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International Journal of Forecasting 32 (2016) 1317–1339 Contents lists available atScienceDirect International Journal of Forecasting journal homepage:www.elsevier.com/locate/ijforecast Global equity market volatility spillovers: A broader role for the United States Daniel Buncic a, ∗ ,Katja I.M. Gisler b a Institute of Mathematics & Statistics, University of St. Gallen, Switzerland b School of Economics & Political Science, University of St. Gallen, Switzerland a r t i c l e i n f o Keywords:
Realized volatility HAR modelling and forecasting Augmented HAR model US volatility information VIX International volatility spillovers a b s t r a c t Rapach et al. (2013) recently showed that U.S. equity market returns contain valuable information for improving return forecasts in global equity markets. In this study, we extend the work of Rapach et al. (2013) and examine whether U.S.-based equity market information can be used to improve realized volatility forecasts in a large cross-section of international equity markets. We use volatility data for the U.S. and 17 foreign equity markets from the Oxford Man Institute’s realized library, and augment our benchmark HAR model with U.S. equity market volatility information for each foreign equity market. We show that U.S. equity market volatility information improves the out-of-sample forecasts of realized volatility substantially in all 17 foreign equity markets that we consider. Not only are these forecast gains highly significant, they also produce out-of-sample R2 values of between 4.56% and 14.48%, with 9 being greater than 10%. The improvements in out- of-sample forecasts remain statistically significant for horizons up to one month ahead. A substantial part of these predictive gains is driven by forward-looking volatility, as captured by the VIX.
© 2016 International Institute of Forecasters. Published by Elsevier B.V. All rights reserved. ‘ . . .
since the US equity market is the world’s largest, investors likely focus more intently on this market, so that information on macroeconomic fundamentals relevant for equity markets worldwide diffuses gradually from the US market to other countries’ markets .’ [Rapach,Strauss,&Zhou,2013, p. 1635] 1. Introduction In a recent influential paper,Rapachet al.(2013) showed that the equity market returns of the United States ∗ Correspondence to: Institute of Mathematics and Statistics, Bo- danstrasse 6, 9000 St.Gallen, Switzerland.
E-mail addresses: [email protected](D. Buncic), [email protected](K.I.M. Gisler).
URL: http://www.danielbuncic.com(D. Buncic). (US) have significant predictive power for forecasting eq- uity returns in a large cross-section of international eq- uity markets. This predictive power is attributed to the leading role played by the US in generating relevant macroeconomic and financial information for both US and non-US investors.Rapachet al.(2013) argue that information frictions cause information to diffuse only gradually from the US to other equity markets around the world, leading to lagged US returns having predic- tive content. The US is the world’s largest economy, is a large and important trading partner for many coun- tries, and has the world’s largest equity market in terms of market capitalization. Thus, when forming investment decisions, investors who take a global investment per- spective are focused intently on not only developments in macroeconomic and financial fundamentals in the http://dx.doi.org/10.1016/j.ijforecast.2016.05.001 0169-2070/ ©2016 International Institute of Forecasters. Published by Elsevier B.V. All rights reserved. 1318 D. Buncic, K.I.M. Gisler / International Journal of Forecasting 32 (2016) 1317–1339 US, but also the formation of expectations and risk premia that arise from this process. 1 The objective of this study is to provide a first compre- hensive analysis of the predictive content of US-based eq- uity market volatility information for volatility forecasts in a large cross-section of 17 international (non-US) equity markets. For this purpose, we use daily realized volatility data from the Oxford Man Institute’s realized library and augment the well-known and widely used heterogeneous autoregressive (HAR) model ofCorsi(2009) with (lagged) daily, weekly and monthly US realized volatility and VIX HAR components. We utilize the HAR model ofCorsi (2009) as our benchmark realized volatility model for mea- suring the contribution of US-based volatility information to realized volatility forecasts in international equity mar- kets, employing standard in-sample and out-of-sample evaluation criteria. In this context, our study can be viewed as an extension of the work ofRapachet al.(2013), but with our analysis focussing on the role of the US as a source of relevant volatility information. We find US-based volatility information to play an overwhelmingly strong role for all 17 international equity markets that we consider. Our study is related to a growing body of volatility spillover literature. The literature on spillovers in interna- tional equity markets goes back to the research ofEunand Shim(1989),Hamaoet al.(1990) andLinet al.(1994). More recent studies include those ofBaurandJung(2006) and Savva,Osborn,andGill(2009), among others. The contri- butions to this body of literature typically differ in their definitions of the interdependence measure adopted and the modelling approaches used. For example,Hamaoet al.
(1990) use a generalized autoregressive conditional het- eroscedastic (GARCH) type of model to analyze spillovers across three major equity markets. They define spillovers as impacts from foreign stock markets on the conditional mean and variance of daytime returns in the subsequently traded markets. Their results show evidence of volatility spillovers across these markets. Similarly,Linet al.(1994) employ a signal extraction model with GARCH innovations in order to analyze spillover effects between two major eq- uity markets. In contrast toHamaoet al.(1990),Linet al.
(1994) do not find any evidence of spillover effects be- tween the two stock markets, but rather attribute their 1 The New York Stock Exchange (NYSE) is by far the largest equity market in the world, with a market capitalization of over 21 Trillion US Dollars (as of the end of 2014). The second is the NASDAQ, with a market capitalization of around 7 Trillion. Tokyo and London are the next biggest, with market capitalizations of around 4 Trillion. Moreover, an abundance of economic and financial data are released every day.
These range from soft survey data related to durable goods, inventories, employment reports, the ISM (manufacturing) index, PMIs (purchasing manager indices) and the like, to hard data releases related to jobless claims, home sales, residential construction, personal income and outlays, PPI, CPI, employment and GDP figures. International financial agents and the financial media focus on these releases intently. Also, in terms of a calender (or trading) day timeline, it is the last (or one of the last) equity markets to close. As market participants begin work on a given day, they naturally look at important financial and economic developments in the US first. The dominant role of the US market as a source of both return and volatility transmission in international equity markets has been documented in numerous multi-country studies (see for exampleBecker, Finnerty,&Friedman,1995;Engle,1990;Hamao,Masulis,&Ng,1990; King&Wadhwani,1990;Lin,Engle,&Ito,1994, and others). contradictory results to non-synchronous trading and stale quotes at opening time. On the other hand,EunandShim (1989) apply a vector autoregressive model (VAR) to stock market returns from nine international markets, and use simulated responses to trace the spillover effects of inter- national stock market shocks. They find that US stock mar- ket shocks are transmitted to the other markets quickly, but not vice versa, highlighting the dominant role of the US. More recently,Savvaet al.(2009) use a dynamic con- ditional correlation (DCC) model to analyze return and volatility spillovers across four major stock markets. Their results show that both domestic stock prices and volatili- ties are subject to spillover effects. In fact, they find more evidence of spillovers from Europe to the US than the other way around. However, this finding might be attributable to the pseudo-closing approach that they use in order to avoid synchronous trading. With the increasing availability of high-frequency data, the literature on volatility spillovers has again gained mo- mentum (seeBonato,Caporin,&Ranaldo,2013;Diebold &Yilmaz,2014,2016;Dimpfl&Jung,2012;Fengler& Gisler,2015, among others).DieboldandYilmaz(2014) model realized volatility as a vector autoregressive process and define volatility spillovers based on a multiple-step- ahead forecast-error variance decomposition. Their results suggest that there are strong realized volatility spillovers across financial institutions, particularly during crises. Us- ing a similar approach,FenglerandGisler(2015) extend the results ofDieboldandYilmaz(2012,2014)by including realized covariances in the spillover transmission mech- anism. They show that realized covariance spillovers are substantial, and allow for an earlier detection of the recent financial and debt-ceiling crises that are attributable to a flight-to-quality phenomenon.Bonatoet al.(2013), on the other hand, define spillovers as the dependence of real- ized covariance on cross-lag realized covariances. They model realized covariance matrix as a Wishart autoregres- sive process, and find that sector and currency covariance spillovers improve the forecasting performance. Similarly, DimpflandJung(2012) model realized volatility and re- turn spillovers around the globe in a structural VAR frame- work. They find significant return and realized volatility spillovers that also result in forecast improvements. 2 In summary, the spillover literature has analyzed return and volatility spillovers in international stock markets exten- sively. However, the literature to date has not analyzed the predictive content of US realized volatility information for realised volatility forecasts in a large cross-section of inter- national equity markets. Our study aims to fill this gap. Although our study is related to the volatility spillover literature, we intentionally avoid the use of (structural) VAR approaches for modelling the information flow from the US to international equity markets. Standard structural VAR models require assumptions on the causal ordering 2 The modelling of spillover effects also plays a much broader role in the financial stability literature. For instance, given the role of US volatility and an interconnected world, it may be important to account for US-based information when designing macro-prudential stress tests, especially for Eastern European countries. See for instanceBuncicandMelecky(2013) for a recent study as to how this could be implemented. D. Buncic, K.I.M. Gisler / International Journal of Forecasting 32 (2016) 1317–1339 1319 if impulse responses or forecast error variance decompo- sitions are used as measures of spillovers. This may not be a problem for smaller VARs, where the causal order- ing is known from the underlying assumption as to which equity market generates the most important information (i.e., the US), for instance, or in cases where the causal or- dering is based on the chronological structure of the mar- kets that are analysed (see for exampleDimpfl&Jung, 2012;Knaus,2014). Nevertheless, since we are considering realized volatility data for 17 international equity markets, it becomes much more difficult to justify the ordering of the countries in a structural VAR. Also, overlapping trading hours mean that one can only analyze up to three different international stock markets (i.e., over three different trad- ing/time zones). Moreover, estimating unrestricted VARs with large numbers of variables is highly inefficient, lead- ing to poor out-of-sample forecast performances. Thus, we prefer to examine the role of the US as a source of volatility information within our proposed simple augmented HAR modelling framework.
Using daily realized volatility data for the US and 17 international equity markets, covering the period from January 3, 2000, to November 13, 2015, we find the US eq- uity market to play a strong role internationally as a source of volatility information. Our in-sample results show that US-based volatility information is jointly highly significant.
The daily and weekly HAR components of the log VIX, to- gether with the daily and monthly HAR components of the US realized volatility, are the most important sources of volatility information from the US. For some equity mar- kets, such as the All Ordinaries and the EURO STOXX 50, the parameter estimate on the daily US HAR component has a larger magnitude than its own daily HAR component, suggesting that the previous day’s high frequency volatil- ity information from the US is more important than its own lagged volatility. Moreover, our in-sample analysis shows that the low frequency volatility component from the US has a negative effect on the realized volatility in non-US equity markets. This finding is consistent across all 17 of the international equity markets that we consider.
Our out-of-sample analysis shows that one-step-ahead forecasts from the augmented HAR model with US volatility information are highly significant, yieldingClark andWest(2007) adjusted t-statistics of at least 8.4 and as high as 15.7, indicating rather strong rejections of the null hypothesis of no forecast improvement. The one-step- ahead out-of-sample R2 values range from 4.56% (Hang Seng) to 14.84% (All Ordinaries), and are above 10% for nine of the 17 equity markets that we analyse. Thus, the forecast improvements are not only highly statistically significant, but also sizeable economically. To put these magnitudes into perspective,PattonandSheppard(2015) recently documented improvements in out-of-sample R2 values of in the order of 2.5%–3% by splitting the volatility into bad and good volatility states (in addition to various other considerations related to leverage and signed jumps). Thus, improvements in excess of 10% are substantial. Our out- of-sample analysis also shows that forecast improvements are experienced consistently over the full out-of-sample period, and are not driven purely by individual episodes.
The forecast improvements for the multiple-step-ahead horizon remain highly significant (at the 1% level) for all 17 international equity markets at the five-day-ahead (one week) and 10-day-ahead (two week) horizons, and start to deteriorate at the 22-day-ahead (one month) horizon.
The improvements in the 22-day-ahead forecasts remain significant and produce positive out-of-sample R2 values for 12 of the 17 equity markets that we analyse. Overall, our results show that US-based volatility data are most informative for forecasts of realized volatility for the All Ordinaries index and all of the European equity markets that are included in our comparison.
The remainder of the paper is organised as follows.
In Section2, we outline how realized volatility is mod- elled and how we extend the standard HAR model ofCorsi (2009) by augmenting it with US-based information about equity market volatility. The data that we use in the study are described in detail in Section3. In Section4, we eval- uate the importance of US-based volatility information for the determination of volatility in 17 international (non-US) equity markets, by means of in-sample and out-of-sample evaluations. In Section5, we provide an analysis of the ro- bustness of our findings. Lastly, we conclude the study.
2. Modelling the volatility This section outlines the modelling approach that we use to assess the role played by US equity market volatility information in improving realized volatility forecasts in a large cross-section of international equity markets. Before describing the empirical model that we employ for modelling and forecasting realized volatility in international equity markets, we first briefly describe the background that links empirical realized volatility to its theoretical counterpart, integrated volatility.
2.1. Theoretical framework Let p t denote the logarithm (log) of an asset price at time t. The log asset price is assumed to be a continuous- time diffusion process that is driven by Brownian motion, with the dynamics described by the following stochastic differential equation:
dp t= µ tdt +σ tdW t, (1) where µ tis a predictable and locally bounded drift term, σ t is a càdlàg volatility process that is bounded away from zero, and W tis a standard Brownian motion. The quadratic variation (QV) process of p t is given by 3 :
QV t= t 0 σ 2 s ds . (2) In the absence of jumps, as is the case in our setting in Eq.
