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The purpose of this paper is to investigate dynamics of causal interactions and financial risk contagion among BRICS stock markets under rare events.

Two methods are adopted: The new causal inference technique, namely, the Liang causality analysis based on information flow theory and the dynamic causal index (DCI) are used to measure the financial risk contagion.

The causal relationships among the BRICS stock markets estimated by the Liang causality analysis are significantly stronger in the mid-periods of rare events than in the pre- and post-periods. Moreover, different rare events have heterogeneous effects on the causal relationships. Notably, under rare events, there is almost no significant Liang's causality between the Chinese and other four stock markets, except for a few moments, indicating that the former can provide a relatively safe haven within the BRICS. According to the DCIs, the causal linkages have significantly increased during rare events, implying that their connectivity becomes stronger under extreme conditions.

The obtained results not only provide important implications for investors to reasonably allocate regional financial assets, but also yield some suggestions for policymakers and financial regulators in effective supervision, especially in extreme environments.

This paper uses the Liang causality analysis to construct the causal networks among BRICS stock indices and characterize their causal linkages. Furthermore, the DCI derived from the causal networks is applied to measure the financial risk contagion of the BRICS countries under three rare events.

The purpose of the paper is to better measure the risks and volatility of the Bitcoin market by using the proposed novel risk measurement model.

The joint regression analysis of value at risk (VaR) and expected shortfall (ES) can effectively overcome the non-elicitability problem of ES to better measure the risks and volatility of financial markets. And because of the incomparable advantages of the long- and short-term memory (LSTM) model in processing non-linear time series, the paper embeds LSTM into the joint regression combined forecasting framework of VaR and ES, constructs a joint regression combined forecasting model based on LSTM for jointly measuring VaR and ES, i.e. the LSTM-joint-combined (LSTM-J-C) model, and uses it to investigate the risks of the Bitcoin market.

Empirical results show that the proposed LSTM-J-C model can improve forecasting performance of VaR and ES in the Bitcoin market more effectively compared with the historical simulation, the GARCH model and the joint regression combined forecasting model.

The proposed LSTM-J-C model can provide theoretical support and practical guidance to cryptocurrency market investors, policy makers and regulatory agencies for measuring and controlling cryptocurrency market risks.

A novel risk measurement model, namely LSTM-J-C model, is proposed to jointly estimate VaR and ES of Bitcoin. On the other hand, the proposed LSTM-J-C model provides risk managers more accurate forecasts of volatility in the Bitcoin market.

This paper aims to better jointly estimate Value at Risk (VaR) and expected shortfall (ES) by using the joint regression combined forecasting (JRCF) model.

Combining different forecasting models in financial risk measurement can improve their prediction accuracy by integrating the individual models’ information. This paper applies the JRCF model to measure VaR and ES at 5%, 2.5% and 1% probability levels in the Chinese stock market. While ES is not elicitable on its own, the joint elicitability property of VaR and ES is established by the joint consistent scoring functions, which further refines the ES’s backtest. In addition, a variety of backtesting and evaluation methods are used to analyze and compare the alternative risk measurement models.

The empirical results show that the JRCF model outperforms the competing models. Based on the evaluation results of the joint scoring functions, the proposed model obtains the minimum scoring function value compared to the individual forecasting models and the average combined forecasting model overall. Moreover, Murphy diagrams’ results further reveal that this model has consistent comparative advantages among all considered models.

The JRCF model of risk measures is proposed, and the application of the joint scoring functions of VaR and ES is expanded. Additionally, this paper comprehensively backtests and evaluates the competing risk models and examines the characteristics of Chinese financial market risks.