high λ<sub>max</sub> ≈ 225.</figcaption></figure>

**3.1. Robustness with respect to the hyperparameter selection**

As explained in section 2.2.1, the outcome of the nonlinear filtering procedure crucially depends on the
hyperparameter λ, that indirectly controls the number of changes occurring in the first derivative of the
denoised times series (cf equation (1)). As explained in section 2.2.2, to avoid the arbitrariness of an
expert-eye selection of that hyperparameter, the procedure implements an automated and data driven
hyperparameter tuning, based on the adaptation of the SURE criterion to correlated multivariate data. In
practice, the implementation of that procedure implies the computation of the trace of a large Jacobian
matrix, of size (M × N) × (M × N) (with M components and N samples in time), repetitively at each
iteration. To decrease the corresponding heavy computational burden of the iterative minimization
procedure, the trace computation is replaced in the algorithm by an estimation stemming from an inner
product between directional finite differences against a realization of random vector, consisting of
independent identically distributed Gaussian random variables. This results in randomness in the automated
selection of λ, which may thus be different each time the algorithm is run.

It is proposed in the present work to turn that apparent weakness into a significant additional strength of the procedure. Indeed, the low computational load of the resulting algorithm permits that it can be run a large number of times, providing the user, not with a single estimated λ, but with a vector of estimated λ. This vector of estimated λ is illustrated in histogram form, in figure 2(left) from 200 independent repeated runs of the algorithm. First, it permits a better estimate of the optimal λ, e.g. from the most often observed value, i.e. most likely observation, found here experimentally to be λopt ≃ 185. Second, it permits the assessment of the robustness of the estimated denoised time series against the intrinsic variability of the automatically value of λ, inherited from the random nature of the procedure (cf section 2.2.2).

The proposition is to conduct the robustness assessment by considering the 95%-span [λ<sub>min</sub>, λ<sub>max</sub>] of
estimated λ. Empirically, it is observed that λ<sub>min</sub> ≃ 100 and λ<sub>max</sub> ≃ 225. Then, the denoised times series
Y<sub>λ<sub>opt</sub></sub>, Y<sub>λ<sub>min</sub></sub>, Y<sub>λ<sub>max</sub></sub> are obtained from using these three values of λ in equation (1), and crisis indicators for
each of these three denoised series are obtained. We compare them visually then quantitatively using a
weighted Jaccard index (Jaccard 1901) adapted for weighted data as in Tanimoto (1958) and Bouchard *et al*
(2013). This outputs a number between 0 and 1, quantifying by means of inner products the accordance
between several detection lists. These Jaccard indicators are reported in the legend in figure 2(right), and this
shows that datings are essentially identical for the three values of λ<sub>opt</sub> (red), λ<sub>min</sub> (blue), λ<sub>max</sub> (black).
Robustness is assessed by a large Jaccard index, found to be 0.74, computed from the demanding
coincidences of the three crisis dating lists obtained with λ<sub>opt</sub>, λ<sub>min</sub>, λ<sub>max</sub>.

The proposed procedure thus permits a relevant estimation of the selected hyperparameter λ<sub>opt</sub> and
shows that the estimated crisis indicators are robust, in datings and strengths, with respect to the precise
value of the selected hyperparameter.

**3.2. Crisis detection and dating from correlations**

The results of crisis detection and dating from correlations are presented in figure 3 and summarized in table 2.

