Improving stock market volatility forecasts with complete subset linear and quantile HAR models

• HAR models are used to forecast Realized Volatility (RV) for a portfolio of stocks. • Low-, medium-, and high-frequency components of RV are isolated using wavelet decomposition. • Low-frequency volatility forecasts improve utility and reduce portfolio turnover. • Benefits are robust to direct forecasts of portfolio weights and jump robust RV. • Results stress the distinction between statistical and practical performance metrics in volatility forecasting. This study proposes a wavelet-based approach to forecasting Realized Volatility (RV) and evaluates its economic value within a volatility-timing framework. We apply wavelet decomposition to separate short-, medium-, and long-term components and generate forecasts using Heterogeneous Autoregressive (HAR) models. Forecasts based on the low-frequency component consistently lead to better portfolio outcomes, reducing turnover and enhancing investor utility without increasing risk. These results hold even when portfolio weights are forecast directly after being constructed from RV, or when jump-robust volatility estimates are used. The results highlight the importance of aligning forecast evaluation with practical investment objectives. Forecasts delivering the greatest welfare gains may not minimize conventional statistical loss functions.

This study introduces a realized volatility fuzzy time series (RV‐FTS) model that applies a fuzzy c‐means clustering algorithm to estimate time‐varying latent volatility states and their corresponding membership degrees. These memberships are used to construct a fuzzified volatility estimate as a weighted average of cluster centroids. The final volatility forecast is generated through an exponentially weighted moving average (EWMA) mechanism that combines the most recent fuzzified volatility estimate with the previous forecast, governed by the smoothing parameter . The two hyperparameters are estimated using a rolling‐window cross‐validation approach. Our empirical study is based on volatility forecasts for 14 major stock market indices, covering more than 20 years of data. We predict 1‐ to 22‐day‐ahead volatility and compare the RV‐FTS model with nine standard volatility model benchmarks: generalized autoregressive conditional heteroscedasticity (GARCH), ARFIMA, AR, heterogeneous autoregressive (HAR), EWMA, and random forest models, as well as conditional combination forecasts. We find that, in the short‐term, day‐ahead setting, the RV‐FTS model tends to outperform the benchmark models under the mean squared error loss and performs similarly to the best models under the QLIKE loss. The conditional combination forecast shows that across all markets and multiple forecast horizons, there are periods when the weight of the RV‐FTS model in the conditional combination of eight models reaches 50% or more. The volatility timing strategy also shows that the RV‐FTS model leads to higher cost‐ and risk‐adjusted returns compared with a benchmark volatility model.

Exploiting the heteroskedasticity in measurement error to improve volatility predictions in oil and biofuel feedstock markets

Since Bitcoin price is highly volatile, forecasting its volatility is crucial for many applications, such as risk management or hedging. We study which model is the most suitable for forecasting Bitcoin volatility. We consider several GARCH and two heterogeneous autoregressive (HAR) models and compare them. Since we utilize realized variance estimated from high frequency data as a proxy for true volatility, we can draw sharper conclusions than studies which use only daily data. We find that EGARCH and APARCH perform best among the GARCH models. HAR models based on realized variance perform better than GARCH models based on daily data. Superiority of HAR models over GARCH models is strongest for short-term volatility forecasts.

International tourism is a major source of export receipts for many countries worldwide. Although it is not yet one of the most important industries in Taiwan (or the Republic of China), an island in East Asia off the coast of mainland China (or the People's Republic of China), the leading tourism source countries for Taiwan are Japan, followed by USA, Republic of Korea, Malaysia, Singapore, UK, Germany and Australia. These countries reflect short, medium and long haul tourist destinations. Although the People's Republic of China and Hong Kong are large sources of tourism to Taiwan, the political situation is such that tourists from these two sources to Taiwan are reported as domestic tourists. Daily data from 1 January 1990 to 30 June 2007 are obtained from the National Immigration Agency of Taiwan. The heterogeneous autoregressive (HAR) model is used to capture long memory properties in the data. In comparison with the HAR(1) model, the estimated asymmetry coefficients for GJR(1,1) are not statistically significant for the HAR(1,7) and HAR(1,7,28) models, so that their respective GARCH(1,1) counterparts are to be preferred. These empirical results show that the conditional volatility estimates are sensitive to the long memory nature of the conditional mean specifications. Although asymmetry is observed for the HAR(1) model, there is no evidence of leverage. The quasi-maximum likelihood estimators (QMLE) for the GARCH(1,1), GJR(1,1) and EGARCH(1,1) models for international tourist arrivals to Taiwan are statistically adequate and have sensible interpretations. However, asymmetry (though not leverage) was found only for the HAR(1) model and not for the HAR(1,7) and HAR(1,7,28) models.

Modeling CAC40 volatility using ultra-high frequency data

We analyze the predictability of daily data on the CBOE VIX and SKEW indices, used to capture the average level of risk-neutral risk and downside risk, respectively, as implied by S&P 500 index options. In particular, we use forecast models from the Heterogeneous Autoregressive (HAR) class to test whether and how lagged values of the VIX and of the SKEW may increase the forecasting power of HAR for the SKEW and the VIX. We find that a simple HAR is very hard to beat in out-of-sample experiments aimed at forecasting the VIX. In the case of the SKEW, the benchmarks (the random walk and an AR(1)) are clearly outperformed by HAR models at all the forecast horizons considered and there is evidence that special definitions of the SKEW index based on put options data only yield superior forecasts at all horizons.

