International Journal of Forecasting 32 (2016) 1317–1339
Contents lists available at ScienceDirect 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
article info
Keywords:
Realized volatility
HAR modelling and forecasting Augmented HAR model
US volatility information
VIX
International volatility spillovers
abstract
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
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, Rapach et 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: daniel.buncic@unisg.ch (D. Buncic), katja.gisler@unisg.ch (K.I.M. Gisler).
URL: http://www.danielbuncic.com (D. Buncic).
http://dx.doi.org/10.1016/j.ijforecast.2016.05.001
(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. Rapach et 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
the US
equity
market
is the world’s largest,
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 of Corsi (2009) with (lagged) daily, weekly and monthly US realized volatility and VIX HAR components. We utilize the HAR model of Corsi (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 of Rapach et 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 of Eun and Shim (1989), Hamao et al. (1990) and Lin et al. (1994). More recent studies include those of Baur and Jung (2006) and Savva, Osborn, and Gill (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, Hamao et 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, Lin et al. (1994) employ a signal extraction model with GARCH innovations in order to analyze spillover effects between two major eq- uity markets. In contrast to Hamao et al. (1990), Lin et 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 example Becker, 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, Eun and Shim (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, Savva et 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 (see Bonato, Caporin, & Ranaldo, 2013; Diebold & Yilmaz, 2014, 2016; Dimpfl & Jung, 2012; Fengler & Gisler, 2015, among others). Diebold and Yilmaz (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, Fengler and Gisler (2015) extend the results of Diebold and Yilmaz (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. Bonato et 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, Dimpfl and Jung (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 instance Buncic and Melecky (2013) for a recent study as to how this could be implemented.
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 example Dimpfl & 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, yielding Clark and West (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, Patton and Sheppard (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 Section 2, we outline how realized volatility is mod- elled and how we extend the standard HAR model of Corsi (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 Section 3. In Section 4, 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 Section 5, 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 pt 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:
dpt =μtdt+σtdWt, (1)
where μt is a predictable and locally bounded drift term, σt is a càdlàg volatility process that is bounded away from zero,andWt isastandardBrownianmotion.Thequadratic variation (QV) process of pt is given by3:
t
QVt = σs2ds. (2)
0
In the absence of jumps, as is the case in our setting in Eq.
(1), the term t σ 2 ds in Eq. (2) is known as the integrated 0s
D. Buncic, K.I.M. Gisler / International Journal of Forecasting 32 (2016) 1317–1339 1319
variance (IV) of the process pt .
3 The quadratic variation process of pt is defined as [pt ] = plim
m2m→∞ k=1 (p(tk ) − p(tk−1 )) , where plim denotes convergence in probability,
and0=t0 ≤t1 <···
26 That is, using the notation of Andrews and Monahan (1992), the bandwidth parameter is set to 1.3221 αˆ (2) Tos 1/5 , where the constant
αˆ (2) = 4ρˆ 2 /(1 − ρˆ )4 , and ρˆ 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 in Table 4 that 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 CWt +h sequence. We then ‘re- colour’ again to obtain the HAC variance, using the ratio of the square of the ARMA lag polynomials (see Andrews & Monahan, 1992 for 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 the Appendix.