(1), the term t 0 σ2 s ds in Eq.(2)is known as the integrated variance (IV) of the process p t. 3 The quadratic variation process of p t is defined as [p t] = plim m→∞ m k = 1( p (t k ) − p(t k − 1)) 2 , where plim denotes convergence in probability, and 0 =t 0 ≤ t 1 < · · ·
RV t= m i = 1 r 2 t ,i, (3) and its square root √ RV tis known as the realized volatility .
The general properties of the estimator in Eq.(3)are summarised byAndersen,Bollerslev,Diebold,andLabys (2003).
2.2. Empirical volatility model There exist three broad classes of empirical models for RV. The first belongs to the traditional ARMA and fraction- ally integrated ARMA (ARFIMA) classes of long-memory time series models for RV (seeAndersenet al.,2003;Bail- lie,1996;Baillie,Bollerslev,&Mikkelsen,1996;Comte &Renault,1996,1998, among many others). The second class considers nonlinear time series models, where long- memory patterns in RV are generated spuriously from nonlinear short-memory models with structural breaks or regime switches (see for instance the papers byChen,Här- dle,&Pigorsch,2010;Fengler,Mammen,&Vogt,2015; McAleer&Medeiros,2008, and others). The third belongs to the class of heterogeneous autoregressive (HAR) mod- els for RV, as initially introduced into the realized variance modelling literature byCorsi(2009). We use the HAR model ofCorsi(2009) as our bench- mark RV model for each of the foreign equity markets that we consider. The HAR model has a cascade-type structure, where the volatility at any point in time is constructed as a linear combination of daily, weekly and monthly volatility components. This temporal cascade structure is motivated by the so-called heterogeneous market hypothesis (HMH) ofMülleret al.(1993), where it is assumed that finan- cial markets are populated by heterogeneous agents, each with different endowments, risk profiles, institutional con- straints and information processing capabilities, as well as various other characteristics (seeCorsi,2009, for a more detailed discussion). The defining feature of the HAR model is that each agent has a different time horizon for trading.
The intuition is that the short-term volatility does not mat- ter to a long-term investor, whereas the long-term volatil- ity is still of importance to short-term investors because of its impact on the investment opportunity set.
To formalise the structure of the HAR model for RV, let log RV ( d ) t = log RV t, log RV (w) t =1 5 5 i = 1 log RV t+ 1− iand log RV ( m ) t =1 22 22 i = 1 log RV t+ 1− ibe the daily, weekly, and monthly HAR components. The HAR model is then defined as 4 :
log RV t+ 1 = b 0 + b( d ) log RV ( d ) t + b(w) log RV (w) t + b( m ) log RV ( m ) t + ϵ t+ 1, (4)4 Note here that the original formulation of the HAR model byCorsi (2009) used RV instead of log RV in the HAR specification in Eq.(4). where ϵ t+ 1 is an innovation term. One of the main at- tractions of the HAR model in Eq.(4)is its simplicity.
Once the daily, weekly, and monthly volatility components have been constructed, the HAR model can be estimated by ordinary least squares (OLS) regression. Moreover, the HAR model is an extremely difficult benchmark model to beat in out-of-sample forecast evaluations, due to its par- simonious setup (seeCorsi,Audrino,&Renó,2012, for a recent survey of different types of models for RV that have been evaluated against the HAR model). Since we are in- terested primarily in a real time out-of-sample compari- son of the predictive content of US equity market volatility information on the volatility in other global equity mar- kets, it is necessary to update the model parameters of interest recursively when constructing the forecasts. Un- like AR(FI)MA and other more general nonlinear time se- ries models, which require a numerical optimisation of the likelihood function, and therefore are time consuming to estimate, as well as frequently being numerically unstable, the HAR model in Eq.(4)can be estimated efficiently and accurately by standard OLS procedures.
At this point, we should also highlight the fact that the HAR model ofCorsi(2009) has undergone numerous re- finements since its initial introduction. For instance, some recent evidence suggests that separating the quadratic variation process in Eq.(2)into continuous and jump com- ponent parts can lead to better out-of-sample forecasts (see for instanceAndersenet al.,2007;Corsiet al.,2010; Corsi&Renó,2012). Moreover, allowing for nonlinear and asymmetric effects in the HAR model, such as the leverage effect, also seems to be beneficial for out-of-sample fore- casting (seeBollerslev,Litvinova,&Tauchen,2006;Chen &Ghysels,2011;Corsi&Renó,2012;Patton&Sheppard, 2015, among others). Nevertheless, in spite of these find- ings, we want to abstract from the inclusion of such refine- ments of the HAR model in this study, and instead focus our attention solely on the role of the US as a source of information in relation to international asset price volatil- ity, and, most importantly, on the question of whether this information can be exploited in order to improve fore- casts of the realized volatility in other global equity mar- kets. 5 In studies using S&P 500 RV data, the improvements Nevertheless, there has been a shift toward modelling the log RV series.
In the words ofAndersen,Bollerslev,andDiebold(2007,p. 704): ‘ from a modeling perspective, the logarithmic realized volatilities are more amenable to the use of standard time series procedures ’. Moreover, log transformed RV data are much closer to being normally distributed, and there is also no need to impose any non-negativity restrictions on the fitted and forecasted volatilities. We will therefore followCorsi,Pirino,andRenó (2010),CorsiandRenó(2012) and many others in the empirical RV literature and use log RV in the HAR model. 5 Evidently, as the number of regressors grows, one could also make the modelling of the HAR more flexible by using either a time-varying parameter model, like those that were used byBuncicandMoretto(2015), BuncicandPiras(2016) andGrassi,Nonejad,anddeMagistritis(2014), or a shrinkage estimator such as the lasso for variable selection, as was done byBuncicandMelecky(2014) in a cross-sectional setting. Nevertheless, despite the fact that our econometric modelling could be extended to address additional, potentially important features in the data, this would abstract further from our consideration of the information contained in US volatility data for forecasting the volatility in international equity markets. D. Buncic, K.I.M. Gisler / International Journal of Forecasting 32 (2016) 1317–1339 1321 in out-of-sample forecast performances seem to be rather marginal relative to the magnitudes that we find at the in- ternational level. For instance, the models considered by PattonandSheppard(2015), which allow the volatility to be split into bad and good volatility components (in ad- dition to various other considerations related to leverage and signed jumps), lead to improvements in the out-of- sample R2 values of around 2.5%–3% points at best (see Ta- ble 6 ofPatton&Sheppard,2015). Another recent study byProkopczuk,Symeonidis,andWeseSimen(in press), which analysed the impacts of jumps on energy-related asset prices, found that jumps do not improve the out-of- sample forecasts of the volatility. Their findings are robust to a number of different jump detection procedures, and also to varying the size of the estimation window.
We assess the value of US equity market volatility data for forecasts of the log RV in other equity markets around the world by augmenting each individual foreign (local) equity market’s benchmark HAR model with US volatility information. This is achieved by adding the log RV and log VIX HAR components from the US as predictor variables. Since the VIX displays a rather strong long-range dependence, it appears to be desirable to apply a HAR- type structure. In fact,Fernandes,Medeiros,andScharth (2014) recently assessed the success of different HAR type models in forecasting the VIX, and found that a pure HAR specification as perCorsi(2009) is very difficult to beat out-of-sample. 6 We specify the following augmentedHAR model for each of the 17 international equity markets that we consider:
log RV t+ 1 = benchmark (local) HAR components of each foreign equity market β 0 + β( d ) log RV ( d ) t + β(w) log RV (w) t + β( m ) log RV ( m ) t + β( d ) VIX log VIX ( d ) t + β(w) VIX log VIX (w) t + β( m ) VIX log VIX ( m ) t US volatility information: VIX HAR components + β( d ) US log RV ( d ) t ,US + β(w) US log RV (w) t ,US + β( m ) US log RV ( m ) t ,US US volatility information: RV HAR components + ϵUS t + 1, (5) where the daily, weekly, and monthly HAR components for US log RV and log VIX, denoted by log RV ( ·) t ,US and log VIX ( ·) t , are defined analogously to the local HAR components used above in Eq.(4). The log VIX tseries is the log of the Chicago Board Options Exchange (CBOE) volatility index (henceforth, VIX for short), β 0 is a standard regression intercept, and ϵUS t + 1 is again a random disturbance term.
Our motivation for including the VIX as an additional source of volatility information in the augmented HAR model in Eq.(5)is as follows. Recall that the VIX measures the volatility implied by option prices on the S&P 500, thus 6 We thank an anonymous referee for suggesting that we also use a HAR structure for the log VIX process. In an earlier version of the paper, we used only log VIX ( d ) t = log VIX tas an additional control variable in the augmented HAR specification in Eq.(5). Adding the weekly and monthly HAR VIX components improved both the overall in-sample fit and the out- of-sample forecast performance of the augmented HAR model in Eq.(5). reflecting investors’ expectations about the stock market volatility over the next month. 7 Thus, the VIX is meant to provide not only a forward looking view on expectedUS eq- uity market volatility, but also a general sense of the risk aversion in the market. A higher value in the VIX is gen- erally taken as an indication that market participants an- ticipate an overall negative economic or financial outlook, and hence have an increased aversion to risk (seeBrun- nermeier,Nagel,&Pedersen,2009, for a discussion). This increased risk aversion is likely to spill over into other in- ternational equity markets, given the dominant position of the US in the world economy as a source of economic and financial information. Moreover, in a recent study,Grassi et al.(2014) documented that the VIX has some predictive power for S&P 500 realized volatility forecasts. We there- fore expect the VIX likewise to contain predictive informa- tion that can be exploited to improve the realized volatility forecasts in other international equity markets.
3. Data We obtain daily volatility data from the publicly- available Oxford-Man Institute’s Quantitative Finance Realized Library ofHeber,Lunde,Shephard,andShep- pard(2009). The Oxford-Man Realized Library uses high- frequency tick data from Reuters DataScope Tick History 8 to construct a whole suite of daily realized measures of the asset price variability, as well as providing the num- ber of transactions, the time span between the first and last observations, the close-to-open return, the local open- ing time, the high–low range, the high–open range, and the opening and closing prices for each series. 9 The library con- tains realized measures for four US and 17 foreign (non-US) equity price indices from January 3, 2000, to the present.
Our sample ends on November 13, 2015.
As our preferred estimator of asset price variation, we use the realized variance sampled at equally-spaced five minute intervals (simply ‘five minute RV’ henceforth). This is the estimator given under the heading ‘*.rv’in the 7 The VIX is computed as the weighted average of the implied volatilities of options on the S&P 500 index for a wide range of strikes, and mainly first and second month expirations. Note here thatChow,Jiang, andLi(2014) recently showed the VIX to be a biased measure of market expectations about the future volatility. Nevertheless, we include the VIX as a regressor in the HAR model in order to account for the potential predictive information that it may have for the volatility in other global equity markets, rather than trying to gauge whether it is an appropriate measure of volatility expectations in the US. 8 http://www2.reuters.com/productinfo/tickhistory/material/ DataScopeTickHistoryBrochure_260707.pdf. 9 The term ‘realized measures’ was coined byLiu,Patton,andSheppard (2015). The various types of realized measures that are included in the library are listed athttp://realized.oxford- man.ox.ac.uk/documentation/ estimators. With regard to the quality of the tick data,Heberet al.(2009) point out that the raw data from Reuters DataScope Tick History are already of a high quality. Nevertheless,Heberet al.(2009) still employ the high frequency data cleaning procedure described in detail athttp:
//realized.oxford- man.ox.ac.uk/documentation/data- cleaningand in the references therein, in order to make the data suitable for econometric analysis. Also, our data are from Library Version 0.2. The url link to the data source ishttp://realized.oxford- man.ox.ac.uk/media/1366/ oxfordmanrealizedvolatilityindices.zip. 1322 D. Buncic, K.I.M. Gisler / International Journal of Forecasting 32 (2016) 1317–1339 Oxford-Man Realized Library for each equity market data block. The choice of the five minute RV is due partly to simplicity and partly to robustness. In a recent extensive study of realized measures,Liuet al.(2015) highlighted the fact that there is little evidence to suggest that the five minute RV is outperformed significantly by any of the other realized measures that are considered in the benchmark comparison. In particular, when working with international equity market data,Liuet al.(2015,p. 294) pointed out that ‘‘ more sophisticated realized measures generally perform significantly worse ’’ than the five minute RV. We therefore use the five minute RV estimator ofHeber et al.(2009) – henceforth simply ‘RV’, to avoid cumbersome and repetitive language – throughout this study. Moreover, to make the magnitudes of the RV measure comparable to those of the VIX, we transform all RV series to annualised volatilities. 10 In total, we have access to realized measures data for 17 international equity markets that are included in the Oxford-Man Realized Library. These are the FTSE 100 (United Kingdom), the Nikkei 225 (Japan), the DAX (Germany), the All Ordinaries (Australia), the CAC 40 (France), the Hang Seng (Hong Kong), the KOSPI (South Korea), the AEX (The Netherlands), the Swiss Market Index (Switzerland), the IBEX 35 (Spain), the S&P CNX Nifty (India), the IPC Mexico (Mexico), the Bovespa (Brazil), the S&P TSX (Canada), the Euro STOXX 50 (Euro area), the FT Straits Times (Singapore), and the FTSE MIB (Italy).