The first major outcome is that the proposed procedure allows to detect and date correctly all the
benchmark global financial crises in the study period referenced in the expert-based reference literature
(Eichengreen and Bordo 2002, Reinhart and Rogoff 2009) (figure 3). This finding is crucial because it
validates the method, which is original in the field of model-based global financial crises datings. The second
major outcome is that the procedure permits to characterize the nature of the crises in terms of changes in
<table><caption>**Table 2.** Synthetic table on the correlations-based crisis detection: [μ, σ, s, γ]. +, 0 or – indicate whether the changes in slope are positive, null or negative. 0– and 0+ indicate very lowly negative and positive values compared to other components. The last three columns indicate whether the specific features associated with the correlation descriptors are met.</caption><thead><tr><th>Date</th><th>[μ, σ, s, γ]</th><th>Event</th><th>Flight to quality</th><th>Downside risk</th><th>Extreme events</th></tr></thead><tbody><tr><td>1973/07-12</td><td>[-, +, +, +]</td><td>A. First oil shock</td><td>X</td><td></td><td></td></tr><tr><td>1976/06</td><td>[0-, +, 0-, 0]</td><td>B. End of the fixed exchange rate system</td><td>X</td><td></td><td></td></tr><tr><td>1987/09</td><td>[-, 0+, 0-, +]</td><td>C. Stock market crash</td><td>X</td><td></td><td></td></tr><tr><td>2007/02</td><td>[-, +, +, -]</td><td>D. Global financial crisis: subprime mortgage phase</td><td>X</td><td>X</td><td>X</td></tr><tr><td>2009/02-09</td><td>[-, +, +, -]</td><td>E. Global financial crisis: global phase</td><td>X</td><td>X</td><td>X</td></tr></tbody></table>

moments, from now on referred to as ‘features’ (table 2), which can be associated with well-established characteristics of financial crises in the broad field of financial macroeconomics.

When crises occur, they interrupt the background pattern of financial integration on the structure of correlations. The observation of 60 years of globalization (cf figure 3) reveals a dominant pattern of financial integration comprising an increase in the mean of correlations, a decrease in standard deviation (mostly in the longest period of rapid financial integration between the 1980s to 2000s), a decrease in skewness, and an increase in kurtosis. The occurrence of crises mostly induces on correlations a decrease in mean, an increase in standard deviation, an increase in skewness, and a decrease in kurtosis.

This dominant pattern is fully consistent with the existing literatures on the measurement of system risk and the modeling of financial crises and extreme events. In particular, the rise in standard deviations is linked to the ‘flight to quality’ phenomenon. This flight from the riskiest to the least risky assets reinforces the divergence of individual dynamics, and therefore of bilateral correlations, in times of crisis (for a more detailed discussion of the definition and measurement of the flight to quality, see Gubareva and Borges 2016). The rise in skewness echoes an emerging ‘downside risk’ literature, associating in particular financial crises with ‘broad, infrequent and abrupt’ downside variations in financial conditions, measured by skewness (Ranciere et al 2008). Finally, the fall in kurtosis can be related with the applications of extreme value theory to the study of financial crises (see, e.g. Rocco 2014). This theory focuses on rare events, and therefore on distribution tails.

The characteristics of the 1973 crisis (A.) consists of, first, a closely consecutive double changepoint; and, second, an impact on two out of the three features. The standard deviation (‘flight to quality’) and skewness (‘downside risk’) increase. No fall in kurtosis (‘extreme events’) is observed. This can be linked to the fact that the 1973 crisis is not exclusively of financial origin, but associated with the first oil shock. For instance, Hamilton (2013) emphasizes that the financial and energy dimensions of this crisis coexist. This dual dimension encompasses a general appreciation of commodity prices, and a high inflation environment resulting in a negative real return on financial assets. This energy dimension to the crisis may explain why the ‘extreme risk’ feature is not met as expected for an ideal-type major financial crisis.

The shift to the flexible exchange rates regime leading to the 1976 crisis (B.) impacts one feature out of three: the standard deviation increases (‘flight to quality’). This crisis ends a transition period in the functioning of the foreign exchange markets. In 1971, the United States made the unilateral decision to suspend the gold convertibility of dollar, and 1973 marks the end of the fixed exchange rate regime. In 1976, there is a consensus among advanced economies in favor of the transition to flexible exchange rates (Bastidon et al 2020), which is formalized in the amendment of the International Monetary Fund’s articles of agreement (Kingston agreement, Obstfeld et al 1985). The result that the 1976 crisis is characterized by the sole feature of ‘flight to quality’ can thus be interpreted as a validation by the markets of this institutional transition. Only peripheral equity markets are impacted, which explains the increase in the standard deviation of correlations.

