Title: Spillover between US equity market and the Euro area during bull and bear market
1. Introduction
In the recent contribution to already a vast literature investigating predictability of excess stock market return (e.g., see Fama and Schwert 1977; Ang and Bekaert 2007; Campbell and Thompson 2008; Rapach, Strauss, and Zhou 2010; Zhu 2013),Rapach et al (2013) propose that stock returns of the US equity market have the predictive power for forecasting stock returns in non-US equity markets. The main reason is that the US equity market takes a world- leading role towards the global equity markets. Because of information frictions, it takes time for non-US equity markets to change according to information gradually spreading from the US equity market.
Though compliant with the most of relevant literature emphasizing that the excess stock returns are predictable, this finding contrasts those reported in Welch and Goyal (2008). Welch and Goyal (2008) conducts a comprehensive exercise testing most popular variables that were proposed earlier in the literature to be good predictors of the excess stock returns. In particular, Welch and Goyal (2008) summarize their findings stating that ‘. . . most models seem unstable or even spurious’.
There are a body of evidences which show that the degree of spillover effect between two markets varies over time (Badhani 2009, Pescetto and Violaris 2003, Savva et al.2009, among others) N Badhani (2009) argues that the change of spillover effect through time can be divided into two different frameworks. One framework is known as “sign asymmetry” or “news asymmetry”. In other words, different types of information can effect the level of spillover. Antoniou, Pescetto and Violaris (2003) find that bad news has larger effect on the stock markets than the good ones. Savva, Osborn and Gill (2009) show that negative shocks have greater effects on equity market than positive ones. The other one is called “phase asymmetry”.Researchers find that the spillover effect increases during the bear market and decreases in the bull market. Longin and Solnik (2001) find that the correlation of equity markets increases when one economy experiences crisis. Bekaert et al. (2005) find contagion during crisis periods and find time variation in world and regional market integration.
Some studies also analyze the interaction between the two spillover frameworks. Most of them conclude that bad news increases volatility more in bull markets than in bear markets, while there is no difference between the volatility effects of good news in bad and good times.(see, for example, Veronesi, 1999; Hameed and Kusnadi, 2006 and Laakkonen and Lanne, 2008).
The above mentioned asymmetry has motivated a large number of different asymmetric volatility models. Koutmos and Booth (1995) using a multivariate EGARCH model to observe an asymmetry in volatility transmission among New York, Tokyo and London stock exchanges. Verma and Verma (2005) using the VAR model to analyze the response asymmetry of Latin American equity market returns to the US market upturns and downturns. Singh et al. (2008) use VAR and GARCH (BEKK) models to examine the price and volatility spillover between India and other markets. Yaya, Gil-Alana and Shittu (2015) use GJR-GARCH model to analyze
asymmetry of individual stock index. However, to the best of my knowledge, the literature to date has not used HAR model to analyze the change of spillover effect from the US equity market to the euro aera in the bull and bear market phase. My study aims to fill this gap
My study is the extend of the work of Buncic and Gisler (2016). Using the augmented HAR model of Buncic and Gisler (2016), I test whether the predictive power of the US equity market to the equity markets of the Euro area change or not during the bull and bear markets.
The remainder of the paper is organized as follows. In Section 2, I describe methodology. The methodology covers two main parts. One part outlines two dating algorithms to identify the bear and bull market phases. The other part presents theoretical framework and empirical model of realized volatility. I choose HAR model of as the standard empirical model Corsi (2009) and the augmented HAR model of Buncic and Gisler (2016). The data that I use in the study are described in detail in Section 3. In Section 4, I use in-sample and out-of-sample tests to evaluate the importance of US-based volatility information for the forest of volatility in the Euro area stock markets Section 5 presents an analysis of the robustness of findings. Lastly, I conclude the study.