For the US, the library contains realized measures for four different equity market indices. These are the Dow Jones Industrial Average (DJIA), the Russel 2000, the Nasdaq 100 and the S&P 500. We use the S&P 500 as our key headline US equity market index. The Nasdaq 100 is a specialized technology industry index, and thus is defined too narrowly to be considered as a valid headline US equity market index. The Russel 2000, on the other hand, is likely to be too sensitive to volatility movements induced by the small cap nature of the index. From our point of view, only the DJIA qualifies as a viable alternative to the S&P 500, as it is an index that is focused on widely by the financial media, thus providing broad headline information about the performance of US equities. Nevertheless, one evident shortcoming of the DJIA is that it is composed of only 30 blue chip stocks, and we therefore find it to be too narrowly focused as well. Thus, our preference is to use the S&P 500 as our key equity market index for the US. 11 The VIX data that we include in the augmented HAR model in Eq.(5)are obtained from the St. Louis Fed FRED2 database. 1210 This is done by taking the five minute RV series, re-scaling it by 100 2 × 252, and taking the square root to be interpreted as the annualised volatility (in percentage terms). That is, the annualized realized volatility is equal to: (‘ ∗ .rv ′ × 100 2 × 252 )1 /2 .
11 However, we would like to stress that, while we have chosen the S&P 500 here, the results that we obtain change very little if we use the DJIA as the US headline index instead. The results based on the DJIA are available from the authors upon request. 12 The url of the database ishttp://research.stlouisfed.org/fred2/. The FRED mnemonic for the VIX is VIXCLS, and it contains daily closing prices (16:15 EST) of the Chicago Board Options Exchanges (CBOEs) volatility index. Table 1provides standard summary statistics on all of the (log transformed) RV and VIX data that are used in our study. In addition to the summary statistics inTable 1, we also show time series and autocorrelation function (ACF) and partial ACF (PACF) plots inFigs. 1and2, to provide further information about the data series that we use. The first to fifth columns ofTable 1show the equity index, the corresponding country, the full sample period, the number of observations T, and the percentage of missing entries (%Miss). The percentages of missing entries were obtained by matching the dates from the Oxford-Man Realized Library to official trading dates data from Bloomberg. In columns six to twelve, we list standard sample statistics such as the mean, median (Med), standard deviation (Std.dev), skewness (Skew) and kurtosis (Kurt), as well as the minimum (Min) and maximum (Max) of each series. The last six columns (grouped in threes) provide the first to third order ACF and PACF (ACF(1–3) and PACF(1–3), respectively). We can see from the third column ofTable 1 that there are some differences with respect to the actual first data points across the various equity markets that are available. For all but two series, the first data point is on either the 3rd or the 4th of January 2000. For the S&P TSX (Canada), the sample starts on May 2, 2002, and for the S&P CNX Nifty (India) it starts on July 8, 2002. 13 The end of the sample is either the 12th or 13th of November 2015 for all series except for FT Straits Times (Singapore), which ends on the 18th of September 2015, due to data availability in the Oxford-Man Realized Library. Looking over the summary statistics inTable 1, one sees that the log RV data are distributed fairly symmetrically, with the means and medians lining up reasonably closely, the skewness being between 0 and 1, and the kurtosis being around 3 for all but four markets. 14 Interestingly, Bovespa and FT Straits Times have the lowest variations, with the standard deviations of log RV being only around 0.37, while those for the remaining series are closer to 0.5.
The ACF and PACF entries inTable 1highlight the well- known long-memory property of volatility data. The most persistent log RV series are the KOSPI (South Korea) and the Swiss Market Index, with first order ACFs of 0.86 and 0.85, respectively, while the least persistent ones are the All Ordinaries and the Bovespa, with values of around 0.67.
The VIX is the most persistent series overall, with an ACF(1) of 0.98. The long-memory property of realized volatility 13 Before July 8, 2002, the availability of realized measures data for the S&P CNX Nifty was extremely sparse. That is, only 100 data entries were available for the 653 entries before July 8, 2002 (553 missing entries). We therefore decided to delete all entries before July 8, 2002, and start the effective sample for the S&P CNX Nifty from July 8, 2002. There are three other equity markets with unusual missing data patterns that deserve mentioning: (1) the All Ordinaries (Australia), where data are missing for 15 consecutive days from July 4, 2014, to July 25, 2014; (2) the FT Straits Times (Singapore), where 43 consecutive entries are missing from January 2, 2008, to March 3, 2008; and (3) the Hang Seng (Hong Kong), which had 168 entries missing out of 300 between September 5, 2008 to November 3, 2009. All missing entries were deleted from the final data set used in the analysis. 14 These exceptions are the log RV series of Bovespa (Brazil) and Euro STOXX 50, which are close to 5, and S&P CNX Nifty (India) and IPC Mexico (Mexico), with kurtosis values of 4.1 and 3.6, respectively, thus showing somewhat heavier tails than a Gaussian random variable. D. Buncic, K.I.M. Gisler / International Journal of Forecasting 32 (2016) 1317–1339 1323Table 1 Summary statistics of (log) RV and (log) VIX data. Equity index Country Full sample period T%Miss Mean Med Std.dev Skew Kurt Min Max ACF(1–3) PACF(1–3) FTSE 100 United Kingdom 04.01.2000–13.11.2015 3986 0.63 2.41 2.37 0.51 0.47 3.14 1.13 4.68 0.84 0.81 0.79 0.84 0.34 0.21 Nikkei 225 Japan 04.01.2000–13.11.2015 3844 1.38 2.61 2.60 0.43 0.30 3.53 1.29 4.50 0.78 0.72 0.69 0.78 0.30 0.17 DAX Germany 03.01.2000–13.11.2015 4017 0.47 2.78 2.75 0.51 0.36 3.13 1.27 4.80 0.84 0.80 0.77 0.84 0.33 0.20 All Ordinaries Australia 04.01.2000–13.11.2015 3966 1.29 2.13 2.09 0.48 0.51 3.51 0.57 4.13 0.67 0.66 0.64 0.68 0.38 0.23 CAC 40 France 03.01.2000–13.11.2015 4040 0.45 2.70 2.70 0.48 0.31 3.18 1.16 4.73 0.83 0.79 0.77 0.83 0.33 0.19 Hang Seng Hong Kong 03.01.2000–13.11.2015 3643 7.47 2.54 2.51 0.41 0.60 3.88 1.27 4.65 0.72 0.70 0.68 0.72 0.38 0.23 KOSPI South Korea 04.01.2000–13.11.2015 3907 0.46 2.65 2.63 0.51 0.32 2.93 1.33 4.81 0.86 0.83 0.81 0.86 0.35 0.19 AEX The Netherlands 03.01.2000–13.11.2015 4039 0.50 2.60 2.55 0.50 0.48 3.18 0.87 4.56 0.84 0.81 0.78 0.84 0.33 0.19 Swiss Market Index Switzerland 04.01.2000–13.11.2015 3972 0.48 2.45 2.36 0.46 0.89 3.82 1.45 4.63 0.85 0.82 0.80 0.85 0.34 0.20 IBEX 35 Spain 03.01.2000–13.11.2015 4005 0.47 2.74 2.78 0.49 −0.04 2.88 1.17 4.61 0.84 0.80 0.78 0.84 0.32 0.21 S&P CNX Nifty India 08.07.2002–13.11.2015 3295 0.97 2.73 2.67 0.48 0.75 4.09 1.20 5.38 0.75 0.70 0.67 0.75 0.31 0.19 IPC Mexico Mexico 03.01.2000–13.11.2015 3967 0.76 2.42 2.36 0.49 0.62 3.58 1.07 4.74 0.68 0.64 0.61 0.68 0.33 0.20 Bovespa Brazil 03.01.2000–12.11.2015 3879 1.31 3.01 2.98 0.37 0.70 4.92 1.61 4.87 0.67 0.61 0.56 0.67 0.29 0.15 S&P TSX Canada 02.05.2002–13.11.2015 3379 0.68 2.15 2.08 0.52 0.84 4.07 0.75 4.56 0.79 0.76 0.73 0.79 0.35 0.20 Euro STOXX 50 Euro Area 03.01.2000–13.11.2015 4017 1.27 2.76 2.73 0.50 0.10 5.09 −1.07 5.11 0.77 0.72 0.70 0.77 0.32 0.21 FT Straits Times Singapore 03.01.2000–18.09.2015 3839 2.71 2.33 2.30 0.38 0.60 3.70 1.45 4.29 0.81 0.77 0.75 0.81 0.33 0.22 FTSE MIB Italy 03.01.2000–12.11.2015 4000 0.68 2.65 2.63 0.49 0.30 2.97 1.42 4.75 0.83 0.79 0.76 0.83 0.33 0.19 S&P 500 United States 03.01.2000–13.11.2015 3964 0.73 2.53 2.50 0.53 0.47 3.36 1.02 4.94 0.80 0.76 0.73 0.80 0.34 0.18 VIX (log) United States 03.01.2000–13.11.2015 3992 0.00 2.96 2.92 0.37 0.65 3.30 2.29 4.39 0.98 0.97 0.96 0.98 0.08 0.08 Notes: The table reports standard summary statistics of the 18 annualised (log) realized volatility series of the various equity markets and the (log) VIX. Columns one to five show the equity index, the corresponding country, the full sample period, the sample size T, and the percentage of missing trading days (%Miss). Columns six to twelve show common sample statistics, namely the mean, median (Med), standard deviation (Std.dev), skewness (Skew) and kurtosis (Kurt), as well as the minimum (Min) and maximum (Max) of the series. The last six columns (grouped into blocks of three) provide the first- to third-order autocorrelation function (ACF) and partial ACF (PACF). The full available sample is from January 3, 2000, to November 13, 2015. 1324 D. Buncic, K.I.M. Gisler / International Journal of Forecasting 32 (2016) 1317–1339Fig. 1.
Time series evolution of (log) RV and (log) VIX over the full available sample period for each series.
data is also visible from the ACF and PACF plots inFig. 2.
For instance, it is evident that the log RV series of Bovespa displays the shortest memory, while the Nikkei 225 has the most hyperbolic-looking ACF decay pattern. Finally, the time series plots of the log RV and log VIX inFig. 1show that the movement in volatility across equity markets is rather homogenous for major events such as the Lehman Brothers collapse in September 2008.
One last, and potentially important, point that we would like to stress here is that the Oxford-Man Realized Library only uses intraday data collected over the official (local) trading hours of the respective equity markets of in- terest. That is, no variation due to overnight price changes is considered in the construction of the realized measures.
Since we are using information from the US at time tto forecast the (log) realized volatility in all other foreign eq- uity markets at time t+ 1 (and further ahead), there is no overlap in the official trading hours between the US mar- ket’s previous day closing and the foreign market’s current day opening. 1515 The official trading hours of the New York Stock Exchange (NYSE) are from 9:30 to 16:00 Eastern Standard Time (EST), which is 14:30–21:00 Coordinated Universal Time (UTC) in (northern hemisphere) winter. Of 4. Assessing the value of US volatility information We begin our assessment of the importance of US eq- uity market volatility data and its usefulness for improv- ing the modelling and forecasting of the realized volatility in other international equity markets by looking at the in- sample contribution of US-based volatility information to the model. We then extend the analysis by using standard forecast evaluation techniques to determine whether these in-sample gains carry over into the out-of-sample forecast environment.
Before evaluating the in-sample fit of the augmented HAR model in Eq.(5), it will be convenient to condense the representation of the model somewhat. For this purpose, the foreign equity markets that we include, the first one to open the next day is the Australian Securities Exchange (ASX) in Sydney at 10:00 Australian Eastern Standard Time (AEST), which is 00:00 UTC. During (northern hemisphere) summer, the UTC closing time for the US market is 20:00 UTC, while the ASX in Sydney opens at 23:00 UTC. Hence, there is a three-hour gap between New York closing and Sydney opening. Also, the switches to and from Daylight Saving Time (DST) do not occur on the same days. For the US, DST is ‘ on’ from March to November, while for Australia, DST is ‘ off’ from April to October. However, this is immaterial for our discussion, as it does not cause any overlap in trading hours. D. Buncic, K.I.M. Gisler / International Journal of Forecasting 32 (2016) 1317–1339 1325Fig. 2.
Autocorrelation function (ACF) and partial ACF (PACF) plots of all (log) RV series.
let us define x t = 1 log RV ( d ) t log RV (w) t log RV ( m ) t as the ( 1 × 4 ) vector of HAR components (including an intercept term) of the foreign equity market of interest, and let the two (1 × 3 ) vectors containing US equity volatility information be denoted by xVIX t = log VIX ( d ) t log VIX (w) t log VIX ( m ) t and xUS t = log RV ( d ) t ,US log RV (w) t ,US log RV ( m ) t ,US .
We further define y t+ 1 = log RV t+ 1. Then, we can express the augmented HAR model in Eq.(5)in the following compact form: y t+ 1 = local volatility info x tβ +xVIX t β VIX + xUS t β US US volatility info + ϵUS t + 1, (6) where β= β 0 β( d ) β (w) β( m ) ′ , and β VIX = β ( d ) VIX β(w) VIX β( m ) VIX ′ and β US = β ( d ) US β(w) US β( m ) US ′ are the corresponding (4 × 1), ( 3 × 1) and (3 × 1) dimensional foreign and US parameter vectors, respectively. Similarly, the local equity market’s 1326 D. Buncic, K.I.M. Gisler / International Journal of Forecasting 32 (2016) 1317–1339 HAR model in Eq.(4)can be written compactly as:
y t+ 1 = x tb + ϵ t+ 1, (7) where x t is as defined above, b= b 0 b( d ) b(w) b( m ) ′ , and ϵ t+ 1 is an error term.