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 Tos MSFE Rel-MSFE R2os CW-stat p-value
Forecast horizon h = 5
FTSE 100 Nikkei 225 DAX
All Ordinaries CAC 40
Hang Seng
KOSPI
AEX
Swiss Market Index IBEX 35
S&P CNX Nifty IPC Mexico Bovespa
S&P TSX
Euro STOXX 50 FT Straits Times FTSE MIB
United Kingdom Japan
Germany Australia
France
Hong Kong South Korea
The Netherlands Switzerland Spain
India
Mexico
Brazil
Canada
Euro Area Singapore
Italy
26.03.2002–13.11.2015 07.05.2002–13.11.2015 22.03.2002–13.11.2015 04.04.2002–13.11.2015 27.03.2002–13.11.2015 02.05.2002–13.11.2015 08.05.2002–13.11.2015 28.03.2002–13.11.2015 05.04.2002–13.11.2015 10.04.2002–13.11.2015 27.09.2004–13.11.2015 03.04.2002–13.11.2015 29.04.2002–12.11.2015 08.07.2004–13.11.2015 21.03.2002–13.11.2015 11.04.2002–18.09.2015 03.04.2002–12.11.2015
3348 0.0377 0.8826 3163 0.0438 0.9123 3366 0.0412 0.9105 3307 0.0397 0.8515 3381 0.0382 0.8768 2997 0.0311 0.9440 3231 0.0350 0.9618 3381 0.0403 0.8853 3325 0.0316 0.9031 3349 0.0377 0.9084 2651 0.0472 0.9512 3328 0.0459 0.8873 3242 0.0371 0.9539 2782 0.0465 0.9191 3365 0.0494 0.8696 3229 0.0211 0.9166 3348 0.0394 0.9247
0.1174 10.2683 0.0877 6.9091 0.0895 9.7591 0.1485 9.6919 0.1232 10.3148 0.0560 4.8934 0.0382 5.5161 0.1147 10.7585 0.0969 9.5770 0.0916 9.3822 0.0488 7.2642 0.1127 9.1160 0.0461 5.7047 0.0809 7.7913 0.1304 9.6366 0.0834 6.4696 0.0753 9.5552
0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
Forecast horizon h = 10
FTSE 100 Nikkei 225 DAX
All Ordinaries CAC 40
Hang Seng
KOSPI
AEX
Swiss Market Index IBEX 35
S&P CNX Nifty IPC Mexico Bovespa
S&P TSX
Euro STOXX 50 FT Straits Times FTSE MIB
United Kingdom Japan
Germany Australia
France
Hong Kong South Korea
The Netherlands Switzerland Spain
India
Mexico
Brazil
Canada
Euro Area Singapore
Italy
11.04.2002–13.11.2015 21.05.2002–13.11.2015 09.04.2002–13.11.2015 18.04.2002–13.11.2015 12.04.2002–13.11.2015 16.05.2002–13.11.2015 22.05.2002–13.11.2015 15.04.2002–13.11.2015 19.04.2002–13.11.2015 24.04.2002–13.11.2015 11.10.2004–13.11.2015 17.04.2002–13.11.2015 14.05.2002–12.11.2015 22.07.2004–13.11.2015 08.04.2002–13.11.2015 25.04.2002–18.09.2015 17.04.2002–12.11.2015
3338 0.0390 0.9311 3153 0.0429 0.9433 3356 0.0427 0.9455 3297 0.0382 0.8972 3371 0.0407 0.9205 2987 0.0287 0.9559 3221 0.0349 0.9879 3371 0.0436 0.9236 3315 0.0351 0.9463 3339 0.0391 0.9491 2641 0.0454 1.0046 3318 0.0418 0.9262 3232 0.0362 0.9812 2772 0.0456 0.9407 3355 0.0498 0.9221 3219 0.0204 0.9540 3338 0.0412 0.9637
0.0689 8.0319 0.0567 5.4364 0.0545 7.2399 0.1028 7.8199 0.0795 8.0689 0.0441 4.5394 0.0121 3.4735 0.0764 8.4435 0.0537 7.7335 0.0509 7.1849
−0.0046 4.2106 0.0738 7.6535 0.0188 3.9384 0.0593 6.6146 0.0779 7.7728 0.0460 5.3990 0.0363 6.8724
0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0003 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
Forecast horizon h = 22
FTSE 100 Nikkei 225 DAX
All Ordinaries CAC 40
Hang Seng
KOSPI
AEX
Swiss Market Index IBEX 35
S&P CNX Nifty IPC Mexico Bovespa
S&P TSX
Euro STOXX 50 FT Straits Times FTSE MIB
United Kingdom Japan
Germany Australia
France
Hong Kong South Korea
The Netherlands Switzerland Spain
India
Mexico
Brazil
Canada
Euro Area Singapore
Italy
16.05.2002–13.11.2015 25.06.2002–13.11.2015 14.05.2002–13.11.2015 23.05.2002–13.11.2015 17.05.2002–13.11.2015 21.06.2002–13.11.2015 28.06.2002–13.11.2015 20.05.2002–13.11.2015 24.05.2002–13.11.2015 30.05.2002–13.11.2015 22.11.2004–13.11.2015 22.05.2002–13.11.2015 19.06.2002–12.11.2015 26.08.2004–13.11.2015 13.05.2002–13.11.2015 31.05.2002–18.09.2015 22.05.2002–12.11.2015
