Uncovering asymmetries in the relationship between fear and the stock market using a hidden co-integration approach

Accepted Manuscript Title: Uncovering asymmetries in the relationship between fear and the stock market using a hidden co-integration approach Authors: Fotini Economou, Yannis Panagopoulos, Ekaterini Tsouma PII: DOI: Reference:

S0275-5319(17)30418-X http://dx.doi.org/doi:10.1016/j.ribaf.2017.07.116 RIBAF 806

To appear in:

Research in International Business and Finance

Received date: Accepted date:

22-6-2017 5-7-2017

Please cite this article as: Economou, Fotini, Panagopoulos, Yannis, Tsouma, Ekaterini, Uncovering asymmetries in the relationship between fear and the stock market using a hidden co-integration approach.Research in International Business and Finance http://dx.doi.org/10.1016/j.ribaf.2017.07.116 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Uncovering asymmetries in the relationship between fear and the stock market using a hidden co-integration approach

Fotini Economou1,*, Yannis Panagopoulos2 and Ekaterini Tsouma3

1 2

Centre of Planning and Economic Research (KEPE), 11, Amerikis str., 106 72 Athens, Greece, . Centre of Planning and Economic Research (KEPE) , email: [email protected].

3

Centre of Planning and Economic Research (KEPE), email: [email protected].

*

Corresponding author: Tel. +302103676418, email: [email protected]

Abstract In this paper we implement the methodological approach of hidden co-integration in order to examine the relationship between the fear index and the stock market index. To this end, we employ daily data of the S&P 500, FTSE 100 and DAX 30 stock market indices and their respective implied volatility indices, i.e. CBOE VIX, FTSE Volatility Index and VDAX New Volatility Index, for the 2000-2014 period. Our empirical results indicate overall asymmetry in the reaction of the fear indicator to stock market innovations for the US market. For the UK and Germany our findings suggest the existence of specific types of asymmetry concerning mainly the size and the time span of the adjustment process. The findings have important implications for asset allocation, active investment and hedging strategies.

Keywords: fear index; stock market index; hidden co-integration; asymmetry. JEL Classification: G12, G15

1. Introduction According to Read (2009), “fear is an essential animal emotion”, which is associated with anxiety or aggression that evolved in order to serve survival. When it comes to investments, fear is associated with risk, i.e. uncertainty regarding an outcome and lack of control, and may 1

lead to emotional responses that cannot be adequately predicted or modeled. Understanding investors’ sentiment is of particular importance, more so since it has been found to increase the probability of stock market crises (Schmeling, 2009; Zouaoui et al., 2011). Even though it is difficult to capture or measure emotions, there are several proxies for investors’ sentiment, being either direct when derived from surveys or indirect when the employed variables are highly correlated with investors’ sentiment (Brown and Cliff, 2004; Baker and Wurgler, 2007). The Chicago Board Options Exchange implied volatility index (CBOE VIX) is generally recognized as the “investor fear gauge”, i.e. a measure of fear (Whaley, 2000), and its response to the stock market returns has been closely examined.1 Empirical evidence has documented a negative asymmetric return-volatility relationship which refers to the fact that negative stock market returns are associated with greater positive changes in volatility compared to the negative volatility changes that occur when stock market returns increase (Giot, 2005). In other words, investors are more sensitive to market losses than market gains.2 According to Low (2004), this asymmetry indicates another form of loss aversion. Extreme downward price movements are associated with rapidly increasing risk, while extreme upward price movements are associated with modest risk reduction. As a result, fear is striking quickly in down markets, while market exuberance is slowly building in up markets. Kling and Gao (2008) also confirm that the reaction of institutional investors’ sentiment to negative market returns is more sensitive compared to positive returns. This finding is not surprising taking into consideration loss aversion and Prospect Theory (Kahneman and Tversky, 1979). Moreover, Camerer (2005) assumes that loss-aversion is “often an exaggerated emotional fear reaction”, while Lopes (1987), studying risk-taking behavior between hope and fear, states that “fear feeds risk-aversion”.

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Recent literature confirms the appropriateness of VIX as an investor fear gauge indicating that it may serve as a fear measure for other equity markets as well (Sarwar, 2012). 2 It has to be mentioned that the reverse relationship also holds. Durand et al. (2011) document two-way causality between VIX and risk premia (market risk premium and Fama-French (1993) factors) and Smales (2016) documents a two-way causality between VIX and stock market returns employing vector autoregression analysis.

2

This paper contributes and, hence, enriches the existing literature by implementing a methodological approach that was recently employed in other economic contexts3, in order to examine the asymmetric relationship between the fear index and the stock market index. The economic intuition behind this approach is that the relation between some economic variables might not be the same whenever they increase or decrease (Schorderet, 2003). The particular advantage of this approach lies in the use of positive and negative decomposed fear and stock market indexes that may allow us to test for the existence and the type of asymmetric relations. We examine three developed stock markets –the US, the UK and Germany– that offer an interesting setting for analysis providing comparable implied volatility indices. Apart from that, our choice is motivated by the prominent role of the US market and the aim to conduct a comparative analysis with the most important markets in continental Europe. To this end we use daily data of the S&P 500, FTSE 100 and DAX 30 stock market indices and their respective implied volatility indices i.e. CBOE VIX index, FTSE Volatility Index and VDAX New Volatility Index for the 2000-2014 period. Most studies attribute the estimated negative relationship either to the leverage effect or to the volatility feedback hypothesis. In the first case, as the firm’s value falls its equity value becomes a larger share of the total value and the volatility of equity is expected to increase (Black, 1976), while in the second case positive volatility shocks cause negative returns (Poterba and Summers, 1986; Campbell and Hentschel, 1992). Aboura and Chevallier (2016) examined the time-varying leverage and feedback effects in the US market. According to their empirical results, the dynamic leverage effect has a significant impact on the equity markets, while the volatility of volatility has a significant impact only on the dynamic feedback effect. Recent literature mostly focuses on the implied volatility employing the VIX index for the US market. Several studies have tried to explain the negative asymmetric impact of the stock 3

The hidden co-integration approach has been employed to examine the relationship between unemployment and output (Koutroulis et al., 2016), wholesale and retail interest rates (Panagopoulos and Spiliotis, 2015), energy production and economic growth (Tiwari et al., 2015), stock prices and mutual fund units (Alexakis et al., 2013), government spending and terms of trade volatility (Hatemi and Irandoust, 2012).