4.1. In-sample evaluation We fit the HAR model in Eq.(6)to three sample periods, in order to gauge the magnitude and significance of the es- timated parameters in Eq.(6). We first estimate the model over the full data set available, then also consider the two sub-periods leading up to and following the Lehman Broth- ers collapse on September 15, 2008. Estimation results for the full period are shown inTable 2, while estimation re- sults for the two sub-periods are provided in Table A.1 and Table A.2 in theAppendix. In each table, the first column shows the foreign equity index of interest and the second the time period over which the model in Eq.(6)was fit- ted, while the remaining columns show the set of 10 point estimates of the augmented HAR model parameters, and two χ2 test statistics of a joint test of significance of all US parameters being different from zero (null hypothe- sis is H 0 : [ β VIX ; β US ] = 0 6× 1) and only those of the US log RV series being different from zero (null hypothesis is H 0:
β US = 0 3× 1). In square brackets below the parame- ter estimates ( χ2 -test statistics), we show two-sided (one- sided) p-values computed using a heteroskedasticity and autocorrelation consistent (HAC) variance/covariance ma- trix estimator. 16 In addition to the tabulated in-sample estimation results, we also provide graphical representations of the β parameter estimates (excluding the intercept) over the three sub-periods inFig. 3. Each plot inFig. 3shows point estimates (very thin blue line) and the corresponding 95% confidence intervals (light blue shading) for the full sample period. We then superimpose the point estimates obtained from the pre and post Lehman Brothers collapse periods (thick red and thin black lines, respectively) on these plots. The most notable in-sample fitting results can be summarised as follows. 17 First, US equity market volatility data from the previous trading day are highly informative.
A formal test of the null hypothesis H 0 : [ β VIX ; β US ] = 0 6× 1 is rejected strongly by the data for all equity markets of interest. The values of the χ2 US - test statistic are between 95.09 (lowest) for the Hang Seng and 511.43 (highest) for the AEX over the full sample period. As the 1% upper tail critical value of a χ2 random variable with six degrees of freedom is 16.81, we can see that these are fairly strong rejections. When assessing the significance of the log RV US predictors separately from the log VIX, as is shown in the last column ofTable 2, we can see that these remain highly significant, with p-values of less than 1% in general, 16 We use a standard Bartlett kernel and aNeweyandWest(1994) rule of thumb bandwidth set equal to 4 (T /100 )2 /9 .
17 We would like to emphasize here that we offer only a descriptive assessment of the in-sample results, and are not interested in a structural interpretation of these estimates per se. As was pointed out by a referee, the parameter estimates that we report in the tables are reduced form estimates, and care must be taken to not interpret them as structural ones. with the only exceptions being the Bovespa and KOSPI series, which are not affected significantly by lagged US RV information. Second, it is evident from the plots inFig. 3that the estimates are quite stable over the three sample periods, generally remaining inside (or at least close to) the 95% confidence intervals (CI) of the full sample period esti- mates. 18 Looking at the magnitudes of the parameter es- timates, we can see that the daily and weekly log VIX t components are highly significant. Moreover, the daily log VIX tcomponent is positive, while the weekly compo- nent is negative. The negative sign on the ˆ β (w) VIX coefficient is somewhat surprising, as it suggests that the weekly VIX component has a negative effect on the high frequency daily volatility component of the foreign equity market of interest. 19 Furthermore, we find a mixed picture in terms of significance for the US HAR components. That is, we find the daily US HAR component to be highly significant for all of the foreign equity indices except for the three North and South American indices and the KOSPI. These results are consistent with the χ2 US - and χ2 RV -test statistic results, which show that the Bovespa and KOSPI in partic- ular are not affected significantly by lagged US RV infor- mation. Moreover, due to the general trading hour overlap between the three North and South American equity mar- kets and the NYSE, most of the US-based equity market volatility information is probably transferred to the three North and South American equity indices on the same trad- ing day, potentially being responsible for the insignificant daily US HAR component. In addition to the daily compo- nent, we also find the monthly US HAR component to be significantly different from zero at the 1% level for all Eu- ropean indices and the All Ordinaries.
4.2. Out-of-sample forecast evaluation Given the strong in-sample evidence of the importance of lagged US-based equity market volatility information for the determination of the international equity market volatility, we now assess the value of this information within an out-of-sample forecast environment. Below, we begin by outlining the general prediction setting and evaluation criteria that we use, then proceed to present the out-of-sample forecast evaluation results.
4.2.1. Prediction setting We follow the standard literature on realized volatil- ity forecasting (seeAndersenet al.,2007;Corsi&Renó, 18 Note here that we have plotted the confidence interval for the full sample period, which will contain much tighter intervals than the smaller pre and post Lehman Brothers collapse periods, due to the larger number of observations in the absence of any severe structural breaks. Thus, if these intervals include the point estimates of the two subperiods most of the time, we can take this as indicating that no substantial structural breaks have influenced the parameter estimates. 19 It should be clear here that the two weekly components are correlated, due to the cumulative construction of these series. Although it may seem that the negative sign could be attributed to this correlation, one would also expect to see highly inflated standard errors with multi- collinearity issues, resulting in largely insignificant point estimates.
However, this is not the case here. Thus, we do not believe that the opposite sign structure is driven purely by the correlatedness between these two components. D. Buncic, K.I.M. Gisler / International Journal of Forecasting 32 (2016) 1317–1339 1327Table 2 Augmented HAR model parameter estimates over the full sample period. Equity index Sample period ˆ β ˆ β VIX ˆ β US χ2 US -stat χ2 RV -stat FTSE 100 04.02.2000–13.11.2015 − 0.1103 0.1319 0.4250 0.3176 1.1338 −0.7812 −0.1645 0.0796 −0.0071 −0.1289 468.78 29.49 United Kingdom [0.0165] [0.0000] [0.0000] [0.0000] [0.0000] [0.0000] [0.0248] [0.0000] [0.8676] [0.0027] [0.0000] [0.0000] Nikkei 225 07.02.2000–13.11.2015 0.1301 0.2971 0.3302 0.2779 0.8267 −0.6297 −0.1994 0.1023 −0.0566 0.0033 252.75 30.85 Japan [0.0078] [0.0000] [0.0000] [0.0000] [0.0000] [0.0000] [0.0085] [0.0000] [0.1349] [0.9364] [0.0000] [0.0000] DAX 03.02.2000–13.11.2015 − 0.0252 0.1723 0.4360 0.3063 1.1279 −0.8449 −0.1554 0.0915 −0.0364 −0.1002 393.11 23.93 Germany [0.5620] [0.0000] [0.0000] [0.0000] [0.0000] [0.0000] [0.0387] [0.0000] [0.4017] [0.0339] [0.0000] [0.0000] All Ordinaries 07.02.2000–13.11.2015 0.0838 0.0368 0.3720 0.5229 0.6772 −0.3721 −0.2950 0.2708 −0.0646 −0.1933 452.67 180.45 Australia [0.0766] [0.0603] [0.0000] [0.0000] [0.0000] [0.0028] [0.0002] [0.0000] [0.1530] [0.0001] [0.0000] [0.0000] CAC 40 03.02.2000–13.11.2015 − 0.0391 0.1431 0.4785 0.2525 1.2060 −0.9286 −0.1060 0.0866 −0.0203 −0.1160 484.65 30.08 France [0.3208] [0.0000] [0.0000] [0.0000] [0.0000] [0.0000] [0.1089] [0.0000] [0.6045] [0.0033] [0.0000] [0.0000] Hang Seng 03.02.2000–13.11.2015 0.0970 0.1322 0.4103 0.3765 0.6160 −0.5222 −0.0708 0.0470 0.0429 −0.0737 95.09 13.77 Hong Kong [0.0349] [0.0000] [0.0000] [0.0000] [0.0000] [0.0000] [0.3784] [0.0120] [0.2621] [0.0726] [0.0000] [0.0032] KOSPI 07.02.2000–13.11.2015 0.0734 0.2893 0.4128 0.2499 0.8755 −0.8524 −0.0339 0.0102 0.0333 −0.0102 207.64 3.50 South Korea [0.0944] [0.0000] [0.0000] [0.0000] [0.0000] [0.0000] [0.6525] [0.5439] [0.3432] [0.8033] [0.0000] [0.3210] AEX 03.02.2000–13.11.2015 − 0.0576 0.1460 0.4822 0.2636 1.2354 −0.9707 −0.1128 0.1020 −0.0124 −0.1327 511.43 38.96 The Netherlands [0.1557] [0.0000] [0.0000] [0.0000] [0.0000] [0.0000] [0.0884] [0.0000] [0.7501] [0.0011] [0.0000] [0.0000] Swiss Market Index 04.02.2000–13.11.2015 − 0.0089 0.1783 0.4877 0.2555 1.0473 −0.8159 −0.1286 0.0463 0.0194 −0.1058 422.33 16.04 Switzerland [0.7973] [0.0000] [0.0000] [0.0000] [0.0000] [0.0000] [0.0422] [0.0039] [0.5450] [0.0038] [0.0000] [0.0011] IBEX 35 04.02.2000–13.11.2015 − 0.0290 0.1889 0.4884 0.2546 1.0647 −0.8797 −0.0357 0.0634 −0.0185 −0.1336 382.94 25.34 Spain [0.4776] [0.0000] [0.0000] [0.0000] [0.0000] [0.0000] [0.5867] [0.0015] [0.6204] [0.0004] [0.0000] [0.0000] S&P CNX Nifty 07.08.2002–13.11.2015 0.1624 0.2663 0.3800 0.2817 0.5777 −0.5863 −0.0159 0.1513 −0.0210 −0.0873 200.38 58.48 India [0.0065] [0.0000] [0.0000] [0.0000] [0.0000] [0.0000] [0.8650] [0.0000] [0.6721] [0.1279] [0.0000] [0.0000] IPC Mexico 03.02.2000–13.11.2015 0.0470 0.1420 0.2148 0.5839 1.1118 −0.6908 −0.3059 −0.0024 0.1191 −0.2130 269.58 17.41 Mexico [0.4040] [0.0000] [0.0000] [0.0000] [0.0000] [0.0000] [0.0006] [0.9273] [0.0124] [0.0000] [0.0000] [0.0006] Bovespa 04.02.2000–12.11.2015 0.3182 0.2504 0.3621 0.2467 0.7810 −0.4710 −0.2896 −0.0054 −0.0134 0.0367 165.31 0.73 Brazil [0.0000] [0.0000] [0.0000] [0.0000] [0.0000] [0.0000] [0.0003] [0.7893] [0.7403] [0.4167] [0.0000] [0.8664] S&P TSX 05.06.2002–13.11.2015 − 0.0835 0.1531 0.3783 0.4155 1.1951 −0.8073 −0.1860 −0.0080 0.0150 −0.1636 273.36 20.91 Canada [0.0741] [0.0000] [0.0000] [0.0000] [0.0000] [0.0000] [0.0310] [0.7308] [0.7640] [0.0021] [0.0000] [0.0001] Euro STOXX 50 03.02.2000–13.11.2015 − 0.0223 0.0790 0.4335 0.3447 1.2180 −0.8929 −0.1431 0.1798 −0.0364 −0.1910 393.92 60.33 Euro Area [0.6539] [0.0054] [0.0000] [0.0000] [0.0000] [0.0000] [0.0885] [0.0000] [0.4811] [0.0007] [0.0000] [0.0000] FT Straits Times 03.02.2000–18.09.2015 0.1254 0.2262 0.3737 0.3219 0.5266 −0.3673 −0.1998 0.0849 −0.0390 0.0236 163.59 40.46 Singapore [0.0035] [0.0000] [0.0000] [0.0000] [0.0000] [0.0000] [0.0016] [0.0000] [0.2057] [0.4809] [0.0000] [0.0000] FTSE MIB 03.02.2000–12.11.2015 − 0.0271 0.1790 0.4731 0.2643 1.1828 −1.0343 −0.0089 0.0635 0.0012 −0.1292 437.33 22.45 Italy [0.5304] [0.0000] [0.0000] [0.0000] [0.0000] [0.0000] [0.9004] [0.0006] [0.9750] [0.0014] [0.0000] [0.0001] Notes: The table reports OLS regression estimates of the augmented HAR model parameters in Eq.(6)for each foreign equity index. Columns one and two show the equity indices and the corresponding full sample fitting periods. Columns three to 12 show the OLS parameter estimates, together with (two-sided) p-values, computed using heteroskedasticity and autocorrelation (HAC) robust standard errors, in square brackets below the estimates. The last two columns show the χ2 -test statistics ( χ2 US -stat and χ2 RV -stat) of a joint significance test with null hypotheses of H 0: [ β VIX ; β US ] = 0 6× 1 and H 0:
β US = 0 3× 1, respectively, with corresponding (HAC-based) p-values in brackets below. 1328 D. Buncic, K.I.M. Gisler / International Journal of Forecasting 32 (2016) 1317–1339Fig. 3.