3314 0.0461 3129 0.0486 3332 0.0500 3273 0.0410 3347 0.0478 2963 0.0281 3197 0.0390 3347 0.0524 3291 0.0437 3315 0.0437 2617 0.0494 3294 0.0424 3208 0.0399 2748 0.0490 3331 0.0553 3195 0.0242 3314 0.0461
0.9628 0.9782 0.9783 0.9381 0.9485 0.9978 1.0362 0.9518 1.0065 0.9686 1.0602 0.9858 1.0095 0.9675 0.9515 1.0026 0.9785
0.0372 5.9062 0.0218 4.3789 0.0217 5.5171 0.0619 6.1348 0.0515 6.3290 0.0022 3.4927
−0.0362 0.8022 0.0482 6.3770 −0.0065 4.4579 0.0314 5.9966 −0.0602 −0.2516 0.0142 5.4811 −0.0095 2.9363 0.0325 5.1390 0.0485 6.5706 −0.0026 4.4795 0.0215 4.8580
0.0000 0.0000 0.0000 0.0000 0.0000 0.0002 0.2112 0.0000 0.0000 0.0000 0.5993 0.0000 0.0017 0.0000 0.0000 0.0000 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 in Table 3. The p-values corresponding to the CW-statistic are computed from HAC robust standard errors, where we conduct a pre-whitening step using an ARMA (1, 1) model for the CWt +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 follow Andrews and Monahan (1992) and choose the bandwidth optimally with an AR(1) as the approximating model, then re-colour to obtain the HAC standard errors of the CWt+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) theimportanceofotherEuropeanand/orAsianequity markets’ realized volatilities.
All of the tables and figures required to support our dis- cussion below are provided in the Appendix. 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 in Table 2 that
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 βˆ VIX and βˆ US coefficients, as
well as the χ 2 and χ 2 -statistics. We determine how much US RV
of the out-of-sample forecast improvement is driven by the VIX predictor alone by removing xVIX from the augmented
t
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 Anearlierversionofthispaperalsoassessedtheimpactofincreasing 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.
R2 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 R2os values. Again, the only exception here is the All Ordi- naries series, which yields an R2os 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 R2os values are rather small and/or negative. The R2os for the All Ordinaries stays sizeable, at 7.00%, followed by the Nikkei 225 and the IPC Mexico, with R2os 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 not driven 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 all of 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 Wethankananonymousrefereeforpointingthisouttous.
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.
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 in Eq. (21) with a European t ,FC
To clarify what we do, let xVIX t,FC
= log VIX(d) t,FC
log VIX(w) t,FC
INDEX) (NIKKEI VOL INDEX) (KOSPI 200 VOL INDEX)]}.
We then construct forecasts from Eq. (21) for all 17 in-
ternational equity markets, using f VIX in place of xVIX for t ,EU t ,FC
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 f VIX (or f VIX ) t ,EU 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 in Fig. 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 f VIX in place of f VIX produces consistently worse forecasts; to t ,ASIA t ,EU
conserve space, the results are not reported here, but they are available upon request.
log VIX(m) be a (1 × 3) vector of local VIX HAR com- t ,FC
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:
local volatility info VIX
US yt+1=xtβ+xt βVIX+xtβUS
US volatility info
forward-looking local volatility info
+ xVIXβVIX +εUS , (21) t,FC FC t+1
where βVIX = βVIX(d) βVIX(w) βVIX(m)′ is a (3 × 1) vector FC FC FC FC
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.
