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The BEKK GARCH class of models presents a popular set of tools for applied analysis of dynamic conditional covariances. Within this class the analyst faces a range of model choices that trade off flexibility with parameter parsimony. In the most flexible unrestricted BEKK the parameter dimensionality increases quickly with the number of variables. Covariance targeting decreases model dimensionality but induces a set of nonlinear constraints on the underlying parameter space that are difficult to implement. Recently, the rotated BEKK (RBEKK) has been proposed whereby a targeted BEKK model is applied after the spectral decomposition of the conditional covariance matrix. An easily estimable RBEKK implies a full albeit constrained BEKK for the unrotated returns. However, the degree of the implied restrictiveness is currently unknown. In this paper, we suggest a Bayesian approach to estimation of the BEKK model with targeting based on Constrained Hamiltonian Monte Carlo (CHMC). We take advantage of suitable parallelization of the problem within CHMC utilizing the newly available computing power of multi-core CPUs and Graphical Processing Units (GPUs) that enables us to deal effectively with the inherent nonlinear constraints posed by covariance targeting in relatively high dimensions. Using parallel CHMC we perform a model comparison in terms of predictive ability of the targeted BEKK with the RBEKK in the context of an application concerning a multivariate dynamic volatility analysis of a Dow Jones Industrial returns portfolio. Although the RBEKK does improve over a diagonal BEKK restriction, it is clearly dominated by the full targeted BEKK model.

Existing multivariate generalized autoregressive conditional heteroskedasticity (GARCH) models either impose strong restrictions on the parameters or do not guarantee a well-defined (positive-definite) covariance matrix. I discuss the main multivariate GARCH models and focus on the BEKK model for which it is shown that the covariance and correlation is not adequately specified under certain conditions. This implies that any analysis of the persistence and the asymmetry of the correlation is potentially inaccurate. I therefore propose a new Flexible Dynamic Correlation (FDC) model that parameterizes the conditional correlation directly and eliminates various shortcomings. Most importantly, the number of exogenous variables in the correlation equation can be flexibly augmented without risking an indefinite covariance matrix. Empirical results of daily and monthly returns of four international stock market indices reveal that correlations exhibit different degrees of persistence and different asymmetric reactions to shocks than variances. In addition, I find that correlations do not always increase with jointly negative shocks implying a justification for international portfolio diversification.

The study uses the multivariate GARCH-BEKK model (which was first proposed by Baba et al. (1990) and then further developed by Engle and Kroner (1995)) to examine the return and volatility spillover between India and four leading Asian (namely, China, Japan, Singapore and Hong Kong) and two global (namely, the United Kingdom and the United States) equity markets.

The study employs a multivariate GARCH-BEKK model to quantify return correlation and volatility transmission across the pre- and post-2008 global financial crisis periods (apart from other conventional time series modelling like cointegration, Granger causality using vector error correction model (VECM)).

The results show a tendency of the Indian stock market index to move along with the US and Hong Kong market indices. The decrease in the value of the co-integration coefficient during the recession was explained by reduced investor confidence in developing countries. The result further shows a clear distinction in terms of volatility spillover between the Asian market vis-a-vis US and UK markets. Volatility transmission from India to Asian markets was found to be significantly higher as compared to the US and UK. So also, the study’s results show a puzzling result giving us comparable co-integration ranks for phase 2 (expansion) and phase 3 (slow-down) of the business cycle in most cases.

In Granger causality testing, the results were unable to ascertain the difference between phase 2 (expansion) and phase 3 (slowdown). However, the multivariate GARCH (MGARCH)-BEKK model showed a clear reduction in volatility transmission to NIFTY50 (is the flagship index on the National Stock Exchange of India Ltd. (NSE)) as India entered slow-down. This shows that the Indian economy does go through different business cycles, and the changes in parameters hence prove hypothesis 3 to be true with respect to volatility transmission to India from International markets.