Plots of all parameter estimates from the augmented HAR model over the full period and the pre and post Lehman Brothers collapse periods. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.) 2012, and others), and implement a ‘ direct’forecasting ap- proach. 20 That is, we define the (normalised) h-period log 20 SeeChevillon(2007),ChevillonandHendry(2005),Clementsand Hendry(1996),Marcellino,Stock,andWatson(2006), andPesaran,Pick, andTimmermann(2011), among others, for a motivation, evaluation and comparison of the direct forecasting approach to iterated forecasts. RV series as:
y ( h ) t =1 h h j = 1 y t− j+ 1 = 1 h h j = 1 log RV t− j+ 1, (8) and re-formulate the predictive relations in Eqs.(6)and (7)for the general h-step-ahead long-horizon regression D. Buncic, K.I.M. Gisler / International Journal of Forecasting 32 (2016) 1317–1339 1329 setting as:
y ( h ) t = x t− hβ ( h ) + xVIX t − hβ ( h ) VIX + xUS t − hβ ( h ) US + ϵUS t (9) y ( h ) t = x t− hb ( h ) + ϵ t, (10) then compute h-step-ahead forecasts as ˆ y US t + h|t = x tˆ β ( h ) + xVIX t ˆ β ( h ) VIX + xUS t ˆ β ( h ) US (11) ˆ y t+ h|t = x tˆ b ( h ) . (12) The hsuperscripts on the β( h ) • and b( h ) terms (and their estimates) indicate that these are from the h-period off- set (or long-horizon predictive) regressions in Eqs.(9) and(10). 21 The forecast errors that correspond to the predictions in Eqs.(11)and(12)are defined as:
ˆ e US t + h|t = y( h ) t + h − ˆ y US t + h|t (13) ˆ e t+ h|t = y( h ) t + h − ˆ y t+ h|t .
(14) Mean squared forecast errors (MSFEs) are computed as:
MSFE =1 T os T t = T is ( ˆ e t+ h|t ) 2 (15) MSFE (US )= 1 T os T t = T is ( ˆ e US t + h|t ) 2 , (16) respectively, for the two models. The terms T os and T is de- note the numbers of out-of-sample and in-sample obser- vations, where T os = T− T is − h+ 1, and Tis the full sample size. We use the first 500 observations, corresponding to approximately two years of data, as the in-sample fitting period. We consider 500 observations to be large enough to give reasonably precise estimates of all of the parameters required to initialise the out-of-sample forecasts. FollowingCorsiandRenó(2012),Neely,Rapach, Tu,andZhou(2014),Rapachet al.(2013) and others, we then use an expanding window (or recursive) forecasting scheme, where we add an extra observation to the 500 in-sample data points and then re-estimate the models to produce recursively updated parameter estimates and forecasts. Overall, this gives us a minimum of around 2600 data points that can be used to conduct a statistically meaningful out-of-sample forecast evaluation. We should stress again here that we use rather large in-sample fitting and out-of-sample evaluation periods, in order to ensure that our general conclusions regarding the improvements in forecast performances are not sensitive to the choices of these two windows.
4.2.2. Evaluation criteria We assess the out-of-sample forecast performance of the augmented HAR model in Eq.(6)by following the approaches ofCorsiandRenó(2012) and the recent lit- erature on forecasting the equity premium (seeCamp- bell&Thompson,2008;Neelyet al.,2014;Rapach 21 We report in-sample estimation results from the long-horizon regressions on the full sample in Table A.3, Table A.4, and Table A.5 in theAppendix. et al.,2013, and many others), and evaluate the fore- casts in terms of theClarkandWest(2007) mean squared forecast error (MSFE) adjusted t-statistic (denoted CW- statistic) and theCampbellandThompson(2008) out-of- sample R2 (denoted R2 os henceforth). 22 Since the augmented HAR model in Eq.(6)nests the standard HAR model in Eq.(7), we utilize theClarkandWest(2007) MSFE-adjusted t -statistic, which corrects for the bias that arises when the DM test is used to compare nested models.
Following the suggestion byClarkandWest(2007, p. 294), the simplest way to compute the MSFE-adjusted t -statistic is to form the sequence:
CW t+ h = (ˆ e t+ h|t ) 2 − (ˆ e US t + h|t ) 2 DM t+ h + ˆ y t+ h|t − ˆ y US t + h|t 2 adj t+ h .
(17) The DM t+ h term in the first part of Eq.(17)is the standard DieboldandMariano(1995) sequence that is computed to test for (unconditional) superior predictive ability. The second term, adj t+ h, is an adjustment term that arises due to the nested nature of the models being compared, and performs a bias correction (seeClark&West,2007, for more details). The CW-statistic (for horizon h) is then computed as:
CW-statistic = CW V ar ( CW ), (18) where CW =T− 1 os T t = T is CW t+ h and Var ( CW )is the variance of the sample mean, which can be obtained simply as the HAC-robust t-statistic on the intercept term from a regression of CW t+ h on a constant. 23 The CW-statistic implements a test of the null hypoth- esis that the MSFE of the benchmark HAR model, which does not include US equity market volatility information, is equal to that of the augmented HAR model’s forecast in Eq.
(6), against the one-sided alternative hypothesis that the benchmark’s MSFE is greater than that of the augmented HAR model. Hence, a rejection of the null hypothesis sug- gests that the forecasts from the augmented HAR model are significantly smaller than those from the benchmark HAR model on average. It should be highlighted again here that the CW-statistic is particularly suitable in the given con- text, as it is designed for the comparison of nested(fore- casting) models. Our benchmark model yields the standard HAR model, which can be obtained from the augmented HAR model by restricting [β VIX ; β US ] in Eq.(6)to 0 6× 1.
The R2 os ofCampbellandThompson(2008) is computed as follows. Let MSFE (US )be the MSFE from the augmented HAR model including US volatility information, and let 22 Note here that we are performing simple pairwise forecast compar- isons between the augmented and benchmark HAR models for each for- eign equity market’s log RV series, rather than comparing forecasts from many models. Thus, aDieboldandMariano(1995) (DM) type test of un- conditional predictive ability is sufficient for our purpose of assessing the contribution of US-based volatility information to each foreign equity market’s volatility forecasts. 23 See also the discussion in Section 2.1 ofDiebold(2015) for more background on this in the context of the traditional Diebold–Mariano (DM) statistic. 1330 D. Buncic, K.I.M. Gisler / International Journal of Forecasting 32 (2016) 1317–1339 Table 3 One-step-ahead out-of-sample forecast evaluation results (expanding window). Equity index Country Out-of-sample period T os MSFE Rel-MSFE R2 os CW-stat p-value FTSE 100 United Kingdom 14.03.2002–13.11.2015 3356 0.0582 0.8772 0.1228 14.7187 0.0000 Nikkei 225 Japan 22.04.2002–13.11.2015 3171 0.0656 0.9071 0.0929 11.3231 0.0000 DAX Germany 12.03.2002–13.11.2015 3374 0.0612 0.8905 0.1095 15.3556 0.0000 All Ordinaries Australia 21.03.2002–13.11.2015 3315 0.0899 0.8552 0.1448 15.4533 0.0000 CAC 40 France 15.03.2002–13.11.2015 3389 0.0554 0.8598 0.1402 15.5369 0.0000 Hang Seng Hong Kong 19.04.2002–13.11.2015 3005 0.0621 0.9544 0.0456 8.4199 0.0000 KOSPI South Korea 25.04.2002–13.11.2015 3239 0.0539 0.9277 0.0723 11.9394 0.0000 AEX The Netherlands 18.03.2002–13.11.2015 3389 0.0564 0.8585 0.1415 15.5860 0.0000 Swiss Market Index Switzerland 22.03.2002–13.11.2015 3333 0.0440 0.8763 0.1237 14.5667 0.0000 IBEX 35 Spain 27.03.2002–13.11.2015 3357 0.0550 0.8935 0.1065 14.6317 0.0000 S&P CNX Nifty India 15.09.2004–13.11.2015 2659 0.0810 0.9336 0.0664 10.0383 0.0000 IPC Mexico Mexico 19.03.2002–13.11.2015 3336 0.0928 0.9210 0.0790 11.4184 0.0000 Bovespa Brazil 17.04.2002–12.11.2015 3250 0.0617 0.9521 0.0479 11.1674 0.0000 S&P TSX Canada 24.06.2004–13.11.2015 2790 0.0800 0.9153 0.0847 10.0294 0.0000 Euro STOXX 50 Euro Area 11.03.2002–13.11.2015 3373 0.0822 0.8645 0.1355 13.0201 0.0000 FT Straits Times Singapore 01.04.2002–18.09.2015 3237 0.0377 0.9317 0.0683 10.1945 0.0000 FTSE MIB Italy 20.03.2002–12.11.2015 3356 0.0599 0.8894 0.1106 15.7027 0.0000 Notes: The table reports the one-step-ahead out-of-sample forecast evaluation results using an expanding estimation window for the 17 foreign equity markets that we consider. We produce the first out-of-sample forecast using an initial 500 (in-sample) data points, then expand this window. Columns one to four show the equity markets of interest, the corresponding country, the out-of-sample evaluation period and the number of out-of-sample observations T os . Columns five to seven then show the mean squared forecast errors (MSFEs) of the benchmark HAR model (without US volatility information), the relative MSFE (Rel-MSFE), computed as MSFE (US )/MSFE, where MSFE (US )and MSFE are from the augmented and benchmark HAR models respectively, and the CampbellandThompson(2008) out-of-sample R2 (R 2 os ), computed as R2 os = 1− MSFE (US )/MSFE. The last two columns report the Clark–West (CW) test statistics and the corresponding one-sided asymptotic p-values.
MSFE denote the mean squared forecast error from the benchmark HAR model. Then, the R2 os comparing the performances of the two forecasts is defined as:
R 2 os = 1− MSFE (US ) MSFE .
(19) Intuitively, the R2 os statistic in Eq.(19)measures the pro- posed model’s MSFE reduction relative to the benchmark model. If R2 os > 0, the proposed model performs better than the benchmark model, while R2 os < 0 suggests that the benchmark model performs better.
In addition to the CW-statistic ofClarkandWest(2007) and the out-of-sample R2 ofCampbellandThompson (2008), we also compute the cumulative difference be- tween the squared forecast errors of the two HAR models over the out-of-sample period. This cumulative difference (denoted cumSFE) is used commonly in the forecasting literature as a tool for highlighting the predictive perfor- mance over time of the proposed model relative to the benchmark model (seeGoyal&Welch,2008;Rapachet al., 2013, among many others). In our setting, this difference is defined as cumSFE t+ h = t τ = T is (ˆ e τ+ h|τ )2 − (ˆ e US τ + h|τ )2 , ∀ t = T is , . . . , T− G. (20) The cumSFE sequence allows us to analyse the changes over time in the forecast performances of the two models.
A value of the cumSFE series that is above zero indicates that the cumulative sum of the squared forecast errors of the benchmark model is larger than that of the proposed augmented HAR forecasts, indicating that the benchmark’s forecasts are less accurate. Moreover, an upward-sloping cumSFE sequence means that the proposed augmented HAR model produces consistentlybetter predictions than the benchmark HAR model (i.e., without US volatility information).
4.2.3. Forecast evaluation results One-step-ahead results .Table 3presents the one-step- ahead out-of-sample forecast evaluation results for all 17 international equity markets that we consider, using an ex- panding (recursive) estimation window, with the initial in- sample period consisting of T is = 500 observations. The first four columns inTable 3show the foreign equity in- dex of interest, the corresponding country, the actual out- of-sample evaluation period, and the effective number of out-of-sample observations T os that are used. Columns five to seven show the MSFE of the benchmark HAR model, the relative MSFE (denoted Rel-MSFE and computed as MSFE (US )/ MSFE), and the R2 os ofCampbellandThompson (2008). The last two columns then show theClarkand West(2007) MSFE-adjusted t-statistic (CW-statistic) and the corresponding one-sided p-values.
The evaluation results inTable 3show the strong positive effect of information about US equity market volatility on the out-of-sample forecasts of log RV in global equity markets. The CW-statistic is in excess of 8.4 for all 17 international equity markets that we consider, resulting in p-values that are effectively zero. The out-of-sample R2 values ofCampbellandThompson(2008) are as high as 14.48%, 14.15%, 14.02% and 13.55% for the All Ordinaries, AEX, CAC 40 and Euro STOXX 50, respectively, with the two lowest values, 4.79% and 4.56%, being recorded for the Bovespa and Hang Seng. Note here that these R2 os magnitudes are considerable. To put them in perspective, compare them to those reported byPattonandSheppard (2015), which allow the volatility to be split into bad and good volatility components (in addition to various other considerations related to leverage and signed jumps) and which yield R2 os improvements of (only) about 2.5%–3% D. Buncic, K.I.M. Gisler / International Journal of Forecasting 32 (2016) 1317–1339 1331 points at best, at the one-step-ahead horizon (see Table 6 ofPatton&Sheppard,2015). Although it is difficult to compare our findings to theirs directly, since we consider different information sets, it should still be clear that the forecast improvements from augmenting the benchmark local HAR model with US-based volatility information are substantial. Note here also that we are using sample sizes of at least 2600 observations for the out-of-sample evaluation periods, and up to 3300. Thus, our test results are not sensitive to, or driven by, ‘ small sample issues’.
We provide additional evidence of the strength of our out-of-sample forecast results by examining the evolution over time of the cumulative difference between the squared forecast errors from the augmented HAR model and those of the benchmark HAR model. This cumSFE series (at the one-step-ahead horizon, h= 1), as defined in Eq.(20), is plotted as the thin blue line inFig. 4. Recall that the cumSFE series is defined such that an increasing value indicates an improvement in the augmented HAR model’s predictive performance relative to the benchmark HAR model (i.e., the benchmark HAR model produces larger one-step-ahead out-of-sample forecast errors). In addition to the expanding (recursive) window based cumSFE series shown inFig. 4, we also compute the cumSFE series based on forecasts from a rolling window scheme, i.e., one that constructs the forecasts using a fixed-length estimation window of 500 observations when rolling through the out- of-sample period. This series is plotted as the thick orange line inFig. 4. Our intention here is to provide a visual indication that our expanding (recursive) window based out-of-sample forecast evaluation results are broadly similar to those obtained from a rolling window based set- up, and therefore are not sensitive to this choice.
