The post-pandemic acceleration of digital tools has reshaped how small and medium-sized enterprises (SMEs) in Latin America compete, yet the resulting maturity gains remain poorly understood in individual emerging-market economies. Ecuador exemplifies this gap: SMEs account for over ninety per cent of the formal business register but appear in few econometric studies, most based on convenience samples. This study asks how strongly digital adoption is associated with the digital maturity of Ecuadorian SMEs and which tools best predict e-commerce engagement. The analysis draws on the National Digital Skills Survey (ENHD, Encuesta Nacional de Habilidades Digitales), administered by Ecuador’s Ministry of Telecommunications (MINTEL), covering 847 SMEs in 2021. A ten-indicator Digital Adoption Index (DAI) is related to the ENHD maturity score through least squares regression with robust standard errors, and a binary logit model with multicollinearity diagnostics identifies the tools that predict e-commerce adoption. The DAI explains over half of the variance in digital maturity, robust to firm-size and sector controls. Website presence exerts the largest marginal effect on e-commerce adoption, followed by cloud services. Top adopters score twice as high as non-adopters, consistent with cumulative advantage. Policy implications include subsidised digital-foundations bundles for laggard micro-enterprises.
This study investigates the short-term and long-term impacts of gross domestic product (GDP), inflation, foreign capital flows, trade balance and interest rate on stock market performance in Saudi Arabia for the period 1990-2023. The autoregressive distributed lag (ARDL) approach and error correction model (ECM) are employed to empirically examine the short-run and long-run relationships. The ARDL-ECM technique is effective for analyzing cointegration and assessing adjustment processes. Additionally, impulse response function (IRF) analysis based on the vector autoregression (VAR) model, estimated using these macroeconomic indicators, is applied in this paper. This study provides novel insights and addresses emerging gaps in the literature concerning Saudi Arabia as a developing economy. The long-term relationship in the bounds test results confirms its existence. In the long run, inflation and interest rate exert a statistically significant negative effect on stock market performance, while the trade balance has a significant positive impact. GDP and foreign capital inflows do not exhibit statistically significant long-run effects. Short-run dynamics indicate persistence in stock market performance along with significant effects from inflation and interest rate changes, while GDP and foreign capital inflows remain statistically insignificant in the long-run scenario. Forecast error variance decomposition (FEVD) results show that approximately 68.5% of the variation in market performance is explained by its own shocks, followed by foreign capital flows (16.3%) and inflation (8.4%). While foreign capital flow does not exhibit statistical significance in the ARDL long-run estimates, its contribution in variance decomposition highlights its role as an important source of external shocks. These findings are relevant to various stakeholders, including investors and policymakers. Additionally, policy emphasis should be placed on controlling inflation and maintaining stable interest rates while improving trade balance conditions. Although foreign capital flow does not show a direct long-run effect, its role in influencing market variability suggests the need for a stable and well-regulated investment environment.
This study examines the moderating and threshold effects of institutional quality on the relationship between external debt and economic growth using panel data from 39 lower- and upper-middle-income countries over the period of 1996–2023. To address econometric challenges commonly found in previous studies—including endogeneity, cross-sectional dependence, and slope heterogeneity—the analysis employs a dynamic common correlated effect (DCCE) estimator and a dynamic panel threshold model (DPTM). The DCCE results indicate that external debt exerts a negative and statistically significant effect on economic growth, whereas institutional quality has a positive effect. Furthermore, the interaction between external debt and institutional quality is positive and significant, suggesting that stronger institutions mitigate the adverse growth effects of external debt. The threshold analysis reveals significant institutional quality thresholds across income groups. For the full sample, the estimated threshold value of institutional quality is 2.74. Disaggregated results indicate thresholds of 1.80 for lower-middle-income countries (LMICs) and 4.33 for upper-middle-income countries (UMICs). Although external debt continues to exert a negative impact on growth in both regimes, the magnitude of this adverse effect declines once institutional quality surpasses the threshold levels. These findings highlight the critical role of institutional quality in shaping the debt–growth nexus. Strengthening governance structures—including improving transparency, rule of law, and fiscal accountability—can help mitigate the growth-reducing effects of external debt and improve countries’ capacity to manage debt sustainability in developing economies.
