Exploiting Spillovers to Forecast Crashes

• Author: Philip Hans Franses,Francine Gresnigt,Erik Kole
• DOI: http://doi.org/10.1002/for.2434
• Published date: 01 December 2017
• Date: 01 December 2017

Journal of Forecasting,J. Forecast. 36, 936–955 (2017)

Published online 18 August 2016 in Wiley Online Library (wileyonlinelibrary.com)DOI: 10.1002/for.2434

Exploiting Spillovers to Forecast Crashes

FRANCINE GRESNIGT,ERIK KOLE AND PHILIP HANS FRANSES

ABSTRACT

We develop Hawkes models in which events are triggered through self-excitation as well as cross-excitation. We

examine whether incorporating cross-excitation improves the forecasts of extremes in asset returns compared to only

self-excitation. The models are applied to US stocks, bonds and dollar exchange rates. We predict the probability

of crashes in the series and the value at risk (VaR) over a period that includes the financial crisis of 2008 using a

moving window. A Lagrange multiplier test suggests the presence of cross-excitationfor these series. Out-of-sample,

we find that the models that include spillover effects forecast crashes and the VaR significantly more accurately than

the models without these effects. Copyright © 2016 John Wiley & Sons, Ltd..

KEY WORDS Hawkes processes; extremal dependence; value-at-risk; financial crashes; spill-over

INTRODUCTION

We develop Hawkes models in which events are triggered through self- as well as cross-excitement. Exploiting the

cross-sectional dependence between financial series, we aim to improve the forecasts of extremes and their sizes.

It has already been shown that Hawkes models contribute to the identification and prediction of extreme events

in financial markets.1Currently, research in finance pays much attention to the modeling of extremal dependence

between financial markets, though with an in-sample focus. We extend these studies on contagion, as we examine

whether incorporating this dependence improves forecasts. For a wide range of assets, we find that Hawkes models

with spillover effects forecast the probability of crashes and the value at risk (VaR) significantly more accurately than

models without.

Traders, regulators of financial markets and risk management benefit greatly from the accurate forecasts of

extreme price movements in financial markets. Nowadays,a large literature focuses on modeling extremal dependence

between financial markets.2This topic gained interest since the financial crisis of 2008 as this crisis demonstrated the

consequences of the cohesion between the behavior of the prices in the financial markets. For example, on 29 Septem-

ber, 15 October and 1 December in 2008 the S&P 500, the Dow Jones Industrial Average (DJI) and the NASDAQ all

suffered top 20 percentage losses. Furthermore, on 29 September the euro/dollar rate and the pound/dollar rate also

dropped by a large amount, while the US bond market boomed. On 16 October, just 1 day after the major US stock

markets crashed, and on the 1 December both currencies fell again sharply. Moreover, 4 days after these dates US

bond prices shifted significantly. These joint extremes demonstrate the overlap of periods in which financial markets

are subject to tension with extreme price movements as a result.

A Hawkes model uses an inhomogeneous Poisson process to model the occurrence of events above a certain

threshold. The event rate increases with the arrival of a newevent, whereas the event rate decays with the time passed

since an event. As the probability of events increases when an event occurs, the Hawkes process is called a self-

exciting process. Characteristics typically observed in processes that fit Hawkes models are the clustering of extremes

and serial dependence.

Earthquakes, for which the Hawkes models were originally designed, exhibit clustering behavior in space as well

as in time (Ogata, 1998). Similarities between the behavior of stock market returns around crashes and the dynamics

of earthquake sequences have been noted in the so-called econophysics literature, in which physics models are applied

to economics.3This literature indicates that it is likely that speculative bubbles which lead to crashes in the stock

market, whether or not triggered by an exogenous factor, are caused by the positive herding behavior of investors.

As the positive herding behavior of investors is locally self-enforcing, instability in the financial markets grows

endogenously. Such a self-excitation can also be observed in seismic behavior around earthquake sequences, where

an earthquake usually generates aftershocks, which in turn can generate new aftershocks, and so on. Earthquakes and

stock returns therefore share characteristics typically observable as the clustering of extremes.

Correspondence to: Francine Gresnigt, Econometric Institute, Erasmus School of Economics, Erasmus University Rotterdam, Burgemeester

Oudlaan 50, Room H8-28, PO Box 1738, 3000DR Rotterdam, The Netherlands. E-mail: gresnigt@ese.eur.nl

1See, among others, Bowsher (2002, 2007); Chavez-Demoulin et al. (2005); Hewlett (2006); Bauwens and Hautsch (2009); Herrera and Schipp

(2009); Embrechts et al. (2011); Chavez-Demoulin and McGill (2012); Bormetti et al. (2013); Aït-Sahalia et al. (2014, 2015); Gourieroux et al.

(2014); Grothe et al. (2014); Gresnigt et al. (2015a).

2See, among others, Longin and Solnik (2001); Poon et al. (2003, 2004); Bekaert et al. (2011); Grothe et al. (2014); Aït-Sahalia et al. (2015).

