This study performs an empirical investigation into the price volatility characteristics of China’s stock market utilizing GARCH-family methodologies. Daily data of the Shanghai Composite Index from 2010 to 2024 are employed, and an ARMA-GARCH-type framework is constructed to capture the volatility structure of the return series. The analysis begins with stationarity and ARCH effect tests on the original series. The results indicate significant volatility clustering and conditional heteroskedasticity, justifying the application of GARCH modeling. Subsequently, alternative specifications are estimated, incorporating diverse distributional assumptions for the residuals, to assess their comparative goodness-of-fit. The results show that the EGARCH(1,1) model under the skewed t-distribution provides the best fit, effectively capturing both the asymmetric and heavy-tailed features of the return volatility. Ljung-Box and ARCH-LM tests on the standardized residuals confirm the adequacy of the model fit. The empirical findings suggest that volatility in China’s stock market returns exhibits strong persistence and heightened sensitivity to negative shocks. These results offer quantitative insights for investor risk management and regulatory market surveillance, and also provide a methodological foundation for future research on multivariate volatility modelling.

Financial markets show persistent volatility, creating barriers to achieving exact financial predictions. The forecasting of multivariate financial data requires forecasting models like the Vector Autoregressive (VAR) model for modeling linear dependencies, the Long Short-Term Memory (LSTM) model for modeling non-linear patterns, and the Generalized Autoregressive Conditional Heteroscedastic (GARCH) model that is capable of modeling volatility clustering. Each of these models fails to handle complete data complexity on its own, as they specialize in unique properties of the data. Recent studies have been carried out that enhance forecasting accuracy by combining two models. The first case is the VAR-GARCH model, which can model linear and volatility clustering aspects but fails to model non-linear dependencies. Another case is the LSTM-GARCH model that can explain non-linear dependencies and volatility patterns, but fails to explain linear dependencies. A third instance is the VAR-LSTM model that can explain the linear and volatility aspects, but fails to model the non-linear patterns. However, there is a need to have a model that can combine the three models to explain the linear, non-linear, and volatility aspects in financial time series data collectively. This research fills this gap by combining VAR, LSTM, and GARCH into a VAR-LSTM-GARCH hybrid model, which provides improved forecasting. This study uses historical five-year daily data for VIX, US Dollar Index, and S&P 500 E-mini futures obtained from Yahoo Finance. The model-building process involves constructing a VAR (9) model selected using AIC criteria to reveal linear dependencies. The residuals from the VAR are used to train an LSTM model to capture nonlinear trends. The residuals of the LSTM are then used to fit an M-GARCH (1, 1) model, which generates volatility cluster estimates. The VAR-LSTM-GARCH hybrid model demonstrates superior performance with substantial improvements across all evaluation metrics compared to individual models, showing consistently lower prediction errors and enhanced forecasting accuracy. The progressive three-stage modeling approach demonstrates that each component contributes incrementally to forecasting performance, with the incorporation of volatility modeling through GARCH being particularly effective in enhancing predictive accuracy. The research suggests using this hybrid model for volatility prediction on multiple portfolios and emphasizes future development of real-time diagnostic processes. The new approach delivers an advanced instrument that helps financial analysts work efficiently by effectively capturing the complex interdependencies in multivariate financial time series data.

The need to provide an acceptable model and forecast for stock prices of solid minerals in Nigeria is valuable for investors and analysts. It will empower them to better understand and manage the associated risks in stock price movements. This study aimed to model and forecast the volatility of stock prices of solid minerals, like gold, tin, and zinc. The data utilized in this study was sourced from the Central Bank of Nigeria and Nigeria Stock Exchange. It is the monthly stock prices for selected solid minerals like; Gold, Tin, and Zinc. Multivariate GARCH models such as the VECH, BEKK, Diagonal VECH and Diagonal BEKK model were employed to provide the needed multivariate volatility modeling. The findings revealed that, on average, investors experienced positive returns, and a non-symmetric distribution. It was also discovered that intricate patterns exist within the volatility dynamics of these stocks. Volatility clustering, ARCH effects, and the persistence of volatility shocks over time was identified, emphasizing the non-random nature of stock returns volatility. It is recommended that investors and analysts carefully consider the implications of volatility clustering, ARCH effects, and persistence in volatility shocks when making investment decisions in the stock market, particularly regarding gold, tin, and zinc stocks.

This study examines the financial performance of Lodging Real Estate Investment Trusts (REITs) during the COVID-19 pandemic, focusing on the impact of the March 2020 market crash. While existing research primarily addresses general financial markets, we investigate the lodging REIT index and its correlation with seven major financial indices across the U.S. and Europe. The analysis highlights Lodging REITs’ sensitivity to recession-related shocks by using daily data from January 1, 2019, to December 31, 2021, applying risk-adjusted performance measures and a two-stage multivariate volatility EGARCH model. The findings reveal significant increases in conditional volatility in response to negative shocks, persistent correlation dynamics, and quicker mean-reversion behavior compared to other indices. These results underscore Lodging REITs’ resilience and unique characteristics during periods of market uncertainty, offering actionable insights for investors and policymakers.