This article analyses trading-performance patterns in a stock market simulation conducted with 134 second-year students at the University of Mons (Belgium) on 11 December 2025. Participants had a virtual capital of 100,000 euros and were free to trade CAC 40 securities without any restrictions on the number or volume of transactions. An academic incentive scheme, combining a participation bonus and bonuses for the three best portfolios, created a tournament-style environment with continuous ranking feedback. This feature is considered as part of the experimental context rather than as a separately identified causal mechanism. We estimate a quadratic model linking performance to activity, measured by the number of mean-centered transactions to reduce the collinearity between the first-degree term and its square, and control exposure via the average percentage of cash in the portfolio, portfolio variability (measured as the standard deviation of portfolio value) and the average trade size. Breusch-Pagan and White tests indicate heteroscedasticity, justifying a robust inference. The results highlight a convex relationship between activity and performance: the marginal association is initially negative but becomes positive above a model-implied upper-tail level corresponding to approximately 46 transactions. This value should not be interpreted as a behavioral level or as a trading rule. The percentage of cash in the portfolio and the average trade size are negatively associated with performance, while the portfolio variability does not show a statistically significant association with performance. Overall, the results indicate heterogeneous trading patterns rather than a single activity-performance profile.
Regulatory risk measurement under Basel III's Fundamental Review of the Trading Book places Expected Shortfall (ES) at the center of market risk capital, yet the fourth-order Edgeworth expansion, still widely used for Value-at-Risk (VaR) and ES calculations, can produce negative densities in the tail regions where these measures concentrate, while saddlepoint approximations preserve positivity but face their own limits in heavy-tailed and sub-Gaussian settings. Whether either method delivers reliable tail estimates in the rare-disaster regimes documented in the empirical consumption-disaster literature therefore remains an open question. We address it by comparing the two approximations across 648 rare-disaster parameter combinations and five additional distributional families (Student-t, Hansen skewed-t, generalised error distribution (GED), two-sided jump mixture, and generalised hyperbolic), and by deriving a closed-form characterisation of the Edgeworth validity envelope. We establish three core findings. First, the validity envelope is bounded above by a sharp kurtosis ceiling at gamma 2=4 and laterally by a non-monotone skewness boundary peaking at |gamma 1 & lowast;,max| approximate to 0.685 at gamma 2 approximate to 2.533; 87.5% of the rare-disaster grid falls outside it. Second, accuracy is threshold-dependent: Edgeworth dominates at moderate quantiles, saddlepoint at extreme quantiles, with negative-density regions inflating Edgeworth ES error from 6.20% inside the envelope to 47.04% outside it. Third, these results reconcile only when point probability, density validity, and integrated-tail accuracy are treated as distinct accuracy criteria. The findings have direct implications for ES-based regulatory capital in heavy-tailed regimes and motivate a regime-conditional rather than universal approximation choice.
In past decades, the macroeconomic stability of India has been tested repeatedly by major global disruptions, including oil price shocks, the 2008 global financial crisis and the COVID-19 pandemic. Analysing how macroeconomic variables respond to these shocks is essential for evaluating external vulnerability and policy resilience in emerging economies. Our study provides a comprehensive empirical investigation of the dynamic responses of wholesale price inflation, industrial output, oil prices and exchange rates in India by employing monthly data from January 1993 to December 2024. To examine long-run equilibrium relationships along with short-run adjustment dynamics, the present study employs co-integration analysis within a Vector Error Correction Model (VECM) framework. Further, we applied impulse response functions and forecast error variance decomposition to track volatility spillover mechanisms. Quantile regression and ARCH-GARCH models were further estimated to account for distributional heterogeneity and time-varying volatility. The findings of our study suggested stable long-run linkages among the selected variables, where oil price shocks emerged as a key external source of macroeconomic fluctuations. Short-run dynamics suggested that shocks in oil prices are transmitted primarily through inflation and exchange rate channels and then affect industrial output. Distributional estimates revealed the effects were stronger during stress periods, indicating tail risks that were not captured by the mean-based models. Lastly, volatility analysis confirmed persistent clustering, especially during phases of crisis. Overall, the findings suggest that India's macroeconomic system remains externally sensitive, with adjustment mechanisms that operate gradually but come under strain during global disruptions. These results underscore the importance of energy risk management and crisis-responsive macroeconomic stabilisation policies.
With significant market unsureness, “static” methods fail to account for economic uncertainty, may be less precise and, accordingly, less helpful when selecting investment alternatives. Methods that take into account the current economic situation and allow for adapting the alternative selection to external uncertainty are becoming more relevant. One of such methods is the fuzzy set theory. This article addresses the mathematical framework of such an approach for the economic analysis of investment project selection. A step-by-step scheme for implementing the fuzzy set method for investment projects is presented. Studies performed on the example of three investment alternatives give grounds for asserting the compatibility and feasibility of using two methods (the fuzzy set method may be partly based on the results of pairwise comparisons of experts according to the Saaty method) and confirmation or refutation of previous intuitive decisions of investors based on a comprehensive analysis of the criterion composition and the use of mathematical grounded technique.
