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37 pages, 4086 KiB  
Article
Should South Asian Stock Market Investors Think Globally? Investigating Safe Haven Properties and Hedging Effectiveness
by Md. Abu Issa Gazi, Md. Nahiduzzaman, Sanjoy Kumar Sarker, Mohammad Bin Amin, Md. Ahsan Kabir, Fadoua Kouki, Abdul Rahman bin S Senathirajah and László Erdey
Economies 2024, 12(11), 309; https://doi.org/10.3390/economies12110309 (registering DOI) - 15 Nov 2024
Abstract
In this study, we examine the critical question of whether global equity and bond assets (both green and non-green) offer effective hedging and safe haven properties against stock market risks in South Asia, with a focus on Bangladesh, India, Pakistan, and Sri Lanka. [...] Read more.
In this study, we examine the critical question of whether global equity and bond assets (both green and non-green) offer effective hedging and safe haven properties against stock market risks in South Asia, with a focus on Bangladesh, India, Pakistan, and Sri Lanka. The increasing integration of global financial markets and the volatility experienced during recent economic crises raise important questions regarding the resilience of South Asian markets and the potential protective role of global assets. Drawing on methods like VaR and CVaR tail risk estimators, the DCC-GJR-GARCH time-varying connectedness approach, and cost-effectiveness tools for hedging, we analyze data spanning from 2014 to 2022 to assess these relationships comprehensively. Our findings demonstrate that stock markets in Bangladesh experience lower levels of downside risk in each quantile; however, safe haven properties from the global financial markets are effective for Bangladeshi, Indian, and Pakistani stock markets during the crisis period. Meanwhile, the Sri Lankan stock market neither receives hedging usefulness nor safe haven benefits from the same marketplaces. Additionally, global green assets, specifically green bond assets, are more reliable sources to ensure the safest investment for South Asian investors. Finally, the portfolio implications suggest that while traditional global equity assets offer ideal portfolio weights for South Asian investors, global equity and bond assets (both green and non-green) are the cheapest hedgers for equity investors, particularly in the Bangladeshi, Pakistani, and Sri Lankan stock markets. Moreover, these results hold significant implications for investors seeking to optimize portfolios and manage risk, as well as for policymakers aiming to strengthen regional market resilience. By clarifying the protective capacities of global assets, particularly green ones, our study contributes to a nuanced understanding of portfolio diversification and financial stability strategies within emerging markets in South Asia. Full article
16 pages, 1956 KiB  
Article
The GARCH-EVT-Copula Approach to Investigating Dependence and Quantifying Risk in a Portfolio of Bitcoin and the South African Rand
by Thabani Ndlovu and Delson Chikobvu
J. Risk Financial Manag. 2024, 17(11), 504; https://doi.org/10.3390/jrfm17110504 - 8 Nov 2024
Viewed by 405
Abstract
This study uses a hybrid model of the exponential generalised auto-regressive conditional heteroscedasticity (eGARCH)-extreme value theory (EVT)-Gumbel copula model to investigate the dependence structure between Bitcoin and the South African Rand, and quantify the portfolio risk of an equally weighted portfolio. The Gumbel [...] Read more.
This study uses a hybrid model of the exponential generalised auto-regressive conditional heteroscedasticity (eGARCH)-extreme value theory (EVT)-Gumbel copula model to investigate the dependence structure between Bitcoin and the South African Rand, and quantify the portfolio risk of an equally weighted portfolio. The Gumbel copula, an extreme value copula, is preferred due to its versatile ability to capture various tail dependence structures. To model marginals, firstly, the eGARCH(1, 1) model is fitted to the growth rate data. Secondly, a mixture model featuring the generalised Pareto distribution (GPD) and the Gaussian kernel is fitted to the standardised residuals from an eGARCH(1, 1) model. The GPD is fitted to the tails while the Gaussian kernel is used in the central parts of the data set. The Gumbel copula parameter is estimated to be α=1.007, implying that the two currencies are independent. At 90%, 95%, and 99% levels of confidence, the portfolio’s diversification effects (DE) quantities using value at risk (VaR) and expected shortfall (ES) show that there is evidence of a reduction in losses (diversification benefits) in the portfolio compared to the risk of the simple sum of single assets. These results can be used by fund managers, risk practitioners, and investors to decide on diversification strategies that reduce their risk exposure. Full article
(This article belongs to the Special Issue Digital Economy and the Role of Accounting and Finance)
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<p>Scatter plots of simulated data of Gumbel copula (<math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mfenced open="|" close="|" separators="|"> <mrow> <mn>2</mn> </mrow> </mfenced> </mrow> </semantics></math>) with positive dependence (<b>top left</b>), no dependence (<math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mo>±</mo> <mn>1</mn> </mrow> </semantics></math>) (<b>top right</b> and <b>bottom right</b>), and negative dependence (<b>bottom left</b>).</p>
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<p>The empirical mixture model (semi-parametric) CDF and PDF plots of the Bitcoin standardised residuals.</p>
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<p>Diagnostic plots for the upper tails of the Bitcoin standardised residuals fitted using GPD.</p>
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<p>Diagnostic plots for the lower tails of the Bitcoin residuals fitted using GPD.</p>
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<p>The empirical mixture model (semi-parametric) CDF and PDF plots of the Rand standardised residuals.</p>
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<p>Diagnostic plots for the upper tails of the Rand residuals fitted using GPD.</p>
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<p>Diagnostic plots for the lower tails of the Rand residuals fitted using GPD.</p>
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<p>A scatter plot of the joint uniform marginal variates <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>u</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msub> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mrow> <mi>u</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> </mrow> </semantics></math> for the bivariate rates of Bitcoin and the Rand, respectively.</p>
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25 pages, 314 KiB  
Article
Heterogeneous Responses of Energy and Non-Energy Assets to Crises in Commodity Markets
by Dimitrios Vortelinos, Angeliki Menegaki, Ioannis Passas, Alexandros Garefalakis and Georgios Viskadouros
Energies 2024, 17(21), 5438; https://doi.org/10.3390/en17215438 - 31 Oct 2024
Viewed by 360
Abstract
In this study, we investigate the heterogeneity in energy and non-energy commodities by analyzing their four realized moments: returns, realized volatility, realized skewness and realized kurtosis. Utilizing monthly data, we examine two commodity categories over various crisis periods. This study examines a comparative [...] Read more.