Examining the cumSFE series shown inFig. 4, we can summarise the most interesting results from these plots as follows. First, the cumSFE is (nearly) uniformly above zero for the entire out-of-sample evaluation period and for all 17 foreign equity markets that we consider. The main exceptions are the Bovespa index for Brazil and the Hang Seng index for Hong Kong, which do not appear to be above zero consistently until about October 2007, but which increase steadily thereafter. Second, the cumSFE series increases (nearly) monotonically for all series over the full out-of-sample period. There are occasional instances of ‘ flattening off ’ for some of the 17 equity markets, occurring largely around the time period between September 2008 and June 2010. Nonetheless, if one was to draw a hypothetical straight line from the beginning of the out- of-sample period until its end in November 13, 2015, one would find the cumSFE series to line up to such a straight line fairly closely. This highlights the consistent and steady improvement over time that is provided by the inclusion of US volatility information when forecasting the volatility in other international equity markets.
Third, it is interesting to observe that the cumulative improvement of the augmented HAR model over the benchmark HAR is strongest for the All Ordinaries, the Euro STOXX 50, the CAC 40, the DAX and the AEX, and weakest for the Hang Seng, the FT Straits Times and the Bovespa equity indices. Overall, it is clear that, apart from the All Ordinaries, the European equity indices benefit the most from the inclusion of US-based equity market volatility information. The strong improvement in the log RV forecasts of the All Ordinaries makes sense because of the narrow time gap between NYSE closing in the US and ASX opening in Australia. Nonetheless, it is somewhat surprising to see here that the predictive improvement is much weaker for the other four Asian equity indices, namely the Nikkei 225, the Hang Seng, the FT Straits Times, and the KOSPI, where the trading gap is similarly short as for the All Ordinaries. 24 Of these four equity indices, the Nikkei 225 shows the largest forecast improvements when US volatility information is included, though the improvements are considerably smaller than for the All Ordinaries index. Looking at the one-step-ahead out-of-sample forecast evaluation results presented inTable 3, we can see that, in general, the predictability pattern in the European equity markets is fairly homogenous across the eight indices that we include. The improvement in the out-of-sample R2 of CampbellandThompson(2008) is between 10.65% (IBEX 35) and 14.15% (AEX). The improvements for the three North and South American equity indices are smaller than for the European equity markets overall, with the Brazilian Bovespa showing the smallest gain (4.79%). In regard to this result, we conjecture that the general trading hour overlap between these markets and the NYSE means that most of the US-based equity market volatility information is transferred on the same trading day. The NYSE is open from 14:30 to 21:00 UTC (during winter). The IPC Mexico and S&P TSX trade over the same hours as the NYSE, while the Bovespa is open from 13:00 to 20:00 UTC. The HAR components of the respective foreign equity markets seem to absorb and carry most of the relevant volatility information in real time, thereby reducing the importance of lagged US volatility information. 25 Overall, our results highlight the strong out-of-sample predictive content of US volatility information for volatility forecasts in a broad range of international equity markets.
Multi-step-ahead results . Our multi-step-ahead out-of- sample forecast evaluation results are presented inTa- ble 4. We followCorsiandRenó(2012) and construct (nor- malised) multi-period log RV forecasts for horizons of h= 5, 10 and 22 steps ahead, as defined in Eq.(8).Table 4is split into three parts, with each part corresponding to one of the three forecast horizons that we consider. The column entries inTable 4contain the same information as the one- step-ahead evaluation results reported inTable 3. Before discussing the multi-step-ahead forecast evalu- ation results, we would like to stress that we take partic- ular care when computing the HAC standard errors that 24 Both the Nikkei 225 and the KOSPI open at 00:00 UTC during summer, the same as the All Ordinaries, while the FT Straits Times and Hang Seng open at 01:00 and 01:20, respectively. 25 In a somewhat different context,Nikkinen,Mohammed,Petri,and Äjiö(2006) found that Latin American countries are not affected by US news announcements, which highlights the fact that they are less integrated with the US. Also, in the news effect and announcement literature,Brand,Buncic,andTurunen(2010) showed that European equity and bond markets react less to news from the US, such as initial unemployment claims, after conditioning on ECB announcements. 1332 D. Buncic, K.I.M. Gisler / International Journal of Forecasting 32 (2016) 1317–1339Fig. 4.
Time series evolution of the cumulative difference between the squared one-step-ahead forecast errors from the benchmark HAR model and those from the augmented HAR model (cumSFE). The thin (blue) lines show the results computed on an expanding estimation window, using an initial in-sample fitting period of 500 observations. The thick (orange) lines show the corresponding rolling window (fixed T is = 500) results. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.) are needed to construct the p-values of the CW-statistic.
It is well known that h-step-ahead forecast errors follow at least an MA( h− 1) process. When computing the dif- ferences of the squared forecast errors from the two com- peting models in order to construct the CW-statistic, the CW t+ h sequence itself will be autocorrelated for h> 1.
This autocorrelation can be sizable for large values of h. We employ a pre-whitening step, using an ARMA (1 ,1 ) as the approximating model for the CW t+ h sequence so as to re- duce the initial autocorrelation in the series, then apply a quadratic spectral (QS) kernel-based non-parametric HAC estimator to the residuals from the ARMA (1 ,1 ) model.
FollowingAndrewsandMonahan(1992), we choose the bandwidth optimally, with an AR(1) as the approximating model for the ARMA (1 ,1 ) (pre-whitened) residuals, then re-colour to obtain the required HAC standard errors. 2626 That is, using the notation ofAndrewsandMonahan(1992), the bandwidth parameter is set to 1 .3221 ˆ α (2 ) T os 1/5 , where the constant ˆ α (2 ) = 4ˆ ρ 2 /( 1− ˆ ρ )4 , and ˆ ρ is the AR(1) parameter estimate obtained from an AR(1) regression of the (pre-whitened) residual series obtained We can see from the multi-step-ahead forecast evalua- tion results inTable 4that the forecast improvements rela- tive to the benchmark HAR model remain highly significant for all 17 international equity markets at the 5-day-ahead (one week), 10-day-ahead (two week), and 22-day-ahead (one month) horizons. At the 22-day horizon, the forecast improvements are only insignificant for the KOSPI and the S&P CNX Nifty. 27 To summarise the out-of-sample forecast evaluating results that we have presented in this section, it is clear that including lagged US equity market volatility information leads to substantial improvements in the out-of-sample predictions of volatility in all 17 of the international equity markets that we analyze. Moreover, this improvement has from the ARMA (1 ,1 ) model fitted to the CW t+ h sequence. We then ‘ re- colour ’ again to obtain the HAC variance, using the ratio of the square of the ARMA lag polynomials (seeAndrews&Monahan,1992for more details of the exact computations). 27 For more information about the influences of the different predictor variables, consult the long-horizon predictive regression results reported in Table A.3, Table A.4, and Table A.5 in theAppendix. D. Buncic, K.I.M. Gisler / International Journal of Forecasting 32 (2016) 1317–1339 1333 Table 4 Multiple-step-ahead out-of-sample forecast evaluation results (expanding window). Equity index Country Out-of-sample period T os MSFE Rel-MSFE R2 os CW-stat p-value Forecast horizon h= 5 FTSE 100 United Kingdom 26.03.2002–13.11.2015 3348 0.0377 0.8826 0.1174 10.2683 0.0000 Nikkei 225 Japan 07.05.2002–13.11.2015 3163 0.0438 0.9123 0.0877 6.9091 0.0000 DAX Germany 22.03.2002–13.11.2015 3366 0.0412 0.9105 0.0895 9.7591 0.0000 All Ordinaries Australia 04.04.2002–13.11.2015 3307 0.0397 0.8515 0.1485 9.6919 0.0000 CAC 40 France 27.03.2002–13.11.2015 3381 0.0382 0.8768 0.1232 10.3148 0.0000 Hang Seng Hong Kong 02.05.2002–13.11.2015 2997 0.0311 0.9440 0.0560 4.8934 0.0000 KOSPI South Korea 08.05.2002–13.11.2015 3231 0.0350 0.9618 0.0382 5.5161 0.0000 AEX The Netherlands 28.03.2002–13.11.2015 3381 0.0403 0.8853 0.1147 10.7585 0.0000 Swiss Market Index Switzerland 05.04.2002–13.11.2015 3325 0.0316 0.9031 0.0969 9.5770 0.0000 IBEX 35 Spain 10.04.2002–13.11.2015 3349 0.0377 0.9084 0.0916 9.3822 0.0000 S&P CNX Nifty India 27.09.2004–13.11.2015 2651 0.0472 0.9512 0.0488 7.2642 0.0000 IPC Mexico Mexico 03.04.2002–13.11.2015 3328 0.0459 0.8873 0.1127 9.1160 0.0000 Bovespa Brazil 29.04.2002–12.11.2015 3242 0.0371 0.9539 0.0461 5.7047 0.0000 S&P TSX Canada 08.07.2004–13.11.2015 2782 0.0465 0.9191 0.0809 7.7913 0.0000 Euro STOXX 50 Euro Area 21.03.2002–13.11.2015 3365 0.0494 0.8696 0.1304 9.6366 0.0000 FT Straits Times Singapore 11.04.2002–18.09.2015 3229 0.0211 0.9166 0.0834 6.4696 0.0000 FTSE MIB Italy 03.04.2002–12.11.2015 3348 0.0394 0.9247 0.0753 9.5552 0.0000 Forecast horizon h= 10 FTSE 100 United Kingdom 11.04.2002–13.11.2015 3338 0.0390 0.9311 0.0689 8.0319 0.0000 Nikkei 225 Japan 21.05.2002–13.11.2015 3153 0.0429 0.9433 0.0567 5.4364 0.0000 DAX Germany 09.04.2002–13.11.2015 3356 0.0427 0.9455 0.0545 7.2399 0.0000 All Ordinaries Australia 18.04.2002–13.11.2015 3297 0.0382 0.8972 0.1028 7.8199 0.0000 CAC 40 France 12.04.2002–13.11.2015 3371 0.0407 0.9205 0.0795 8.0689 0.0000 Hang Seng Hong Kong 16.05.2002–13.11.2015 2987 0.0287 0.9559 0.0441 4.5394 0.0000 KOSPI South Korea 22.05.2002–13.11.2015 3221 0.0349 0.9879 0.0121 3.4735 0.0003 AEX The Netherlands 15.04.2002–13.11.2015 3371 0.0436 0.9236 0.0764 8.4435 0.0000 Swiss Market Index Switzerland 19.04.2002–13.11.2015 3315 0.0351 0.9463 0.0537 7.7335 0.0000 IBEX 35 Spain 24.04.2002–13.11.2015 3339 0.0391 0.9491 0.0509 7.1849 0.0000 S&P CNX Nifty India 11.10.2004–13.11.2015 2641 0.0454 1.0046 −0.0046 4.2106 0.0000 IPC Mexico Mexico 17.04.2002–13.11.2015 3318 0.0418 0.9262 0.0738 7.6535 0.0000 Bovespa Brazil 14.05.2002–12.11.2015 3232 0.0362 0.9812 0.0188 3.9384 0.0000 S&P TSX Canada 22.07.2004–13.11.2015 2772 0.0456 0.9407 0.0593 6.6146 0.0000 Euro STOXX 50 Euro Area 08.04.2002–13.11.2015 3355 0.0498 0.9221 0.0779 7.7728 0.0000 FT Straits Times Singapore 25.04.2002–18.09.2015 3219 0.0204 0.9540 0.0460 5.3990 0.0000 FTSE MIB Italy 17.04.2002–12.11.2015 3338 0.0412 0.9637 0.0363 6.8724 0.0000 Forecast horizon h= 22 FTSE 100 United Kingdom 16.05.2002–13.11.2015 3314 0.0461 0.9628 0.0372 5.9062 0.0000 Nikkei 225 Japan 25.06.2002–13.11.2015 3129 0.0486 0.9782 0.0218 4.3789 0.0000 DAX Germany 14.05.2002–13.11.2015 3332 0.0500 0.9783 0.0217 5.5171 0.0000 All Ordinaries Australia 23.05.2002–13.11.2015 3273 0.0410 0.9381 0.0619 6.1348 0.0000 CAC 40 France 17.05.2002–13.11.2015 3347 0.0478 0.9485 0.0515 6.3290 0.0000 Hang Seng Hong Kong 21.06.2002–13.11.2015 2963 0.0281 0.9978 0.0022 3.4927 0.0002 KOSPI South Korea 28.06.2002–13.11.2015 3197 0.0390 1.0362 −0.0362 0.8022 0.2112 AEX The Netherlands 20.05.2002–13.11.2015 3347 0.0524 0.9518 0.0482 6.3770 0.0000 Swiss Market Index Switzerland 24.05.2002–13.11.2015 3291 0.0437 1.0065 −0.0065 4.4579 0.0000 IBEX 35 Spain 30.05.2002–13.11.2015 3315 0.0437 0.9686 0.0314 5.9966 0.0000 S&P CNX Nifty India 22.11.2004–13.11.2015 2617 0.0494 1.0602 −0.0602 −0.2516 0.5993 IPC Mexico Mexico 22.05.2002–13.11.2015 3294 0.0424 0.9858 0.0142 5.4811 0.0000 Bovespa Brazil 19.06.2002–12.11.2015 3208 0.0399 1.0095 −0.0095 2.9363 0.0017 S&P TSX Canada 26.08.2004–13.11.2015 2748 0.0490 0.9675 0.0325 5.1390 0.0000 Euro STOXX 50 Euro Area 13.05.2002–13.11.2015 3331 0.0553 0.9515 0.0485 6.5706 0.0000 FT Straits Times Singapore 31.05.2002–18.09.2015 3195 0.0242 1.0026 −0.0026 4.4795 0.0000 FTSE MIB Italy 22.05.2002–12.11.2015 3314 0.0461 0.9785 0.0215 4.8580 0.0000 Notes: The table reports the multi-step-ahead out-of-sample forecast evaluation results for the 17 international equity markets that we consider. Forecasts for horizons h= 5,10 and 22 are shown in the top, middle and bottom panels, respectively. The target variable is the (normalised) multi-period log RV, as defined in Eq.(8). The columns are the same as those described inTable 3. The p-values corresponding to the CW-statistic are computed from HAC robust standard errors, where we conduct a pre-whiteningstep using an ARMA (1 ,1 ) model for the CW t+ h sequence in order to reduce the initial autocorrelation in the series, then apply a quadratic spectral (QS) kernel based non-parametric HAC estimator to the ARMA (1 ,1 ) residuals. We followAndrewsandMonahan (1992) and choose the bandwidth optimally with an AR(1) as the approximating model, then re-colourto obtain the HAC standard errors of the CW t+ h sequence.
a lasting impact, affecting forecasts as far as one month ahead. The equity markets that are impacted most by the US volatility information are the Australian All Ordinaries index and all of the European equity indices in our sample.