The General Data Protection Regulation (GDPR), adopted by the European Union in 2018, aims to enhance consumer trust and market efficiency by strengthening data protection. The concurrent stringent compliance requirements raise operational costs and could reshape competition by favoring larger firms with greater regulatory capacity. While the GDPR reduces data-related risks and promotes global digital trade through its extraterritorial reach, the potential advantage to larger firms could incentivize strategic responses such as mergers and acquisitions (M&A) to consolidate market power. Given the rapid expansion of Chinese digital firms in e-commerce, social media, and cloud services across the EU, this study examines how the GDPR has affected their cross-border M&A activities between 2014 and 2021. Based on difference-in-difference analysis, the study finds that the GDPR did not have a statistically significant impact on the number or value of mergers and acquisitions by Chinese digital firms in the EU in the short term. This suggests that firms may enhance their institutional adaptability by strengthening their compliance capabilities. However, institutional and cultural differences pose long-term entry barriers for the firms. The study contributes by highlighting how firms adjust internationalization strategies under stringent regulatory regimes, offering policy-relevant insights for governments and regulatory authorities.
Suppose that we have a statistical model with q unknown parameters w, and an estimate w<^>, based on a sample of size n. A basic question is: what is the covariance of the estimate? The covariance is needed for the Central Limit Theorem (CLT). This gives a first approximation for the distribution of w<^>. But what if qn=n increases with n? How fast can it increase and the CLT still hold? An answer has so far only been given for the sample mean. The same is true for the Edgeworth expansions. These are expansions in powers of n-1/2 for the density and distribution of w<^>. For fixed q, these expansions are important, as they show how small n can be for the CLT to apply. When it does, they can greatly improve the accuracy of the CLT. I give conditions that allow for the Edgeworth expansions to remain valid when qn=q increases with n. Earlier Edgeworth expansions when qn=q increases, have only been done for a sample mean, and only for a 2nd order Edgeworth expansion. In contrast, I consider a very large class of estimates, the class of non-lattice standard estimates. An estimate is said to be a standard estimate if its mean converges to its true value as n increases, and for r >= 1, its rth order cumulants have magnitude n1-r and can be expanded in powers of n-1. For this class of estimates, I show that the Edgeworth expansions hold if qn grows as a power of n less than 1/6. That is, I give these expansions in powers of n-1/2qn3. This large class of estimates has a huge range of potential applications, as estimates of high dimension are common in nearly all areas of applied statistics. The most important type of standard estimate is when w<^> is a smooth function of a sample mean, of dimension p say. When either or both qn=q and pn=p increase with n, I give conditions on their growth for the Edgeworth expansions for w<^> to remain valid: the eighth power of p times the sixth power of q cannot grow as fast as n. This holds for fixed q=qn if pn grows less than a power of n less than 1/8. This appears to be the first time when Edgeworth expansions have been given when not one, but two dimensions, are allowed to increase to infinity with n. This gives two different pathways for allowing an increase in dimensionality. When q=1, I give 5th order Edgeworth-Cornish-Fisher expansions for the standardized distribution and its quantiles of any smooth function of a sample mean of dimension pn, when pn is a power of n less than 1/2. However for the special case when this function is linear, there is no restriction whatever on how fast pn can increase! If also the components of the sample mean are independent, then these expansions are in powers of (np)-1/2. I also give a method that greatly reduces the number of terms needed for the 2nd and 3rd order terms in the Edgeworth expansions, that is, for the 1st and 2nd order corrections to the CLTs. I also extend these results to the case where w<^>is an element of Rq is a function of several independent sample means, each of dimension increasing with n, with total dimension p.
This article examines two-state proportional hazard rate models with unobserved heterogeneity specific to each state, a framework that is especially relevant for labor market transitions. To make estimation feasible in large longitudinal datasets, we implement hshaz2s, a Stata routine that uses analytical expressions for the gradient vector and Hessian matrix of the log-likelihood function through the dual second-order moment (d2 ml) method. The empirical application estimates a discrete-time duration model for transitions between employment and unemployment using Spanish labor market microdata for young low-skilled workers over 2000-2019. The results show that apprenticeship contracts are associated with lower exit rates from employment than other temporary contracts, but not with faster transitions from unemployment back into employment. The estimates also reveal substantial state-specific unobserved heterogeneity, with a large latent group characterized by persistent spells in both states. Analytical second-order information also markedly reduces convergence time under richer heterogeneity structures. Overall, the article makes this class of two-state hazard models operational for applied research and provides new evidence on apprenticeship and temporary contracts in Spain.
