GARCH Research Articles

Mapping Risk–Return Linkages and Volatility Spillover in BRICS Stock Markets through the Lens of Linear and Non-Linear GARCH Models

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.

Observable or latent Markov chains for score-driven regime-switching volatility?

ABSTRACT We study the statistical and forecasting performances of two regime-switching Beta-t-EGARCH (exponential generalized autoregressive conditional heteroscedasticity) models, i.e. observable-switching (OS) Beta-t-EGARCH and Markov-switching (MS) Beta-t-EGARCH. Both are non-path-dependent score-driven regime-switching volatility models, and their regime-switching specifications can be related to corresponding non-path-dependent Markov-switching GARCH (MS-GARCH) specifications. We present the estimation procedures for OS-Beta-t-EGARCH and MS-Beta-t-EGARCH. We use data on the weekly log-returns of the Standard & Poor’s 500 (S&P 500) index and a random sample of 50 stocks from the S&P 500 from March 1986 to July 2024 ( T = 2 , 000 ). The out-of-sample forecasting window is from May 2005 to July 2024 ( T f = 1 , 000 ). We compare the in-sample statistical and out-of-sample density forecasting performances of Beta-t-EGARCH, OS-Beta-t-EGARCH, and MS-Beta-t-EGARCH. We find that the statistical and density forecasting performances of OS-Beta-t-EGARCH are superior to MS-Beta-t-EGARCH, motivating its practical use by investors and risk managers.

DYNAMIC LINKS BETWEEN COMMODITY MARKETS & PAKISTANI SECTORAL INDICES: IMPLICATIONS FOR PORTFOLIO DIVERSIFICATION

This research aimed to examine the inter-connections among international commodities markets and sectoral indices of Pakistan over the most recent period ranging from July 19, 2016, to December 29, 2023. This time period is selected based on recent important events of global significance including COVID-19 pandemics and recent geopolitical crisis emerged due to Russia-Ukraine war in Feb, 2022. This study is utilizing Generalized Autoregressive Conditional Heteroscedasticity (GARCH) model for the deeper inquiry of interplay among the variables. The findings of the study indicate significant spillovers among variables with varying degrees of significance. The results show that various sectors of Pakistan economy showed asymmetric behavior indicating that some sectors were the risk transmitters while other were risk recipients. Moreover, volatility linkages are time-varying with heightened transmissions during turbulent periods. The results provide the significant information in reaching the desired conclusion and making decisions. The findings of study thus had important implications for investors (individual & institutional), fund managers and policy makers while devising strategies for diversification and risk mitigation in the Pakistani context especially in stressful periods.

PENERAPAN MODEL ASYMMETRIC POWER AUTOREGRESSIVE CONDITIONAL HETEROSCEDASTICITY (APARCH) TERHADAP HARGA MINYAK MENTAH DUNIA

Heteroscedasticity poses a challenge in ARIMA modeling by causing residual variance to be non-constant, leading to less efficient estimates. This issue often arises in time series data due to volatility, which measures data fluctuation over time. To address heteroscedasticity, models like ARCH and GARCH incorporate variance changes into forecasting. However, they lack the ability to capture asymmetry, the difference in impact between good and bad news on volatility. The APARCH model, on the other hand, addresses this by modeling volatility with asymmetry elements. Daily world crude oil prices, known for high volatility, serve as a case study for this research. By employing the APARCH model, the study aims to forecast these prices accurately. Results indicate that the APARCH(1,1) model outperforms the best GARCH model, ARCH(2), as it yields a smaller Mean Absolute Percentage Error (MAPE) of 6.033487. This highlights the superior accuracy of APARCH in forecasting data with heteroscedasticity issues, particularly in the context of daily crude oil prices.

Unveiling change-point detection with ARMA-GARCH models

This paper delves into the asymptotic theory of a Cumulative Sum (CUSUM)-type test designed for detecting change points in the variance of AutoRegressive Moving-Average models, incorporating Generalized Autoregressive Conditional Heteroscedasticity innovations. We demonstrate that, under the null hypothesis (no change), the CUSUM test statistic converges to the supremum of a standard Brownian bridge. Using a Monte Carlo simulation, we highlight the strong performance of our proposed test when compared to the methods by Song and Kang. Furthermore, we provide a real-world data analysis to showcase the practical application of our test.