In this study, we investigate the heterogeneity in energy and non-energy commodities by analyzing their four realized moments: returns, realized volatility, realized skewness and realized kurtosis. Utilizing monthly data, we examine two commodity categories over various crisis periods. This study examines a comparative approach to descriptive statistics across different crisis periods and the full sample and assesses the out-of-sample significance of heteroscedasticity using GARCH models. The findings reveal significant heterogeneity in both energy and non-energy commodities, with energy commodities exhibiting higher average returns and volatility. Crisis periods markedly influence the statistical properties of these commodities. GARCH models outperform ARMA models in forecasting realized moments, particularly for non-energy commodities. This research contributes to the literature by providing robust evidence of heterogeneity in commodity markets and highlights the importance of considering all four realized moments in such analyses. Full article
(This article belongs to the Section C: Energy Economics and Policy)
16 pages, 6519 KiB  
Article
Market Volatility vs. Economic Growth: The Role of Cognitive Bias
by Neha Parashar, Rahul Sharma, S. Sandhya and Apoorva Joshi
J. Risk Financial Manag. 2024, 17(11), 479; https://doi.org/10.3390/jrfm17110479 - 24 Oct 2024
Viewed by 819
Abstract
This study aims to investigate the interaction between market volatility, economic growth, and cognitive biases over the period from April 2006 to March 2024. Market volatility and economic growth are critical indicators that influence economic stability and investment behavior. Financial market volatility, defined [...] Read more.
This study aims to investigate the interaction between market volatility, economic growth, and cognitive biases over the period from April 2006 to March 2024. Market volatility and economic growth are critical indicators that influence economic stability and investment behavior. Financial market volatility, defined by abrupt and erratic changes in asset values, can have a big impact on the expansion and stability of the economy. According to conventional economic theory, there should be an inverse relationship between market volatility and economic growth since high volatility can discourage investment and erode trust. Market participants’ cognitive biases are a major aspect that complicates this connection. Due to our innate susceptibility to cognitive biases, including herd mentality, overconfidence, and loss aversion, humans can make poor decisions and increase market volatility. These prejudices frequently cause investors to behave erratically and irrationally, departing from reasonable expectations and causing inefficiencies in the market. Cognitive biases have the capacity to sustain feedback loops, which heighten market turbulence and may hinder economic expansion. Similarly, cognitive biases have the potential to cause investors to misread economic indicators or ignore important details, which would increase volatility. This study uses the generalized autoregressive conditional heteroskedasticity (GARCH) model on GDP growth data from the US, the UK, and India, alongside S&P 500, FTSE 100, and NIFTY 50 data sourced from Bloomberg, to examine evidence of these biases. The results show evidence of the predictive nature of market fluctuations on economic performance across the markets and highlight the substantial effects of cognitive biases on market volatility, disregarding economic fundamentals and growth, emphasizing the necessity of considering psychological factors in financial market analyses and developing strategies to mitigate their adverse effects. Full article
(This article belongs to the Special Issue Globalization and Economic Integration)
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<p>GARCH model fit vs. the standardized residuals of SPX, UKX, and NIFTY.</p>
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<p>Market volatility vs. economic growth (US).</p>
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<p>Market volatility vs. economic growth (UK).</p>
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<p>Market volatility vs. economic growth (India).</p>
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15 pages, 468 KiB  
Article
Evaluating Volatility Using an ANFIS Model for Financial Time Series Prediction
by Johanna M. Orozco-Castañeda, Sebastián Alzate-Vargas and Danilo Bedoya-Valencia
Risks 2024, 12(10), 156; https://doi.org/10.3390/risks12100156 - 30 Sep 2024
Viewed by 748
Abstract
This paper develops and implements an Autoregressive Integrated Moving Average model with an Adaptive Neuro-Fuzzy Inference System (ARIMA-ANFIS) for BTCUSD price prediction and risk assessment. The goal of these forecasts is to identify patterns from past data and achieve an understanding of the [...] Read more.
This paper develops and implements an Autoregressive Integrated Moving Average model with an Adaptive Neuro-Fuzzy Inference System (ARIMA-ANFIS) for BTCUSD price prediction and risk assessment. The goal of these forecasts is to identify patterns from past data and achieve an understanding of the future behavior of the price and its volatility. The proposed ARIMA-ANFIS model is compared with a benchmark ARIMA-GARCH model. To evaluated the adequacy of the models in terms of risk assessment, we compare the confidence intervals of the price and accuracy measures for the testing sample. Additionally, we implement the diebold and Mariano test to compare the accuracy of the two volatility forecasts. The results revealed that each volatility model focuses on different aspects of the data dynamics. The ANFIS model, while effective in certain scenarios, may expose one to unexpected risks due to its underestimation of volatility during turbulent periods. On the other hand, the GARCH(1,1) model, by producing higher volatility estimates, may lead to excessive caution, potentially reducing returns. Full article
(This article belongs to the Special Issue Volatility Modeling in Financial Market)
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<p>Typical ANFIS structure. Adapted from <a href="#B13-risks-12-00156" class="html-bibr">Jang</a> (<a href="#B13-risks-12-00156" class="html-bibr">1993</a>).</p>
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<p>Architecture for an ANFIS with four rules.</p>
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<p>BTC/USD price vs forecast from ARIMA model.</p>
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<p>Scatterplot for squared returns.</p>
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<p>Predictions in the testing sample with ANFIS and GARCH.</p>
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<p>Predictions and confidence intervals in the testing sample with ANFIS.</p>
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<p>Predictions and confidence intervals for the testing sample using GARCH(1,1).</p>
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26 pages, 2996 KiB  
Article
Mapping Risk–Return Linkages and Volatility Spillover in BRICS Stock Markets through the Lens of Linear and Non-Linear GARCH Models
by Raj Kumar Singh, Yashvardhan Singh, Satish Kumar, Ajay Kumar and Waleed S. Alruwaili
J. Risk Financial Manag. 2024, 17(10), 437; https://doi.org/10.3390/jrfm17100437 - 29 Sep 2024
Viewed by 857
Abstract
This paper explores the influence of the risk–return relationship and volatility spillover on stock market returns of emerging economies, with a particular focus on the BRICS countries. This research is undertaken in a context where discussions on de-dollarization and the expansion of BRICS [...] Read more.