The Properties of Realized Correlation: Evidence from the Greek Equity Market

In the international monetary system, gold plays a significant role. Predicting gold prices is a useful and unique skill for anybody. As a result, improving one's ability to anticipate gold futures is critical. The study presented in this paper relates to gold futures predictions, based on heterogeneous autoregressive (HAR) theory, and Heterogeneous Autoregressive model of Realized Volatility (HAR-RV model), coupled with gold's daily trade volume and CBOE Volatility Index (VIX) to create three unique models: Heterogeneous Autoregressive model of Realized Volatility and Trading volume (HAR-RV-T model), Heterogeneous Autoregressive model of Realized Volatility and Volatility Index (HAR- RV-VIX model), and Heterogeneous Autoregressive model of Realized Volatility, Trading Volume, and Volatility Index (HAR-RV-T&VIX model). This paper mainly explores a method to predict the volatility of gold futures. Improve the ability to forecasting the volatility of gold prices is obviously conducive to effectively play the futures, including hedging, risk management, price analysis, and other tasks. The research concludes that adding trading volume and sentiment indicator contributes to a more robust HAR model and performs better on forecasting.

We apply heterogeneous autoregressive (HAR) models—including nine univariate, two multivariate and three combination models—to high-frequency data to predict the one-day forward volatilities of two strategically linked commodities, gold and silver. We provide evidence that it is difficult to beat the benchmark HAR model using univariate models and that, a much better strategy is to average the forecasts from many models. In addition, the forecasts are not improved by using volatilities from strategically linked commodities; thus, no volatility spillovers are detected. Interestingly, when the two strategically linked commodities are modelled together using the generalized HAR model, the forecasts are comparable to those of combination forecast models.

We propose moving average threshold heterogeneous autoregressive (MAT‐HAR) models as a novel combination of heterogeneous autoregression (HAR) and threshold autoregression (TAR). The MAT‐HAR has multiple groups of lags of a target series, and a threshold term can appear in each group. The threshold is a moving average of lagged target series, which guarantees time‐varying thresholds and simple estimation via least squares. We show via Monte Carlo simulations that the MAT‐HAR has sharp in‐sample and out‐of‐sample performance. An empirical application on the industrial production of Japan suggests that significant threshold effects exist, and the MAT‐HAR has a higher forecast accuracy than the HAR.

This study examines the role of after‐hours information in forecasting Chinese commodity futures volatility, exploiting the introduction of a night session that potentially facilitates real‐time responses to information originating in overseas markets. We generate timely forecasts on future daytime realized volatility for 10 commodity futures, using heterogeneous autoregressive (HAR) models augmented with and without past nights' realized variance measures. Our results reveal significant predictive power, both in‐sample and out‐of‐sample, associated with the night‐time realized volatility across markets. In contrast, the inclusion of daily squared overnight returns as an alternative measure provides limited improvements. Furthermore, we document the empirical merit of separately considering the jump and continuous components in the night‐session price variation, with its superior performance being most pronounced over long forecasting horizons. The improved statistical accuracy is also shown to be economically meaningful for a risk‐averse investor and remains robust to changes in the identification, estimation, and forecasting procedure.

A long-memory and non-linear realized volatility model class is proposed for direct Value-at-Risk (VaR) forecasting. This model, referred to as RNN-HAR, extends the heterogeneous autoregressive (HAR) model, a framework known for efficiently capturing long memory in realized measures, by integrating a Recurrent Neural Network (RNN) to handle the non-linear dynamics. Quantile loss-based generalized Bayesian method with Sequential Monte Carlo is employed for model estimation and sequential prediction in RNN-HAR. The empirical analysis is conducted using daily closing prices and realized measures with around 12 years of data till 2022, covering 31 market indices. The proposed model’s one-step-ahead VaR forecasting performance is compared against a basic HAR model and its extensions. The results demonstrate that the proposed RNN-HAR model consistently outperforms all other models considered in the study. The implementation code of the HAR-RNN model is publicly available on GitHub: https://github.com/chaowang-usyd/RNN-HAR.

In this study, we investigate the impacts of common information between oil futures and the United States stock markets on forecasting oil volatility using the multivariate heterogeneous autoregressive realized volatility model. We have several noteworthy findings. First, the in‐sample estimation results show that the negative returns and the ‘bad’ jumps have larger influence on oil futures volatility than positive returns and ‘good’ jumps, respectively. Second, the out‐of‐sample results further indicate that our multivariate heterogeneous autoregressive (HAR) model can generate higher forecast accuracy than individual HAR model for the realized volatility‐type models, which highlights the importance of incorporating volatility information from the stock Market. Third, our findings are robust across the direction‐of‐change test and different windows. Furthermore, from the perspective of the investors' allocation portfolio in the future, the use of common information can also bring greater gains.

This study evaluates the performance of four models for predicting daily realized volatility of S&P 500 index (SPX), Dow Jones Industry Average (DJIA), Canadian dollar (CAD/USD) and British Pound (USD/GBP) exchange rates. The competing models include a Simple Regression Model (SRM), Stochastic Volatility model with Lagged inter-temporal dependence (SVL), Stochastic Volatility model with Contemporaneous dependence (SVC), and a Heterogeneous Autoregressive (HAR) model. The main purpose is to examine whether allowing asymmetric relationships between return and volatility and leptokurtosis, or modelling the long-memory behaviour of volatility, would result in an improvement in forecast accuracy. Different approaches are considered when constructing daily realized volatility. Employing realized volatility in the in-sample estimation, the procedure is straightforward. The famous Diebold and Mariano's (1995) robust tests are applied to investigate whether the competing models provide equally accurate forecasts. ...