Portfolio Diversification Opportunities between Europe and India do Exist

In this article, a humble effort has been made to check the volatility transmission from European stock markets (FTSE100, DAX30, CAC40 and SMI20) to the Indian stock market (NIFTY50) to determine whether there is any opportunity for portfolio managers to diversify their portfolio. Various statistical tools and econometric models, such as vector autoregression (VAR) and dynamic conditional correlation (DCC)-multivariate generalised autoregressive conditional heteroscedasticity (MGARCH), are used to detect the volatility transmission from the European market to the Indian market. The FTSE100 and the NIFTY50 have been found to have bidirectional causality. VAR results suggest that only the DAX30 and SMI20 (Germany and Switzerland) markets have an impact on the Indian market. Finally, using DCC-MGARCH, the researcher found that information spillover takes place from all European countries to India over a longer period of time, but portfolio diversification between the London market (FTSE) and the Indian market (NIFTY) and German market (DAX) with Indian market (NIFTY) is possible over a shorter period of time. JEL Codes: C01, G1, G17

Predictive performance of denoising algorithms in S&P 500 and Bitcoin returns

This paper aims to investigate the effectiveness of both the Hankel matrix and the Wavelet denoise methods to address the issue of denoising volatility in datasets, specifically squared returns, and to enhance the accuracy of post-sample forecasting. When converting data from levels to volatility, noise levels are amplified, and the forecasting models suffer from structural bias, leading to reduced accuracy gains. The main challenge in denoising operations is preserving systemic impulses while extracting sporadic and unpredictable components. While conventional denoising algorithms struggle to distinguish systemic patterns from noise, the Hankel matrix approach surpasses them in performance, resulting in improved predictive accuracy. Additionally, this paper explores the Wavelet denoise method, which is found to be less effective than the Hankel matrix approach for GARCH models. The empirical work showcases the denoising process and the accuracy improvements achieved in various datasets for GARCH models. By implementing the Hankel matrix approach, simulations reveal a notable accuracy gain of over 10%.

Research on the volatility forecasting model of KOSPI index returns using AR(M)-GARCH(P,Q) model

In this study, we estimated the volatility of the KOSPI index returns and analyzed volatility trends. The data used in the study consisted of monthly observations from January 2005 to December 2022, and the KOSPI index raw data was transformed into log returns. The volatility estimation model used the AR(m)-GARCH(p,q) model, which combines the autoregressive error model and the GARCH model that explains the persistence of volatility at low orders. The goodness of fit of the model was confirmed using the Portmanteau Q-test and LM-test. Applying the autoregressive error model revealed significant autocorrelation in the log returns of the KOSPI index at lags 3 and 6. Residual analysis indicated that the residuals followed white noise, but the squared residuals exhibited heteroscedasticity. Therefore, after fitting the autoregressive error model, we applied the GARCH model and conducted residual analysis, finding both the residuals and squared residuals significant at a 5% significance level. The volatility forecasting results indicated a continuous increase in volatility. The findings of this study are expected to provide important implications for policymakers responsible for risk management in the Korean stock market.

Beyond GARCH: Intraday Insights Into the Exchange Rate and Stock PriceVolatility Dynamics in Borsa Istanbul Sectors

This study investigated the impact of exchange rate volatility on sectoral stockvolatility by employing the intraday volatility measure directly calculated fromthe original data, using daily data from 27 Borsa Istanbul sectors between April 29, 2003, and April 25, 2023. In the literature, GARCH models are commonly usedtostudy the volatility spillovers between exchange rates and stock prices, typicallyusing aggregate data. However, the GARCH family models provide inefficient andbiased estimates if they are misspecified. Moreover, using aggregate-level data maylead to biased and misleading conclusions. The research used intraday volatilitymeasures to overcome the shortcomings of GARCH models. The ordinary least squares (OLS), GARCH (1,1) methods, and Garman and Klass (1980) volatilityestimator are used. The empirical results showed that the estimates fromeachmethod vary significantly, and these disparities in the results might be due tomisspecification in GARCH (1,1) models. The intraday volatility model estimationresults showed that although stock price volatilities in all sectors are positively andsignificantly affected by exchange rate volatility, their magnitudes varysignificantly. Taken together, this implies the presence of vast heterogeneities inthe responses of sectoral stock price volatilities to exchange rate volatility. The results encourage policymakers to pay special attention to these heterogeneities toprevent capital flights and underinvestment. Additionally, the findings assist investors in making more effective decisions by helping themadapt their investment strategies to factor in exchange rate fluctuations and mitigate the impact of unexpected events in the exchange rate market

INTEREST RATES AND INFLATION AS DETERMINANT FACTORS OF SAVING IN COMMERCIAL BANKS OF NORTH MACEDONIA

This study aims to analyze the effect of the Central Bank of North Macedonia (National Bank), which is in North Macedonia presented by National Central Bank (BI) Rate and inflation rate on the savings rate at commercial banks in North Macedonia by using the analysis tools of autoregressive conditional heteroscedasticity (ARCH) and generalized autoregressive conditional heteroscedasticity (GARCH). The analysis results show that the BI Rate and inflation significantly affect saving rates at commercial banks in North Macedonia. The importance of being a source of investment funds to support development activities can be a consideration for the government to formulate a policy for allocating investment from these savings funds for productive sectors. Practically, it is hoped that the distribution of savings through investment activities can encourage economic growth and increase economic development in North Macedonia.

