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In recent years, the frequent occurrence of air pollution incidents has seriously affected people's health and life. Therefore, PM , as the main pollutant, is an important research object of air pollution at present. Effectively improving the prediction accuracy of PM volatility makes the PM prediction content perfect, which is an important aspect of PM concentration research. The volatility series has an inherent complex function law, which drives the volatility movement. When machine learning algorithms such as LSTM (Long Short-Term Memory Network) and SVM (Support Vector Machine) are used for volatility analysis, a high-order nonlinear form is used to fit the functional law of the volatility series, but the time-frequency information of the volatility has not been utilized. Based on EMD (Empirical Mode Decomposition) technique, GARCH (Generalized AutoRegressive Conditional Heteroskedasticity) model and machine learning algorithms, a new hybrid PM volatility prediction model is proposed in this study. This model realizes time-frequency characteristic extraction of volatility series through EMD technology, and integrates residual and historical volatility information through GARCH model. The simulation results of the proposed model are verified by comparing the samples of 54 cities in North China with the benchmark models. The experimental results in Beijing showed that MAE (mean absolute deviation) of hybrid-LSTM decreased from 0.00875 to 0.00718 compared with LSTM, and hybrid-SVM based on the basic model SVM also significantly improved generalization ability, and its IA (index of agreement) improved from 0.846707 to 0.96595, showing the best performance. The experimental results show that the hybrid model is superior to other considered models in terms of prediction accuracy and stability, which verifies that the hybrid system modeling method is suitable for PM volatility analysis.

We first propose a reduced-form model in discrete time for S&P 500 volatility showing that the forecasting performance can be significantly improved by introducing a persistent leverage effect with a long-range dependence similar to that of volatility itself. We also find a strongly significant positive impact of lagged jumps on volatility, which however is absorbed more quickly. We then estimate continuous-time stochastic volatility models that are able to reproduce the statistical features captured by the discrete-time model. We show that a single-factor model driven by a fractional Brownian motion is unable to reproduce the volatility dynamics observed in the data, while a multifactor Markovian model fully replicates the persistence of both volatility and leverage effect. The impact of jumps can be associated with a common jump component in price and volatility. This article has online supplementary materials.

In the field of business research method, a literature review is more relevant than ever. Even though there has been lack of integrity and inflexibility in traditional literature reviews with questions being raised about the quality and trustworthiness of these types of reviews. This research provides a literature review using a systematic database to examine and cross-reference snowballing. In this paper, previous studies featuring a generalized autoregressive conditional heteroskedastic (GARCH) family-based model stock market return and volatility have also been reviewed. The stock market plays a pivotal role in today’s world economic activities, named a “barometer” and “alarm” for economic and financial activities in a country or region. In order to prevent uncertainty and risk in the stock market, it is particularly important to measure effectively the volatility of stock index returns. However, the main purpose of this review is to examine effective GARCH models recommended for performing market returns and volatilities analysis. The secondary purpose of this review study is to conduct a content analysis of return and volatility literature reviews over a period of 12 years (2008–2019) and in 50 different papers. The study found that there has been a significant change in research work within the past 10 years and most of researchers have worked for developing stock markets.

We provide a framework for integration of high-frequency intraday data into the measurement, modeling, and forecasting of daily and lower frequency return volatilities and return distributions. Building on the theory of continuous-time arbitrage-free price processes and the theory of quadratic variation, we develop formal links between realized volatility and the conditional covariance matrix. Next, using continuously recorded observations for the Deutschemark/Dollar and Yen/Dollar spot exchange rates, we find that forecasts from a simple long-memory Gaussian vector autoregression for the logarithmic daily realized volatilities perform admirably. Moreover, the vector autoregressive volatility forecast, coupled with a parametric lognormal-normal mixture distribution produces well-calibrated density forecasts of future returns, and correspondingly accurate quantile predictions. Our results hold promise for practical modeling and forecasting of the large covariance matrices relevant in asset pricing, asset allocation, and financial risk management applications.

GARCH-type models have been widely used for forecasting crude oil price volatility, but often ignore the structural changes of time series, which may lead to spurious volatility persistence. Therefore, this paper focuses on the smooth and sharp structural changes in crude oil price volatility, i.e., smooth shift and regime switching, respectively, and investigates which structural change based GARCH models have better performance for forecasting crude oil price volatility. The empirical results indicate that, first, the flexible Fourier form (FFF) GARCH-type models considering smooth shift can accurately model structural changes and yield superior fitting and forecasting performance to traditional GARCH-type models. Second, the Markov regime switching (MRS) GARCH model incorporating regime switching exhibits superior fitting performance compared to the single-regime GARCH-type models, but it does not necessarily beat the counterparts for forecasting. Finally, the FFF-GARCH-type models outperform MRS-GARCH for forecasting crude oil price volatility and portfolio performance.