This paper explores the influence of the risk–return relationship and volatility spillover on stock market returns of emerging economies, with a particular focus on the BRICS countries. This research is undertaken in a context where discussions on de-dollarization and the expansion of BRICS membership are gaining momentum, making it a novel and distinct exercise compared to prior studies. Utilizing econometric techniques to investigate daily market returns from 1 April 2008 to 31 March 2023, a period that witnessed major events like the global financial crisis, the COVID-19 pandemic, and the Russia–Ukraine conflict, linear and non-linear models like ARCH, GARCH, GARCH-M, EGARCH, and TGARCH, are employed to assess stock return volatility behaviour, assuming a Gaussian distribution of error terms. The diagnostic test confirms that the distribution is non-normal, stationary, and heteroscedastic. The key findings indicate a lack of the risk–return relationship across all BRICS stock markets, except for South Africa; a more pronounced effect of unpleasant news over pleasant news; a slow mean-reverting process in volatility; the EGARCH model is the best fit model as evidenced by a higher log likelihood and lower Akaike information criterion and Schwardz information criterion parameters; and finally, the presence of significant bidirectional and unidirectional spillover effects in the majority of instances. These findings are valuable for investors, regulators, and policymakers in enhancing returns and mitigating risk through portfolio diversification and informed decision making. Full article
(This article belongs to the Special Issue Risk Management in Capital Markets)
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<p>Trend analysis of GDP, population, trade and investment.</p>
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<p>Time Series Graphs of Closing Price Indices.</p>
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<p>Volatility Clustering of Stock Returns.</p>
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<p>News Impact Curves (ARCH Model).</p>
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<p>News Impact Curves (GARCH Model).</p>
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<p>New Impact Curves (GARCH Mean Model).</p>
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<p>New Impact Curves (EGARCH Model).</p>
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<p>Asymmetric News Impact Curves (TGARCH Model).</p>
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32 pages, 552 KiB  
Article
Bayesian Lower and Upper Estimates for Ether Option Prices with Conditional Heteroscedasticity and Model Uncertainty
by Tak Kuen Siu
J. Risk Financial Manag. 2024, 17(10), 436; https://doi.org/10.3390/jrfm17100436 - 29 Sep 2024
Viewed by 498
Abstract
This paper aims to leverage Bayesian nonlinear expectations to construct Bayesian lower and upper estimates for prices of Ether options, that is, options written on Ethereum, with conditional heteroscedasticity and model uncertainty. Specifically, a discrete-time generalized conditional autoregressive heteroscedastic (GARCH) model is used [...] Read more.
This paper aims to leverage Bayesian nonlinear expectations to construct Bayesian lower and upper estimates for prices of Ether options, that is, options written on Ethereum, with conditional heteroscedasticity and model uncertainty. Specifically, a discrete-time generalized conditional autoregressive heteroscedastic (GARCH) model is used to incorporate conditional heteroscedasticity in the logarithmic returns of Ethereum, and Bayesian nonlinear expectations are adopted to introduce model uncertainty, or ambiguity, about the conditional mean and volatility of the logarithmic returns of Ethereum. Extended Girsanov’s principle is employed to change probability measures for introducing a family of alternative GARCH models and their risk-neutral counterparts. The Bayesian credible intervals for “uncertain” drift and volatility parameters obtained from conjugate priors and residuals obtained from the estimated GARCH model are used to construct Bayesian superlinear and sublinear expectations giving the Bayesian lower and upper estimates for the price of an Ether option, respectively. Empirical and simulation studies are provided using real data on Ethereum in AUD. Comparisons with a model incorporating conditional heteroscedasticity only and a model capturing ambiguity only are presented. Full article
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<p>Time series plots for adjusted close prices and log returns.</p>
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<p>SACF and SPACF plots for log returns.</p>
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24 pages, 1437 KiB  
Article
Bitcoin, Fintech, Energy Consumption, and Environmental Pollution Nexus: Chaotic Dynamics with Threshold Effects in Tail Dependence, Contagion, and Causality
by Melike E. Bildirici, Özgür Ömer Ersin and Yasemen Uçan
Fractal Fract. 2024, 8(9), 540; https://doi.org/10.3390/fractalfract8090540 - 18 Sep 2024
Viewed by 912
Abstract
The study investigates the nonlinear contagion, tail dependence, and Granger causality relations with TAR-TR-GARCH–copula causality methods for daily Bitcoin, Fintech, energy consumption, and CO2 emissions in addition to examining these series for entropy, long-range dependence, fractionality, complexity, chaos, and nonlinearity with a [...] Read more.
The study investigates the nonlinear contagion, tail dependence, and Granger causality relations with TAR-TR-GARCH–copula causality methods for daily Bitcoin, Fintech, energy consumption, and CO2 emissions in addition to examining these series for entropy, long-range dependence, fractionality, complexity, chaos, and nonlinearity with a dataset spanning from 25 June 2012 to 22 June 2024. Empirical results from Shannon, Rényi, and Tsallis entropy measures; Kolmogorov–Sinai complexity; Hurst–Mandelbrot and Lo’s R/S tests; and Phillips’ and Geweke and Porter-Hudak’s fractionality tests confirm the presence of entropy, complexity, fractionality, and long-range dependence. Further, the largest Lyapunov exponents and Hurst exponents confirm chaos across all series. The BDS test confirms nonlinearity, and ARCH-type heteroskedasticity test results support the basis for the use of novel TAR-TR-GARCH–copula causality. The model estimation results indicate moderate to strong levels of positive and asymmetric tail dependence and contagion under distinct regimes. The novel method captures nonlinear causality dynamics from Bitcoin and Fintech to energy consumption and CO2 emissions as well as causality from energy consumption to CO2 emissions and bidirectional feedback between Bitcoin and Fintech. These findings underscore the need to take the chaotic and complex dynamics seriously in policy and decision formulation and the necessity of eco-friendly technologies for Bitcoin and Fintech. Full article
(This article belongs to the Special Issue Fractional-Order Dynamics and Control in Green Energy Systems)
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<p>Bitcoin prices, daily % change. Source: Yahoo Finance, <a href="https://finance.yahoo.com/quote/BTC-USD/history/" target="_blank">https://finance.yahoo.com/quote/BTC-USD/history/</a> (accessed on 29 August 2024).</p>
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<p>Energy consumption of Bitcoin network, daily % change. Source: Cambridge Center for Alternative Finance, Cambridge University, <a href="https://ccaf.io/cbnsi/cbeci" target="_blank">https://ccaf.io/cbnsi/cbeci</a> (accessed on 29 August 2024).</p>
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<p>STOXX global fintech index, daily % change. Source: STOXX Global Fintech Index, <a href="https://stoxx.com/index/stxftgr/" target="_blank">https://stoxx.com/index/stxftgr/</a> (accessed on 29 August 2024).</p>
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<p>Global CO<sub>2</sub> emissions, daily % change. Source: Global Monitoring Laboratory, National Oceanic and Atmospheric Admin., <a href="https://gml.noaa.gov/ccgg/trends/data.html" target="_blank">https://gml.noaa.gov/ccgg/trends/data.html</a> (accessed on 29 August 2024).</p>
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<p>Tsallis entropy for varying entropic index q and Shannon entropy with fixed q = 1.</p>
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39 pages, 1816 KiB  
Article
The Interplay of Dietary Habits, Economic Factors, and Globalization: Assessing the Role of Institutional Quality
by Mohammad Naim Azimi, Mohammad Mafizur Rahman and Tek Maraseni
Nutrients 2024, 16(18), 3116; https://doi.org/10.3390/nu16183116 - 15 Sep 2024
Viewed by 1308
Abstract
Background: Dietary habits are pivotal for population health and well-being, yet remain a pressing global issue, particularly in Sub-Saharan Africa (SSA), where economic instability and institutional challenges exacerbate dietary problems. Despite extensive research, there is a notable gap in the literature regarding the [...] Read more.