The weakest results are obtained for the South American equity markets, and some of the Asian markets. 1334 D. Buncic, K.I.M. Gisler / International Journal of Forecasting 32 (2016) 1317–1339 5. Robustness checks In this section, we address some pertinent concerns in relation to the robustness of our out-of-sample forecast evaluation results. 28 In particular, we address questions related to:
(i)most of the out-of-sample forecasting power coming from the log VIX, (ii)the role of each foreign equity market’s own forward- looking volatility information, and (iii)the importance of other European and/or Asian equity markets’ realized volatilities.
All of the tables and figures required to support our dis- cussion below are provided in theAppendix. Here, we sim- ply summarize the main findings of the robustness checks.
In all of the evaluations that we present, our reference model is the augmented HAR model, which uses only US volatility information. To conserve space and to improve the readability of the supporting figures and tables that are provided, we only report out-of-sample evaluation results based on an expanding (recursive) estimation window, us- ing the first 500 data points for the initial in-sample fitting.
In all figures, we draw the reference results from the aug- mented HAR model in Eq.(6)with a blue line, to facilitate the comparison to previously plotted results.
5.1. Does the VIX drive all of the forecast improvement results?
It is evident from the results reported inTable 2that the log VIX HAR components capture a substantial part of the overall in-sample improvement when both forward- looking and backward-looking US volatility information are included in the prediction model. This can be seen from the relative magnitude of the ˆ β VIX and ˆ β US coefficients, as well as the χ2 US and χ2 RV -statistics. We determine how much of the out-of-sample forecast improvement is driven by the VIX predictor alone by removing xVIX t from the augmented HAR in Eq.(6)and repeating the out-of-sample forecast evaluations, again against the benchmark HAR model in Eq.(7)as before. These evaluation results are reported in Table A.6, Figure A.1 and Table A.7.
Overall, we can see that the VIX plays an important role in predicting the volatility in all 17 of the interna- tional equity markets that we consider. Nevertheless, the predictive performance is heterogeneous, and depends on both the forecast horizon and the foreign equity market being analysed. For instance, at the one-step-ahead hori- zon, we can see from Table A.6 that the forecast improve- ments remain highly significant for all 17 international equity markets. The lowest CW-statistic recorded now drops to around four (S&P TSX), while the largest one is still over 16 (All Ordinaries). However, the out-of-sample 28 An earlier version of this paper also assessed the impact of increasing the size of the in-sample fitting period to 1000 observations and using the Dow Jones Industrial Average as the headline US equity index on the out- of-sample results. Overall, our findings are not affected by these choices.
These additional results are available upon request. R 2 values are uniformly lower, with some of them being as low as 0.69% and 0.84% for the Bovespa and S&P TSX one- step-ahead forecasts, though that for the All Ordinaries is still rather high, at 12.59%. Comparing the cumSFE series of the full augmented HAR model (blue line) to that of the one that only includes US RV HAR components as regres- sors (brown line), plotted in Figure A.1, we can see that, apart from the All Ordinaries, and also the FT Straits Times, the S&P CNX Nifty and the Hang Seng to a lesser extent, the slopes of the cumSFE series are subdued considerably, with those of the Bovespa and S&P TSX in particular remaining rather flat over the entire out-of-sample period. For most of the other international equity market indices, the VIX HAR components account for approximately half of the cumu- lative predictive gains. One can see from the longer forecast horizon evaluation results reported in Table A.7 that the performance of the augmented HAR model without the VIX HAR components diminishes quickly. Although the CW-statistic remains sig- nificant at the 1% level for all 17 equity markets at the five-day-ahead horizon, the overall improvement in the forecasts is noticeably weaker, resulting in much smaller R 2 os values. Again, the only exception here is the All Ordi- naries series, which yields an R2 os of 9.53%. The improve- ments deteriorate further for the 10- and 22-day-ahead prediction horizons. Nevertheless, 14 of the 17 forecast improvements remain significant at the 1% level for the 10-day-ahead horizon, though some of the R2 os values are rather small and/or negative. The R2 os for the All Ordinaries stays sizeable, at 7.00%, followed by the Nikkei 225 and the IPC Mexico, with R2 os values of around 3.3%. At the 22-day- ahead horizon, only the All Ordinaries and the Nikkei 225 retain significant and sizable predictive improvements in terms of out-of-sample R2 values.
In summary, we can conclude that the improvements in forecasts up to one week ahead are significant and sizeable, and, with the exception of the Bovespa and S&P TSX equity indices, are notdriven solely by the VIX HAR components.
Nevertheless, it is clear that the amount of predictive in- formation contained in the VIX when forecasting volatil- ity in international equity markets is large, and becomes increasingly important when constructing longer horizon predictions.
5.2. Controlling for other forward-looking volatility We have seen that a substantial part of the out-of- sample predictive gains for some of the 17 foreign equity markets is due to forward-looking volatility information, which we capture by including the (US) VIX in the aug- mented HAR model in Eq.(6). Two questions that arise are whether the S&P 500 option implied volatility index (VIX) captures allof the relevant forward-looking volatil- ity information for all markets, and how informative a for- eign equity market’s own option implied forward-looking volatility information is. 29 To assess the importance of forward-looking volatility information, as contained in the option implied volatility indices of each foreign equity 29 We thank an anonymous referee for pointing this out to us. D. Buncic, K.I.M. Gisler / International Journal of Forecasting 32 (2016) 1317–1339 1335 market’s own VIX series, we obtain VIX data for 13 eq- uity indices from Bloomberg and add local VIX HAR com- ponents to Eq.(6)as additional predictors. 30 The 13 equity markets for which VIX data are available are listed below. Entry Region Equity market’s volatility indexCountry Data begins1 Oceania ASX200 VOL INDEXAustralia 03.01.2008 2 Asia HSI VOL INDEX Hong Kong03.01.2001 3 NIKKEI VOL INDEXJapan 05.01.2001 4 KOSPI 200 VOL INDEXSouth Korea 03.01.2003 5 INDIA NSE VOL INDEXIndia 02.11.2007 6 Europe VDAX NEW Germany 03.01.1992 7 VAEX AEX VOL The Netherlands04.01.2000 8 CAC40 VOL INDEXFrance 04.01.2000 9 FTSE100 VOL INDEXUnited Kingdom 05.01.2000 10 EURO50 VIX Euro Area 04.01.1999 11 Americas SP TSX60 VOL INDEX Canada 02.10.2009 12 MEXICO VOL INDEXMexico 29.03.2004 13 CBOE BRAZIL ETF VOL INDEXBrazil 17.03.2011To clarify what we do, let xVIX t ,FC = log VIX ( d ) t ,FC log VIX (w) t ,FC log VIX ( m ) t ,FC be a (1 × 3) vector of local VIX HAR com- ponents, where the standard daily, weekly and monthly components are computed as before. We assess the value added by including local forward-looking volatility in- formation in addition to the US predictors by modifying Eq.(6)to:
y t+ 1 = local volatility info x tβ +xVIX t β VIX + xUS t β US US volatility info + forward-looking local volatility info x VIX t ,FC βVIX FC + ϵUS t + 1, (21) where βVIX FC = β VIX (d ) FC βVIX (w) FC βVIX (m ) FC ′ is a (3 × 1) vector of parameters that captures the impact of the foreign country’s local VIX information. All of the other terms in Eq.(21)are as defined previously.
As is evident from the list of VIX indices above, we do not have any option implied volatility data available for the Swiss Market Index, the IBEX 35, the FT Straits Times and the FTSE MIB. Also, the available (local) VIX data for some of the equity markets that we include do not go as far back as our RV data (i.e., to the beginning of 2000). 30 Bloomberg also has VIX indices for Russia and South Africa, but these are not listed here because they are not used in our analysis. Also, there are two VDAX indices: an old version, with the mnemonic VDAX VSMI, and a new version. We use the new version, with mnemonic VDAX NEW.Rather than shortening the out-of-sample evaluation pe- riod or excluding these equity markets from the robust- ness analysis, we decided to replace the local market’s VIX HAR predictor vector xVIX t ,FC in Eq.(21)with a European (or Asian) VIX HAR ‘ factor’ predictor vector, which we de- note by fVIX t ,EU (or fVIX t ,ASIA for Asia). That is, let X(EU )= log {[(VDAX NEW ) (VAEX AEX VOL ) (CAC40 VOL INDEX ) ( FTSE100 VOL INDEX ) (EURO50 VIX )]} be the (T × 5 ) log- transformed data matrix consisting of all of the European VIX indices listed under entries 6–10 above. Then, the Eu- ropean VIX HAR factor is defined as the (1 × 3 ) vector f VIX t ,EU = [ fVIX t ,EU ( d ) fVIX t ,EU (w) fVIX t ,EU ( m ) ] , where fVIX t ,EU is the first principal component of X(EU ), with the daily, weekly and monthly HAR components (i.e., fVIX t ,EU ( d ) , f VIX t ,EU (w) , and fVIX t ,EU ( m ) ) computed as before. Similarly, for fVIX t ,ASIA , the first principal component is extracted from X(ASIA )= log{[(HSI VOL INDEX ) (NIKKEI VOL INDEX ) (KOSPI 200 VOL INDEX )]} .
We then construct forecasts from Eq.(21)for all 17 in- ternational equity markets, using fVIX t ,EU in place of xVIX t ,FC for the Swiss Market Index and the IBEX 35, FT Straits Times, FTSE MIB, All Ordinaries, S&P CNX Nifty, Bovespa and S&P TSX indices. 31 These are then compared to the forecasts constructed from the augmented HAR model in Eq.(6), which only includes US volatility information in addition to the local HAR RV components.
Before presenting and discussing these results, we would like to emphasise here that, since we are inter- ested chiefly in the out-of-sample predictive performance of US volatility information for each of the 17 international equity markets that we consider, we only report the out- of-sample evaluation results. Also, when constructing fore- casts from factor-based regression models, it is common to extract the factors recursively when rolling through the out-of-sample period so as to avoid concerns related to look-ahead biases; that is, using future data when con- structing the factors at time t. In order to tilt the out-of- sample prediction results using the European (or Asian) VIX HAR factor in favour of the local VIX model in Eq.(21), we use the full sample data to compute fVIX t ,EU (or fVIX t ,ASIA ) once and then roll through the out-of-sample data, instead of extracting the factor recursively. This should work in favour of the eight equity markets listed above for which no VIX data are available and the factor-based approach is used.
Initially, we again present a visual assessment of the out-of-sample forecast gains by plotting the cumSFE se- quences of our augmented HAR model of Eq.(6)and the model that adds local VIX information (or a European VIX factor) to the predictor set, as defined in Eq.(21)in Figure A.2. 32 As before, both sequences are again computed rela- tive to the local HAR model given in Eq.(7), with the fore- cast horizon being one step ahead. That is, the blue lines in Figure A.2 are the same as the blue lines plotted inFig. 4. 31 Since the beginning date of the KOSPI 200 VOL INDEX is in 2003, the sample and forecasting periods for the Asian countries are shortened correspondingly. 32 Using fVIX t ,ASIA in place of fVIX t ,EU produces consistently worse forecasts; to conserve space, the results are not reported here, but they are available upon request. 1336 D. Buncic, K.I.M. Gisler / International Journal of Forecasting 32 (2016) 1317–1339 The green lines in Figure A.2 are the cumSFE sequences of our augmented HAR model that adds local forward-looking volatility information to the augmented HAR model as de- fined in Eq.(21)(the legend entry is + FC VIX HAR). 33 Recall that the model in Eq.(21), which adds local forward- looking volatility information to the predictor set, pro- duces consistently better out-of-sample forecasts if the green line in Figure A.2 is consistently above the blue one.
As we can see from Figure A.2, this is only the case for the Nikkei 225, Hang Seng, KOSPI and Euro STOXX 50, and very slightly for the DAX. Visually, the improvement seems to be strongest for the Hang Seng series, which is driven largely by a single episode that occurred at around the time of the Lehman Brothers’ collapse in September/October 2008.