The results show that for all countries, the volatility transmitted to India increases significantly going from phase 1 (recession) to phase 2 (expansion) and reduces again once the countries enter slow-down in phase 3 (slowdown). This shows that during expansion shocks and impulses in international markets affect the Indian markets significantly, supporting the increase in co-integration in phase 2 (expansion). During expansion, developing markets like India become profitable for investors, due to the high growth rate when compared to developed countries. This implies that a significant amount of capital enters Indian markets, which is susceptible to the volatility of international markets. The volatility transmission from India to the US and UK was insignificant in phase 1 (recession and recovery) and phase 3 (slow-down) showing a weak linkage between the markets during volatile time periods.

The outbreak of the coronavirus disease 2019 (COVID-19) pandemic is an unprecedented shock to the BRICS (Brazil, Russia, India, China, South Africa) economy and their financial markets have plummeted significantly due to it. This paper adds to the recent literature on contagion due to spillover by uniquely examining the presence of pairwise contagion or volatility transmissions in stock markets returns of India, Brazil, Russia, China and USA prior to and during COVID-19 pandemic period.

In this study, the generalised autoregressive conditional heteroskedasticity (GARCH) by Bollerslev (1986) under diagonal parameterization is used to estimate multivariate GARCH framework also known as BEKK (Baba EngleKraft and Kroner) model on stock market returns of BRIC nations and the US.

The empirical results show that the model captures the volatility spillovers and display statistical significance for own past mean and volatility with both short- and long-run persistence effects. Own volatility spillovers (Heatwave phenomenon) have been found to be highest for the US, China and Brazil compared to Russia and India. The coefficients indicate persistence of volatility for each country in terms of its own past errors. The highest and long-term spillover effect is found between US and Russia. The results recommend that Russia is least vulnerable to outside shocks. Finally after examining the pairwise results, it is suggested that the BRIC countries stock indices have exhibited volatility spillover due to the COVID-19 pandemic.

The study may be extended to include other emerging market economies under a dynamic framework.

Researchers and policymakers may draw useful insights on cross-market interdependencies regarding the spillovers in BRIC countries' stock markets. It also helps design international portfolio diversification strategies and in constructing optimal portfolios during COVID and in a post-COVID world.

COVID-19 has been an improbable event in the history of the world which can have a large impact on the financial economies across the emerging countries. This event can be deemed to be informative enough to measure the co-movements of the equity markets amongst cross-country return series, which has not been investigated so far for BRIC nations.

This paper attempts to model the transmission of volatility between producer and consumer prices in the fresh potato, tomato, and cucumber markets in Greece.

The transmission mechanism of the price volatility is modeled using the most popular multivariate GARCH models while taking into consideration possible asymmetries in the transmission process. The models utilized are the DVECH and the BEKK models from the VECH family, as well as the CCC model and the DCC model from the family of conditional correlation models. The possible asymmetric effects are evaluated using asymmetric GARCH models, as well as estimating volatility impulse responses for independent shocks.

The results reveal that, in the tomato and cucumber markets, which are regulated by the Common Market Organization of fruits and vegetables, producers are less vulnerable to volatility shocks transmitted from consumers. In contrast, in the non-regulated potato market, producers are affected by spillover effects from consumers.

This study is limited to a few commodities (i.e. potatoes, tomatoes, and cucumbers); however, it could be extended to additional commodities.

The results of this study show that, if producers are organized on a cooperative basis within the regulative framework of the CAP, like the tomato and cucumber producers, their place in the food supply chain is strengthened, although the CAP targets more market-oriented agricultural markets that are more exposed to world prices.

The present study attempts to understand the transmission of volatility between producer and consumer prices in the fresh potato, tomato, and cucumber markets in Greece, which is not apparent from previous studies. Furthermore, the volatility clusters that are identified in the present study are associated with certain CAP reforms.

The purpose of this paper is to analyze the Brexit effect – pre-Brexit and post-Brexit referendum periods – on the co-movements between the British pound (GBP), the euro (EUR) and the yen (JPY) against the US dollar (USD).

To ascertain the asymmetric behavior of dynamic correlations, the authors use the dynamic conditional correlation (DCC) model, the asymmetric dynamic conditional correlation (A-DCC) model and the diagonal BEKK model assuming Gaussian and Student’s t distribution. Several dummy variables have been included in order to identify the main periods related to Brexit.

