Bias in the estimation of non‐linear transformations of the integrated variance of returns

Volatility models such as GARCH, although misspecified with respect to the data‐generating process, may well generate volatility forecasts that are unconditionally unbiased. In other words, they generate variance forecasts that, on average, are equal to the integrated variance. However, many applications in finance require a measure of return volatility that is a non‐linear function of the variance of returns, rather than of the variance itself. Even if a volatility model generates forecasts of the integrated variance that are unbiased, non‐linear transformations of these forecasts will be biased estimators of the same non‐linear transformations of the integrated variance because of Jensen's inequality. In this paper, we derive an analytical approximation for the unconditional bias of estimators of non‐linear transformations of the integrated variance. This bias is a function of the volatility of the forecast variance and the volatility of the integrated variance, and depends on the concavity of the non‐linear transformation. In order to estimate the volatility of the unobserved integrated variance, we employ recent results from the realized volatility literature. As an illustration, we estimate the unconditional bias for both in‐sample and out‐of‐sample forecasts of three non‐linear transformations of the integrated standard deviation of returns for three exchange rate return series, where a GARCH(1, 1) model is used to forecast the integrated variance. Our estimation results suggest that, in practice, the bias can be substantial. Copyright © 2006 John Wiley & Sons, Ltd.     

Forecasting volatility

In this paper, we investigate the time series properties of S&P 100 volatility and the forecasting performance of different volatility models. We consider several nonparametric and parametric volatility measures, such as implied, realized and model‐based volatility, and show that these volatility processes exhibit an extremely slow mean‐reverting behavior and possible long memory. For this reason, we explicitly model the near‐unit root behavior of volatility and construct median unbiased forecasts by approximating the finite‐sample forecast distribution using bootstrap methods. Furthermore, we produce prediction intervals for the next‐period implied volatility that provide important information about the uncertainty surrounding the point forecasts. Finally, we apply intercept corrections to forecasts from misspecified models which dramatically improve the accuracy of the volatility forecasts. Copyright © 2006 John Wiley & Sons, Ltd.     

Realized Volatility Forecast of Stock Index Under Structural Breaks

We investigate the realized volatility forecast of stock indices under the structural breaks. We utilize a pure multiple mean break model to identify the possibility of structural breaks in the daily realized volatility series by employing the intraday high‐frequency data of the Shanghai Stock Exchange Composite Index and the five sectoral stock indices in Chinese stock markets for the period 4 January 2000 to 30 December 2011. We then conduct both in‐sample tests and out‐of‐sample forecasts to examine the effects of structural breaks on the performance of ARFIMAX‐FIGARCH models for the realized volatility forecast by utilizing a variety of estimation window sizes designed to accommodate potential structural breaks. The results of the in‐sample tests show that there are multiple breaks in all realized volatility series. The results of the out‐of‐sample point forecasts indicate that the combination forecasts with time‐varying weights across individual forecast models estimated with different estimation windows perform well. In particular, nonlinear combination forecasts with the weights chosen based on a non‐parametric kernel regression and linear combination forecasts with the weights chosen based on the non‐negative restricted least squares and Schwarz information criterion appear to be the most accurate methods in point forecasting for realized volatility under structural breaks. We also conduct an interval forecast of the realized volatility for the combination approaches, and find that the interval forecast for nonlinear combination approaches with the weights chosen according to a non‐parametric kernel regression performs best among the competing models. Copyright © 2014 John Wiley & Sons, Ltd.     

Daily volatility forecasts: reassessing the performance of GARCH models

Volatility plays a key role in asset and portfolio management and derivatives pricing. As such, accurate measures and good forecasts of volatility are crucial for the implementation and evaluation of asset and derivative pricing models in addition to trading and hedging strategies. However, whilst GARCH models are able to capture the observed clustering effect in asset price volatility in‐sample, they appear to provide relatively poor out‐of‐sample forecasts. Recent research has suggested that this relative failure of GARCH models arises not from a failure of the model but a failure to specify correctly the ‘true volatility’ measure against which forecasting performance is measured. It is argued that the standard approach of using ex post daily squared returns as the measure of ‘true volatility’ includes a large noisy component. An alternative measure for ‘true volatility’ has therefore been suggested, based upon the cumulative squared returns from intra‐day data. This paper implements that technique and reports that, in a dataset of 17 daily exchange rate series, the GARCH model outperforms smoothing and moving average techniques which have been previously identified as providing superior volatility forecasts. Copyright © 2004 John Wiley & Sons, Ltd.     