Optimal Short-Term Inflation Rate Forecasting Model in Kenya: in Depth Analysis

Purpose: The Kenya economy experienced an increase in the price of basic commodities, increased unemployment rates and consequently reduced real wage levels due to job cuts. An accurate forecast of this unprecedented inflation rate could have cushioned the Kenyan population from its effects. Uncertainty over future inflation forecasting has caused detrimental and negative impact not only globally but also in the Kenyan economy. Combining several techniques of forecasting is an instinctual way to improve prediction performance as the limitations of one method are compensated by the strength of the other model. The general objective of the study was to develop an optimal model of forecasting short term inflation rate in Kenya. The specific objectives were to establish models of forecasting short term inflation rate using SARIMA, GARCH and hybrid SARIMA-GARCH family Models, select optimal model amongst the three models and Predict 12 months ahead inflation rate using the optimal model. Scope of the study from 2005 January to 2024 July. The study was anchored on monetary theory of inflation, Keynesian theory of inflation as well as rationale expectation theory of inflation. Methodology: Data was sourced from KNBS and CBK. The study was guided by positivism research philosophy. Explanatory research design was used in this study. Target population was 230 monthly observations. Sample and sampling techniques used was time series. PACF, ACF Dfuller, Kpss test, Philips perron test indicated the data was stationary after 1ST differencing. Findings: Statistic validation test results indicated SARIMA’s (1,1,1)(1,1,)12 adjusted 2 R was perfectly 1 indicating all variations in squared residuals were explained by lagged residuals. P- value for lagged squared residuals were significant at (0.00). F-statisticvalue was significant (0.020) suggesting overall model fit. Model stability test AR roots polynomial lied outside the unit circle. eGarch(1,1) model with lowest aic -0.534 was best in Garch Family models. Hybrid Sarima((1,1,1),(1,1,1) eGARCH(1,1) was identified. Comparison of Sarima(1,1,1)(1,1,1)12, eGARCH (1,1) and Hybrid Sarima((1,1,1),(1,1,1)12 eGARCH(1,1) using forecast accuracy revealed that hybrid model was the optimal model with lowest MAE 0.166 and RMSE 0.259. Diagnostic checks The Ljung- Box test (LL) and Q2 indicated non-significant autocorrelation p- value were greater than 05% indicating that the models residuals were white noise. The DOF/GED parameter (4.144466*, 1.101958, 6.977499) represented the degrees of freedom for the tdistribution and the coefficients where significant at 0.05% meaning the model’s assumptions for normal distributions were met. The coefficients of AR Normal 0.048140, T-Student −0.031854, GED −0.02396 and MA Normal 0.055167 ,T-Student 0.059645, GED 0.051390 terms were significant at 5%. Recommendations: The results implied that the model predicted a decrease in the Kenya’s inflation rate for the next 12 months. The study recommended that inflation rate would be hovering below an average rate of 10 within the next 12 months with high volatility up to July 2025 and policy makers should use this prediction for planning in order to maintain Kenya’s macroeconomic stability. Policy makers were also advised to use the hybrid model to forecast short term inflation rate 12 months ahead and in future years. The benefits of the study were added knowledge of hybridizing to researchers and to policy makers. This study therefore provided better inflation forecasts (Kenya) to be used for strategical planning for short -term effects of inflation by the government.

Stock price prediction using combined GARCH-AI models

The non-linear and non-stationary nature of financial time series data poses significant challenges for standalone statistical and neural network methods. While predictive modeling in finance often focuses on volatility, there is a notable lack of research on predicting actual stock prices, particularly in the African market. This study addresses this gap by utilizing Airtel stock data from Yahoo Finance, spanning June 28, 2019, to May 8, 2024. The research employs the GARCH model to extract statistical properties, which are then combined with historical prices and fed into LSTM, GRU, and Transformer models leading to GARCH-LSTM, GARCH-GRU, GARCH-Transfomer hybrid models. These hybrid models are benchmarked against standalone LSTM, GRU and Transfomer models using RMSE, MAE, MAPE, and R-squared metrics. Results indicate that hybrid models, especially GARCH-LSTM, significantly outperform standalone models. This integration of GARCH with advanced AI models offers a more robust framework for stock price prediction, enhancing accuracy and reliability in forecasting future prices.