Background: Dietary habits are pivotal for population health and well-being, yet remain a pressing global issue, particularly in Sub-Saharan Africa (SSA), where economic instability and institutional challenges exacerbate dietary problems. Despite extensive research, there is a notable gap in the literature regarding the direct and interactive effects of institutional quality and inflationary shocks on dietary habits. Methods: This study delves into these complex interplays across 44 SSA nations from 2002 to 2022. Employing an innovative entropy method (EM) and the generalized autoregressive conditional heteroskedasticity (GARCH) modeling, the study introduces an inclusive institutional quality index and an inflationary shock predictor as crucial determinants of dietary habits in the literature. Results: The results from the panel-corrected standard error (PCSE) method and feasible generalized least squares (FGLS) model reveal that per capita GDP, school enrollment rate, government expenditures, globalization index, and urbanization are positively associated with population dietary habits, while inflationary shock, food insecurity, and unemployment rate exert negative influences. Notably, institutional quality acts as a catalyst, amplifying the positive effects of the former group and absorbing the negative impacts of the latter on population dietary habits. Additionally, a dynamic panel causality analysis confirms a bidirectional causality nexus between population dietary habits and all variables, except for inflationary shocks, which demonstrate a unidirectional causality link. Conclusions: These findings carry significant policy implications, underscoring the complex dynamics between institutional quality, inflationary shocks, and dietary habits in the region. The bidirectional causality highlights the need for holistic interventions that address economic, social, and institutional factors simultaneously. Moreover, the unidirectional causality of inflationary shocks on dietary habits suggests that stabilizing inflation is critical to protecting dietary habits. These results provide critical insights for policymakers to design targeted interventions aimed at improving nutrition, bolstering institutional frameworks, and ensuring public health resilience in the face of economic and social shocks. Full article
(This article belongs to the Section Nutrition and Public Health)
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<p>Historical global food price index from 1990 to 2024. Source: UN-FAO [<a href="#B5-nutrients-16-03116" class="html-bibr">5</a>]; depicted by authors.</p>
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<p>Constructed GS and InQ for SSA. Source: authors’ depiction.</p>
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<p>Research process. Notes: CSD: cross-sectional dependence, SH: slope heterogeneity, CIPS: cross-sectionally augmented Im, Pesaran, and Shin, PCSE: panel-corrected standard error, FGLS: feasible generalized least square.</p>
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<p>Summary of the results. Source: authors’ depiction.</p>
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52 pages, 6746 KiB  
Article
COVID-19 and Uncertainty Effects on Tunisian Stock Market Volatility: Insights from GJR-GARCH, Wavelet Coherence, and ARDL
by Emna Trabelsi
J. Risk Financial Manag. 2024, 17(9), 403; https://doi.org/10.3390/jrfm17090403 - 9 Sep 2024
Viewed by 708
Abstract
This study rigorously investigates the impact of COVID-19 on Tunisian stock market volatility. The investigation spans from January 2020 to December 2022, employing a GJR-GARCH model, bias-corrected wavelet analysis, and an ARDL approach. Specific variables related to health measures and government interventions are [...] Read more.