Daily FX Volatility Forecasts: Can the GARCH(1,1) Model be Beaten using High‐Frequency Data?

Volatility forecasting remains an active area of research with no current consensus as to the model that provides the most accurate forecasts, though Hansen and Lunde (2005) have argued that in the context of daily exchange rate returns nothing can beat a GARCH(1,1) model. This paper extends that line of research by utilizing intra‐day data and obtaining daily volatility forecasts from a range of models based upon the higher‐frequency data. The volatility forecasts are appraised using four different measures of ‘true’ volatility and further evaluated using regression tests of predictive power, forecast encompassing and forecast combination. Our results show that the daily GARCH(1,1) model is largely inferior to all other models, whereas the intra‐day unadjusted‐data GARCH(1,1) model generally provides superior forecasts compared to all other models. Hence, while it appears that a daily GARCH(1,1) model can be beaten in obtaining accurate daily volatility forecasts, an intra‐day GARCH(1,1) model cannot be. Copyright © 2011 John Wiley & Sons, Ltd.     

Volatility Forecasting via MIDAS,HAR and their Combination: An Empirical Comparative Study for IBOVESPA

In this paper we compare several multi‐period volatility forecasting models, specifically from MIDAS and HAR families. We perform our comparisons in terms of out‐of‐sample volatility forecasting accuracy. We also consider combinations of the models' forecasts. Using intra‐daily returns of the BOVESPA index, we calculate volatility measures such as realized variance, realized power variation and realized bipower variation to be used as regressors in both models. Further, we use a nonparametric procedure for separately measuring the continuous sample path variation and the discontinuous jump part of the quadratic variation process. Thus MIDAS and HAR specifications with the continuous sample path and jump variability measures as separate regressors are estimated. Our results in terms of mean squared error suggest that regressors involving volatility measures which are robust to jumps (i.e. realized bipower variation and realized power variation) are better at forecasting future volatility. However, we find that, in general, the forecasts based on these regressors are not statistically different from those based on realized variance (the benchmark regressor). Moreover, we find that, in general, the relative forecasting performances of the three approaches (i.e. MIDAS, HAR and forecast combinations) are statistically equivalent. Copyright © 2014 John Wiley & Sons, Ltd.     

A Multiplicative Error Model with Heterogeneous Components for Forecasting Realized Volatility

To forecast realized volatility, this paper introduces a multiplicative error model that incorporates heterogeneous components: weekly and monthly realized volatility measures. While the model captures the long‐memory property, estimation simply proceeds using quasi‐maximum likelihood estimation. This paper investigates its forecasting ability using the realized kernels of 34 different assets provided by the Oxford‐Man Institute's Realized Library. The model outperforms benchmark models such as ARFIMA, HAR, Log‐HAR and HEAVY‐RM in within‐sample fitting and out‐of‐sample (1‐, 10‐ and 22‐step) forecasts. It performed best in both pointwise and cumulative comparisons of multi‐step‐ahead forecasts, regardless of loss function (QLIKE or MSE). Copyright © 2015 John Wiley & Sons, Ltd.     

Forecasting realized volatility of oil futures market: A new insight

In this study we propose several new variables, such as continuous realized semi‐variance and signed jump variations including jump tests, and construct a new heterogeneous autoregressive model for realized volatility models to investigate the impacts that those new variables have on forecasting oil price volatility. In‐sample results indicate that past negative returns have greater effects on future volatility than that of positive returns, and our new signed jump variations have a significantly negative influence on the future volatility. Out‐of‐sample empirical results with several robust checks demonstrate that our proposed models can not only obtain better performance in forecasting volatility but also garner larger economic values than can the existing models discussed in this paper.     