D. Buncic, K.I.M. Gisler / International Journal of Forecasting 32 (2016) 1317–1339 1335
(or Asian) VIX HAR ‘factor’ predictor vector, which we de- note by f VIX (or f VIX for Asia). That is, let X(EU) =
Entry Region Equity market’s Country Data begins volatility index
1 Oceania
2
3 Asia
ASX200 VOL INDEX
HSI VOL INDEX
NIKKEI VOL INDEX
KOSPI 200 VOL INDEX
INDIA NSE VOL INDEX
VDAX NEW VAEX AEX VOL
CAC40 VOL INDEX FTSE100 VOL INDEX EURO50 VIX
SP TSX60 VOL INDEX MEXICO VOL INDEX
CBOE BRAZIL ETF VOL INDEX
Australia
Hong Kong Japan
South Korea India
Germany The Netherlands France
United Kingdom Euro Area
Canada Mexico Brazil
03.01.2008
03.01.2001
05.01.2001
03.01.2003
02.11.2007
03.01.1992 04.01.2000
04.01.2000 05.01.2000
04.01.1999 02.10.2009
29.03.2004 17.03.2011
4 5
6 7
8
9
10 11
12 13
Europe
Americas
t ,EU t ,ASIA
log{[(VDAXNEW) (VAEXAEXVOL) (CAC40VOLINDEX)
(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
fVIX = [fVIX(d) fVIX(w) fVIX(m)], where fVIX is the first t,EU t,EU t,EU t,EU t,EU
principal component of X(EU), with the daily, weekly and
monthly HAR components (i.e., f VIX (d), f VIX (w), and f VIX (m)) t,EU t,EU t,EU
computed as before. Similarly, for f VIX
t ,ASIA
, the first principal component is extracted from X(ASIA) = log{[(HSI VOL
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 R2os (h) = [R2os computed from Eq. (21)—R2os computed from Eq. (6) ], where h denotes the forecast horizon that is being evaluated, i.e., h = 1, 5, 10, 22. When R2os (h) > 0, there is an increase in R2os from adding local VIX information to the predictor set in the augmented HAR model. The statistical significance is examined again within a Clark and 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 R2os (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 R2os (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 R2os values out of 17 at the 22-day-ahead horizon. Most of the non-negative increases in R2os 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 R2os values. At the one-step-ahead horizon, only around three to four equity markets improve significantly and achieve sizeable gains, with the improvements in R2os 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 of Clark and West (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 Section 5.2, we again extract the factors from the full sample period only once, rather than recursively as new information becomes available.
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
fRV = [fRV (d) fRV (w) fRV (m)], where fRV is the t,EU t,EU t,EU t,EU t,EU
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 hence- t ,ASIA
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 f RV and f RV to t ,EU t ,ASIA
it as regressors. That is, we form the predictive regression model:
VIX HAR + f(EU) HAR). The light green line in Figure A.3
shows the improvement when both the European RV factor
HAR vector f RV and the Asian RV factor HAR vector f RV
t ,EU 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
38
local volatility info
forward-looking local volatility info
worse out-of-sample forecasts.
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 R2os 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 Section 5.2, any forecast gains that are statistically significant at the one-step-ahead horizon disappear fairly quickly as h increases, with the 22-day- ahead forecast even for the Euro STOXX 50 resulting in a rather small and statistically insignificant improvement.
D. Buncic, K.I.M. Gisler / International Journal of Forecasting 32 (2016) 1317–1339 1337
y
+xUSβ
+ xVIX βVIX t+1 t t VIX t US t,FC FC
This can be seen most
= x β +xVIXβ
US volatility info
augmented HAR as inEq. (6)
+fRV βf +fRV βf +εUS ,
(22)
t,EU EU t,ASIA ASIA t+1
other RV info
where f RV and f RV are the (1 × 3)-dimensional Eu-
t ,EU t ,EU
ropean and Asian RV HAR factor vectors defined above,
and βfEU and βfASIA 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 Section 5.2; that is, informally by the magni-
tude of the R2os(h), statistically using the Clark and West
(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 f RV is included in Eq. (21) t ,EU
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.
Overall, we conclude this last robustness check with the finding that, once we condition on local VIX information, as described in Section 5.2, 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 of Rapach et 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 of Corsi (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
2
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 by Rapach et al. (2013), but also for forecasts of the volatility in 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 at http://dx.doi.org/10.1016/j.ijforecast.2016. 05.001.
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Daniel Buncic is Assistant Professor of Quantitative Economics at the Institute of Mathematics and Statistics in the School of Economics and Political Sciences at the University of St. Gallen. He holds a Ph.D. in Economics from the University of New South Wales in Sydney, Australia. He has published in various journals, including the Journal of the European Economic Association, the Journal of Banking and Finance and the Journal of Financial Stability. He is currently interested in forecasting of financial assets, particularly, commodities, exchange rates and the equity premium using various forecast averaging and aggregation methods. He has previously worked on issues related to stress testing and financial stability issues in general, as well as macro-econometric modeling and its use in policy analysis.
Katja I.M. Gisler holds a Masters in Quantitative Economics and Finance from the University of St. Gallen and is currently a Ph.D. student in the School of Economics and Political Sciences at the University of St. Gallen. Her research interests are in areas of realized volatility modeling, volatility spillovers and financial econometrics in general. She has recently published a paper on volatility spillovers in the Journal of International Money and Finance.
D. Buncic, K.I.M. Gisler / International Journal of Forecasting 32 (2016) 1317–1339 1339