For the DAX, KOSPI and Euro STOXX 50, the improvement appears to be rather marginal, while it is noticeable from about the end of 2011 onwards for the Nikkei 225. To formally gauge the magnitude of any potential out- of-sample forecast improvements, we compute the gain in out-of-sample R2 from adding local VIX information to the augmented HAR model. That is, we define 1R2 os ( h ) = [ R 2 os computed from Eq.(21)— R2 os computed from Eq.(6) ] , where hdenotes the forecast horizon that is being evaluated, i.e., h= 1,5 ,10 ,22. When 1R2 os ( h ) > 0, there is an increase in R2 os from adding local VIX information to the predictor set in the augmented HAR model. The statistical significance is examined again within aClarkand West(2007) MSFE-adjusted t-test environment, since we are examining the predictive gain from adding local VIX information to the augmented HAR model in Eq.(6); that is, the augmented HAR model of Eq.(6)is nested in the model with local VIX information in Eq.(21). Table A.8 reports the predictive gains in terms of 1R2 os ( h ) for h= 1 ,5 ,10 ,22 in the last four columns. To avoid cluttering the table with extra columns showing the magnitudes of the CW-statistics, we have merely added asterisks next to the 1 R2 os ( h ) entries that yield significant CW-statistics. We use standard asterisk notation to denote significance at the 1%, 5%, and 10% levels, respectively. 34 We can see from the evaluation results that are reported in Table A.8 that the change in the out-of-sample R2 values as a result of including local VIX HAR components in the augmented HAR model of(6)is negative for six of the 17 equity markets at the one-step-ahead horizon that we consider. The importance of this predictor variable deteriorates further with an increasing h, producing only three positive 1R2 os values out of 17 at the 22-day-ahead horizon. Most of the non-negative increases in 1R2 os are rather small in magnitude, with notable exceptions being the improvements recorded for the Hang Seng (3.13%), the Euro STOXX 50 (1.43%), the Nikkei 225 and KOSPI (each around 1.4%) and the DAX (0.9%), at the one-step-ahead 33 Note that everything in these plots is kept as in previous figures in order to facilitate comparisons. Some of the countries have different beginning dates for the out-of-sample evaluation, due to the lack of available VIX data, so the plots for the Nikkei 225, the Hang Seng, the KOSPI and the IPC Mexico are shifted somewhat, due to the later out-of- sample starting periods. 34 That is, ∗ , ∗∗ and ∗∗∗ denote significance at the 10%, 5% and 1% levels, respectively. horizon. Moreover, these improvements are statistically significant. From the long-horizon evaluations, it is evident that the improvements remain consistently significant, though small at times, up to 22 days ahead for the Euro STOXX 50, and up to 10 steps ahead for the KOSPI index, the DAX, Hang Seng and the IBEX 35, with the later two being only weakly significant at the 10-day-ahead horizon.
To summarise our results as to the robustness to local volatility information, we conclude that for most of the equity markets there is only a very slight improvement in out-of-sample performances, with six (for h= 1) or more (for h> 1) markets in fact producing negative 1R2 os values. At the one-step-ahead horizon, only around three to four equity markets improve significantly and achieve sizeable gains, with the improvements in R2 os being 1%–3%.
However, these gains deteriorate with the forecast horizon, and are unimportant at h= 22. Overall, we conclude that adding local VIX data to the augmented HAR model in Eq.(6)yields small or no forecast gains, with the most notable exception of the Euro STOXX 50, and, at shorter horizons, the Nikkei 225, the DAX and the Hang Seng VIX information. 35 5.3. Controlling for RV information in other European and/or Asian equity markets As a last robustness check, we analyze whether the realized volatility in other European and/or Asian equity markets includes relevant information that could be utilised to improve out-of-sample forecast performance.
Rather than selecting a few dominant European or Asian equity markets and then adding their HAR component vec- tors into Eq.(21)one at a time as extra control variables for each of the international equity markets that we consider, we again prefer to extract a common RV factor from the European and Asian realized volatility information using principal components. 36 To formalise this, let Y(EU )= log {[ RV(DAX) RV(CAC 40) RV(AEX) RV(Swiss Market Index) RV(IBEX 35) RV(Euro STOXX 50) RV(FTSE MIB) ]}denote 35 The Euro STOXX 50 and the DAX seem to have the most liquid and mature VIX indices. At this point, it is not clear why the other equity markets’ VIX indices are not informative for long horizon forecasts at least, even if short horizons are affected the most by spillover effects. 36 Specifying, say, 11 European and Asian markets to be used one at a time as controls for the potentially relevant RV information contained in these other equity markets would seem feasible here. However, this raises the possibility of too many statistical tests being carried out, an issue which is known as the ‘ multiple comparisons’ problem in the statistics literature. As a solution, one could use a Bonferroni type of correction when evaluating the out-of-sample performance; that is, adjust the significance level of the test based on the number of additional tests that are constructed. However, it is not clear to us whether this is justified when implementing the MSFE-adjusted t-test on the nested model comparisons ofClarkandWest(2007). Moreover, it would be cumbersome to report the evaluation results in an informative way, as one would have 11 prediction evaluations for each international equity market and forecast horizon. In order to stay within the same testing environment and simplify the presentation of the results, we extract a European and an Asian RV factor, rather than including additional RV predictors one at a time as extra controls. As was done with the VIX series in Section5.2, we again extract the factors from the full sample period only once, rather than recursively as new information becomes available. D. Buncic, K.I.M. Gisler / International Journal of Forecasting 32 (2016) 1317–1339 1337 the (T × 8) vector of log-transformed RV data for all Eu- ropean equity indices that are available to us. The Euro- pean RV HAR factor is then defined as the (1 × 3) vector f RV t ,EU = [ fRV t ,EU ( d ) fRV t ,EU (w) fRV t ,EU ( m ) ] , where fRV t ,EU is the first principal component of Y(EU ), with the daily, weekly and monthly HAR components computed as be- fore. The Asian RV factor (denoted by fRV t ,ASIA hence- forth) is computed as the first principal component from Y (ASIA )= log {[RV(Nikkei 225) RV(Hang Seng) RV(KOSPI) RV(FT Straits Times) ]}.37 We assess the importance of RV information in other European and/or Asian equity markets by taking the aug- mented HAR model that adds local VIX data as predictors, defined in Eq.(21), and further adding fRV t ,EU and fRV t ,ASIA to it as regressors. That is, we form the predictive regression model:
y t+ 1 = local volatility info x tβ +xVIX t β VIX + xUS t β US US volatility info augmented HAR as inEq.(6) + forward-looking local volatility info x VIX t ,FC βVIX FC + fRV t ,EU βf EU + fRV t ,ASIA βf ASIA other RV info + ϵUS t + 1, (22) where fRV t ,EU and fRV t ,EU are the (1 × 3)-dimensional Eu- ropean and Asian RV HAR factor vectors defined above, and βf EU and βf ASIA are corresponding (3 × 1) parameter vectors that capture their influence on the international RV. Improvements in out-of-sample forecasts relative to our augmented HAR model are examined in the same set- ting as in Section5.2; that is, informally by the magni- tude of the 1R2 os ( h ), statistically using theClarkandWest (2007) MSFE-adjusted t-test, and visually from plots of the cumSFE sequence. These evaluation results are reported in Figure A.3 and Table A.9. Figure A.3 shows the incremen- tal improvement in the cumSFE from including other RV information, in the form of a European RV factor HAR and an Asian RV factor HAR, in addition to the predictor set of the augmented HAR with local VIX information defined in Eq.(21). The blue line in Figure A.3 again shows the cumSFE of the augmented HAR as a reference point, as was done before. The red line shows the cumSFE when only the Eu- ropean RV factor HAR vector fRV t ,EU is included in Eq.(21) in addition to local VIX information (legend entry +FC 37 For the European RV data, the first three principal components explain 89.1745%, 4.6748%, and 1.8267%, respectively, of the variation in Y(EU ). Thus, using only the first principal component seems to be justified, as it explains nearly 90% of the variation in the data. For the Asian RV data, these values are 68.9685%, 15.4803%, and 11.2654%, respectively.
These results are less clear as to whether one factor is enough to capture all of the important movements in Y(ASIA ). We address the issue that more than one factor may be driving Y(ASIA )by also performing forecast evaluations using a HAR structure on the first two factors, together with equally weighted and R2 weighted linear combinations. The latter was performed in order to keep the number of additional regressors added small, so as to minimise overfitting and the ensuing poor out-of-sample performance. However, the forecasts in all of these assessments were always worse than those based on the first PC only. VIX HAR +f(EU) HAR). The light green line in Figure A.3 shows the improvement when both the European RV factor HAR vector fRV t ,EU and the Asian RV factor HAR vector fRV t ,ASIA are added to Eq.(21)as predictors (legend entry +f(ASIA) HAR).
From the reported results, we can summarise the effect of adding other RV information into the predictor set in the form of factors as follows. First, examining the time series evolution of the cumSFE sequence visually, one can see that there is no improvement, or at the very best only a very mild improvement, in the out-of-sample forecasts as a result of adding other RV information that is not already realised from the addition of the local VIX series assessed earlier. In fact, the results for some equity markets worsen (see for instance the cumSFE series for the Bovespa, the IPC Mexico, the S&P CNX Nifty, and the All Ordinaries index). Second, conditioning on the Asian RV factor HAR generally produces marginally worse out-of-sample forecasts. 38 This can be seen most clearly from the FT Straits Times and the IPC Mexico, and also somewhat more mildly from the DAX, the Euro STOXX 50, the Swiss Market Index and the KOSPI series.
For these equity markets, the loss in precision from adding irrelevant predictors worsens the out-of-sample forecast performance. Third, it is evident from the multiple-horizon statistical comparison in Table A.9 that any gains in R2 os over the benchmark augmented HAR model and their significance levels are very similar to those obtained by only incorporating local VIX information, as was done in Eq.(21), see Table A.8. Moreover, again in line with the results in Section5.2, any forecast gains that are statistically significant at the one-step-ahead horizon disappear fairly quickly as hincreases, with the 22-day- ahead forecast even for the Euro STOXX 50 resulting in a rather small and statistically insignificant improvement.
Overall, we conclude this last robustness check with the finding that, once we condition on local VIX information, as described in Section5.2, including additional RV data in the predictor set does not add any further information to improve the out-of-sample forecasts of RV in international equity markets.
6. Conclusion This study extends the work ofRapachet al.(2013) and investigates whether US-based equity market volatility information has predictive value for volatility forecasts in a large cross-section of international equity markets. We assess the role of the US by augmenting the benchmark HAR model ofCorsi(2009) with daily, weekly and monthly US RV and log VIX HAR components, and evaluating the in-sample and out-of-sample contributions of this information to realized volatility in international equity markets. We find the US to play a strong role as a 38 We have also include current time, i.e., t+ 1, factors for Asia when forming forecasts for the European and North and South American indices, but the difference between using lagged or current time information is immaterial, producing the same statistical conclusions in terms of significance levels. 1338 D. Buncic, K.I.M. Gisler / International Journal of Forecasting 32 (2016) 1317–1339 source of relevant volatility information, being particularly important for the Australian and all of the European equity markets that we consider, with a sizeable part of this relevant volatility information coming from forward- looking (or implied) volatility.
Using a large out-of-sample forecast evaluation period, we find that the volatility forecasts for all 17 equity markets improve substantially and are highly statistically significant when US volatility information is included in the predictor set. The daily out-of-sample R2 values range between 4.56% (Hang Seng) and 14.48% (All Ordinaries), and are above 10% for nine of the 17 equity markets that we analyse. Moreover, our results show that the Australian and all of the European equity markets benefit the most from the inclusion of US-based volatility information, while the South and North American equity markets and some of the Asian markets benefit the least. An assessment of the forecast performance over time shows that this improvement in predictive performance is consistent over the entire out-of-sample period, and is not driven solely by a few individual events. Moreover, we show that the improvements remain significant for forecast horizons of up to 22 days ahead for 15 of the 17 equity markets, still yielding sizable out-of-sample R2 values of around 5%–6% for four of the 17 equity markets that we include.
One interesting finding from our in-sample analysis is that the low frequency US volatility component has a negative effect. That is, the parameter estimates on the weekly log VIX HAR component are negative and highly significant for all 17 equity markets. The values that we obtain range from −1.03 to −0.37, with the majority being in the range −0.9 to −0.8. The monthly US RV HAR component is significantly negative for 12 of the 17 equity markets, with values largely ranging from −0.20 to −0.10.
So far, there does not seem to have been any discussion in the literature as to why this negative effect occurs, and what economic forces lie behind it, particularly with regard to the weekly log VIX HAR component.
In summary, our analysis confirms that the US plays a leading role as a source of equity market information.
This role is important not only for international equity return forecasts, as documented byRapachet al.(2013), but also for forecasts of the volatilityin international equity markets.
Acknowledgments We are grateful to Francis Diebold, Adrian Pagan, Francesco Ravazzolo, Valentyn Panchenko, Dave Rapach, Paul Söderlind, Angelo Ranaldo, Francesco Audrino, Matthias Fengler, Lorenzo Camponovo, Davide La Vecchia, Jeroen Rombouts, Kamil Yilmaz, Giampiero Gallo, Victor Todorov and Jonathan Wright, as well as seminar partic- ipants at the University of St. Gallen, the University of Pennsylvania, and the 9th International Conference on Computational and Financial Econometrics (CFE 2015) in London for helpful discussions and comments on earlier drafts of the paper. Katja Gisler gratefully acknowledges financial support from the Swiss National Science Founda- tion through grants 144033 and 161796. Appendix A. Supplementary data Supplementary material related to this article can be found online athttp://dx.doi.org/10.1016/j.ijforecast.2016.
05.001.
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