Modeling Realized Volatility Dynamics with a Genetic Algorithm

The heterogeneous autoregressive model of realized volatility (HAR‐RV) is inspired by the heterogeneous market hypothesis and characterizes realized volatility dynamics through a linear function of lagged daily, weekly and monthly realized volatilities with a (1, 5, 22) lag structure. Considering that different markets can have different heterogeneous structures and a market's heterogeneous structure can vary over time, we build an adaptive heterogeneous autoregressive model of realized volatility (AHAR‐RV), whose lag structure is optimized with a genetic algorithm. Using nine common loss functions and the superior predictive ability test, we find that our AHAR‐RV model and its extensions provide significantly better out‐of‐sample volatility forecasts for the CSI 300 index than the corresponding HAR models. Furthermore, the AHAR‐RV model significantly outperforms all the other models under most loss functions. Besides, we confirm that Chinese stock markets' heterogeneous structure varies over time and the (1, 5, 22) lag structure is not the optimal choice. Copyright © 2016 John Wiley & Sons, Ltd.     

Forecasting exchange rate volatility: a multiple horizon comparison using historical,realized and implied volatility measures

Recent studies suggest realized volatility provides forecasts that are as good as option‐implied volatilities, with improvement stemming from the use of high‐frequency data instead of a long‐memory specification. This paper examines whether volatility persistence can be captured by a longer dataset consisting of over 15 years of intra‐day data. Volatility forecasts are evaluated using four exchange rates (AUD/USD, EUR/USD, GBP/USD, USD/JPY) over horizons ranging from 1 day to 3 months, using an expanded set of short‐range and long‐range dependence models. The empirical results provide additional evidence that significant incremental information is found in historical forecasts, beyond the implied volatility information for all forecast horizons. Copyright © 2008 John Wiley & Sons, Ltd.     

Garch forecasting performance under different distribution assumptions

This paper investigates the forecasting performance of the Garch (1, 1) model when estimated with NINE different error distributions on Standard and Poor's 500 Index Future returns. By utilizing the theory of realized variance to construct an appropriate ex post measure of volatility from intra‐day data it is shown that allowing for a leptokurtic error distribution leads to significant improvements in variance forecasts compared to using the normal distribution. This result holds for daily, weekly as well as monthly forecast horizons. It is also found that allowing for skewness and time variation in the higher moments of the distribution does not further improve forecasts. Copyright © 2006 John Wiley & Sons, Ltd.     

An outlier robust GARCH model and forecasting volatility of exchange rate returns

Since volatility is perceived as an explicit measure of risk, financial economists have long been concerned with accurate measures and forecasts of future volatility and, undoubtedly, the Generalized Autoregressive Conditional Heteroscedasticity (GARCH) model has been widely used for doing so. It appears, however, from some empirical studies that the GARCH model tends to provide poor volatility forecasts in the presence of additive outliers. To overcome the forecasting limitation, this paper proposes a robust GARCH model (RGARCH) using least absolute deviation estimation and introduces a valuable estimation method from a practical point of view. Extensive Monte Carlo experiments substantiate our conjectures. As the magnitude of the outliers increases, the one‐step‐ahead forecasting performance of the RGARCH model has a more significant improvement in two forecast evaluation criteria over both the standard GARCH and random walk models. Strong evidence in favour of the RGARCH model over other competitive models is based on empirical application. By using a sample of two daily exchange rate series, we find that the out‐of‐sample volatility forecasts of the RGARCH model are apparently superior to those of other competitive models. Copyright © 2002 John Wiley & Sons, Ltd.     

