Long memory versus structural breaks in modeling and forecasting realized volatility

We explore the possibility of structural breaks in the daily realized volatility of the Deutschemark/Dollar, Yen/Dollar and Yen/Deutschemark spot exchange rates with observed long memory behavior. We find that structural breaks in the mean can partly explain the persistence of realized volatility. We propose a VAR-RV-Break model that provides superior predictive ability when the timing of future breaks is known. With unknown break dates and sizes, we find that a VAR-RV-I(d) long memory model provides a robust forecasting method even when the true financial volatility series are generated by structural breaks.     

Mean Reversion in Net Discount Ratios: A Study in the Context of Fractionally Integrated Models

This article introduces a new alternative to the ongoing debate about stationarity and mean reversion of the net discount ratio. Modeling the net discount ratio as a fractionally integrated (I(d)) process, we apply recently developed frequency domain estimation procedures and find evidence that the net discount ratio is an I(d) process with 1/2 ≤d < 1. Although nonstationary, such series behave like stationary processes in one interesting respect; they are mean‐reverting. We present results from a simulation experiment suggesting that the finding of a nonstationary, but mean‐reverting net discount ratio generally supports the validity of current practice in estimating economic damages in personal injury litigation. Moreover, if recognized and accounted for, the presence of long memory in the net discount ratio even offers the potential to significantly improve forecasts of the present value of future earnings.     

Long memory in log-range series: Do structural breaks matter?

This paper examines whether the observed long memory behavior of log-range series is to some extent spurious and whether it can be explained by the presence of structural breaks. Utilizing stock market data we show that the characterization of log-range series as long memory processes can be a strong assumption. Moreover, we find that all examined series experience a large number of significant breaks. Once the breaks are accounted for, the volatility persistence is eliminated. Overall, the findings suggest that volatility can be adequately represented, at least in-sample, through a multiple breaks process and a short run component.     

Uncovering the US term premium: An alternative route

The estimates of the US term premium crucially depend upon the ex-ante decision on whether the short-term rate is either an I(0) or an I(1) process. In this paper we estimate a fractionally integrated (I(d)) model which simultaneously determines both the order of integration of the short-term rate and the associated term premium. We show that the term premium experienced a sharp increase from essentially zero in mid-2007 to almost 3% in 2009. We also show that unemployment and term premium dynamics exhibit a very significant positive co-movement.     

Dynamic factor long memory volatility

In this paper, we develop a long memory orthogonal factor (LMOF) multivariate volatility model for forecasting the covariance matrix of financial asset returns. We evaluate the LMOF model using the volatility timing framework of Fleming et al. [J. Finance, 2001, 56, 329–352] and compare its performance with that of both a static investment strategy based on the unconditional covariance matrix and a range of dynamic investment strategies based on existing short memory and long memory multivariate conditional volatility models. We show that investors should be willing to pay to switch from the static strategy to a dynamic volatility timing strategy and that, among the dynamic strategies, the LMOF model consistently produces forecasts of the covariance matrix that are economically more useful than those produced by the other multivariate conditional volatility models, both short memory and long memory. Moreover, we show that combining long memory volatility with the factor structure yields better results than employing either long memory volatility or the factor structure alone. The factor structure also significantly reduces transaction costs, thus increasing the feasibility of dynamic volatility timing strategies in practice. Our results are robust to estimation error in expected returns, the choice of risk aversion coefficient, the estimation window length and sub-period analysis.     

Forecasting daily variability of the S&P 100 stock index using historical,realised and implied volatility measurements

The increasing availability of financial market data at intraday frequencies has not only led to the development of improved volatility measurements but has also inspired research into their potential value as an information source for volatility forecasting. In this paper, we explore the forecasting value of historical volatility (extracted from daily return series), of implied volatility (extracted from option pricing data) and of realised volatility (computed as the sum of squared high frequency returns within a day). First, we consider unobserved components (UC-RV) and long memory models for realised volatility which is regarded as an accurate estimator of volatility. The predictive abilities of realised volatility models are compared with those of stochastic volatility (SV) models and generalised autoregressive conditional heteroskedasticity (GARCH) models for daily return series. These historical volatility models are extended to include realised and implied volatility measures as explanatory variables for volatility. The main focus is on forecasting the daily variability of the Standard & Poor's 100 (S&P 100) stock index series for which trading data (tick by tick) of almost 7 years is analysed. The forecast assessment is based on the hypothesis of whether a forecast model is outperformed by alternative models. In particular, we will use superior predictive ability tests to investigate the relative forecast performances of some models. Since volatilities are not observed, realised volatility is taken as a proxy for actual volatility and is used for computing the forecast error. A stationary bootstrap procedure is required for computing the test statistic and its p-value. The empirical results show convincingly that realised volatility models produce far more accurate volatility forecasts compared to models based on daily returns. Long memory models seem to provide the most accurate forecasts.     