Evidence from observational studies plays a central role in shaping public policy in health, education, and financial regulation, where randomized experiments are rarely feasible. Propensity score matching (PSM) is a widely used method to approximate fair comparisons between treatment and control groups. Incorporating machine learning into the estimation of propensity scores can strengthen prediction and enhance the credibility of findings. However, stronger predictive models create a “predictability paradox”. As predictive accuracy improves, estimated propensity scores for treated and control units become more distinct when treatment assignment is strongly predictable from observed covariates, revealing limited overlap between groups. In the limit, near-perfect prediction produces near-complete separation between groups, rendering traditional matching infeasible and confining inference to a narrow subset of units near the boundary of the propensity score distribution, a setting analogous to a regression discontinuity design (RDD). Researchers thus face perverse incentives to use weaker models for statistically significant but spurious results. These dynamics jeopardize the reliability of evidence for policy. To safeguard decision-making, we propose a simple reform: require that studies using PSM disclose model error rates, including false positive and false negative rates, along with information on overlap and effective sample size.
We propose a fully nonparametric empirical autoregressive copula framework for univariate time series, designed to capture nonlinear and asymmetric serial dependence while exactly preserving the empirical marginal distribution. The method decouples marginal behavior from temporal dependence by (i) constructing a shape-preserving empirical marginal via monotone interpolation and mapping observations to the unit interval, and (ii) estimating the lag-lead dependence through a nonparametric conditional AR(1) copula density on (0,1)2. To ensure stable estimation near the boundaries, we employ reflection-based kernel methods that mitigate edge effects and yield well-behaved conditional densities on the unit support. Forecasts are obtained from the implied conditional predictive density: we compute point forecasts either as conditional modes (maximum a posteriori) on the copula scale or as conditional means, and then back-transform exactly using the empirical quantile function, guaranteeing marginal fidelity and support-respecting predictions. Empirically, we evaluate the approach on three CBOE volatility indices (VIX, VXD, and RVX) and benchmark it against linear ARMA models, copula-based parametric competitors, and state-space/heteroskedasticity baselines (Local level, TVP-AR, and ARMA-GARCH). The results highlight that modeling the full conditional transition density nonparametrically can deliver competitive-often best or near-best-forecast accuracy across horizons, particularly in the presence of pronounced volatility regimes and asymmetric adjustments.
The relationship between capital structure and firm market valuation remains a central yet unresolved question in corporate finance, with outcomes shaped critically by industry-specific asset structures and financing environments. This study investigates how capital structure influences market valuations across two structurally divergent sectors in India, the asset-light information technology (IT) industry and the asset-intensive automobile industry, using balanced panel data for 14 firms in each sector over 2005–2024. Fixed effect and random effect panel regression models are employed to isolate the direct effect of leverage on earnings per share (EPS), with model selection determined by the Hausman specification test. Complementing these estimations, the Graphical Lasso is applied to recover a sparse conditional dependence network among key financial variables, an approach particularly suited to this research question, as capital structure, profitability, tangibility, and growth are jointly determined, rendering pairwise correlations insufficient for identifying genuine financial linkages. The findings establish that debt exerts a positive and statistically significant effect on market valuations in both sectors, but through distinct economic channels: moderate leverage amplifies profitable growth signals in IT firms, while tax shield benefits drive valuation in automobile firms, constrained by asset tangibility and debt-servicing thresholds. These results support trade-off theory in the automobile sector and pecking order logic in the IT sector, underscoring that sector-specific financing strategies yield superior valuation outcomes compared to universally applied capital structure prescriptions.