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market returns on the VIX index providing a behavioral explanation. Low (2004) was the first to provide a behavioral explanation of loss-aversion based on Kahneman and Tversky (1979). Hibbert et al. (2008) extend Low’s (2004) research. They examine the relationship between CBOE VIX (NASDAQ volatility index) changes and S&P 500 (NASDAQ) returns on a daily and intraday basis for the 1998-2006 (2001-2006) period. The authors indicate the superiority of the VIX index compared to alternative volatility measures documenting a significantly negative asymmetric relationship that can be better explained by the behavioral explanations they provide based on the representativeness heuristic, the affect heuristic, and extrapolation. Moreover, Park (2011) attributes the asymmetric relationship to the asymmetric herding behavior in the stock market, providing an alternative behavioral explanation. According to Smales (2014, 2016), VIX is also negatively related to news sentiment with this relationship being asymmetric and more intense after the release of negative news. In the same lines, Badshah (2013) confirms the superiority of the behavioral explanations (affect and representativeness heuristics) to fundamental explanations. Testing for different implied volatility indices in different markets, the author identifies that the VIX displays the highest asymmetric return-volatility relationship with its respective stock market index (compared to VSTOXX for the DOW JONES EURO STOXX 50 index; VDAX for the DAX 30 index; and VXN for the NASDAQ 100 index). Siriopoulos and Fassas (2008) also document a negative asymmetric relationship between VFTSE and FTSE-100. On the other hand, Gonzalez and Novales (2009) identify a strong negative contemporaneous relationship between VDAXNEW and DAX 30 index which is not asymmetric. Dichtl and Drobetz (2012) conclude on a negative VDAX-DAX relation, being more pronounced between negative DAX returns and the returns on VDAX, than in the case of positive DAX returns. Tallau (2012) also finds a stronger rise in VDAX following negative DAX returns, as compared to the fall in VDAX resulting from positive DAX returns. Finally, Velev et al. (2016) examine 12 stock indices and their respective implied volatility indices from the US, European Union and Asia Pacific 4

and document an asymmetric short-term volatility behavior in the case of negative price changes. However, this asymmetry disappears in the long-run in line with the idea of long term market equilibrium.Our empirical results indicate overall asymmetry in the reaction of the fear indicator to stock market innovations for the US market. For the UK and Germany our findings suggest the existence of specific types of asymmetry concerning mainly the size and the time span of the adjustment process. Emotions such as greed and fear may significantly affect stock market prices (Westerhoff, 2004). The examination of fear in the stock market can shed light into investors’ sentiment during up and down market days, especially during periods of market crises when investors are more likely to fall prey to their emotions. According to Chira et al. (2013), increases in market fear (implied market volatility) have a significant impact on bank stocks’ returns during the global financial crisis. Moreover, Galariotis et al. (2016a) document that investors’ sentiment was an important determinant for Credit Default Swaps spread during the subprime crisis with spill-over effects. As a result, our findings have important implications for asset allocation, active investment and hedging strategies. The rest of the paper is structured as follows: Section 2 presents the dataset and the methodology employed in order to test for the relationship between the fear and the stock market index, Section 3 presents and discusses the empirical results and Section 4 concludes.

2. Data and methodology 2.1 Data In order to examine the asymmetric relationship between the fear and the stock market index, we employ daily data of the S&P 500, FTSE 100 and DAX 30 stock market indices and their respective implied volatility indices i.e. CBOE VIX, FTSE Volatility Index and VDAX New

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Volatility Index, for the 2000-2014 period.4 All data series were obtained from the Thomson Reuters Datastream database. The employed implied volatility indices are all constructed based on the CBOE VIX methodology and are publically available.5 Figures 1-3 present the evolution of the variables in both raw and first difference form, as well as their cumulative positive/negative changes for the US, the UK and the German stock markets, while Table 1 reports the relevant descriptive statistics for the whole sample period. Panel A reports the results for the USA, Panel B for the UK and Panel C for Germany. There are 3,771 daily observations for the US, 3,817 observations for the UK and 3,813 for the German market. A closer look at Figures 1-3 indicates that the European markets under examination are more correlated to each other than to the US market. 6 It is also evident that periods with decreasing stock market prices are associated with increasing implied volatility. Moreover, the respective descriptive statistics indicate higher standard deviation of the VIX changes compared to the European markets’ indices. Our dataset covers three developed stock markets. However, there are several differences between the US and the two European markets in their regulatory framework in the financial markets. Gałkiewicz (2015) identifies several differences between the US and the German/EU markets in the regulation of the use of derivatives and leverage by funds. According to the author, the US fund regulation regarding the use of direct leverage is less strict compared to the respective German/EU regulation. Moreover, there are several differences in the shortsales regimes. According to Beber and Pagano (2013), regulators reacted differently to the 2007-2009 crisis regarding short selling bans and restrictions failing to support stock prices, with the exception of the US market. In the same spirit, Elineau (2011) identifies differences

4

The selection of the starting date of our sample was based on data availability of the relevant implied volatility index for the UK stock market. 5 It has to be mentioned though that VFTSE, launched in June 2008 and VDAX-NEW in April 2005. However, historical daily values are available from January 2000 for both indices. 6 In fact, the correlation between the US market returns and the European markets returns is 0.26 and 0.19 for the German and the UK markets respectively, while UK and German returns are highly correlated (+0.64). The correlation of the percentage change of the implied volatility indices is also higher between the European markets (+0.72) compared to the US market (0.42 with the German and 0.34 with the UK market).