This study rigorously investigates the impact of COVID-19 on Tunisian stock market volatility. The investigation spans from January 2020 to December 2022, employing a GJR-GARCH model, bias-corrected wavelet analysis, and an ARDL approach. Specific variables related to health measures and government interventions are incorporated. The findings highlight that confirmed and death cases contribute significantly to the escalation in TUNINDEX volatility when using both the conditional variance and the realized volatility. Interestingly, aggregate indices related to government interventions exhibit substantial impacts on the realized volatility, indicating a relative resilience of the Tunisian stock market amidst the challenges posed by COVID-19. However, the application of the bias-corrected wavelet analysis yields more subtle outcomes in terms of the correlations of both measures of volatility to the same metrics. Our econometric implications bear on the application of such a technique, as well as on the use of the realized volatility as an accurate measure of the “true” value of volatility. Nevertheless, the measures and actions undertaken by the authorities do not exclude fear and insecurity from investors due to another virus or any other crisis. The positive and long-term impact on the volatility of US equity market uncertainty, VIX, economic policy uncertainty (EPU), and the infectious disease EMV tracker (IDEMV) is obvious through the autoregressive distributed lag model (ARDL). A potential vulnerability of the Tunisian stock market to future shocks is not excluded. Government and stock market authorities should grapple with economic and financial fallout and always instill investor confidence. Importantly, our results put mechanisms such as overreaction to public news and (in)efficient use of information under test. Questioning the accuracy of announcements is then recommended. Full article
(This article belongs to the Special Issue Stability of Financial Markets and Sustainability Post-COVID-19)
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<p>Evolution of TUNINDEX stock return (2 January 2020–30 December 2022). Source: The Author.</p>
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<p>Wavelet transform coherence: Tunisian stock return volatility versus COVID-19 cases rate. Notes: The black contour shows where the spectrum is significantly different from red noise at the 5% level. The lighter shade represents the cone of influence, marking high-power areas and indicating the autocorrelation of wavelet power at each scale. The horizontal axis represents time from 3 March 2020 to 30 December 2022, and the vertical axis denotes scale bands with daily frequency. Arrows to the right (left) indicate in-phase (out-of-phase) relationships, meaning a positive (negative) connection. If arrows move right and up (down), the first variable “m” (cases rate) drives (follows), while if arrows move left and up (down), variable “n” (TUNINDEX volatility) leads (lags). This visualization helps understand the dynamic relationships between variables. Source: The Author.</p>
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<p>Wavelet transform coherence: Tunisian stock return volatility versus COVID-19 death rate. Notes: The black contour identifies areas where the spectrum is statistically significant at the 5% level compared to red noise. The lighter shade designates the cone of influence, outlining high-power regions. Time, ranging from 23 March 2020 to 30 December 2022, is represented on the horizontal axis, while the vertical axis denotes scale bands with daily frequency. Source: The Author.</p>
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<p>Wavelet transform coherence: Tunisian stock return volatility versus stringency index. Notes: The black contour identifies areas where the spectrum is statistically significant at the 5% level compared to red noise. The lighter shade designates the cone of influence, outlining high-power regions. Time, ranging from 3 January 2020 to 30 December 2022, is represented on the horizontal axis, while the vertical axis denotes scale bands with daily frequency. Source: The Author.</p>
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<p>Wavelet transform coherence: Tunisian stock return volatility versus containment health index. Notes: The black contour identifies areas where the spectrum is statistically significant at the 5% level compared to red noise. The lighter shade designates the cone of influence, outlining high-power regions. Time, ranging from 3 January 2020 to 30 December 2022, is represented on the horizontal axis, while the vertical axis denotes scale bands with daily frequency. Source: The Author.</p>
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<p>Wavelet transform coherence: Tunisian stock return volatility versus economic support index. Notes: The black contour identifies areas where the spectrum is statistically significant at the 5% level compared to red noise. The lighter shade designates the cone of influence, outlining high-power regions. Time, ranging from 3 January 2020 to 30 December 2022, is represented on the horizontal axis, while the vertical axis denotes scale bands with daily frequency. Source: The Author.</p>
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<p>Wavelet transform coherence: Tunisian stock return volatility versus government response index. Notes: The black contour identifies areas where the spectrum is statistically significant at the 5% level compared to red noise. The lighter shade designates the cone of influence, outlining high-power regions. Time, ranging from 3 January 2020 to 30 December 2022, is represented on the horizontal axis, while the vertical axis denotes scale bands with daily frequency. Source: The Author.</p>
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<p>Wavelet transform coherence: Tunisian stock return volatility versus school closing index. Notes: The black contour identifies areas where the spectrum is statistically significant at the 5% level compared to red noise. The lighter shade designates the cone of influence, outlining high-power regions. Time, ranging from 3 January 2020 to 30 December 2022, is represented on the horizontal axis, while the vertical axis denotes scale bands with daily frequency. Source: The Author.</p>
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<p>Wavelet transform coherence: Tunisian stock return volatility versus workplace closing. Notes: The black contour identifies areas where the spectrum is statistically significant at the 5% level compared to red noise. The lighter shade designates the cone of influence, outlining high-power regions. Time, ranging from 3 January 2020 to 30 December 2022, is represented on the horizontal axis, while the vertical axis denotes scale bands with daily frequency. Source: The Author.</p>
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<p>Wavelet transform coherence: Tunisian stock return volatility versus public events canceling. Notes: The black contour identifies areas where the spectrum is statistically significant at the 5% level compared to red noise. The lighter shade designates the cone of influence, outlining high-power regions. Time, ranging from 3 January 2020 to 30 December 2022, is represented on the horizontal axis, while the vertical axis denotes scale bands with daily frequency. Source: The Author.</p>
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<p>Wavelet transform coherence: Tunisian stock return volatility versus public gathering restrictions. Notes: The black contour identifies areas where the spectrum is statistically significant at the 5% level compared to red noise. The lighter shade designates the cone of influence, outlining high-power regions. Time, ranging from 3 January 2020 to 30 December 2022, is represented on the horizontal axis, while the vertical axis denotes scale bands with daily frequency. Source: The Author.</p>
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<p>Wavelet transform coherence: Tunisian stock return volatility versus public transport closure. Notes: The black contour identifies areas where the spectrum is statistically significant at the 5% level compared to red noise. The lighter shade designates the cone of influence, outlining high-power regions. Time, ranging from 3 January 2020 to 30 December 2022, is represented on the horizontal axis, while the vertical axis denotes scale bands with daily frequency. Source: The Author.</p>
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<p>Wavelet transform coherence: Tunisian stock return volatility versus stay-at-home requirements. Notes: The black contour identifies areas where the spectrum is statistically significant at the 5% level compared to red noise. The lighter shade designates the cone of influence, outlining high-power regions. Time, ranging from 3 January 2020 to 30 December 2022, is represented on the horizontal axis, while the vertical axis denotes scale bands with daily frequency. Source: The Author.</p>
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<p>Wavelet transform coherence: Tunisian stock return volatility versus internal movement restrictions. Notes: The black contour identifies areas where the spectrum is statistically significant at the 5% level compared to red noise. The lighter shade designates the cone of influence, outlining high-power regions. Time, ranging from 3 January 2020 to 30 December 2022, is represented on the horizontal axis, while the vertical axis denotes scale bands with daily frequency. Source: The Author.</p>
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<p>Wavelet transform coherence: Tunisian stock return volatility versus international travel control. Notes: The black contour identifies areas where the spectrum is statistically significant at the 5% level compared to red noise. The lighter shade designates the cone of influence, outlining high-power regions. Time, ranging from 3 January 2020 to 30 December 2022, is represented on the horizontal axis, while the vertical axis denotes scale bands with daily frequency. Source: The Author.</p>