Aim: To develop and empirically evaluate a quantum-like tensor state model for bivariate time series, focusing on the joint dynamics of timber harvesting and residential construction across Polish regions. Methodology: Two related processes were encoded in a four-dimensional complex Hilbert space as a tensor product state whose residual dynamics were governed by a parametrised two-qubit unitary operator acting as a nonlinear second-stage correction to pooled OLS forecasts. The model was estimated on panel data for 16 voivodeships over 2005–2025 and compared with linear, autoregressive, polynomial, and tree-based residual benchmarks using MSE and MAE. Results: The tensor model substantially reduced global forecast errors for harvesting amplitudes relative to all benchmarks and achieved competitive performance for construction, with statistically significant gains over linear and low-order nonlinear specifications and win rates above 80–90% of regions for harvesting, while its accuracy was broadly comparable to a shallow regression tree for construction. Implications and recommendations: The findings indicate that quantum-inspired tensor state models can serve as practically useful tools for forecasting and interpreting time-varying cross-sector dependence in regional panels, supporting planning and risk assessment in forestry construction systems. Future research should extend the framework to multisector settings, richer entangling kernels, and partially regionalised operators, and explore applications to other domains with nonlinear, time-varying co-movement. Originality/value: This study provides one of the first applications of a quantum-like two-qubit tensor state model to classical economic time series, demonstrating that a low-dimensional unitary evolution can yield forecast accuracy at least comparable to strong classical benchmarks while offering a compact, interpretable representation of joint dynamics and entanglement-like interactions between sectors.
Aim: By implementing different policies and intervening in the economy, governments have a significant impact on the state of affairs. The purpose of this study was to investigate how government economic policies affect Nigeria's economic growth. Methodology: The study made use of secondary data sources. Secondary data were gathered from international organizations, statistical agencies, and official publications, including the World Bank, the Central Bank of Nigeria, and the National Bureau of Statistics. This study used quantitative analysis as its only method. The quantitative analysis involved econometric techniques, such as regression (multiple) analysis and panel modelling, to determine the statistical significance and direction of the relationships between the government economic policies and the measures of economic development in Nigeria. Results: An inverse relationship was found between the real gross domestic product (RGDP), tariffs (TARR), and interest rates (INTR). RGDP had a significant negative correlation (-0.6749) with INTR and a moderately negative correlation (-0.5774) with TARR. The interest rate (INTR) and tariff (TARR) had a positive correlation (0.7233), suggesting a potential association between the variables. There was a fairly positive association (0.7278) between sectoral support (SECSUPP) and the exchange rate (EXR). Implications and recommendations: According to the study, efforts should be made to bolster the beneficial effects that have been identified, such as advancements in technology or infrastructure, trade policies that could strengthen the exchange rate, and sector-specific government programmes that could strengthen the exchange. Originality/value: The study contributes to the discussion on the effectiveness of government economic policies in the context of a developing economy by integrating the analysis of multiple policy instruments ─ tariffs, interest rates, sectoral support, and the exchange rate ─ within a single econometric model for Nigeria.
Aim: This paper identifies a forward-looking Taylor-type rule with time-varying coefficients for Poland and derives the implied optimal policy-rate path for the National Bank of Poland over the period 2010-2024. It assesses how expectations, a time-varying neutral rate, and parameter recalibration alter the evaluation of monetary stance relative to a constant-parameter benchmark. Methodology: A New Keynesian-structured Bayesian SVAR was estimated on quarterly data, and conditional forecasts were used to obtain expected inflation and the expected output gap. The optimal policy-rate path and rule coefficients were recovered through constrained numerical optimisation under the effective lower bound by minimising the Equilibrium Monetary Policy Gap (EMPG), defined as the squared deviation of the policy rate from the neutral interest rate. Results: The forward-looking time-varying rule generated a substantially lower EMPG than the classic Taylor rule and indicated a marked divergence after 2022 between the conventional interest-rate gap and the broader monetary policy gap. Implications and recommendations: Static Taylor rules may misstate the policy stance when the neutral rate and reaction coefficients change over time. Originality/value: The paper provides novel evidence for Poland by jointly combining a forward-looking Taylor-rule framework, time-varying parameters, and an explicit neutral-rate benchmark within a New Keynesian BSVAR model.
This study explores the impact of internationalization on the financing decisions and finance costs of Chinese enterprises listed in Hong Kong, extending the pecking order theory to an international context. Utilizing data from 785 companies from 2010 to 2020, the research investigates how the degree of internationalization influences corporate finance strategies, with a focus on the mediating role of the pecking order and the moderating effects of international business factors. The findings reveal that while broader internationalization increases finance costs, deeper internationalization reduces them. Legal distance is found to negatively moderate this relationship, whereas the structure of the financial system positively influences it. The results suggest that multinational enterprises with extensive overseas resource allocation demonstrate greater flexibility in financing decisions, particularly in foreign markets characterized by strong investor protection and efficient direct finance mechanisms. Managers should be cautious about pursuing wide geographic expansion without adequate operating depth because a broad but shallow international presence may increase financing frictions. By contrast, deeper resource commitment abroad can strengthen financing flexibility and improve access to lower-cost funds, especially when institutional conditions in the financing market are favorable.