We propose a decomposition of the realized covariance matrix into components based on the signs of the underlying high-frequency returns, and we derive the asymptotic properties of the resulting realized semicovariance measures as the sampling interval goes to zero. The first-order asymptotic results highlight how the same-sign and mixed-sign components load differently on economic information related to stochastic correlation and jumps. The second-order asymptotic results reveal the structure underlying the same-sign semicovariances, as manifested in the form of co-drifting and dynamic “leverage” effects. In line with this anatomy, we use data on a large cross-section of individual stocks to empirically document distinct dynamic dependencies in the different realized semicovariance components. We show that the accuracy of portfolio return variance forecasts may be significantly improved by exploiting the information in realized semicovariances.

Over the past years, cryptocurrencies have drawn substantial attention from the media while attracting many investors. Since then, cryptocurrency prices have experienced high fluctuations. In this paper, we forecast the high-frequency 1 min volatility of four widely traded cryptocurrencies, i.e., Bitcoin, Ethereum, Litecoin, and Ripple, by modeling volatility to select the best model. We propose various generalized autoregressive conditional heteroscedasticity (GARCH) family models, including an sGARCH(1,1), GJR-GARCH(1,1), TGARCH(1,1), EGARCH(1,1), which we compare to a multivariate DCC-GARCH(1,1) model to forecast the intraday price volatility. We evaluate the results under the MSE and MAE loss functions. Statistical analyses demonstrate that the univariate GJR-GARCH model (1,1) shows a superior predictive accuracy at all horizons, followed closely by the TGARCH(1,1), which are the best models for modeling the volatility process on out-of-sample data and have more accurately indicated the asymmetric incidence of shocks in the cryptocurrency market. The study determines evidence of bidirectional shock transmission effects between the cryptocurrency pairs. Hence, the multivariate DCC-GARCH model can identify the cryptocurrency market’s cross-market volatility shocks and volatility transmissions. In addition, we introduce a comparison of the models using the improvement rate (IR) metric for comparing models. As a result, we compare the different forecasting models to the chosen benchmarking model to confirm the improvement trends for the model’s predictions.

This study investigates the impact of economic policy uncertainty (EPU) on the volatility of European Union (EU) carbon futures prices and whether it has predictive power for the volatility of carbon futures prices. The GARCH-MIDAS model is applied for evaluating the impact of different EPU indexes on the price volatility of European Union Allowance (EUA) futures. We then compare the predictive power for the volatility of the two GARCH-MIDAS models based on different EPU indexes and six GARCH-type models. Our empirical results show that the GARCH-MIDAS models, which exhibit superior out-of-sample predictive ability, outperform GARCH-type models. The results also indicate that EPU has noticeable effect on the volatility of EUA futures. Specifically, the forecast accuracy of the EU EPU index is significantly higher than that of the global EPU index. Robustness checks further confirm that the EPU index (especially the EPU index of the EU) has strong predictive power for EUA futures prices. Additionally, using the volatility forecasting methods that GARCH-MIDAS models combine with the EPU index, investors can construct their portfolios to realize economic returns.

Volatility is usually a proxy indicator for market variation or tendency, containing essential information for investors and policy-makers. This paper proposes a novel hybrid deep neural network model (HDNN) with temporal embedding for volatility forecasting. The main idea of our HDNN is that it encodes one-dimensional time-series data as two-dimensional GAF images, which enables the follow-up convolution neural network (CNN) to learn volatility-related feature mappings automatically. Specifically, HDNN adopts an elegant end-to-end learning paradigm for volatility forecasting, which consists of feature embedding and regression components. The feature embedding component explores the volatility-related temporal information from GAF images via the elaborate CNN in an underlying temporal embedding space. Then, the regression component takes these embedding vectors as input for volatility forecasting tasks. Finally, we examine the feasibility of HDNN on four synthetic GBM datasets and five real-world Stock Index datasets in terms of five regression metrics. The results demonstrate that HDNN has better performance in most cases than the baseline forecasting models of GARCH, EGACH, SVR, and NN. It confirms that the volatility-related temporal features extracted by HDNN indeed improve the forecasting ability. Furthermore, the Friedman test verifies that HDNN is statistically superior to the compared forecasting models.

This study comprehensively investigates the predictability of uncertainty indices for oil market volatility, employing multiple machine learning models based on a large set of uncertainty indices. Empirical findings demonstrate the efficiency of machine learning models for predicting oil futures volatility using uncertainty indices. The results are consistent across various robustness checks and special circumstances. This study highlights the need to combine the efficiency of machine learning models with as much information from uncertainty indices as possible to capture the dynamics of the oil market, which is essential for energy fields to confront future fierce situations and crises.