The 1987 stock markets crash (C.) also involves one out of three features. The standard deviation increases weakly (‘flight to quality’). Although ‘the crash was far more traumatic than any of the possible scenarios’ (Greenspan 2004), this is a short-lived crisis, whose resolution under the leadership of the US central bank is considered by market participants to be particularly relevant. This characteristic is fully compatible with the absence of a significant effect on skewness, associated with the ‘downside risk’ feature, which is a predictor of lasting crises; and kurtosis, associated with the ‘extreme events’ feature.

Finally the detection shows a changepoint in 2007 (D.), and a closely consecutive double changepoint in 2009 (E.), which reflects the exceptional magnitude of the so-called Global Financial Crisis of 2007–2009. We indeed identify the two phases: the subprime mortgage (2007) and global crises (2009) identified by, e.g.
<figure><img src="image_6.png" /><figcaption>**Figure 3.** CrossCorrelation-based crisis detections and datings. The top four plots report times series related to the first four moments used as a summary of the Correlation distribution, to which their denoised piecewise linear versions are superimposed (cf equation (1)). The two bottom plot report respectively change points and crisis indicators. Colors (blue, orange, yellow and violet) quantify the contributions of each moment (mean, standard deviation, skewness and kurtosis, respectively) to the change Point and crisis indicators. Letters in bottom plot, and dashed lines in all plots, correspond to the detected crises, as summarized in table 2 and discussed in the text. The times used for these plots are the reassigned times $\tilde{t}_R(t)$ (cf equation (3)).</figcaption></figure>

Mishkin (2011). The subprime mortgage and global phases of the Global Financial Crisis both impact all three features of ‘flight to quality’, ‘downside risk’ and ‘extreme events’. This crisis is therefore archetypal of global financial crises. This is reminiscent of the consensus in the existing literature that the Global financial crisis of 2007–2009 has no comparison in contemporary history other than the crisis of 1929 (Caggiano et al 2017).

In short, the use of the distribution of correlations thus permits, first, to provide a relevant crisis dating that encompasses all the episodes listed in the expert-based literature; and second, to draw a typology in reference to the ideal-type major crisis that involves the three features (flight to quality, downside risk, extreme events). Over the six last decades, the ideal-type major crisis is the Global financial crisis of 2007–2009, entailing the three features. The other crises, of lesser magnitudes, are associated with only some
<figure><img src="image_7.png" /></figure>

of these features. The first oil crisis (1973) involves two features (flight to quality and downside risk). It partly originates outside the financial sector (energy crisis). The end of the fixed exchange rate regime (1976) involves only one feature (flight to quality). This is the final episode of the turbulence that begins in 1971 with the end of the gold convertibility of the dollar. The equity markets ratified this change of institutional framework. The 1987 crisis, which was particularly short-lasting, also involves one feature (flight to quality).

### 3.3. Crisis detection from volatilities

This second set of results is obtained from the multivariate analysis of the sliding estimated volatilities of each of the 32 countries (cf section 2.1.2). Insofar as the results can be disaggregated to the level of country contributions, they are compared with the expert-based reference literature at the level of individual country events. Let us recall that the aim here is to separate the contribution of volatilities from that of correlations in detecting crises, the two measures being frequently combined in aggregate indicators. The robustness of the achieved detection has been assessed with the same procedure as that described in section 3.1 and leads to the same conclusion of mild sensitivity of the obtained results to the selection of the hyperparameter, with $\lambda_{opt} = 343$, $\lambda_{min} = 320$ and $\lambda_{max} = 360$.

For the optimal value of $\lambda$, the crisis detection and indicators are represented in figure 4 and summarized in table 2. Then figure 5 displays the dating obtained with volatilities (top, in blue), compared to the dating obtained from correlations in section 3.2 (bottom, in red). The volatility based detection has the additional advantage of allowing the identification of individual country contributions to crisis detection. These contributions are presented in table 4, and figure 6 displays the time series of volatilities for a selection of countries.