Optimal sampling frequency for volatility forecast models for the Indian stock markets

This paper evaluates the performance of conditional variance models using high‐frequency data of the National Stock Index (S&P CNX NIFTY) and attempts to determine the optimal sampling frequency for the best daily volatility forecast. A linear combination of the realized volatilities calculated at two different frequencies is used as benchmark to evaluate the volatility forecasting ability of the conditional variance models (GARCH (1, 1)) at different sampling frequencies. From the analysis, it is found that sampling at 30 minutes gives the best forecast for daily volatility. The forecasting ability of these models is deteriorated, however, by the non‐normal property of mean adjusted returns, which is an assumption in conditional variance models. Nevertheless, the optimum frequency remained the same even in the case of different models (EGARCH and PARCH) and different error distribution (generalized error distribution, GED) where the error is reduced to a certain extent by incorporating the asymmetric effect on volatility. Our analysis also suggests that GARCH models with GED innovations or EGRACH and PARCH models would give better estimates of volatility with lower forecast error estimates. Copyright © 2008 John Wiley & Sons, Ltd.     

Empirical analysis and forecasting of volatility dynamics in high‐frequency returns with time‐varying components

We study intraday return volatility dynamics using a time‐varying components approach, and the method is applied to analyze IBM intraday returns. Empirical evidence indicates that with three additive components—a time‐varying mean of absolute returns and two cosine components with time‐varying amplitudes—together they capture very well the pronounced periodicity and persistence behaviors exhibited in the empirical autocorrelation pattern of IBM returns. We find that the long‐run volatility persistence is driven predominantly by daily level shifts in mean absolute returns. After adjusting for these intradaily components, the filtered returns behave much like a Gaussian noise, suggesting that the three‐components structure is adequately specified. Furthermore, a new volatility measure (TCV) can be constructed from these components. Results from extensive out‐of‐sample rolling forecast experiments suggest that TCV fares well in predicting future volatility against alternative methods, including GARCH model, realized volatility and realized absolute value. Copyright © 2009 John Wiley & Sons, Ltd.     

Forecasting volatility with noisy jumps: an application to the Dow Jones Industrial Average stocks

Empirical high‐frequency data can be used to separate the continuous and the jump components of realized volatility. This may improve on the accuracy of out‐of‐sample realized volatility forecasts. A further improvement may be realized by disentangling the two components using a sampling frequency at which the market microstructure effect is negligible, and this is the objective of the paper. In particular, a significant improvement in the accuracy of volatility forecasts is obtained by deriving the jump information from time intervals at which the noise effect is weak. Copyright © 2008 John Wiley & Sons, Ltd.     

Constant versus Time‐Varying Beta Models: Further Forecast Evaluation

Recent advances in the measurement of beta (systematic return risk) and volatility (total return risk) demonstrate substantial advantages in utilizing high‐frequency return data in a variety of settings. These advances in the measurement of beta and volatility have resulted in improvements in the evaluation of alternative beta and volatility forecasting approaches. In addition, more precise measurement has also led to direct modeling of the time variation of beta and volatility. Both the realized beta and volatility literature have most commonly been modeled with an autoregressive process. In this paper we evaluate constant beta models against autoregressive models of time‐varying realized beta. We find that a constant beta model computed from daily returns over the last 12 months generates the most accurate quarterly forecast of beta and dominates the autoregressive time series forecasts. It also dominates (dramatically) the popular Fama–MacBeth constant beta model, which uses 5 years of monthly returns. Copyright © 2011 John Wiley & Sons, Ltd.     

Time series forecasting using functional partial least square regression with stochastic volatility,GARCH, and exponential smoothing