Long memory and level shifts in REITs returns and volatility

This paper provides an empirical investigation of the long memory in the returns and volatility of REITs markets of the USA, the UK, Hong Kong, Australia, and Japan. Initially, we subject the series to unit root tests proposed by Saikkonen and Lütkepohl (2002) and Lanne et al. (2002), which allow for a level shift in the data generating process. We confirm the stationarity of the REITs returns in the presence of structural breaks, with the breaks happening during the 2008 and 2009 periods. Second, by employing long memory tests and estimators, a weak long memory is demonstrated in the return series, but a strong evidence is provided in the volatility measures. Then using Smith (2005)'s modified GPH estimator, we find that a short-memory model with a level shift is a viable alternative to a long memory model for the USA, Hong Kong and Japan and not for the UK nor for Australia. Finally, we confirm that the long memory in volatility is real and not caused by shifts in variance for all markets. Our results should be useful to market participants in the REITs markets, whose success depends on the ability to forecast and model REITs price movements.     

Is integration I(d) applicable to observed economics and finance time series?

The method of cointegration in regression analysis is based on an assumption of stationary increments. Stationary increments with fixed time lag are called ‘integration I(d)’. A class of regression models where cointegration works was identified by Granger and yields the ergodic behavior required for equilibrium expectations in standard economics. Detrended finance market returns are martingales, and martingales do not satisfy regression equations. We ask if there exist detrended processes beyond standard regression models that satisfy integration I(d). We show that stationary increment martingales are confined to the Wiener process, and observe that martingales describing finance data admit neither the integration I(d) nor the ergodicity required for long time equilibrium relationships. In particular, the martingales derived from finance data do not admit the time (or ‘space’) translational invariance required for increment stationarity. Our analysis explains the lack of equilibrium observed earlier between FX rates and relative price levels.     

An application of a random level shifts model to the volatility of Peruvian stock and exchange rate returns

The literature has shown that the volatility of stock and forex rate market returns shows the characteristic of long memory. Another fact that is shown in the literature is that this feature may be spurious and volatility actually consists of a short memory process contaminated with random level shifts (RLS). In this paper, we follow recent econometric approaches estimating an RLS model to the logarithm of the absolute value of stock and forex returns. The model consists of the sum of a short-term memory component and a component of level shifts. The second component is specified as the cumulative sum of a process that is zero with probability ‘1-alpha’ and is a random variable with probability ‘alpha’. The results show that there are level shifts that are rare, but once they are taken into account, the characteristic or property of long memory disappears. Also, the presence of General Autoregressive Conditional Heteroscedasticity (GARCH) effects is eliminated when included or deducted level shifts. An exercise of out-of-sample forecasting shows that the RLS model has better performance than traditional models for modelling long memory such as the models ARFIMA (p,d,q).     

Volatility forecasting in the Chinese commodity futures market with intraday data

Given the unique institutional regulations in the Chinese commodity futures market as well as the characteristics of the data it generates, we utilize contracts with three months to delivery, the most liquid contract series, to systematically explore volatility forecasting for aluminum, copper, fuel oil, and sugar at the daily and three intraday sampling frequencies. We adopt popular volatility models in the literature and assess the forecasts obtained via these models against alternative proxies for the true volatility. Our results suggest that the long memory property is an essential feature in the commodity futures volatility dynamics and that the ARFIMA model consistently produces the best forecasts or forecasts not inferior to the best in statistical terms.     