Volatilities yield the detection of eight crises (figure 4), out of which five crises match with the crises detected by correlations (figure 5, table 3):
<figure><img src="image_8.png" /><figcaption>**Figure 5.** Volatility-based versus CrossCorrelation-based crisis detections and datings. The Volatility-based (top) and CrossCorrelation-based (bottom) crisis detections and datings show satisfactory consistency, as quantified by a Jaccard index of ≈ 0.45 (see text). The times used for these plots are the reassigned times, respectively $\tilde{t}_R(t)$ for CrossCorrelation and $\tilde{t}_V(t)$ for Volatility, cf equations (3) and (4).</figcaption></figure>

<table><caption>**Table 3.** Synthetic table on the volatility-based crisis detection. Note that they were obtained by selecting only the crisis with a negative variation in the slope of the mean of volatilities. The second column indicates the magnitude of this slope variation. 0— corresponds to a minor variation, — to an intermediate value, and —— to a major variation.</caption><thead><tr><th>Date</th><th>Amplitude</th><th>Event</th><th>Correlation label</th></tr></thead><tbody><tr><td>1973/06</td><td>—</td><td>1. First oil shock</td><td>A.</td></tr><tr><td>1975/01</td><td>0—</td><td>2. End of the fixed exchange rate system</td><td>B.</td></tr><tr><td>1977/03&11</td><td>—</td><td>3. End of the fixed exchange rate system</td><td>B.</td></tr><tr><td>1984/10</td><td>—</td><td>4. Developing countries debt crisis resolution, Baker plan</td><td></td></tr><tr><td>1989/01&11</td><td>—</td><td>5. Developing countries debt crisis resolution, Brady plan</td><td></td></tr><tr><td>2000/09</td><td>0—</td><td>6. Final emerging crises + advanced countries stock market crisis</td><td></td></tr><tr><td>2006/09-10</td><td>—</td><td>7. Global financial crisis</td><td>D.</td></tr><tr><td>2010/04-05</td><td>—</td><td>8. Global financial crisis</td><td>E.</td></tr></tbody></table>

* 1973/06 (1.) for volatilities matches 1973/04-07 (A.) for correlations,

* 1975/01 (2.) & 1977/03-11 (3.) matches 1976/06 (B.)

* 2006/09-10 (7.) matches 2007/01-02 (D.)

* 2010/04-05 (8.) matches 2009/02-12 (E.)

In addition, three crises are solely identified by volatilities: 1984 (4.), 1989 (5.) and 2000 (6.), and one crisis is solely identified by correlations: 1987 (C.). It was found in the detection by correlations that the crisis of 1987 did not fulfill the 'downside risk' criterion, associated with lasting crises. The fact that this crisis does not appear in the dating by volatilities confirms this result.

For exclusively volatility based crises, our outcomes enrich the stylized facts from the expert-based datings of Eichengreen and Bordo (2002) and Reinhart and Rogoff (2009). Both expert based datings present lists of events by country for a broad sample, sorted by category (currency and banking crises for Eichengreen and Bordo 2002; currency crises, banking crises, hyperinflation, stock market crises, defaults on external debt, defaults on domestic debt for Reinhart and Rogoff 2009). We show that the contributions of
<figure><img src="image_9.png" /></figure>

the countries form a continuum of values (figure 4). Despite the fact that the crises are led by a small number of countries with the highest contributions (table 4, figure 6), this continuum reflects the notion of spillovers of volatility. In what follows we discuss the specific information provided by country contributions, for each crisis and in chronological order.

The First oil shock crisis of 1973 (1., and A.) is both correlations based and volatility based. As for the other crises identified by the volatility-based detection, the inspection of figure 4(bottom) shows that most countries contribute to the crisis indicator, thus suggesting generalized volatility spillovers. In detail, the main contributors (table 4) are the core financial markets: the USA, the UK and the Netherlands account for more than 50% of the contributions, with comparable individual contributions between 16 and 20%. In the case of the UK, we note a particularly strong increase in volatility before the crisis (figure 6). Advanced financial markets of secondary dimension contribute less.