2. Methodology
⚫ Dating of bull and bear market states
Although there is a common consensus that if the stock price continues rising during a period, the stock market is regarded as the bull market; if stock price broadly falls in a period of time, the stock market is the bear market. Unfortunately, there is no standard approach to define the bull and bear market. When considering the dating of the bull and bear market, researchers can be divided into two groups. One group define the bull or bear market based on the cumulative price changes. They argue that when the stock market price increases (decreases) significantly, the stock market experiences the bull (bear) market phrase. For example, if the movement of the stock price is more than 20% to 25% from its previous local peak or trough, the phrase is defined as the bull or bear market phrase. (Pagan and Sossounov 2003, Lunde and Timmermann 2004, see among others). The other group insist that the important determinant of the bull or bear market is the time span. For instance, for a period the stock price has a broad increase (decline) trend more than 5 months, the market can be deemed as the bull (bear) market (Hanna 2018, Li and Zakamulin 2020, among others). The lack of a formal definition means that conclusions of related research exist some uncertainty. To obtain reliable results, great care should be taken to define and identify business cycles.
In my study, I employ two frequently cited dating algorithms: The one is the Lunde and Timmermann dating algorithm which emphasizes cumulative price changes. The other one is the Pagan and Sossounov dating algorithm which focuses on measuring the length of bull or bear market phase.
Pagan and Sossounov(PS) dating algorithm:
The algorithm of Pagan and Sossounov (2003) is a method to seek out the peak or trough of stock price with predetermined size of window. The PS algorithm modifies slightly the formal dating method which is proposed by Bry and Boschan (1971) (BB) with the Dow theory. The
BB program can be broadly divided into two steps: First, using a criterion to find initial turning points which can divide the market phase into the bull and bear market; Second, duration between these turning points need to be measured and to be censored by using a sequence of rules. The main purpose of these rules is to restrict the minimum length of the individual (the bull or bear) market phase and the whole business cycle. Considering the inaccuracy of smoothing two market phase and the classical Dow theory, the PS algorithm modifies the length of the market phase. In the first step, ignoring the smooth of any market phase, Pagan and Sossounov (2003) change 6 months into 8 months as the minimum size of window of individual market phase. In the second step, motivated by the Dow theory, Pagan and Sossounov (2003) remove complete business cycles whose length are less than 16 months. For individual market phases, unless the stock price changes over 20%, these less than 4- month phase should been removed. The whole method can be summarized as follows:
Step one (finding initial turning points):
(a) Regard 8 months length as the standard size of window for individual market phase.
(b) Seek out potential peaks and troughs which are higher and lower than a window of surrounding points.
Step two (censoring operations):
(a) Eliminate business cycles less than 16 months
(b) Eliminate market phases less than 4 months unless the accumulative price change exceeds 20%
Lunde and Timmermann (LT) dating algorithm:
This approach has no duration constraints, it imposes a minimum on the price change. First, set two standard market movement thresholds 𝜆𝑏𝑒𝑎𝑟 and 𝜆𝑏𝑢𝑙𝑙, where 1> 𝜆𝑏𝑒𝑎𝑟>0, 1> 𝜆𝑏𝑢𝑙𝑙>0. Second, choose an initial state phase and turning point. Third, while in the bull phase, the potential stock price peak can exceed current maximum accumulative price change. Once the fall of stock price exceeds the standard threshold 𝜆𝑏𝑒𝑎𝑟, the bear phase will be triggered. All intermediate points need to be allocated into the bear phase. Forth, while in the bear phase, the stock price is allowed to fall down to create a new lowest trough. This situation can extend the current bear phase. When the change of the stock price is more than the standard threshold 𝜆𝑏𝑢𝑙𝑙 in the bear market. The bear market phase immediately switches to the bull market phase. When setting 𝜆𝑏𝑒𝑎𝑟 and 𝜆𝑏𝑢𝑙𝑙,Lunde and Timmermann consider two kind of threshold. The one is symmetric threshold (𝜆𝑏𝑒𝑎𝑟 = 𝜆𝑏𝑢𝑙𝑙=15%). The other one is non- symmetric threshold (𝜆𝑏𝑢𝑙𝑙 = 𝟐𝟎%>𝟏𝟎% = 𝜆𝑏𝑒𝑎𝑟)
⚫ Modelling realized volatility (RV)
Let 𝑝𝑡 denote the logarithmic (log) price of a given asset at time t, suppose 𝑝𝑡 follows a
continuous time diffusion:
𝑑𝑝 = 𝜇 𝑑 + 𝜎 𝑑𝑊 (1) 𝑡𝑡𝑡𝑡𝑡
Where 𝜇𝑡 is a predictable drift component, 𝜎𝑡 is an instantaneous volatility process, and 𝑊 is a standard Brownian motion.