Safe Havens Characteristics of Islamic Stock Indices During Bearish and Bullish Markets: A Case of Developed and Developing Countries

Purpose — This study examines the performance of Islamic and conventional indices during bullish (positive) and bearish (negative) markets while the economic and financial conditions are different. Design/Methodology/Approach — This study uses GARCH (1,1) and wavelet-assisted cross-spectral, cross-correlation analyses to investigate the performance of Islamic and conventional stock indices and study the presence of safe havens, which reflect low volatility in the indices during times of market turbulence, for investors during bearish and bullish periods from 2019–2021. Findings — The paper finds that there is high and persistent volatility in Islamic indices in bullish markets, while in bearish periods the Islamic indices show low volatility persistence; and conventional indices were highly volatile during crisis periods. Also, the impact of news shocks on Islamic indices was intense before COVID-19, while after that news, shocks had a high impact on the conventional indices. The wavelet coherence analysis shows that there is high coherency between Islamic and conventional indices in developed countries but in developing countries, the opposite prevails. Originality/Value — The study is the first to use both Generalised Autoregressive Conditional Heteroskedasticity (GARCH) and wavelet coherence models for examining Islamic and conventional stock indices during bearish and bullish markets, especially in the COVID-19 pandemic era. Practical Implications — The results conclude that there are safe havens present in Islamic indices in both developing and developed countries, especially the United States (US), the United Kingdom (UK), India, and Indonesia, while in other countries the presence is weak. Both GARCH and wavelet coherence models show the same results regarding the presence of volatility and safe havens for investors in different countries.

On the connectedness of stock returns and exchange rates in emerging and frontier markets in Africa

AbstractThe effect of currency volatility on investments in Africa continues to dominate the headlines, especially in the recent period of global crisis and heightened geopolitical tensions. This paper tackles the dynamic relationship between stock returns and exchange rates in nine emerging and frontier African markets. The study departs from the usual VAR and GARCH models and employs a tool that accounts for time–frequency co‐movements and lead/lag relationships between exchange rates and stock returns in Africa. The bivariate wavelet technique applied to daily data from 1 April 2013 to 31 March 2022 established a profound negative co‐movement between stock returns and exchange rates, especially in the medium to long‐term frequencies. With exchange rates dominantly playing a leading role, it presents a case for policy consideration toward currency stability. We employed the partial wavelet approach to determine how stock returns and exchange rates related during the peak period of the COVID‐19 pandemic and found negative co‐movements within the short‐term frequency, revealing that investors preferred the short‐term horizon in times of crisis. However, when the covariate (COVID‐19) was controlled or suppressed, we discovered the health pandemic failed to drive both stock returns and exchange rates.

THE EVOLUTION AND FORECASTING OF SINO-KOREAN COMMODITY TRADE: A THEORETICAL EXPLORATION BASED ON GARCH AND MACHINE LEARNING MODELS

This paper examines the evolution of Sino-Korean commodity trade and explores the application of advanced econometric and machine learning models for forecasting commodity price volatility. With increasing interdependence between China and South Korea in trading key commodities such as crude oil, LNG, iron ore, and rare earth elements, accurate price forecasting has become crucial for managing economic risks and optimizing trade strategies. The study highlights the limitations of traditional econometric models like GARCH, which, while effective at capturing short-term volatility, struggle to account for the complex, nonlinear dynamics present in modern commodity markets. Machine learning models, including LSTM, random forests, and support vector machines, offer a more flexible and accurate approach by incorporating real-time data and adapting to market shifts. The combination of GARCH and machine learning in hybrid models further enhances forecasting accuracy. As both countries transition toward sustainable energy, the role of advanced forecasting tools will be pivotal in maintaining economic stability and fostering deeper trade cooperation. KEYWORDS：Sino-Korean commodity trade,GARCH model, Machine learning, Commodity price volatility, Crude oil forecasting, LNG and rare earth elements

Volatility spillover effect between cryptocurrency and stock market using MGARCH Bekk model

This paper explores the volatility spillover effects between the cryptocurrency market and the Pakistan Stock Exchange (PSX). Utilising data from January 1, 2019, to April 5, 2024, sourced from Investing and Yahoo Finance, the study employs the Multivariate Generalized Autoregressive Conditional Heteroskedasticity (MGARCH) BEKK model to assess the dynamic interactions between these markets. Stationarity tests confirmed the non-stationarity of time series data at their levels, which became stationary after first differencing, ensuring robust econometric analysis. The results indicate significant volatility spillovers from major cryptocurrencies, such as Bitcoin and Ethereum, to the PSX, highlighting a solid interconnectedness between these markets. This suggests that digital asset volatility significantly influences traditional financial systems. The study concludes that integrating cryptocurrencies into global financial markets introduces risks and opportunities for investors and policymakers. The findings underscore the need for market participants to account for these volatility interactions in their risk management strategies. Additionally, policymakers must consider these interlinkages to maintain financial stability. This research contributes to the literature on financial market volatility by emphasising the importance of understanding the impact of emerging digital currencies on traditional stock markets.

