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1 Title Safe Haven and Hedge Currencies for th Markets : A Copula-Based Approach Author(s) Tachibana, Minoru Editor(s) Citation Discussion Paper New Series (1), Issue Date URL Rights

2 Discussion Paper New Series Safe Haven and Hedge Currencies for the US, UK, and Euro Area Stock Markets: A Copula-Based Approach Minoru Tachibana Discussion Paper New Series No March 2017 School of Economics Osaka Prefecture University Sakai, Osaka , Japan

3 Safe Haven and Hedge Currencies for the US, UK, and Euro Area Stock Markets: A Copula-Based Approach Minoru Tachibana March 2017 Abstract This paper uses a copula-based approach to identify safe haven and hedge currencies for the US, UK, and Euro area stock markets over the period We reveal similarities and differences in the determination of safe haven and hedge currencies across the three stock markets. First, UK and Euro area stock markets have had the same set of hedge currencies, among which the CHF has been the most indispensable hedge currency. Second, there has been no safe haven currency for the UK stock market, whereas the Euro area stock market had some safe haven currencies until 2012, reflecting its large volatility relative to the UK market. Third, for the US stock market, the JPY has become the most important currency both as a safe haven and as a hedge since the 2007 global financial crisis. This is probably because the Japanese economy was less affected directly by that crisis and the subsequent European debt crisis than the three economies. Keywords: Safe haven currencies, Hedge currencies, Stock markets, Copulas, Threshold model. JEL Classification: C58, F31, G15. School of Economics, Osaka Prefecture University, Sakai, Japan. address: mtachi@eco.osakafuu.ac.jp

4 1 Introduction Safe haven currency is defined as a currency that appreciates in times of market stress or turmoil. 1 Even if an economy, where its currency has a status of safe haven currency, is not directly affected by a financial crisis, such a safe haven status would generate adverse shocks to the domestic economy through large appreciation of the currency. Thus enriching understanding of the safe haven currency is important not only for international investors but for policymakers and business administrators whose currencies are potentially save haven currencies. Although the literature on safe haven currencies had been sparse until the 2007 global financial crisis, a gradually increasing number of studies have dealt with this subject in recent years. Ranaldo and Söderlind (2010) provide evidence that the CHF, 2 the JPY, and (to a lesser extent) the EUR have significant safe haven properties over the period : these currencies tend to appreciate against the USD when US stock prices decrease, US bond prices increase, and foreign exchange (FX) markets become more volatile. Fatum and Yamamoto (2016) document evidence that the JPY is the safest, the CHF is the second-most safe, and the USD is the third-most safe safe haven currency during the global financial crisis from August 2007 to January Hossfeld and MacDonald (2015) show that the CHF and (to a lesser extent) the USD qualify as safe haven currencies over the period January 1986 to Semtember In this paper, we identify safe haven and hedge 4 currencies among five currencies (USD, GBP, EUR, CHF, JPY) 5 over the period In particular, our main purpose is to 1 This definition is modified in Section 2.1 in accordance with the safe haven literature and our empirical analysis. 2 We use the following currency codes (in parentheses) throughout the paper: the Swiss franc (CHF), the Japanese yen (JPY), the US dollar (USD), the British pound (GBP), and the euro (EUR). 3 Other studies on safe haven currencies include Kaul and Sapp (2006), Fratzscher (2009), McCauly and McGuire (2009), Kohler (2010), Hoffmann and Suter (2010), Habib and Stracca (2012), Botman et al. (2013), Coudert et al. (2014), De Bock and de Carvalho Filho (2015), and Grisse and Nitschka (2015). 4 Hedge currency is defined as a currency that is negatively correlated with another asset on average. As with a safe haven currency, this definition is modified in Section 2.1 in accordance with our empirical analysis. 5 We also examined additional five currencies (Australian dollar, Canadian dollar, New Zealand dollar, Norwegian krone, and Swedish krona). The results showed that they were neither safe haven nor hedge 1