Findings show a negative impact of the pre-Brexit referendum period on the correlation between GBP and EUR, while there is no significant effect on GBP–JPY and EUR–JPY pairs. The loss of correlation in the GBP–EUR pairing has not recovered during the post-Brexit referendum period, which could be attributed to the uncertainty about the final impact of Brexit on British and Eurozone economies.

The loss of correlation in the GBP–EUR pair has important implications for individual investors, portfolio managers and traders with respect to hedging activities, international trading and investment strategies.

The results are the first to address how Brexit has impacted on the co-movements between exchange rates using different multivariate models that allow for correlations to change over time.

Capital markets thrive on information, and the information revolution has transformed these markets all over the world. Investors can now keep track of the movements of capital markets in real-time and they react to the flow of information from around the world. One of the concerns of stock market investors is whether the markets operate efficiently, independently, and with sound fundamentals. However, real market movements tend to exhibit a link as is evident from recent market movements across the world.

The assessment of interdependence between stock markets is an important aspect of international portfolio management. The aim of this chapter is to examine the shock and volatility spillover between the Standard and Poor’s 500 (S&P500) index from the United States (US) Stock Exchange and the Istanbul Stock Exchange 100 (BIST100) index from the Stock Exchange Istanbul.

S&P500 index, which is the most important index representing US markets, and BIST100 index, which is the index representing the Turkish market, were used as variables in this study. In the analysis, the causality in variance test was applied to determine the volatility spillover between these two markets. Later, multivariate GARCH (MGARCH) models were used to measure the volatility spillover in the markets. VAR(1)-GARCH (1,1)-Diagonal BEKK model was applied to the daily data to determine the shock and volatility spillover in the markets.

As a result of the variance causality test, it was found that there is a bi-directional volatility spillover between S&P500 index and BIST100 index. When the return spillover between the markets is examined, a one-way spillover from the S&P500 index to the BIST100 index emerged. Diagonal BEKK model results show that each market is affected by its own news (unexpected shocks) and volatility. Furthermore, the volatility is persistent for both markets. These findings demonstrate that the US market and the Turkish market interact with each other.

The purpose of this paper is to examine the conditional correlations and spillovers of volatilities across CEE markets, namely, Hungary, Poland, the Czech Republic, Romania and Croatia, in the post-2007 financial crisis period.

The authors use five-dimensional GARCH-BEKK alongside with the CCC and DCC models.

The estimation results of the three models generally demonstrate that the correlations between these markets are particularly significant. Also, own-volatility spillovers are generally lower than cross-volatility spillovers for all markets.

These results recommend that investors should take caution when investing in the CEE equity markets as well as diversifying their portfolios so as to minimize risk.

Unlike the previous studies in this field, this paper is the first study using multivariate GARCH-BEKK alongside with CCC and DCC models. The study makes an outstanding contribution to the existing literature on spillover effects and conditional correlations in the CEE financial stock markets.

Guar Gum (GG) is used in Shale oil exploration. Excessive price increase in the Guar futures market had a spillover impact on Guar spot prices and affected Guar export from India as Shale oil producers started exploring alternate sources. In this paper, the role of excessive speculation in the futures market, and its adverse impact on the guar-based agri-business ecosystem have been empirically explored.

Volatility spillover dynamics between WTI crude oil and Guar futures have been explored using bivariate-Granger Causality, BEKK–GARCH models with Wavelet multi-resolution analysis. The wavelet-based models capture the multi-scale features of mean and volatility spillover to identify the effect of heterogenous investment behavior in the time and frequency domain.

The results provide evidence that excessive speculation in futures markets increases spot market volatility. The results also suggest that the excess presence of short-term investors can destabilize the futures market.

The purpose of the commodity futures market is to support price discovery and risk management. However, speculative practices can destabilize these purposes leading to the failure of the business ecosystem.

The novelty of this paper is twofold. First, it explores the economic linkages between the spot and futures market and tests whether the presence of heterogeneous traders affects the economic linkages. Second, it models the impact of short-term speculative investment on the destabilization of the spot market.