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between the US and European markets regarding short selling regulation after the global financial crisis. Differences also exist in the trading procedures that may also affect asset pricing (see Galariotis et al., 2016b). Finally, even though the implied volatility derivatives have attracted investors’ interest over the past few years, the US and the European markets display differences in their variety and trading volume. In fact, there is a wide variety of products linked to the CBOE VIX index, such as VIX futures, options, and exchange traded products, that make volatility trading accessible to a broader range of investors in order to form hedging strategies (Park, 2016).

2.2 Hidden cointegration methodology 2.2.1 Hypotheses and CECMs Various methodologies have been applied in order to investigate the relationship between the fear and the stock market indexes.7 In this paper we apply the Granger and Yoon (G&Y) (2002) hidden co-integration (HC) approach, which seems to be well suited for a deeper investigation of the way in which fear and the stock market respond to shocks and, hence, of their dynamic relationship. The main advantage of implementing the G&Y HC methodology is that it allows for the existence of distinct co-integrating relationships between subcomponents of time series. This applies even when co-integration between aggregate time series is not identified. HC, as a simple example of non-linear cointegration makes it, hence, possible to investigate long-run relationships among disaggregated data series, revealing valuable information and hidden structure in the components of the underlying aggregate data series, which would be otherwise lost or unnoticed.

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Applied methodologies include regression analysis (e.g. Low (2004) uses two- and four-partition asymmetric regressions, Giot (2005) estimates regressions including dummy variables to highlight the effect of positive and negative returns, Hibbert et al. (2008) apply different regression models, and in particular models with returns segregated into positive and negative changes, Whaley (2009) applies regression analysis including a return term conditional on the market going down, Smales (2016) estimates an OLS specification including an additional term conditional on the market declining), quantile-regression models (Badshah (2013)), and GARCH processes (Velev et al. (2016)), to name a few.

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In other words, it enables the investigation of potential different degrees of downward or upward rigidities in the aggregate series, by testing for cointegrating relationships among their non-stationary components. These are given by the cumulative sums of positive and negative changes in the aggregate series. If any of these components are found to be co-integrated, then the data series are said to be hidden co-integrated. Suppose X t and Yt are two random walk time series, with: X t  X t 1   t = X 0  i 1  i

(1)

Yt  Yt 1   t = Y0  i 1 i

(2)

t

t

where, X 0 , Y 0 denote initial values and  i and  i mean zero white noise disturbance terms. 



Positive and negative shocks are then defined as follows:  i  max(  i , d ) ,  i  min( i , d ) ,

i   max( i , d ) and i   min(i , d ) , where d stands for an unknown threshold value (with d =0 as the most popular choice). Note that:  i   i   i and i  i  i . The above equations can be transformed to:  _ X t  X t 1   t  X 0  i 1  i  i 1  i , t

t

and

(3)

  Yt  Yt 1  t  Y0  i 1i  i 1i . t

t

(4)

Simplifying to X t  1  i , X t  1  i , Yt   1i , Yt   1i , one obtains t



t





t

t







X t  X 0  X t  X t and Yt  Y0  Yt  Yt , and it follows that X t   t , X t   t , 



Yt   t , and Yt    t . In terms of empirical application, first differences are obtained for both X t and Yt , the observations are sorted according to positive and negative movements ( X t , X  , Yt  , Y  ) and the cumulative sums of positive (and negative) changes at a given time for all variables are calculated ( X t   X t , X t   X t , Yt    Yt , 

Yt    Yt ). 

8





Having obtained the two components of the aggregate series, alternative long-run cointegration hypotheses can be formulated and corresponding Crouching Error Correction Models (CECMs) can be derived, according to G&Y. More specifically, and as outlined also by Honarvar (2009), Alexakis et al. (2013) and Panagopoulos and Spiliotis (2015), the prevailing four alternative long-run hypotheses between the pre-selected pairs of X t and Yt are linked to four different dynamic CECMs implied by the existence or non-existence of HC (Table 2). Recall that, as G&Y notice, CECMs can be considered as standard error correction models, except for the fact that they show long-run equilibrium relationships and short-run dynamics of stationary components of data series, rather than data series themselves.

This feature offers better insights into the asymmetry issue both on the long run and short run level and provides additional data flexibility by allowing for cumulative, positive and/or negative, long-run estimators to be embedded in the CECM structure. In addition, it includes the implementation of a CECM, which is not limited to two regimes. Practically, it allows us to investigate all possible combinations of co-integration between data components (Honarvar, 2009), hence, being richer than the standard asymmetric Error Correction Model (ECM) and more flexible than the Threshold Error Correction Model (TECM)8. Note that the CECM also looks closer to the LSE–Hendry general to specific (GETS)9 approach than to the TECM, since it presupposes an ex-ante disaggregation between positive and negative components of the data and then a cumulative aggregation of these two parts in the Data Generation Process (DGP). 2.2.2 Model coefficients and testing interpretations The above described methodological approach allows specific reference to be made to the estimated coefficients, while it enables the interpretation of testing results, depending on the 8 9

See, for example, Enders and Granger (1998), Enders and Siklos (2001), etc. See, for example, Bachmeier and Griffin (2003) and Rao and Rao (2008), etc.

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valid hypothesis and the corresponding CECMs models. Note, for example, that the CECM model applying to hypothesis 2 includes one error correction term, while the respective one corresponding to hypothesis 3 includes two error correction terms. In addition, on the basis of the distinction between positive and negative components of the examined variables and the resulting distinct coefficients, several comparisons can be conducted and significant results can be derived on the potential existence and type of asymmetry effects. In the present analysis, we define the general long-run relationships underlying the cointegration analysis and the general form of the CECMs corresponding to hypotheses 2 and 3 (Table 3). Starting with the long run, the coefficients on the –aggregate or two decomposed– independent variables, 𝜑 , 𝜑1 and 𝜑2 in the equations underlying the cointegration analysis (equations (8) to (10)) are seen to reflect the positive/negative long-run elasticity among the fear index and the stock market. Under the assumption of cointegration, the size of the longrun elasticity or rigidity describes the extent to which an increase/decrease of the stock market is transmitted to the decrease/increase of fear. In addition, when HC is confirmed in both directions, the obtained long-run elasticities/rigidities using disaggregated (positive and negative) components allow us to implement new rigidity tests, i.e. whether 𝜑1 = 𝜑2 = 1 and compare the implied effects. In the CECMs (equations (11) to (13)), the  0 ,  0,1 and  0, 2 coefficients mirror the instantaneous elasticities between the fear and the stock market indices, accounting for the cotemporaneous effects or short-run adjustments. The  ,  1 and  2 coefficients, namely the coefficients on the error correction terms, measure the speed of reaction of the fear indicator to any positive/negative changes in the stock market, in other words the speed of the medium-run adjustment process. They describe how fast an increase/decrease in the stock market in the previous period will be passed through in the short run in order for the ex ante long-run equilibrium relationship to be restored. Testing, in a next step, whether  1   2 intuitively 10