𝑡
The quadratic variation (QV) process of 𝑝𝑡 is given by:
𝑄𝑉 =∫𝑡𝜎2𝑑𝑠 (2) 𝑡0𝑠
The term ∫𝑡 𝜎2𝑑𝑠 is the integrated variance (IV) of the process 𝑝 . 0𝑠𝑡
According to definition, the simplest consistent estimator of QV is the realized volatility(RV). Suppose that on a given trading day, I partition the interval [0, t] and define the grid of observation times. The subintervals between two adjacent observation times are equidistant. The observed object is the logarithmic (log) price of any given asset. The classical RV estimator in Eq.(2) is definded as:
𝑅𝑉=∑𝑚 𝑟2 (3) 𝑡 𝑖=1 𝑡,𝑖
And its square root √𝑅𝑉 is the realized volatility (RV). 𝑡
Several salient properties of realized volatility have been identified in the literature:
(1) the unconditional distribution of daily returns exhibits excess kurtosis;
(2) daily returns are not autocorrelated (except for the first order, in some cases);
(3) daily returns that are standardized by the realized variance measure are almost Gaussian; (4) the unconditional distribution of realized variance and volatility is distinctly non-normal, and is extremely right-skewed;
(5) realized volatility does not seem to have a unit root, but there is strong evidence of fractional integration.
On the other hand, the natural logarithm of the volatility has the following empirical regularities:
(1) the logarithm of realized volatility is close to normal;
(2) the logarithm of realized volatility displays long-range dependence.
⚫ Empirical volatility model
HAR model
The Heterogeneous Autoregressive (HAR) model was proposed by Corsi (2009) as an alternative to model and forecast realized volatilities, and is inspired by the Heterogeneous Market Hypothesis of Müller et al. (1993) and the asymmetric propagation of volatility between long and short horizons. The main point of the Heterogeneous Market Hypothesis is that market participants can be divided into different groups, each group has different time horizons in analysis of past events and news and in trading goals. Corsi (2009) defines the partial volatility as the volatility generated by a certain market component, and the model is an additive cascade of different partial volatilities (generated by the actions of different types of market participants). At each level of the cascade (or time scale), the unobserved volatility process is assumed to be a function of the past volatility at the same time scale and the expectation of the next period values of the longer term partial volatilities (due to the
asymmetric propagation
1 ∑5 𝑙𝑜𝑔𝑅𝑉 , 𝑙𝑜𝑔𝑅𝑉(𝑚) = 5 𝑖=1 𝑡+1−𝑖 𝑡
of
1 ∑22 22 𝑖=1
volatility). Let 𝑙𝑜𝑔𝑅𝑉(𝑑) = 𝑙𝑜𝑔𝑅𝑉 , 𝑙𝑜𝑔𝑅𝑉(𝑤) = 𝑡𝑡𝑡
𝑙𝑜𝑔𝑅𝑉 be the daily, weekly, and monthly HAR 𝑡+1−𝑖
components. The HAR model is then defined as
𝑙𝑜𝑔𝑅𝑉 =𝑏 +𝑏(𝑑)𝑜𝑔𝑅𝑉(𝑑)+𝑏(𝑤)𝑜𝑔𝑅𝑉(𝑤)+𝑏(𝑚)𝑜𝑔𝑅𝑉(𝑚)+𝜖 (4)
𝑡+10 𝑡 𝑡 𝑡𝑡+1 Where 𝜖𝑡+1 is an innovation term.