3See, among others, Sornette (2003); Weber et al. (2007); Petersen et al. (2010); Baldovin et al. (2011, 2012a,b); Bormetti et al. (2013).

Copyright © 2016 John Wiley & Sons, Ltd.

Exploiting Spillovers to Forecast Crashes 937

Like earthquake sequences, financial series seem to cluster in a dimension other than the time dimension.4Extreme

stock returns across markets are found to be more correlated than small returns (Bae et al., 2003). They occur more

frequently at the same time than expected under the assumption of a normal dependence structure (Mashal and

Zeevi, 2002; Hartmann et al. 2004; Sun et al. 2009). This suggests that financial markets experience stress at the

same time. For example, using the multivariate generalized autoregressive conditional heteroskedasticity (MGARCH)

framework, volatility spillover effects between stock markets have been detected in numerous studies.5Interpreting

volatility as a measure for the tension, these findings indicate that stress from financial markets pours over into other

financial markets.

In this paper we extend Hawkes models to include contagion in financial markets. In the models we allow extreme

events in one financial market to trigger both the occurrence and the magnitude of extreme events in other markets.

Studies on Hawkes models for financial series in a multivariate setting include Bormetti et al. (2013), Grothe et al.

(2014) and Aït-Sahalia et al. (2015). Unlike these and other in-sample studies on financial contagion, we explicitly

examine the effects of cross-excitation on out-of-sample forecasts. We assess its added value for the probability

forecast of an extreme event and for VaR. Thereby we extend Chavez-Demoulin et al. (2005), Herrera and Schipp

(2009) and Chavez-Demoulin and McGill (2012), who showed using VaR and expected shortfall that univariate

Hawkes models contribute to the modeling and prediction of risk in finance. Moreover, Byström (2004) finds that

conditional models based on extreme value theory give particularly accurate VaR measures, which are superior to

traditional approaches such as GARCH for both standard and more extreme VaR quantiles.

In somewhat more detail, we use the Lagrange multiplier (LM) principle (see Breusch and Pagan, 1980; Engle,

1982; Hamilton, 1996) as in Gresnigt et al. (2015b) to test whether the spillovers contribute to the model fit. The

correctness of the model specifications is further evaluated by means of the residual analysis methods as proposed in

Ogata (1988). We assess the quality of the probability forecasts by the quadratic and log probability scores and the test

of Diebold and Mariano (1995) and with an asymmetric loss function as proposed by Van Dijk and Franses (2003).

For the evaluation of the VaR forecasts we use the unconditional coverage, independence and conditional coverage

test of Christoffersen (1998), a dynamic quantile test in the line of Engle and Manganelli (2004) and the test designed

by Diebold and Mariano (1995) based on an asymmetric loss function as in Giacomini and Komunjer (2005).

We apply the models to extreme losses in three stock markets and to extreme losses and gains in the US bond

market and two exchange rates. Hence, for the stock market our analysis focuses on long positions, whereas we

consider both long and short positions for bond and FX markets. We forecast the probability of crashes and the VaR

over a period that includes the financial crisis of 2008 using a moving window. Estimating the models, we find that

on average the LM tests reject the absence of cross-excitation. Performing residual analysis, the fit of the models for

the various series is good. Out-of-sample, models with spillover effects provide significantly more accurate forecasts

of the occurrence of an extreme return and of the VaR than the models without spillover effects for almost all series.

We conclude that including cross-sectional dependence improves the forecasts of crashes, and hence cross-sectional

dependence should not be ignored.

The rest of our paper is organized as follows. In the next section we give a brief introduction on Hawkes models.

Furthermore, we propose an extension of the univariate Hawkes model, which incorporates cross-excitation, and we

discuss an LM test for dependence across series. In the third section we apply the models and the LM test to US

stocks, bonds and exchange rate data. The fourth section contains an in-sample assessment of the models by means

of a residual analysis. The models are evaluated out-of-sample on the basis of the prediction of the probability of an

extreme and the VaR in the fifth section. The sixth section concludes.

HAWKES MODELS

Univariate model

The Hawkes model is a branching model. Each event can trigger subsequent events, and these can again trigger sub-

sequent events. The model is based on the mutually self-exciting Hawkes point process, which is an inhomogeneous

Poisson process. For the Hawkes process, the intensity at which events arrive at time tdepends on the history of

events prior to time t.

Consider an event process .t1;m

1/, ..., .tN;m

N/,wheretidefines the time and mithe mark of event i.Let

HtD¹.ti;m

i/Wti

ºrepresent the entire history of events up to time t. The conditional intensity of jump arrivals

following a Hawkes process is represented by

.tjÂIHt/DC1X

iWti

g.t ti;m

i/(1)

4See, among others, Eun and Shim (1989); Fischer and Palasvirta (1990); King and Wadhwani (1990); Lin et al. (1994); Connolly and Wang

(2003).

5See, among others, Hamao et al. (1990); Bae and Karolyi (1994); Koutmos and Booth (1995); Booth et al. (1997); Kanas (1998).

Copyright © 2016 John Wiley & Sons, Ltd. J. Forecast. 36, 936–955 (2017)