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<p>Wavelet transform coherence: Tunisian stock return volatility versus public information gathering. Notes: The black contour identifies areas where the spectrum is statistically significant at the 5% level compared to red noise. The lighter shade designates the cone of influence, outlining high-power regions. Time, ranging from 3 January 2020 to 30 December 2022, is represented on the horizontal axis, while the vertical axis denotes scale bands with daily frequency. Source: The Author.</p>
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<p>Wavelet transform coherence: Tunisian realized volatility versus the COVID-19 cases rate. Notes: The black contour shows where the spectrum is significantly different from red noise at the 5% level. The lighter shade represents the cone of influence, marking high-power areas, indicating autocorrelation of wavelet power at each scale. The horizontal axis represents time from 3 March 2020 to 30 December 2022, and the vertical axis denotes scale bands with daily frequency. Arrows to the right (left) indicate in-phase (out-of-phase) relationships, meaning a positive (negative) connection. If arrows move right and up (down), the first variable “m” (cases rate) drives (follows), while if arrows move left and up (down), variable “n” (TUNINDEX volatility) leads (lags). This visualization helps understand the dynamic relationships between variables. Source: The Author.</p>
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<p>Wavelet transform coherence: Tunisian realized volatility versus COVID-19 death rate. Notes: The black contour identifies areas where the spectrum is statistically significant at the 5% level compared to red noise. The lighter shade designates the cone of influence, outlining high-power regions. Time, ranging from 23 March 2020 to 30 December 2022, is represented on the horizontal axis, while the vertical axis denotes scale bands with daily frequency. Source: The Author.</p>
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<p>Wavelet transform coherence: Tunisian realized volatility versus stringency index. Notes: The black contour identifies areas where the spectrum is statistically significant at the 5% level compared to red noise. The lighter shade designates the cone of influence, outlining high-power regions. Time, ranging from 3 January 2020 to 30 December 2022, is represented on the horizontal axis, while the vertical axis denotes scale bands with daily frequency. Source: The Author.</p>
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<p>Wavelet transform coherence: Tunisian realized volatility versus containment health index. Notes: The black contour identifies areas where the spectrum is statistically significant at the 5% level compared to red noise. The lighter shade designates the cone of influence, outlining high-power regions. Time, ranging from 3 January 2020 to 30 December 2022, is represented on the horizontal axis, while the vertical axis denotes scale bands with daily frequency. Source: The Author.</p>
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<p>Wavelet transform coherence: Tunisian realized volatility versus economic support index. Notes: The black contour identifies areas where the spectrum is statistically significant at the 5% level compared to red noise. The lighter shade designates the cone of influence, outlining high-power regions. Time, ranging from 3 January 2020 to 30 December 2022, is represented on the horizontal axis, while the vertical axis denotes scale bands with daily frequency. Source: The Author.</p>
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<p>Tunisian realized volatility versus government response index. Notes: The black contour identifies areas where the spectrum is statistically significant at the 5% level compared to red noise. The lighter shade designates the cone of influence, outlining high-power regions. Time, ranging from 3 January 2020 to 30 December 2022, is represented on the horizontal axis, while the vertical axis denotes scale bands with daily frequency. Source: The Author.</p>
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<p>Wavelet transform coherence: Tunisian realized volatility versus school closing index. Notes: The black contour identifies areas where the spectrum is statistically significant at the 5% level compared to red noise. The lighter shade designates the cone of influence, outlining high-power regions. Time, ranging from 3 January 2020 to 30 December 2022, is represented on the horizontal axis, while the vertical axis denotes scale bands with daily frequency. Source: The Author.</p>
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<p>Wavelet transform coherence: Tunisian realized volatility versus workplace closing. Notes: The black contour identifies areas where the spectrum is statistically significant at the 5% level compared to red noise. The lighter shade designates the cone of influence, outlining high-power regions. Time, ranging from 3 January 2020 to 30 December 2022, is represented on the horizontal axis, while the vertical axis denotes scale bands with daily frequency. Source: The Author.</p>
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<p>Wavelet transform coherence: Tunisian stock return volatility versus public events canceling. Notes: The black contour identifies areas where the spectrum is statistically significant at the 5% level compared to red noise. The lighter shade designates the cone of influence, outlining high-power regions. Time, ranging from 3 January 2020 to 30 December 2022, is represented on the horizontal axis, while the vertical axis denotes scale bands with daily frequency. Source: The Author.</p>
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<p>Wavelet transform coherence: Tunisian realized volatility versus public gathering restrictions. Notes: The black contour identifies areas where the spectrum is statistically significant at the 5% level compared to red noise. The lighter shade designates the cone of influence, outlining high-power regions. Time, ranging from 3 January 2020 to 30 December 2022, is represented on the horizontal axis, while the vertical axis denotes scale bands with daily frequency. Source: The Author.</p>
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<p>Wavelet transform coherence: Tunisian realized volatility versus public transport closure. Notes: The black contour identifies areas where the spectrum is statistically significant at the 5% level compared to red noise. The lighter shade designates the cone of influence, outlining high-power regions. Time, ranging from 3 January 2020 to 30 December 2022, is represented on the horizontal axis, while the vertical axis denotes scale bands with daily frequency. Source: The Author.</p>
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<p>Wavelet transform coherence: Tunisian realized volatility versus stay-at-home requirements. Notes: The black contour identifies areas where the spectrum is statistically significant at the 5% level compared to red noise. The lighter shade designates the cone of influence, outlining high-power regions. Time, ranging from 3 January 2020 to 30 December 2022, is represented on the horizontal axis, while the vertical axis denotes scale bands with daily frequency. Source: The Author.</p>
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<p>Wavelet transform coherence: Tunisian realized volatility versus internal movement restrictions. Notes: The black contour identifies areas where the spectrum is statistically significant at the 5% level compared to red noise. The lighter shade designates the cone of influence, outlining high-power regions. Time, ranging from 3 January 2020 to 30 December 2022, is represented on the horizontal axis, while the vertical axis denotes scale bands with daily frequency. Source: The Author.</p>
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<p>Wavelet transform coherence: Tunisian realized volatility versus international travel control. Notes: The black contour identifies areas where the spectrum is statistically significant at the 5% level compared to red noise. The lighter shade designates the cone of influence, outlining high-power regions. Time, ranging from 3 January 2020 to 30 December 2022, is represented on the horizontal axis, while the vertical axis denotes scale bands with daily frequency. Source: The Author.</p>
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<p>Wavelet transform coherence: Tunisian realized volatility versus public information gathering. Notes: The black contour identifies areas where the spectrum is statistically significant at the 5% level compared to red noise. The lighter shade designates the cone of influence, outlining high-power regions. Time, ranging from 3 January 2020 to 30 December 2022, is represented on the horizontal axis, while the vertical axis denotes scale bands with daily frequency. Source: The Author.</p>
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17 pages, 935 KiB  
Article
Analyzing the Selective Stock Price Index Using Fractionally Integrated and Heteroskedastic Models
by Javier E. Contreras-Reyes, Joaquín E. Zavala and Byron J. Idrovo-Aguirre
J. Risk Financial Manag. 2024, 17(9), 401; https://doi.org/10.3390/jrfm17090401 - 7 Sep 2024
Cited by 1 | Viewed by 749
Abstract
Stock market indices are important tools to measure and compare stock market performance. The Selective Stock Price (SSP) index reflects fluctuations in a set value of financial instruments of Santiago de Chile’s stock exchange. Stock indices also reflect volatility linked to high uncertainty [...] Read more.