constitutes a speed symmetry test and allows direct inference to potential distinct medium-run adjustment processes of components, in terms of speed. Given the implementation of hypothesis 3, an additional step for the investigation of potential asymmetries concerns the construction of the two mean adjusted lag operators, which can be derived from equation (13). Analytically, these two operators are defined as  1  (1   0,1 ) / 1 , and  2  (2   0,2 ) /  2 . They measure the time required for the remaining value to be transmitted and the pass through process to be completed, i.e. the value left after the instant pass-through effect of the process. Potential differences (similarities) between the respective calibrated coefficients would indicate asymmetry (symmetry) in the time span of the adjustment process. 3. Model implementation and empirical results 3.1 Co-integration and CECM analysis results In the present section, we implement the HC approach for the cases of the fear and stock market indices of the USA (noted VIX and S&P500), the UK (noted VFTSE and FTSE100) and Germany (noted VDAX and DAX30). In the first stage of the empirical analysis, we conduct ADF unit root tests for the aggregate and decomposed data series, in order to derive the order of integration of the variables. According to the test results, in most cases we cannot reject the hypothesis of a unit root in the level series, with the exception of the aggregate fear indicators VIX, VFTSE and VDAX. When conducting tests for the series in first differences, the unit root hypothesis can be rejected in all cases.10 In concluding that the majority of the underlying series are integrated of order one, the next step is to search for co-integration. Co-integration analysis We investigate the existence of a long-run relation between the underlying variables in aggregate and decomposed form, by applying Engle-Granger co-integration tests. Cointegration and HC test results for all three countries under investigation are offered in Table 10

The detailed results of the conducted unit root tests are available upon request.

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4.11 The provided evidence on co-integration includes some interesting observations, since it signals the existence of both similarities and differences in the results among the examined countries. More specifically, and according to the testing results for the case of the aggregate variables, the hypothesis that the series are not co-integrated can be rejected in the case of the co-integrating equation with the fear indicators, namely VIX, VFTSE and VDAX, as the dependent variables.

In proceeding with the tests for the disaggregated series, as far as the USA is concerned, we uncover evidence of HC in the case of the investigated relation between the positive component of the fear index and the negative component of the stock market index. This means, for example, that a fall in the stock market index would lead to an increase in the fear index in the long-run. In contrast, we do not discover evidence of HC in the relation between the negative component of the fear index and the positive component of the stock market index. As a result, it follows that when the stock market index is increasing it does not share a common stochastic trend with the fear index. In other words, the presented evidence favors the argument of potential different degrees of downward and upward rigidities in the aggregate series. Consequently, this finding, presents a significant first indication of asymmetry, since it appears that there exists a long-run linkage between the positive component of the fear index and the negative component of the stock market index, while the same does not hold in the opposite case. 12 The inferred evidence enforces the argument in favor of the implementation of the HC approach, which enables a deeper analysis of relevant hidden structure. Having established the existence of HC among one pair of the decomposed series, the validity of hypothesis 2 (as presented in Table 2) arises. In other words, since we find evidence on the existence of a long 11

The estimated long-run equations for all the aggregate and disaggregated series are not reported for the sake of brevity but are available upon request. The inclusion of a trend leads to similar results. Note that in all cases, both the negative sign and statistical significance of the coefficients on the stock market indices confirm the kind of relation between fear and stock market as expected by economic theory and evidenced by empirical findings. The relevant coefficients are given in Table 6. 12 Our results are robust when implementing the Schorderet (2003) auxiliary HC model.

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run relation between VIX(+) and S&P500(-), but not between VIX(-) and S&P500(+), it follows that the CECM corresponding to hypothesis 2 applies. Interesting insights are offered by the cointegration analysis with reference to the UK and Germany. In contrast to the results for the US, while we find indications on the existence of co-integration on an aggregate level, no evidence on the existence of HC in either direction is detected when using decomposed data series. As a result, in this first step of our analysis for the UK and Germany, we do not obtain clear-cut evidence on whether the stock market and the fear index are subject to common shocks. However, it seems that if they do, there are (yet) no indications for a more pronounced link between one specific pair of disaggregated components, as opposed to the other, and also no signs implying different degrees of upward or downward rigidities in the aggregate series. Consequently, in order to exhaust the range of all possible combinations we go into a deeper investigation and proceed in the cases of both the UK and Germany with the implementation of the more enriched CECM corresponding to hypothesis 3 (as presented in Table 2). This analysis makes it possible to identify any existing relevant non-linear structure.