Accurate minimum capital risk requirements: A comparison of several approaches

In this paper we estimate, for several investment horizons, minimum capital risk requirements for short and long positions, using the unconditional distribution of three daily indexes futures returns and a set of short and long memory stochastic volatility and GARCH-type models. We consider the possibility that errors follow a t-Student distribution in order to capture the kurtosis of the returns’ series. The results suggest that accurate modelling of extreme observations obtained for long and short trading investment positions is possible with an autoregressive stochastic volatility model. Moreover, modelling futures returns with a long memory stochastic volatility model produces, in general, excessive volatility persistence, and consequently, leads to large minimum capital risk requirement estimates. Finally, the models’ predictive ability is assessed with the help of out-of-sample conditional tests.     

True or spurious long memory in European non-EMU currencies

We examine the Croatian Kuna, the Czech Koruna, the Hungarian Forint, the Polish Złoty, the Romanian Leu, and the Swedish Krona whether their Euro exchange rates volatility exhibits true or spurious long memory. Recent research reveals long memory in foreign exchange rate volatility and we confirm this finding for these currency pairs by examining the long memory behavior of squared residuals by means of the V/S test. However, by using the ICSS approach we also find structural breaks in the unconditional variance. Literature suggests that structural breaks might lead to spurious long memory behavior. In a refined test strategy, we distinguish true from spurious long memory for the six exchange rates. Our findings suggest that Czech Koruna and Hungarian Forint only feature spurious long memory, while the rest of the series have both structural breaks and true long memory. Lastly, we demonstrate how to extend existing models to jointly model both properties yielding superior fit and better Value-at-Risk forecasts. The results of our work help to avoid misspecification and provide a better understanding of the properties of the foreign exchange rate volatility.     

The economic importance of rare earth elements volatility forecasts

We compare the suitability of short-memory models (ARMA), long-memory models (ARFIMA), and a GARCH model to describe the volatility of rare earth elements (REEs). We find strong support for the existence of long-memory effects. A simple long-memory ARFIMA (0, d, 0) baseline model shows generally superior accuracy both in- and out-of-sample, and is robust for various subsamples and estimation windows. Volatility forecasts produced by the baseline model also convey material forward-looking information for companies in the REEs industry. Thus, an active trading strategy based on REE volatility forecasts for these companies significantly outperforms a passive buy-and-hold strategy on both an absolute and a risk-adjusted return basis.     

Relative forecasting performance of volatility models: Monte Carlo evidence

Abstract

A Monte Carlo (MC) experiment is conducted to study the forecasting performance of a variety of volatility models under alternative data-generating processes (DGPs). The models included in the MC study are the (Fractionally Integrated) Generalized Autoregressive Conditional Heteroskedasticity models ((FI)GARCH), the Stochastic Volatility model (SV), the Long Memory Stochastic Volatility model (LMSV) and the Markov-switching Multifractal model (MSM). The MC study enables us to compare the relative forecasting performance of the models accounting for different characterizations of the latent volatility process: specifications that incorporate short/long memory, autoregressive components, stochastic shocks, Markov-switching and multifractality. Forecasts are evaluated by means of mean squared errors (MSE), mean absolute errors (MAE) and value-at-risk (VaR) diagnostics. Furthermore, complementarities between models are explored via forecast combinations. The results show that (i) the MSM model best forecasts volatility under any other alternative characterization of the latent volatility process and (ii) forecast combinations provide systematic improvements upon most single misspecified models, but are typically inferior to the MSM model even if the latter is applied to data governed by other processes.     

Multiple-days-ahead value-at-risk and expected shortfall forecasting for stock indices,commodities and exchange rates: Inter-day versus intra-day data