Stock market indices are important tools to measure and compare stock market performance. The Selective Stock Price (SSP) index reflects fluctuations in a set value of financial instruments of Santiago de Chile’s stock exchange. Stock indices also reflect volatility linked to high uncertainty or potential investment risk. However, economic shocks are altering volatility. Evidence of long memory in SSP time series also exists, which implies long-term persistence. In this paper, we studied the volatility of SSP time series from January 2010 to September 2023 using fractionally heteroskedastic models. We considered the Autoregressive Fractionally Integrated Moving Average (ARFIMA) process with Generalized Autoregressive Conditional Heteroskedasticity (GARCH) innovations—the ARFIMA-GARCH model—for SSP log returns, and the fractionally integrated GARCH, or FIGARCH model, was compared with a classical GARCH one. The results show that the ARFIMA-GARCH model performs best in terms of volatility fit and predictive quality. This model allows us to obtain a better understanding of the observed volatility and its behavior, which contributes to more effective investment risk management in the stock market. Moreover, the proposed model detects the influence volatility increments of the SSP index linked to external factors that impact the economic outlook, such as China’s economic slowdown in 2012 and the subprime crisis in 2008. Full article
(This article belongs to the Special Issue Political Risk Management in Financial Markets)
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<p>Selective Stock Price index (<b>top</b>) and log returns (<b>bottom</b>), 2 January 2010 to 30 September 2023.</p>
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<p>Sample autocorrelation function of SSP log-returns (<b>top</b>). Absolute value of SSP log returns (<b>middle</b>) and squares of SSP log returns (<b>bottom</b>).</p>
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<p>Histogram of SSP log returns (2 January 2010 to 30 September 2023).</p>
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<p>Left to right: Histogram of standardized residuals and sample ACF of residuals and square residuals of the GARCH<math display="inline"><semantics> <mrow> <mo>(</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>)</mo> </mrow> </semantics></math> model.</p>
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<p>Left to right: Histogram of standardized residuals, sample ACF of residuals, square residuals of ARFIMA<math display="inline"><semantics> <mrow> <mo>(</mo> <mn>2</mn> <mo>,</mo> <mi>d</mi> <mo>,</mo> <mn>1</mn> <mo>)</mo> </mrow> </semantics></math>-GARCH<math display="inline"><semantics> <mrow> <mo>(</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>)</mo> </mrow> </semantics></math> model.</p>
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<p>Left to right: Histogram of standardized residuals, sample ACF of residuals, square residuals of FIGARCH<math display="inline"><semantics> <mrow> <mo>(</mo> <mn>1</mn> <mo>,</mo> <mi>d</mi> <mo>,</mo> <mn>1</mn> <mo>)</mo> </mrow> </semantics></math> model.</p>
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<p>Estimated volatility under GARCH, ARFIMA-GARCH, and FIGARCH models for SSP log returns (January 2010–September 2023).</p>
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20 pages, 2817 KiB  
Article
The Impact of COVID-19 Pandemic on the Jordanian Stock Market Returns Volatility: Evidence from ASE20
by Nahil Ismail Saqfalhait and Omar Mohammad Alzoubi
Economies 2024, 12(9), 238; https://doi.org/10.3390/economies12090238 - 6 Sep 2024
Viewed by 738
Abstract
This research examines the impact of the COVID-19 pandemic on the volatility behavior of Amman Stock Exchange (ASE) returns using ARMA–GARCH-type models for three sub-periods: pre-COVID-19, during COVID-19, and post-COVID-19. The research finds that volatility persistence is significant across all periods, with the [...] Read more.
This research examines the impact of the COVID-19 pandemic on the volatility behavior of Amman Stock Exchange (ASE) returns using ARMA–GARCH-type models for three sub-periods: pre-COVID-19, during COVID-19, and post-COVID-19. The research finds that volatility persistence is significant across all periods, with the pandemic period showing the highest impact of shocks. Bad news has no statistically significant impact on volatility in the pre-COVID-19 period or during the pandemic, while in the post-pandemic period, good news significantly influences volatility. Additionally, there exist notable changes in the autocorrelation and the shock structure of the AR and MA components. Considering these alterations in the asymmetric effects, the AR and MA components suggest significant shifts in market dynamics, investor sentiments, and economic policies in response to pandemic experiences. Full article
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<p>The ASE20 index over the full data set.</p>
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<p>Returns of the ASE20 index over the full data set.</p>
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<p>Returns of the ASE20 index over the pre-pandemic data set.</p>
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<p>Returns of the ASE20 index over the pandemic data set.</p>
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<p>Returns of the ASE20 index over the post-pandemic data set.</p>
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17 pages, 2569 KiB  
Proceeding Paper
Oil Price Volatility and MENA Stock Markets: A Comparative Analysis of Oil Exporters and Importers
by Khalil Mhadhbi and Ines Guelbi
Eng. Proc. 2024, 68(1), 63; https://doi.org/10.3390/engproc2024068063 - 2 Sep 2024
Viewed by 775
Abstract
This paper explores the transmission of volatility from Brent oil price evolution to the stock returns of 7 MENA countries, encompassing three importers and four exporters, after excluding four initial countries using the ARCH test. Employing the GARCH-BEKK estimation method, we detect this [...] Read more.