The Fourth Industrial Revolution (4IR) has introduced modern, high technologies that are automated, such as precision farming, to enhance agricultural production. However, this comes at the cost of human labor being replaced by machines that are deemed efficient. This study investigated the impact of 4IR automation on agricultural employment in South Africa, spanning from 1990 to 2024. To analyze this, the study employed the Johansen test for cointegration and the vector error correction model to test for long-run and short-run dynamics. Stationarity was achieved, and the Johansen test confirmed cointegration. The vector error correction model results revealed that both long-run and short-run relationships between 4IR automation and agricultural employment exist, indicating that human labor is particularly at risk of being replaced by automation, such as advanced agricultural machinery. The results imply that, although automation improved agricultural productivity, it caused an increase in agricultural unemployment within South Africa. Therefore, to balance the advancement of technology and agricultural employment, the study recommends skills improvement and government intervention for enhancing human labor within the agricultural sector.
Aim: The main object of this study was to present a comparison between GARCH models, i.e. the standard GARCH model, asymmetric GJR-GARCH, and logarithmic EGARCH on exchange rate (IDR/USD) volatility. Comparison of Symmetrical; Asymmetrical; and Logarithmic Models Using GARCH; GJR-GARCH; and EGARCH Method in Forecasting Indonesia–USA Currency Volatility. Methodology: The authors used GARCH, Glosten-Jagannathan-Runkle GARCH (GJR-GARCH) and Exponential GARCH (EGARCH) in estimating and forecasting exchange rate volatility. The variables were IDR/USD, Jakarta Stock Exchange Composite Index (JCI), World Oil Price, and Nominal Broad U.S. Dollar Index, while the data were daily, taken from World Bank, Federal Reserve Economic Data, and Indonesian Stock Exchange during 2006-2025. Results: The results revealed that in the GARCH method, there was high persistence of volatility, and the shocks were of long-lasting duration, but the model was symmetrical. The GJR-GARCH model showed that negative shocks have larger effects than positive shocks on IDR/USD but with problematic negative coefficients. Lastly, in the final comparison it was revealed that the EGARCH specifications were the most reliable in capturing asymmetric volatility dynamics, with strong evidence of leverage effects where negative shocks increase future volatility more than positive shocks. Implications and recommendations: As IDR/USD volatility took a long time to dissipate, and negative shocks had a significantly larger effect than positive shocks, this meant that the market reacted stronger on depreciation rather than on appreciation. Therefore, it was becoming essential for policymakers in Indonesia to provide an asymmetrical policy framework to prevent the negative shocks extending into a prolonged period of distrust by the market towards IDR. One of the actions to be taken was to increase interest rate. This was a preventive action to respond to depreciation, and at the same time acknowledging the asymmetric approach to responding to depreciation. Originality/value: The study compared three GARCH models to examine and forecast IDR/USD volatility, choosing one more statistically and economically reliable which makes this study unique. The findings present a comprehensive and methodologically established comparison that is unbiased and shows each model’s limitations and strengths. The study also provided additional contributions regarding integration of broad variables into the models, with the use of world oil price, JCI, and Nominal Broad US Index as variables.
Aim: This article investigates whether persistent homology and persistence entropy capture structural properties of financial time series beyond variance-based risk measures. Using data from the WIG20 index (2019–2024), the study examines whether topological descriptors reflect intrinsic geometric and temporal organization rather than merely volatility intensity. Methodology: Logarithmic returns are embedded using sliding-window delay coordinates and analysed with Vietoris–Rips persistent homology. Betti numbers, persistence diagrams and rolling 𝐻₁ persistence entropy are computed. Relationships with classical risk diagnostics are evaluated using linear correlations, nonlinear dependence measures, regime comparisons and shuffle-based tests. Results: Persistence entropy shows weak association with volatility and second-moment risk. Stronger relationships appear with higher-order distributional characteristics such as skewness and kurtosis. Volatility-based regimes do not significantly separate entropy, whereas a structural split around the 2022 geopolitical shock reveals a significant increase, indicating a shift in return geometry. Shuffle experiments confirm dependence on temporal ordering. Implications: Persistence entropy captures structural and temporal organization of financial returns and may complement classical econometric risk measures. Originality/value: The study shows that persistent homology reflects structural organization of return dynamics rather than acting as a volatility proxy.