We propose a method for improving the predictive ability of standard forecasting models used in financial economics. Our approach is based on the functional partial least squares (FPLS) model, which is capable of avoiding multicollinearity in regression by efficiently extracting information from the high‐dimensional market data. By using its well‐known ability, we can incorporate auxiliary variables that improve the predictive accuracy. We provide an empirical application of our proposed methodology in terms of its ability to predict the conditional average log return and the volatility of crude oil prices via exponential smoothing, Bayesian stochastic volatility, and GARCH (generalized autoregressive conditional heteroskedasticity) models, respectively. In particular, what we call functional data analysis (FDA) traces in this article are obtained via the FPLS regression from both the crude oil returns and auxiliary variables of the exchange rates of major currencies. For forecast performance evaluation, we compare out‐of‐sample forecasting accuracy of the standard models with FDA traces to the accuracy of the same forecasting models with the observed crude oil returns, principal component regression (PCR), and least absolute shrinkage and selection operator (LASSO) models. We find evidence that the standard models with FDA traces significantly outperform our competing models. Finally, they are also compared with the test for superior predictive ability and the reality check for data snooping. Our empirical results show that our new methodology significantly improves predictive ability of standard models in forecasting the latent average log return and the volatility of financial time series.     

Realized Volatility Forecasting of Agricultural Commodity Futures Using Long Memory and Regime Switching

We investigate the dynamic properties of the realized volatility of five agricultural commodity futures by employing the high‐frequency data from Chinese markets and find that the realized volatility exhibits both long memory and regime switching. To capture these properties simultaneously, we utilize a Markov switching autoregressive fractionally integrated moving average (MS‐ARFIMA) model to forecast the realized volatility by combining the long memory process with regime switching component, and compare its forecast performances with the competing models at various horizons. The full‐sample estimation results show that the dynamics of the realized volatility of agricultural commodity futures are characterized by two levels of long memory: one associated with the low‐volatility regime and the other with the high‐volatility regime, and the probability to stay in the low‐volatility regime is higher than that in the high‐volatility regime. The out‐of‐sample volatility forecast results show that the combination of long memory with switching regimes improves the performance of realized volatility forecast, and the proposed model represents a superior out‐of‐sample realized volatility forecast to the competing models. Copyright © 2016 John Wiley & Sons, Ltd.     

Forecasting and trading currency volatility: an application of recurrent neural regression and model combination

In this paper, we examine the use of non‐parametric Neural Network Regression (NNR) and Recurrent Neural Network (RNN) regression models for forecasting and trading currency volatility, with an application to the GBP/USD and USD/JPY exchange rates. Both the results of the NNR and RNN models are benchmarked against the simpler GARCH alternative and implied volatility. Two simple model combinations are also analysed. The intuitively appealing idea of developing a nonlinear nonparametric approach to forecast FX volatility, identify mispriced options and subsequently develop a trading strategy based upon this process is implemented for the first time on a comprehensive basis. Using daily data from December 1993 through April 1999, we develop alternative FX volatility forecasting models. These models are then tested out‐of‐sample over the period April 1999–May 2000, not only in terms of forecasting accuracy, but also in terms of trading efficiency: in order to do so, we apply a realistic volatility trading strategy using FX option straddles once mispriced options have been identified. Allowing for transaction costs, most trading strategies retained produce positive returns. RNN models appear as the best single modelling approach yet, somewhat surprisingly, model combination which has the best overall performance in terms of forecasting accuracy, fails to improve the RNN‐based volatility trading results. Another conclusion from our results is that, for the period and currencies considered, the currency option market was inefficient and/or the pricing formulae applied by market participants were inadequate. Copyright © 2002 John Wiley & Sons, Ltd.     

Model Uncertainty and Forecast Combination in High‐Dimensional Multivariate Volatility Prediction

In multivariate volatility prediction, identifying the optimal forecasting model is not always a feasible task. This is mainly due to the curse of dimensionality typically affecting multivariate volatility models. In practice only a subset of the potentially available models can be effectively estimated, after imposing severe constraints on the dynamic structure of the volatility process. It follows that in most applications the working forecasting model can be severely misspecified. This situation leaves scope for the application of forecast combination strategies as a tool for improving the predictive accuracy. The aim of the paper is to propose some alternative combination strategies and compare their performances in forecasting high‐dimensional multivariate conditional covariance matrices for a portfolio of US stock returns. In particular, we will consider the combination of volatility predictions generated by multivariate GARCH models, based on daily returns, and dynamic models for realized covariance matrices, built from intra‐daily returns. Copyright © 2015 John Wiley & Sons, Ltd.