Crouching error correction models Since the results of the co-integration/hidden co-integration analysis vary among the countries under investigation and we are bound to proceed with the implementation of different CECMs –relating to the application of distinct hypotheses–, the corresponding evidence is presented separately. Starting with the USA, we implement the relevant dynamic model (corresponding to hypothesis 2) among the alternative CECMs. In its general form, the corresponding CECM, with the differenced positive component of the fear index as the dependent variable, according to G&Y (2002) can be expressed as follows:

VIX (  )  C0  1 * [( res[1])]t 1   0,1 * S&P500 (  )  lags (VIX (  ) , S&P500 (  ) )   t

13

(14)

where the lagged term 𝑟𝑒𝑠[1] denotes residuals from the respective long-run equation (equation (9) in general form). The short-run equation is estimated for ΔVIX(+), including one lagged term of the corresponding residuals, the contemporaneous differenced negative component of the price index and three lags of the relevant differenced disaggregated components. Excluding all non-significant terms, the implementation of the dynamic model leads to the following equation (t-statistics are given in brackets):

VIX (  )  0.02  0.001* (res[1])t 1  0.101* S&P500(  )  0.135 * VIX (  ) t 2 (1.71) (4.89)

(87.56)

(8.40) (15)

 0.003 * S&P500(  ) t 1  0.011* S&P500(  ) t 2   t (2.59)

2

R  0.68

(5.72)

According to the estimated CECM (equation (15)), there is a negative and statistically significant contemporaneous relation, confirming the related well-documented finding in the empirical literature (e.g. Hibbert et al. 2008, Aboura and Wagner 2016, etc.). Furthermore, the coefficient on the error correction term, which indicates the speed of adjustment to the longrun equilibrium, is significant and exhibits the correct sign. Therefore, and according to the intuition in G&Y (2002), it follows that S&P500(-) is a common stochastic trend, responsible for the long-run dynamic behavior of VIX(+).13

We can, hence, once again conclude that the relationship between the fear indicator and the stock index for the case of the USA is not symmetric. In other words, fear seems to respond in an asymmetric way, reacting to negative changes in the stock market index, but not to positive ones. The empirical evidence with respect to the asymmetric response of the VIX index to the stock market is in line with previous literature regarding the US market (Hibbert et al., 2008; Badshah, 2013; Aboura and Wagner, 2016).

13

Note further, that in the equation with the negative component of the fear indicator as the dependent variable, the coefficient of the error correction term is not significant at the 1% or 5% significance level.

14

Proceeding with the analysis for the UK and Germany, the relevant more enriched dynamic CECM (corresponding to hypothesis 3) is applied. Table 5 offers first the respective CECMs in their general form, according to G&Y (2002), with the differenced aggregate fear index as the dependent variable. The models include the relevant residual terms (res[1] and res[2]) obtained from the corresponding long-run equations lagged one period (equations (9) and (10) in general form) the differenced contemporaneous positive and negative components of the price index and three lags of all the differenced disaggregated components. Secondly, the table presents the estimated short-run equations after the elimination of all insignificant terms. Along the lines of the evidence provided in the empirical literature and in accordance with the respective findings for the US, we obtain, for both the UK and Germany, highly significant coefficients on the differenced contemporaneous negative and positive stock market index components. In addition, we obtain significant negative error correction terms in all cases. This important initial finding indicates that the adjustment of the fear index takes place to both positive and negative shocks in the stock market index. There are, however, differences in the degree to which the fear index responds to negative and positive changes in the stock market index. This becomes obvious on the basis of the size of the coefficients on the differenced contemporaneous negative and positive stock market index components and holds for both the UK and Germany. Note, that previous work on the German case by Dichtl and Drobetz (2012) and Tallau (2012), reporting an asymmetric relation between the stock market and the fear index, relies on similar findings based on more/less pronounced coefficients, while Gonzalez and Novales (2009) do not uncover an asymmetric relationship for Germany. The above results offer some significant insights into the way fear responds to shocks in the stock market in the three countries under investigation. However, before drawing any final conclusions on differences and similarities, we resort to the deeper investigation of the findings regarding the estimated coefficients and proceed with their interpretation and the conduction of some tests based on the structure of the relevant models. 15

3.2 Coefficient inference The HC methodological approach allows specific inference to be made based on the estimated coefficients. Their ensuing interpretation and conclusions derived from additional hypothesis testing can significantly enrich the analysis of the relation between the fear index and the stock market index. Recall that they may vary depending on the valid hypothesis and the corresponding CECM models. For example, the CECM model applying to hypothesis 2 includes one error correction term, while the respective one corresponding to hypothesis 3 includes two error correction terms. We summarize the related evidence in Table 6. Starting with the long-run equation coefficients, their relatively small size, alongside with the rejection of the hypothesis that they equal unity, suggests that positive and/or negative changes in the stock market index are not fully transmitted to the fear index. The degree of the detected rigidity appears to be higher in the UK and Germany –as compared to the US– and quite similar between the two countries. Moving on to the short run equation coefficients, it is interesting to observe the significance of the instant effects, which are strongest in the case of the US market, as compared to the respective effects for the UK and Germany, (  0,1 and  0, 2 coefficients). Recall, that due to the detected asymmetry characteristic and the application of hypothesis 2, the single higher coefficient for the US concerns the instantaneous elasticity of the positive component of the fear index to contemporaneous changes in the negative component of the stock market index. For the UK and Germany, the estimated coefficients are significantly lower, indicating less pronounced contemporaneous effects. In other words, the instantaneous response of fear to the stock market is weaker in the UK and Germany, as compared to the US. At the same time, and while the coefficients corresponding to the two pairs of disaggregated variables seem to be quite similar in the cases of the two countries, we uncover some evidence of asymmetry. As already indicated in the above, and based on the size of the respective  0,1 and  0, 2 coefficients, the evidence suggests that the instantaneous elasticity and, thus, the 16

contemporaneous effect of any negative changes in the stock market index is twice as high as the respective one caused by positive changes in the stock market index. This holds for both the UK and Germany. Once again, this seems to be in line with similar findings in the empirical literature, concluding on an asymmetric relation between the stock market and fear in the UK and Germany based on this kind of evidence. To the degree that the fear indicator reacts to any positive/negative changes in the stock market, the size of the  1 and  2 coefficients provides evidence on the speed of adjustment and, hence, on the speed of the medium-run adjustment process. For the USA, any kind of adjustment seems to originate from the response of the positive component of the fear index to changes in the negative component of the stock market index, signaling the characteristic of one-sided rigidity and overall asymmetry in the process of the reaction of the fear index. For the UK and Germany, the corresponding effects seem to be higher in magnitude than in the US. In other words, it takes less time in the UK and Germany for any positive and/or negative change in the stock market in the previous period to be passed through in the short run, in order for the ex ante long-run equilibrium relationship to be restored. At the same time, the practically equal, in size, coefficients for the two countries indicate resemblance concerning the speed of the medium-run adjustment process. An additional point of similarity concerns the results of the applied speed symmetry tests for the UK and Germany. The acceptance of the hypothesis that (𝐻0 : 𝜃1 = 𝜃2 ), in both cases, leads to the conclusion that the fear indicator reacts with the same speed to positive as well as to negative changes in the stock market. Finally, coefficient inference on the calibrated mean adjusted lag operators 𝛾1 and 𝛾2 suggests once again quite similar results for the UK and Germany. At the same time, differences in the magnitude of the two estimated lag operators reveal another type of asymmetry, namely asymmetry in the time span of the adjustment process for both countries. The time required for the remaining value to be transmitted to the fear index following positive changes in the stock market index –and the pass-through process to be completed– 17