The end of the fixed exchange rate system crises of 1975 and 1977 (2. and 3., matching B.) are also both correlations based and volatility based. The volatility-based detection of 1975 and 1977 captures information that complements correlations. While correlations identify the institutional event of the end of the fixed exchange rate system (1976, B.), volatilities identify, ex ante and ex post, the fluctuations of the adjustment process of exchange rates (1975, 2.; and 1977, 3.). In expert-based datings, 1975 (2.) coincides with a clear predominance of currency crises, after the fixed exchange rate system was abandoned in 1973. 9 countries are affected by single currency crises, and 6 by twin or more crises. These crises affect developing economies, but also the United States hit by a single currency crisis related to the central role of the US dollar, and the United Kingdom hit by a joint currency and hyperinflation crisis. 1977 (3.) still coincides predominantly with currency crises, either single or combined. 15 countries are affected, usually developing or medium-size advanced countries. These crises can still be linked to the ex post adjustment to the flexible exchange rate system introduced in 1976. In addition to currency crises, 9 advanced economies, including the United States, are affected by stock market crashes, which are the second dominant type of crisis.

These stylized facts from the expert-based literature are replicated and measured in the contributions to the detection of the crises of 1975 and 1977 (table 4), which are driven by the core financial markets (UK and NL for 62% of the contributions in 1975, UK and USA for 40% of the contributions in 1977). For these three countries, the end of the 1970s corresponds to a return to the low volatilities regime, after a particularly sharp rise in the case of the UK (figure 6). In 1977, we also notice significant contributions from the economies of the European Monetary System whose currencies were regularly attacked, entering a long period of high volatility (figure 6). Italy and France account for 22% and 29% of contributions in 1977.

The 1984 (4.) and 1989 (5.) crises are specifically identified by the volatility-based detection. The main stylized fact of the 1984 crisis (4.) from the expert-based literature is still the predominance of currency
crises, affecting 7 countries by single crises, and 14 countries by twin or more crises, with a preponderance of Latin American countries. Defaults on external or domestic debt are the main single crisis factor, affecting 12 developing countries. Finally, stock market crashes and banking crises affect 4 countries. These stylized facts correspond to the first phase of the extension of the debt crisis in developing economies, revealed by the wave of defaults in 1982. Before the adoption in 1985 of the Baker Plan, which was the first stage of its resolution, the crisis was handled on a case-by-case basis. This explains the persistence of single or combined currency crises. This also explains the occurrence of stock market and banking crises in advanced economies, whose banks are largely exposed to the defaults of developing countries.

Like the 1984 crisis (4.), the 1989 crisis (5.) can be read in the perspective of the resolution of the 1982 debt crisis. In expert-based datings, single currency crises (6 countries) and twin or more currency crises (9 countries) still dominate and almost exclusively affect developing or transition countries. The number of banking crises increases sharply with 9 countries, concerning both advanced economies (4 countries) and developing economies (5 countries). This pattern corresponds to the failure of the Baker Plan, which aimed at resolving the crisis by rolling over the claims; and the adoption of the Brady Plan of 1989, the final stage of the resolution of the 1982 debt crisis, which aimed at reducing the claims. This specific resolution is associated with banking crises in both advanced and developing economies, which confirms that it does not give rise to generalized moral hazard. Both lenders, affected by a significant decline in the market capitalization of their banking sectors; and borrowers, affected by a drastic macroeconomic adjustment, bear their losses (Rogoff 2002).

The contributions to the volatility-based detection of the 1984 (4.) and 1989 (5.) crises highlight the same global nature of the so-called 'debt crisis of developing economies' of the 1980s. Advanced economies of intermediate size are the main contributors, which is consistent with the high volatility regime affecting France and Italy throughout the decade (figure 6). The core economies, USA, UK and NL, whose volatility has normalized, contribute less. The contribution of developing economies is declining over time. In 1985, South Africa is the main contributor. In 1989, only India has a significant positive contribution. Korea and South Africa contribute little or nothing, suggesting their individuation from the previously homogeneous group of developing economies.