The Interplay of Dietary Habits, Economic Factors, and Globalization: Assessing the Role of Institutional Quality.

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. 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. 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. 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.

The Effect of Exchange Rate Volatility on Foreign Tourist Visits in Indonesia: GARCH Analysis

In this era of globalization, currency exchange rates are one of the crucial factors affecting a country's economy. The exchange rate of a country's currency plays an important role in economic stability and international trade, especially in the tourism sector. This study aims to conduct an in-depth analysis to identify patterns of rupiah exchange rate volatility using the Generalized Autoregressive Conditional Heteroscedasticity (GARCH) model, and then examine whether these patterns have a significant correlation with the number of foreign tourist arrivals in Indonesia. In analyzing the data, researchers conducted several classical assumption tests, namely the normality test and the heteroscedasticity test. The test results show that the data used, namely the rupiah exchange rate data and the number of foreign tourist visits, are not normally distributed and heteroscedastic. In this case, it resulted in researchers not use the Pearson correlation test so they used the Kendall-Tau correlation test and Granger Causality test which resulted in the two variables being correlated. Furthermore, GARCH modeling is carried out which provides forecasting on both data to increase in 2024. The results of this study are expected to provide a basis for consideration for stakeholders to determine policies, especially in the financial and tourism sectors.

Multiscale neural dynamics in sleep transition volatility across age scales: a multimodal EEG-EMG-EOG analysis of temazepam effects.

Recent advances in computational modeling techniques have facilitated a more nuanced understanding of sleep neural dynamics across the lifespan. In this study, we tensorize multiscale multimodal electroencephalogram (EEG), electromyogram (EMG), and electrooculogram (EOG) signals and apply Generalized Autoregressive Conditional Heteroskedasticity (GARCH) modeling to quantify interactions between age scales and the use of pharmacological sleep aids on sleep stage transitions. Our cohort consists of 22 subjects in a crossover design study, where each subject received both a sleep aid and a placebo in different sessions. To understand these effects across the lifespan, three evenly distributed age groups were formed: 18-29, 30-49, and 50-66years. The methodological framework implemented here employs tensor-based machine learning techniques to compute continuous wavelet transform time-frequency features and utilizes a GARCH model to quantify sleep signal volatility across age scales. Support Vector Machines are used for feature ranking, and our analysis captures interactions between signal entropy, age, and sleep aid status across frequency bands, sleep transitions, and sleep stages. GARCH model results reveal statistically significant volatility clustering in EEG, EMG, and EOG signals, particularly during transitions between REM and non-REM sleep. Notably, volatility was higher in the 50-66 age group compared to the 18-29 age group, with marked fluctuations during transitions from deep sleep to REM sleep (standard deviation of 0.35 in the older group vs. 0.30 in the 18-29 age group, p < 0.05). Statistical comparisons of volatility across frequency bands, age scales, and sleep stages highlight significant differences attributable to sleep aid use. Mean conditional volatility parameterization of the GARCH model reveals directional influences, with a causality index of 0.75 from frontal to occipital regions during REM sleep transition periods. Our methodological framework identifies distinct neural behavior patterns across age groups associated with each sleep stage and transition, offering insights into the development of targeted interventions for sleep regularity across the lifespan.

Robust estimation of the range-based GARCH model: Forecasting volatility, value at risk and expected shortfall of cryptocurrencies

Traditional volatility models do not work well when volatility changes rapidly and in the presence of outliers. Therefore, two lines of improvements have been developed separately in the existing literature. Range-based models benefit from efficient volatility estimates based on low and high prices, while robust methods deal with outliers. We propose a range-based GARCH model with a bounded M-estimator, which combines these two improvements with a third new improvement: a modified robust method, which adds elasticity in treating the outliers. We apply this model to Bitcoin, Ethereum Classic, Ethereum, and Litecoin and find that it forecasts variances, value at risk, and expected shortfall more accurately than the standard GARCH model, the standard range-based GARCH model, and the GARCH model with the robust estimation. Utilization of high and low prices joined with a novel treatment of outliers makes our model perform well during extreme periods when traditional volatility models fail.