I use the HAR model of Corsi (2009) as the benchmark RV model for the Euro area stock market and the augmented HAR model modified by Buncic and Gisler (2016) to examine the predictive power of the US equity market to the Euro area.
𝑙𝑜𝑔𝑅𝑉 =【𝑏 +𝑏(𝑑)𝑙𝑜𝑔𝑅𝑉(𝑑)+𝑏(𝑤)𝑙𝑜𝑔𝑅𝑉(𝑤)+𝑏(𝑚)𝑙𝑜𝑔𝑅𝑉(𝑚)】+【𝑏(𝑑)𝑙𝑜𝑔𝑅𝑉(𝑑) 𝑡+1 0 𝑡 𝑡 𝑡 𝑉𝐼𝑋 𝑡
+ 𝑏(𝑤)𝑙𝑜𝑔𝑅𝑉(𝑤) + 𝑏(𝑚)𝑙𝑜𝑔𝑅𝑉(𝑚)】 + 【𝑏(𝑑)𝑙𝑜𝑔𝑅𝑉(𝑑) + 𝑏(𝑤)𝑙𝑜𝑔𝑅𝑉(𝑤) 𝑉𝐼𝑋 𝑡 𝑉𝐼𝑋 𝑡 𝑈𝑆 𝑡 𝑈𝑆 𝑡
+ 𝑏(𝑚)𝑙𝑜𝑔𝑅𝑉(𝑚)】 + 𝜖
𝑈𝑆 𝑡 𝑡+1
The part 𝑏(𝑑)𝑙𝑜𝑔𝑅𝑉(𝑑) + 𝑏(𝑤)𝑙𝑜𝑔𝑅𝑉(𝑤) + 𝑏(𝑚)𝑙𝑜𝑔𝑅𝑉(𝑚) presents benchmark (local) HAR 𝑡𝑡𝑡
components of the Euro area equity market; The rest two parts both show the US volatility information.
3. Data
I obtain daily volatility data from the publicly available Oxford-Man Institute’s Quantitative Finance Realized Library of Heber, Lunde, Shephard, and Sheppard (2009). The Oxford-Man Realized Library uses five-minute frequency data from Reuters DataScope Tick History to generate daily realized measures of the asset price variability. The library contains realized measures for US and the euro area equity price indices from January 3, 2000, to the present. The sample ends on June 1, 2020.
The five-minute frequency is a trade-off between accuracy, which is theoretically optimized using the highest possible frequency, and microstructure noise that can arise through the bid- ask bounce, asynchronous trading, infrequent trading, and price discreteness, among other factors (see Madhavan, 2000; Biais et al., 2005 for very useful surveys on this issue).
I choose realized measures data for the US and the euro area equity markets that are included in the Oxford-Man Realized Library. These are the S&P 500 (the US) and the Euro STOXX 50 (Euro area).
For the US equity market, I choose the S&P 500 to represent the US equity market. The VIX is an implied volatility index derived from a number of put and call options on the S&P 500 index with maturities close to the target of 22 trading days and is derived without reference to a restrictive option pricing model. The VIX is constructed to be a general measure of the market’s estimate of average S&P 500 volatility over the subsequent 22 trading days (Blair et al., 2001; Christensen and Prabhala, 1998). , In an empirical research, Grassi et al. (2014) documented that the VIX has some predictive power for S&P 500 realized volatility forecasts. The VIX data that I include in the augmented HAR model in Eq. (5) can be obtained from Yahoo Finance. I choose the stock index the Euro STOXX 50 to show the performance of the euro area stock market.
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