Forecasting multivariate volatilities with exogenous predictors: An application to industry diversification strategies

ABSTRACTThis paper introduces an extended multivariate EGARCH model that overcomes the zero‐return problem and allows for negative news and volatility spillover effects, making it an attractive tool for multivariate volatility modeling. Despite limitations, such as noninvertibility and unclear asymptotic properties of the QML estimator, our Monte Carlo simulations indicate that the standard QML estimator is consistent and asymptotically normal for larger sample sizes (i.e., ). Two empirical examples demonstrate the model's superior performance compared to multivariate GJR‐GARCH and Log‐GARCH models in volatility modeling. The first example analyzes the daily returns of three stocks from the DJ30 index, while the second example investigates volatility spillover effects among the bond, stock, crude oil, and gold markets. Overall, this extended multivariate EGARCH model offers a flexible and comprehensive framework for analyzing multivariate volatility and spillover effects in empirical finance research.

The study applies multivariate FIGARCH and FIAPARCH volatility models with constant and dynamic conditional correlations to analyze the volatility of exchange rates of the US dollar against the euro and the Japanese yen, as well as the prices of gold and oil. The results reveal the presence of long memory in the volatility of all analyzed series, with dual long memory observed in the case of oil. An asymmetric reaction of oil and gold price volatility to "positive" and "negative" shocks is confirmed based on the applied FIAPARCH models, while such asymmetry is not supported for exchange rates. The applied multivariate volatility models reveal dynamic conditional correlations between exchange rates, gold, and oil prices, aiding optimal portfolio construction. The study identifies volatility spillovers from gold prices and the EUR-USD exchange rate to oil prices, as well as from the USD-JPY exchange rate to the EUR-USD exchange rate. Bidirectional spillovers are observed only between gold prices and the EUR-USD exchange rate.

Abstract This article proposes a new predictive model for large-dimensional realized covariance matrices. Using high-frequency data, we estimate daily realized covariance matrices for the constituents of the S&P 500 Index and a set of observable factors. Using a standard decomposition of the joint covariance matrix, we express the covariance matrix of the individual assets similar to a dynamic factor model. To forecast the covariance matrix, we model the components of the covariance structure using a series of autoregressive processes. A novel feature of the model is the use of the data-driven hierarchical clustering algorithm to determine the structure of the idiosyncratic covariance matrix. A simulation study shows that this method can accurately estimate the block structure as long as the number of blocks is small relative to the number of stocks. In an out-of-sample portfolio selection exercise, we find that the proposed model outperforms other commonly used multivariate volatility models in extant literature.

We develop a hybrid model of multivariate volatility that uses recurrent neural networks to capture the conditional variances of latent orthogonal factors in a GO-GARCH framework. Our approach seeks to balance model flexibility with ease of estimation and can be used to model conditional covariances of a large number of assets. The model performs favourably in comparison with relevant benchmark models in a minimum variance portfolio (MVP) scenario.

Univariate and Multivariate Volatility Models for Portfolio Value at Risk

This research examines the impact of the coronavirus index on the returns and volatility of ten major cryptocurrencies during the COVID-19 pandemic. For this purpose, we applied a multivariate volatility GARCH model with an integrated dynamic conditional correlation (DCC) approach to daily cryptocurrency values observed data during the January-December, 2020 period. Moreover, we used the Granger causality test to study return-volume correlations. The findings indicate that cryptocurrency volatility declined after the World Health Organization declared on March 11, 2020, that the coronavirus was a pandemic. Unlike most of the relevant previous studies, we found that the COVID-19 crisis did not have a long-term effect on cryptocurrency returns and volatility but only presented a short-term effect. Our results have implications for investors who need to determine an optimal portfolio for a scenario other than the base.

A practical multivariate approach to testing volatility spillover

This is a review about financial dependencies which merges efforts in econophysics and financial economics during the last few years. We focus on the most relevant contributions to the analysis of asset markets’ dependencies, especially correlational studies, which in our opinion are beneficial for researchers in both fields. In econophysics, these dependencies can be modeled to describe financial markets as evolving complex networks. In particular, we show that a useful way to describe dependencies is by means of information filtering networks that are able to retrieve relevant and meaningful information in complex financial datasets. In financial economics these dependencies can describe asset comovement and spill-overs. In particular, several models are presented that show how network and factor model approaches are related to modeling of multivariate volatility and asset returns, respectively. Finally, we sketch out how these studies can inspire future research and how they contribute to support researchers in both fields to find a better and a stronger common language.