The last event that appears specifically in the volatilities-based detection is the crisis of 2000 (6.). It is interesting to note that this stock market crisis is considered particularly difficult to grasp in terms of its causes, to the point of being a test for early warning indicators (Stock and Watson 2003). The expert-based databases list 19 countries in exclusive stock market crisis, mainly in advanced economies; and 13 countries in combined stock market crisis, mainly emerging. Finally, 13 countries, mainly developing, emerging and transition economies, are affected by other crises of various kinds. This count illustrates that the event combines a one-off stock market crisis in advanced economies and the end of a period of recurrent crises in emerging economies (since the Mexican crisis of 1994) and developing economies (since the outbreak of 1982). It is therefore particularly interesting that it is identified by the volatility-based detection.

Once again, an examination of the contributions to the volatility-based detection of the 2000 crisis (6.) reveals and measures the same main stylized fact as the expert-based literature. The first contributors are emerging economies (Korea and Chile), followed by advanced markets of secondary dimension. The contributions of core financial markets (USA, NL) and large emerging economies (Brazil, South Africa) take intermediate values. After this crisis, the volatilities of advanced economies tend to rise again after a long period of stability at low levels. In contrast, the volatilities of emerging economies converge to low levels, e.g. in a particularly spectacular way for Korea (see figure 6).

Finally, the global financial crisis is identified by the detections of both volatilities and correlations. The hierarchy of contributions to volatility-based detection in 2006 (7.) and 2010 (8.) is clear. The core markets (USA, NL in particular) are among the first contributors. Advanced economies of medium dimension (and some emerging economies) come next. Emerging economies predominate among the lowest contributors, signaling the origin of the crisis, initiated in the US and affecting particularly the most complex financial systems. The position of Norway, which contributed almost at the same level as the US in 2006, can be explained by the size of the real estate bubble preceding the crisis, which was twice that of the US (IMF 2010). Moreover, a striking feature of the crises of 2006 and 2010 is the homogeneity of volatility dynamics of countries (figure 6). This result reinforces the characterization of the global financial crisis as an 'ideal type' in the correlations-based detection.

In sum, crisis detection obtained from volatilities reinforces the interest of a strategy based on the separate treatment of cross-correlation and volatility indicators. Specific volatility-based crises are identified. These specific crises combine financial fragilities in developing and emerging economies on the one hand, and advanced economies on the other hand, through debt relationships (1984, 1989) or by temporal coincidence of multiple crises (2000). The detail of the contributions to volatility-based crises shows that they highlight the same main stylized facts as the expert-based reference dating, namely the driving role of
the economies at the heart of the financial system in the 1975 and 1977 crises; the extension of the debt crisis to creditors in developing economies in the 1984 and 1989 crises; the dual nature of the 2000 crisis; and finally the specific homogeneity of the dynamics of the global financial crisis. The volatility-based automated data-driven procedure we have devised here allows us to go beyond expert-based datings by quantifying and hierarchizing these individual country contributions.

## 4. Conclusion

We propose an original indicator of major global financial crises based on scarce data, since only share prices are required. To construct this indicator, we combine, in an original way, three fields: nonlinear multivariate filtering, financial cliometrics, and financial macroeconomics. Our contribution is twofold: in terms of methodology, and in terms of results.

In terms of methods, we provide an original indicator of global financial crises, derived from nonlinear multivariate filtering. In the existing literature, crisis detections and datings are based on aggregated indicators, which hence do not enable to disentangle the contributions of cross-correlations and volatilities to crises. To address this shortcoming, we perform two crisis detections and datings, based independently on cross-correlations and on volatilities. To identify the respective contribution of cross-correlations and volatilities to the detection of crises, we use respectively the first four moments of the distribution of pairwise correlations and the standard deviations of equity markets returns.