spans over two days (𝛾2 = 2 for the UK and 𝛾2 = 2.2 for Germany), hence much longer than the time span required in the case of negative changes. Note that since the latter practically equals zero it follows that, in the case of negative changes in the stock market, there is no value left after the instant effect of the process to be passed through to the fear index for both the UK and Germany. Overall, the coefficient analysis provides significant insights into specific characteristics of the relationship between the fear and the stock market indices. Our findings could be very useful for asset allocation and active investment strategies that are based on the evolution of the implied volatility indices.

4. Concluding remarks Implied volatility indices constitute useful indicators to investigate the asymmetric response of investors’ sentiment to the stock market. In order to contribute to the related scientific discussion, in this paper we examine the relationship between the well-known implied volatility fear indicators and their respective stock market indices for three developed equity markets, i.e. USA, UK, Germany, using daily data from 2000 to 2014. To this end, we employ the methodological approach of HC which allows for a thorough analysis of the relation between fear and stock market indices that may reveal asymmetric relations. Taken as a whole, the application of the HC and CECMs methodology reveals the existence of significant differences in the way the fear index responds to shocks in the stock market between the US and European markets, but also interesting common features between the UK and German markets. In addition, the evidence provided by the implementation of the dynamic short-run relations is consistent with the respective indications offered by the cointegration analysis.

18

We document an overall asymmetric relationship between the fear indicator and the respective stock market innovations for the US market. This is in line with previous literature indicating that the VIX index better captures investors’ fear than their positive sentiment. We uncover evidence of HC between the positive component of the fear index and the negative component of the stock market index. For the UK and Germany we find less pronounced contemporaneous effects, alongside with a shorter medium-run adjustment process, as compared to the US. At the same time, we detect asymmetry in the size of the contemporaneous effects of positive and negative changes in the stock market on fear, and asymmetry in the time span of the adjustment process for both countries. According to the latter, the time required for the remaining value to be transmitted to the fear index following positive changes in the stock market index –and the pass-through process to be completed– spans over two days, being hence much longer than the time span required in the case of negative changes. Previous research has documented a highly asymmetric relation between the implied volatility index and the stock market index, especially in the upper regression quantiles, with the VIX index displaying higher asymmetry compared to VDAX and VFTSE (Badshah, 2013; Aussenegg et al., 2013). The reported differences in the empirical results could be attributed to their special institutional characteristics (regulatory framework, short-selling regulation, trading procedure and variety of implied volatility derivative products). Understanding investors’ sentiment has important practical implications for asset allocation, active trading and hedging strategies. Sentiment indicators, such as the VIX index, are usually employed to form contrarian investment strategies (Simon and Wiggins, 2001). Moreover, VIX futures (since 2004) and options (since 2006) are used in order to hedge stock market volatility and diversify portfolios, especially under market stress conditions. According to Whaley (2009), the S&P 500 index option market is dominated by hedgers looking for insurance against future negative stock market performance. Consequently, the VIX index 19

provides useful information especially during down market periods when uncertainty is higher, and investors are looking for confirmatory signals regarding the evolution of the market.14 It is reasonable to assume that during periods of market distress, i.e. periods of negative market returns, high volatility and trading volume, investors are expected to be more emotional and behavioral biases could be more pronounced. Under extreme market conditions fear prevails in the market and may have a significant impact on decision making resulting in suboptimal investment decisions. Garcia (2013) also claims that sentiment has a more important role during economic downturns, i.e. during periods of fear and anxiety. As a result, the reported asymmetric relationship between the fear index and the stock market could be useful for hedging strategies under market stress conditions that may destabilize international financial markets. Future research should also focus on the possible asymmetric behavior of alternative sentiment indicators in different countries under different market conditions in order to provide useful information for international asset allocation.

14

According to Rizzi (2008), pessimism is prevalent during crisis periods and investors are looking for (bad) news that will confirm their beliefs.

20

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Kling, G., Gao, L. 2008. Chinese institutional investors’ sentiment. Journal of International Financial Markets, Institutions and Money 18, 374-387. Koutroulis, A., Panagopoulos, Y., Tsouma, E. 2016. Is there an asymmetric response in unemployment rate to changes in output? A hidden cointegration approach. Journal of Economic Asymmetries 13, 81-88. Lopes, L.L. 1987. Between hope and fear: The psychology of risk. Advances in Social Psychology 20, 255-295. Low, C. 2004. The Fear and exuberance from implied volatility of S&P 100 index options. The Journal of Business 77, 527-546. Panagopoulos, Υ., Spiliotis, Α. 2015. Reassessing rigidities and asymmetries in the interest rate pass–through process: A hidden co-integration approach. Credit and Capital Markets 48, 477-500. Park, B.J. 2011. Asymmetric herding as a source of asymmetric return volatility. Journal of Banking & Finance 35, 2657-2665. Park, Y.H. 2016. The effects of asymmetric volatility and jumps on the pricing of VIX derivatives. Journal of Econometrics 192, 313-328. Poterba, J.M., Summers, L.H. 1986. The persistence of volatility and stock market fluctuations. American Economic Review 76, 1142–1151. Rao, B., Rao, G. 2008. Are US gasoline price adjustments asymmetric?. Applied Economics Letters 15, 443–447. Read, C. 2009. The fear factor. What happens when fear grips Wall Street? Palgrave Macmillan. New York. Rizzi, J. V. 2008. Behavioral bias of the financial crisis. Journal of Applied Finance 18, 1-13. Sarwar, G. 2012. Is VIX an investor fear gauge in BRIC equity markets? Journal of Multinational Financial Management 22, 55-65. Schmeling, M. 2009. Investor sentiment and stock returns: Some international evidence. Journal of Empirical Finance 16, 394-408. Schorderet, Y., 2003. Asymmetric Cointegration. Working Paper No 2003.01, University of Geneva. Simon, D., Wiggins, R. 2001. S&P futures returns and contrary sentiment indicators. Journal of Future Markets 21, 447-62. Siriopoulos, C., Fassas, A. 2008. The information content of VFTSE. Available at SSRN: http://ssrn.com/abstract=1307702. Smales, L.A. 2014. News sentiment and the investor fear gauge. Finance Research Letters 11, 122-130. 23