This paper studies the impact of investor attention to oil prices on returns, volatility, and covariances of three exchange traded funds representing oil, gold, and the stock market. For this purpose, we suggest a new multivariate volatility model based on open, high, low, and closing prices that incorporates the impact of investor attention on returns, volatility, and covariances. We find that this model, which incorporates Google searches for “oil prices” as an exogeneous variable, outperforms other considered multivariate volatility models, and demonstrates that Google searches for “oil prices” can explain and forecast covariances between returns of oil, gold, and the stock market.

Analyzing a dynamic relation between RMB exchange rate onshore and offshore during the extreme market conditions

This research examines the correlations between the return volatility of cryptocurrencies, global stock market indices, and the spillover effects of the COVID-19 pandemic. For this purpose, we employed a two-stage multivariate volatility exponential GARCH (EGARCH) model with an integrated dynamic conditional correlation (DCC) approach to measure the impact on the financial portfolio returns from 2019 to 2020. Moreover, we used value-at-risk (VaR) and value-at-risk measurements based on the Cornish–Fisher expansion (CFVaR). The empirical results show significant long- and short-term spillover effects. The two-stage multivariate EGARCH model’s results show that the conditional volatilities of both asset portfolios surge more after positive news and respond well to previous shocks. As a result, financial assets have low unconditional volatility and the lowest risk when there are no external interruptions. Despite the financial assets’ sensitivity to shocks, they exhibit some resistance to fluctuations in market confidence. The VaR performance comparison results with the assets portfolios differ. During the COVID-19 outbreak, the Dow (DJI) index reports VaR’s highest loss, followed by the S&P500. Conversely, the CFVaR reports negative risk results for the entire cryptocurrency portfolio during the pandemic, except for the Ethereum (ETH).

Models for variances and covariances of asset returns are crucial in risk management and asset allocation. Traditionally, these models were based on daily returns. Daily opening, high, low and closing (OHLC) prices have been sometimes used in multivariate volatility models for variances, but not for correlations. We therefore suggest a new version of the Dynamic Conditional Correlation (DCC) model wherein information from daily OHLC prices is utilized in both variance and correlation equations. The model is evaluated for two datasets: five exchange traded funds and five currencies. The results show that in terms of conditional covariance matrix estimates and forecasts the proposed model significantly outperforms, not only the standard DCC model, but also models that incorporate OHLC prices only in the variance equation.

Green stocks are companies environmental protective and friendly. We test Green stock index in Shanghai Stock Exchange and China Securities Index as safe-havens for global investors. Suitable multivariate-SV model and Bayesian method are used to estimate the spillover effect between different assets among local and global markets. We choose multivariate volatility model because it can efficiently simulate the spillover effect by using machine learning MCMC method. The results show that the Environmental Protection Index (EPI) of Shanghai Stock Exchange (SSE) and China Securities Index (CSI) have no significant volatility spillover from Shanghai Stock index, S &P index, gold price, oil future prices of USA and China. During COVID-19 pandemic, we find Green stock index is a suitable safe-haven with low volatility spillover. Green stock indexes has a strongly one-way spillover to the crude oil future price. Environmentally friendly investor can use diversity green assets to provide a low risk investment portfolio in EPI stock market. The DCGCt-MSV model using machine learning of MCMC method is accurate and outperform others in Bayes parameter estimation.

Some events impact volatilities of most assets, asset classes, sectors and countries, causing serious damage to investment portfolios. The magnitude of such shocks is defined as global COVOL which is an abbreviation for global common volatility, a broad measure of all types of global financial risk. This paper introduces a statistical formulation of such events as common volatility innovations in both a multivariate volatility and an asset pricing context. Simulations verify the statistical performance of a simple but novel estimator and of a test to detect global COVOL. Two empirical examples show the events that have had the biggest impact on financial markets. The results are useful for portfolio optimization and risk forecasting.

SummaryWe propose a minimal representation of variance matrices of dimension k, where parameterization and positive-definiteness conditions are both explicit. Then we apply it to the specification of dynamic multivariate volatility processes. Compared to the most parsimonious unrestricted formulation currently available, the required number of covariance parameters (hence processes) is reduced by about a half, which makes them estimable in full parametric generality if needed. Our conditions are easy to implement: there are only k of them, and they are explicit and univariate. To illustrate, we forecast minimum-variance portfolios and show that risk is always reduced (by a factor of 2 to 3 in spite of us using the simplest dynamics) compared to the standard benchmark used in finance, while also improving returns on the investment. Because of our representation, we do not get the usual dimensionality problems of existing unrestricted models, and the performance relative to the benchmark is actually improved substantially as k increases.