This approach has major advantages. First, it is parsimonious in terms of data. Neither the design of the multivariate signals extracted from the database, nor the final selection of the candidate changepoints require non-financial data of high frequency, which is a critical advantage in cliometrics but also in crises detection and qualification in real-time. Secondly, the approach does not require any particular properties of the data. The segmentation procedure of multivariate time series handles incomplete data, which do not need to be piecewise stationary. It does not require constructing aggregate indicators since it handles multidimensional data. Thirdly, the results are fully endogenous. The segmentation is implemented by an original ClioCrises Toolbox.

In terms of results, our first contribution is that the procedure performs a crisis dating that encompasses all the reference episodes listed in the expert-based literature. Our second contribution is to propose a synthetic indicator based on the first four correlation moments, which is fully in line with the emerging literature on 'downside risk' and 'extreme events' that has accelerated since the COVID19 pandemic. Thirdly, we propose a crisis typology. The contribution of each moment to the changepoints permits to characterize the nature and depth of crises. The higher the contribution of each moment, the deeper the financial crisis. The ideal type major crisis, i.e. the Global Financial Crisis, simultaneously gathers a change in the first four moments of correlations.

The crisis detection obtained from volatilities reinforces the interest of a strategy based on the separate treatment of cross-correlation and volatility indicators. If for the most important crises the detection dates by volatilities and correlations coincide, specific volatility-based crises are also identified. These specific crises combine financial fragility in developing and emerging economies on the one hand, and advanced economies on the other hand, through debt relationships (1984, 1989) or by temporal coincidence of multiple crises (2000). Secondly, we show that country contributions form a continuum of values, which gives credit to the notion of spillovers of volatility. Thirdly, the procedure is a major contribution to the expert-based literature. Existing datings list countries in crisis for a given year. The proposed procedure allows us to quantify and hierarchize country contributions to the dating of volatility crises.

Finally, the application we are proposing concerns a broad base of 32 countries at monthly frequency, which the computational efficiency of the method makes it possible to handle, but the method is also applicable to longer series at annual frequency, with a restricted number of countries. The joint nature of crisis detection suggests that it is also applicable to the detection of national banking crises based on individual share prices. In addition, the scope of the proposed indicator is not limited to historical data. The simple, transparent and reproducible nature of the method is also a decisive advantage for the analysis of financial stability by public authorities. The ability to provide a relevant characterization of crises is crucial for governments, central banks and international financial institutions in deciding on the scale and timing of resolution measures to implement. The proposed indicator and ClioCrises toolbox constitute a new monitoring and decision-making tool for public authorities in financial crisis situations.

## Data availability statement

The data that support the findings of this study are openly available at the following URL/DOI: https://fred.stlouisfed.org.
Acknowledgments

The authors thank the financial and technical support of the Rhône-Alpes Complex System Institute (IXXI, www.ixxi.fr), where the present work was achieved, and acknowledge support of the ANR-19-CE48-0002 DARLING Grant, the ANR-23-CE48-0009 OptiMoCSI, and the GraphNEx CHIST-ERA ANR-21-CHR4-0009 Grant.

This paper has been presented to a wide range of scientific communities. We thank the participants of the following conferences. In the field of Central banking: Polish Academy of Sciences, Paris Scientific Center (2021/10); Hochschule der Deutschen Bundesbank (2023/09); in the field of Complex systems: Conference on Complex systems (CCS, Lyon, 2021/10); in the field of Economic history: World Economic History Conference (WEHC, Paris (2022/07); in the field of Econometrics: Society for Economic Measurement (SEM, Calgary, 2022/10), OFCE Seminar (OFCE, Paris, 2023/11). We wish to thank in particular Arnab Bhattacharjee, Christophe Blot, Michael Bordo, Jérôme Creel, Thibaut Duprey, Steven Durlauf, Etienne Farvaque, Berndt Hayo, Erich Keller, Gregory Levieuge, Laslo Lowasz, Mathieu Plane, Nelly Pustelnik, Xavier Ragot, Janette Rutterford, Dimitris Sotiropoulos, Vincent Touzé.

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