Smales, L. A. 2016. Time-varying relationship of news sentiment, implied volatility and stock returns. Applied Economics 48, 4942-4960. Tallau, C. 2012. Zum zusammenhang von DAX-Renditen und dem volatilitätsindex VDAXNEW. BankArchiv: Zeitschrift für das gesamte Bank- und Börsenwesen 6, 378-386. Tiwari, A.K., Apergis, N., Olayeni, O.R. 2015. Renewable and nonrenewable energy production and economic growth in sub-Saharan Africa: a hidden cointegration analysis. Applied Economics 47, 861-882. Velev, J. P., Payne, B. C., Tresl, J., Toledo, W. 2016. Implied volatility around the world: Geographical markets and asset classes. CERGE-EI Working Paper series, 562. Westerhoff, F.H. 2004. Greed, fear and stock market dynamics. Physica A: Statistical Mechanics and its Applications 343, 635-642. Whaley, R.E. 2000. The investor fear gauge. The Journal of Portfolio Management 26, 12-17. Whaley, R.E. 2009. Understanding the VIX. The Journal of Portfolio Management 35, 98105. Zouaoui, M., Nouyrigat, G., Beer, F. 2011. How does investor sentiment affect stock market crises? Evidence from panel data. Financial Review 46, 723-747.

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Figure 1: US stock market and implied volatility (daily data, 2000-2014) 2500

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25

Figure 3: German stock market and implied volatility (daily data, 2000-2014) 12000 10000

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26

Table 1: Descriptive statistics for the US, the UK and the German stock markets, daily observations (2000-2014) Panel A. USA Mean Median Maximum Minimum Std. Dev. Skewness Kurtosis Observations Panel B. UK Mean Median Maximum Minimum Std. Dev. Skewness Kurtosis Observations Panel C. Germany Mean Median Maximum Minimum Std. Dev. Skewness Kurtosis Observations

S&P 500 1,282.31 1,257.54 2,090.57 676.53 269.70 0.71 3.53 3,771 FTSE 100 5,501.02 5,642.62 6,878.49 3,287.04 849.46 -0.39 2.14 3,817 DAX 30 6,049.75 6,068.53 10,087.12 2,202.96 1,729.17 0.15 2.50 3,813

VIX 20.94 18.93 80.86 9.89 9.01 2.01 9.39 3,771 VFTSE 20.55 18.47 75.54 9.10 8.93 1.76 7.47 3,817 VDAX 24.56 21.58 83.23 11.65 10.23 1.70 6.27 3,813

Δ S&P500 0.17 0.79 104.13 -106.62 14.68 -0.26 7.47 3,771 Δ FTSE 100 -0.03 1.47 431.34 -391.06 62.89 -0.19 6.56 3,817 Δ DAX 30 0.84 4.82 518.14 -523.98 84.01 -0.14 5.65 3,813

Δ VIX 0.00 -0.08 16.54 -17.36 1.71 0.57 21.23 3,771 Δ VFTSE 0.00 -0.06 23.30 -14.14 1.65 1.06 24.20 3,817 Δ VDAX 0.00 -0.07 21.92 -15.05 1.58 1.32 22.82 3,813

Notes: This table presents descriptive statistics of the variables in both raw and first difference form.

27

Table 2: Alternative hypotheses and CECMs Hypotheses

CECM

Hypothesis 1(H1): Neither { X t , Yt  } nor { X t , Yt  } are co-

-

integrated → X t and Yt are not co-integrated Assumption:{ X t , Yt  } are co-integrated with

Hypothesis 2(H2): Either { X t , Yt  } or { X t , Yt  } but not

a co-integrating vector of (1,−1) → CECM model:

both, are co-integrated → X t and Yt are subject either to common

Δ X t =  0 

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

positive or common negative shocks, but not both → X t and Yt are not co-integrated, but

 2 ( X t1 - Yt 1 ) + t

have more structure than in H1 Hypothesis 3(H3): Both { X t , Yt  } and { X t , Yt  } are co-

Assumption:{ X t , Yt  } are co-integrated with

i 0

i 1

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[correspondingly for { X , Yt }]

a co-integrating vector of (1,− k ), where → CECM model:

integrated, but with different co-integrating vectors → the common positive and negative shocks of X and Y are not co-integrated → X t and Yt are not co-integrated, but

Δ X t = 0 

have more structure than in H1

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i 0

i 1

  )+ 4 ( X t1 -k Yt 1 )+ t  3 ( X t1 - Yt 1

(6)

[correspondingly for Yt ] Hypothesis 4(H4): Both { X t , Yt  } and { X t , Yt  } are co-

Assumption: a common co-integrating vector (1,−1) exists → X t and Yt have a standard ECM:

integrated with the same co-integrating vectors → The common positive and negative shocks of X t and Yt are co-integrated

Δ Xt = 0 + +

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k4

k3

→ X t and Yt are co-integrated

k2

k1

  Y i Δ Yt i +   X i Δ X ti + 

i 0



i 1

  )+ ( X t1 - Yt 1 )+ t , (7)  ( X t1 - Yt 1

with    3   4 , where the coefficients associated with Δ X ti and Δ X ti (similarly for   Δ Yt i and Δ Yt i ) should be the same