In order to provide reliable Value-at-Risk (VaR) and Expected Shortfall (ES) forecasts, this paper attempts to investigate whether an inter-day or an intra-day model provides accurate predictions. We investigate the performance of inter-day and intra-day volatility models by estimating the AR(1)-GARCH(1,1)-skT and the AR(1)-HAR-RV-skT frameworks, respectively. This paper is based on the recommendations of the Basel Committee on Banking Supervision. Regarding the forecasting performances, the exploitation of intra-day information does not appear to improve the accuracy of the VaR and ES forecasts for the 10-steps-ahead and 20-steps-ahead for the 95%, 97.5% and 99% significance levels. On the contrary, the GARCH specification, based on the inter-day information set, is the superior model for forecasting the multiple-days-ahead VaR and ES measurements. The intra-day volatility model is not as appropriate as it was expected to be for each of the different asset classes; stock indices, commodities and exchange rates.The multi-period VaR and ES forecasts are estimated for a range of datasets (stock indices, commodities, foreign exchange rates) in order to provide risk managers and financial institutions with information relating the performance of the inter-day and intra-day volatility models across various markets. The inter-day specification predicts VaR and ES measures adequately at a 95% confidence level. Regarding the 97.5% confidence level that has been recently proposed in the revised 2013 version of Basel III, the GARCH-skT specification provides accurate forecasts of the risk measures for stock indices and exchange rates, but not for commodities (that is Silver and Gold). In the case of the 99% confidence level, we do not achieve sufficiently accurate VaR and ES forecasts for all the assets.     

Structural change and long-range dependence in volatility of exchange rates: either, neither or both?

In this paper, we test for the existence of long memory and structural breaks in the realized variance process for the DM/US$ and Yen/US$ exchange rates. While long memory is evident in the actual processes, a structural break analysis reveals that this feature is partially explained by unaccounted changes in regime. We then compare the forecasting performance of Markov switching models with that of an ARFIMA model. The results indicate that neglecting the break process is not important for very short term forecasting once it is allowed for a long memory component in the model, but that superior forecasts can be obtained at longer horizons by modelling both long memory and structural change.     

Combining long memory and level shifts in modelling and forecasting the volatility of asset returns

We propose a parametric state space model of asset return volatility with an accompanying estimation and forecasting framework that allows for ARFIMA dynamics, random level shifts and measurement errors. The Kalman filter is used to construct the state-augmented likelihood function and subsequently to generate forecasts, which are mean and path-corrected. We apply our model to eight daily volatility series constructed from both high-frequency and daily returns. Full sample parameter estimates reveal that random level shifts are present in all series. Genuine long memory is present in most high-frequency measures of volatility, whereas there is little remaining dynamics in the volatility measures constructed using daily returns. From extensive forecast evaluations, we find that our ARFIMA model with random level shifts consistently belongs to the 10% Model Confidence Set across a variety of forecast horizons, asset classes and volatility measures. The gains in forecast accuracy can be very pronounced, especially at longer horizons.     

Volatility forecasts and at-the-money implied volatility: a multi-component ARCH approach and its relation to market models

This article explores the relationships between several forecasts for the volatility built from multi-scale linear ARCH processes, and linear market models for the forward variance. This shows that the structures of the forecast equations are identical, but with different dependencies on the forecast horizon. The process equations for the forward variance are induced by the process equations for an ARCH model, but postulated in a market model. In the ARCH case, they are different from the usual diffusive type. The conceptual differences between both approaches and their implication for volatility forecasts are analysed. The volatility forecast is compared with the realized volatility (the volatility that will occur between date t and t + ΔT), and the implied volatility (corresponding to an at-the-money option with expiry at t + ΔT). For the ARCH forecasts, the parameters are set a priori. An empirical analysis across multiple time horizons ΔT shows that a forecast provided by an I-GARCH(1) process (one time scale) does not capture correctly the dynamics of the realized volatility. An I-GARCH(2) process (two time scales, similar to GARCH(1,1)) is better, while a long-memory LM-ARCH process (multiple time scales) replicates correctly the dynamics of the implied and realized volatilities and delivers consistently good forecasts for the realized volatility.     

Forecasting currency volatility: A comparison of implied volatilities and AR(FI)MA models

We compare forecasts of the realized volatility of the pound, mark and yen exchange rates against the dollar, calculated from intraday rates, over horizons ranging from one day to three months. Our forecasts are obtained from a short memory ARMA model, a long memory ARFIMA model, a GARCH model and option implied volatilities. We find intraday rates provide the most accurate forecasts for the one-day and one-week forecast horizons while implied volatilities are at least as accurate as the historical forecasts for the one-month and three-month horizons. The superior accuracy of the historical forecasts, relative to implied volatilities, comes from the use of high frequency returns, and not from a long memory specification. We find significant incremental information in historical forecasts, beyond the implied volatility information, for forecast horizons up to one week.