This paper explores the transmission of volatility from Brent oil price evolution to the stock returns of 7 MENA countries, encompassing three importers and four exporters, after excluding four initial countries using the ARCH test. Employing the GARCH-BEKK estimation method, we detect this transmission from January 2008 to September 2022. The results reveal significant volatility persistence across six stock markets with three importer countries and three exporters. These findings align with Shiller’s theory, indicating high volatility in financial markets. Tunisia’s stock market shows sensitivity to oil market developments, while the Omani market demonstrates volatility transfer from Brent oil prices. However, Morocco’s market exhibits resilience, with no significant transmission from international oil prices. Exporting countries, except the UAE, display significant and positive coefficients, indicating volatility transmission. The study suggests further research into underlying mechanisms and recommends policymakers and investors implement strategies to mitigate volatility effects. Advanced modeling and behavioral insights can enhance risk management strategies. Full article
(This article belongs to the Proceedings of The 10th International Conference on Time Series and Forecasting)
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<p>Trends in the evolution of global Brent oil prices and stock market yields of MENA oil-importing countries.</p>
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<p>Trends in the evolution of global Brent oil prices and stock market yields of MENA oil-importing countries.</p>
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<p>Trends in the evolution of global Brent oil prices and stock market yields of MENA oil-exporting countries.</p>
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20 pages, 933 KiB  
Article
Improving Volatility Forecasting: A Study through Hybrid Deep Learning Methods with WGAN
by Adel Hassan A. Gadhi, Shelton Peiris and David E. Allen
J. Risk Financial Manag. 2024, 17(9), 380; https://doi.org/10.3390/jrfm17090380 - 23 Aug 2024
Viewed by 705
Abstract
This paper examines the predictive ability of volatility in time series and investigates the effect of tradition learning methods blending with the Wasserstein generative adversarial network with gradient penalty (WGAN-GP). Using Brent crude oil returns price volatility and environmental temperature for the city [...] Read more.
This paper examines the predictive ability of volatility in time series and investigates the effect of tradition learning methods blending with the Wasserstein generative adversarial network with gradient penalty (WGAN-GP). Using Brent crude oil returns price volatility and environmental temperature for the city of Sydney in Australia, we have shown that the corresponding forecasts have improved when combined with WGAN-GP models (i.e., ANN-(WGAN-GP), LSTM-ANN-(WGAN-GP) and BLSTM-ANN (WGAN-GP)). As a result, we conclude that incorporating with WGAN-GP will’ significantly improve the capabilities of volatility forecasting in standard econometric models and deep learning techniques. Full article
(This article belongs to the Section Financial Markets)
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<p>Flowchart for the Data Analysis Process.</p>
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<p>Daily Brent Oil Price (2012–2022).</p>
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<p>Oil Price Returns from 2012 to 2022.</p>
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<p>Autocorrelation Function (ACF) for Oil Price return Data.</p>
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<p>Partial Autocorrelation Function (PACF) for Oil Price Data.</p>
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<p>Oil Price Returns.</p>
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<p>Temperature Over Time in Sydney, Australia (2013–2023).</p>
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<p>Autocorrelation Function (ACF) for Temperature Data.</p>
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<p>Partial Autocorrelation Function (PACF) for Temperature Data.</p>
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<p>Annualized Temperature Data.</p>
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<p>Standardized residuals and annualized conditional volatility.</p>
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19 pages, 837 KiB  
Article
Signs of Fluctuations in Energy Prices and Energy Stock-Market Volatility in Brazil and in the US
by Gabriel Arquelau Pimenta Rodrigues, André Luiz Marques Serrano, Gabriela Mayumi Saiki, Matheus Noschang de Oliveira, Guilherme Fay Vergara, Pedro Augusto Giacomelli Fernandes, Vinícius Pereira Gonçalves and Clóvis Neumann
Econometrics 2024, 12(3), 24; https://doi.org/10.3390/econometrics12030024 - 23 Aug 2024
Viewed by 1017
Abstract
Volatility reflects the degree of variation in a time series, and a measurement of the stock performance in the energy sector can help one understand the pattern of fluctuations within this industry, as well as the factors that influence it. One of these [...] Read more.
Volatility reflects the degree of variation in a time series, and a measurement of the stock performance in the energy sector can help one understand the pattern of fluctuations within this industry, as well as the factors that influence it. One of these factors could be the COVID-19 pandemic, which led to extreme volatility within the stock market in several economic sectors. It is essential to understand this regime of volatility so that robust financial strategies can be adopted to handle it. This study used stock data from the Yahoo! Finance API and data from the energy-price database from the US Energy Information Administration to conduct a comparative analysis of the volatility in the energy sector in Brazil and in the United States, as well as of the energy prices in California. The volatility in these time series were modeled using GARCH. The stock volatility regimes, both before and after COVID-19, were identified with a Markov switching model; the spillover index between the energy markets in the USA and in Brazil was evaluated with the Diebold–Yilmaz index; and the causality between the energy stock price and the energy prices was measured with the Granger causality test. The findings of this study show that (i) the volatility regime introduced by COVID-19 is still prevalent in Brazil and in the USA, (ii) the changes in the energy market in the US affect the Brazilian market significantly more than the reverse, and (iii) there is a causality relationship between the energy stock markets and the energy prices in California. These results may assist in the achievement of effective regulation and economic planning, while also supporting better market interventions. Also, acknowledging the persistent COVID-19-induced volatility can help with developing strategies for future crisis resilience. Full article
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<p>Diagram representing the conducted study.</p>
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<p>Aggregated stock-price time series according to their weights. The vertical line represents the date WHO declared COVID-19 a pandemic.</p>
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<p>The once differentiation of the aggregated stock price time series. Horizontal red lines represent the reference at <math display="inline"><semantics> <mrow> <mo>±</mo> <mn>0.25</mn> </mrow> </semantics></math>.</p>
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<p>Distributions of the stock-price time series. (<b>a</b>) Companies operating in Brazil. (<b>b</b>) Companies operating in the USA. (<b>c</b>) Companies operating in California.</p>
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<p>The Markov switching model for the stock market in the aggregated time series. The vertical line represents the date WHO declared COVID-19 a pandemic.</p>
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<p>The normalized and aggregated energy stock price time series when differentiated one and the normalized energy-price cross-correlation heatmap.</p>
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<p>The normalized and aggregated energy stock price time-series when differentiated once and the normalized energy price.</p>
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<p>Comparison of the normalized energy stock price and left shifted energy price. (<b>a</b>) The normalized and aggregated energy stock price time-series when differentiated once and the normalized energy price when left-shifted at 88 days. (<b>b</b>) The normalized and aggregated energy stock price time-series when differentiated once and the normalized energy price when left-shifted at 100 days.</p>
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<p>Granger causality <span class="html-italic">p</span>-values.</p>
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