[correspondingly for Yt ]

28

Table 3: Long-run equations and CECMs corresponding to hypotheses 2 and 3 Long-run equations in general form, aggregate and disaggregated series 𝐹𝐸𝐴𝑅𝐼𝑁𝐷𝐸𝑋𝑡 = 𝑐 + 𝜑 ∗ 𝑆𝑀𝐼𝑁𝐷𝐸𝑋𝑡 + 𝜀𝑡 (+)

= 𝑐1 + 𝜑1 ∗ 𝑆𝑀𝐼𝑁𝐷𝐸𝑋𝑡

(−)

= 𝑐2 + 𝜑2 ∗ 𝑆𝑀𝐼𝑁𝐷𝐸𝑋𝑡

𝐹𝐸𝐴𝑅𝐼𝑁𝐷𝐸𝑋𝑡 𝐹𝐸𝐴𝑅𝐼𝑁𝐷𝐸𝑋𝑡

(8)

(−)

+ 𝜀𝑡

(9)

(+)

+ 𝜀𝑡

(10)

CECMs in general form, hypotheses 2 and 3 Hypothesis 2

FEARINDEX

()

 C 0   * [( res[1])] t 1   0 * SMINDEX (  ) 

lags ( FEARINDEX FEARINDEX

( )

()

(11)

, SMINDEX (  ) )   t

 C 0   * [( res[1])] t 1   0 * SMINDEX (  ) 

lags ( FEARINDEX

( )

(12)

, SMINDEX (  ) )   t

Hypothesis 3

FEARINDEX  C 0   1 * [( res[1])] t 1   2 * [( res[2])] t 1   0,1 * SMINDEX (  )   0,2 * SMINDEX (  )  lags ( FEARINDEX

( )

, FEARINDEX

()

(13)

, SMINDEX

29

()

, SMINDEX

()

) t

Table 4: Co-integration and hidden co-integration tests Series/Country Test statistics Dependent variable Aggregate series/ Decomposed series USA VIX VIX(+) VIX(-) UK VFTSE VFTSE(+) VFTSE (-) Germany VDAX VDAX (+) VDAX (-)

Without trend 1)

t-Statistic , [p-value]

With trend 2)

t-Statistic1), [p-value]2)

-6.04, [0.00] -3.67, [0.02] -1.82, [0.62]

-3.75, [0.05] -4.48, [0.01] -2.70, [0.41]

-4.01, [0.01] -2.61, [0.23] -1.70, [0.68]

-4.01, [0.03] -2.64, [0.44] -1.56, [0.91]

-3.72, [0.02] -3.70, [0.06] -0.67, [0.95] -1.37, [0.94] 0.92, [1.00] -1.02, [0.98] 1) 2) Notes: Engle-Granger tau-statistics. MacKinnon p-values. Lag length is selected by the automatic selection procedure based on the SIC criterion (t-statistics are given in brackets).

30

Table 5: Dynamic CECMs for the UK and Germany Equations in general form UK VFTSE  C 0   1 * [( res[1])] t 1   2 * [( res[2])] t 1   0,1 * FTSE100 (  )   0, 2 * FTSE100 (  )  lags ( VFTSE (  ) , VFTSE (  ) , FTSE100 (  ) , FTSE100 (  ) )   t

(16)

Germany VDAX  C 0   1 * [( res[1])] t 1   2 * [( res[2])] t 1   0,1 * DAX 30 (  )   0, 2 * DAX 30 (  )  lags ( VDAX (  ) , VDAX (  ) , DAX 30 (  ) , DAX 30 (  ) )   t

(17)

Estimated equations UK VFTSE  0.15  0.004 * ( res[1]) t 1  0.005 * ( res[2]) t 1  0.023 * FTSE100 (  ) ( 4.33) ( 2.70)

( 4.06)

( 44.94)

 0.012 * FTSE100 (  )  0.251 * VFTSE t(1)  0.124 * VFTSE t(2)  0.134 * VFTSE t(3) ( 21.22)

( 9.88)

( 4.84)

( 7.17)

 0.091 * VFTSE t(1)  0.146 * VFTSE t(2)  0.084 * VFTSE t(3)  0.003 * FTSE100 (t 1) ( 3.62)

( 5.84)

( 4.04)

 0.003 * FTSE100t

() t 2

() t 1

( 5.00)

 0.005 * FTSE100 ( 7.00)

(18)

( 5.35)

 0.004 * FTSE100 (t 2) 2

( 5.18)

R  0.55

Germany VDAX  0.18  0.005 * ( res[1]) t 1  0.005 * ( res[2]) t 1  0.018 * DAX 30(  ) ( 5.66) ( 4.95)

( 5.85)

( 49.69)

 0.008 * DAX 30(  )  0.084 * VDAX t(1)  0.102 * VDAX t(2)  0.118 * VDAX t(3) ( 20.67)

( 3.46)

( 6.27)

( 7.03)

 0.224 * VDAX t(1)  0.002 * DAX 30(t 1)  0.002 * DAX 30(t 1) (8.84)

(4.96)

( 3.71)

Notes: t-statistics are given in parentheses.

31

2

R  0.56

(19)

Table 6: Estimated coefficients and test results Parameter/ Hypothesis testing Long-run equations

𝜑1 𝜑2

USA

UK

Germany

-0.108 rejected -

-0.023 -0.023 rejected rejected

-0.018 -0.018 rejected rejected

-0.023 -0.012

-0.018 -0.008

-0.004 -0.005 accepted

-0.005 -0.005 accepted

0 2.2

0 2.0

𝐻0 : 𝜑1 = 1 𝐻0 : 𝜑2 = 1 Short-run equations Instantaneous effects -0.101 𝛿0,1 𝛿0,2 Speed of the adjustment process -0.001 𝜃1 𝜃2 𝐻0 : 𝜃1 = 𝜃2 Time span of the adjustment process 𝛾1 𝛾2

Notes: The reported hypotheses testing results (‘rejected’ or ‘accepted’) are based on the conduction of Wald coefficient tests and are related to the corresponding 𝜒2 and probability statistics.

32