5 examine whether safe haven and hedge currencies vary across three major stock markets (US, UK, and Euro area stock markets). To this end, we develop a copula-based approach and apply it to equity-currency pairs. This paper contributes to the literature in two ways. First, we allow for different patterns of the safe haven and hedge behavior across stock markets. Previous work, on the other hand, has implicitly assumed that safe haven and hedge behavior arise only from the US stock market or from a hypothetical world stock market. For instance, Fatum and Yamamoto (2016) and Hossfeld and MacDonald (2015) identify safe haven currencies by evaluating exchange rate responses to the implied volatility of S&P 500 index options, i.e. the VIX (Fatum and Yamamoto, 2016) or to returns on the MSCI world index (Hossfeld and MacDonald, 2015). 6 However, unless stock markets are perfectly correlated, there would be heterogeneity in the safe haven and hedge behavior across stock markets. This paper takes into consideration this possibility particularly by focusing on the US, UK, and Euro area stock markets, which cover a large portion of equity trading volume in the world. Second, we adopt a copula-based approach to characterize the relationship between stock returns and exchange rate changes and to identify safe haven and hedge currencies. To our knowledge, there is no study that uses a copula approach to determining safe haven and hedge currencies. 7 Another two related strands of literature estimate copula models, but their focuses are neither on a safe haven nor on a hedge currency: Reboredo (2013) uses a copula method to assess the safe haven and hedge roles of gold against the US dollar; Ning (2010), Wang et al. (2013), and Reboredo et al. (2016) estimate copula models to analyze the dependence structure between stock returns in a country and exchange rates of this country s currency. 8 In contrast, we focus on the relationship between stock returns in a country and currency and hence we do not report these results. 6 Hossfeld and MacDonald (2015) identify carry funding currencies in addition to safe haven and hedge currencies. Identifying carry funding currencies is a difficult task with the copula-based approach used here and hence it is beyond the scope of the paper. 7 Ranaldo and Söderlind (2010) estimate linear and non-linear factor models, while Hossfeld and MacDonald (2015) and Fatum and Yamamoto (2016) employ threshold regression models. No other studies listed in footnote 3 use a copula approach. 8 For example, Ning (2010) analyzes four equity-currency pairs: the UK stock and the GBP return pair, the 2

6 exchange rates of another country s currency that is a potentially safe haven or hedge currency. One of the advantages of using a copula approach is that it allows us a great deal of flexibility in specifying a model for the joint distribution of variables of interest (Patton, 2009). The copula approach is based on Sklar s theorem by which one can decompose the joint distribution of variables of interest into their respective marginal distributions and a copula function, the latter characterizing dependence structure of the variables. Because the marginal distributions do not need to be the same as each other and the choice of copula is not constrained by the choice of marginal distributions, we can construct a flexible and complex model for the joint distribution. Another advantage is that two kinds of dependence measures computed from copula, tail dependence and concordance (or linear correlation), help us identify safe haven and hedge currencies. More specifically, we can identify safe haven currency based on the estimated lower tail dependence, since it measures the joint probability of two extreme events (one is an extremely large decline in stock prices and the other is an extremely large appreciation of currency). Similarly, we can identify hedge currency by calculating the concordance measure such as Kendall s τ or the linear correlation of copula, as they indicate an average relationship between equity and currency over time. We further develop the copula-based procedure in Reboredo (2013) who studies whether gold qualifies as a safe haven or hedge against the US dollar. In particular, our most novel improvement is that when a time-varying dependence copula is chosen as the best copula, a threshold model and its linear version are incorporated into the copula procedure, which enables us to formally test for safe haven and hedge currencies. Although Fatum and Yamamoto (2016) and Hossfeld and MacDonald (2015) estimate threshold models as well, our approach utilizes the threshold model and its linear version not as a data generating process but as a test equation to determine safe haven and hedge currencies. By doing so, we can preserve advantages of the copula model discussed above. German stock and the DEM (or EUR) return pair, the French stock and the FRF (or EUR) return pair, and the Japanese stock and the JPY return pair (exchange rates are all expressed in USD per local currency). 3

7 To anticipate the main results of the paper, we reveal similarities and differences in the determination of safe haven and hedge currencies across the US, UK, and Euro area stock markets. First, UK and Euro area stock markets have had the same set of hedge currencies, among which the CHF has been the most indispensable hedge currency. Second, there has been no safe haven currency for the UK stock market, whereas the Euro area stock market had some safe haven currencies until 2012, reflecting its large volatility relative to the UK market. Third, for the US stock market, the JPY has become the most important currency both as a safe haven and as a hedge since the 2007 global financial crisis. This is probably because the Japanese economy was less affected directly by that crisis and the subsequent European debt crisis than the three economies. The remainder of the paper proceeds as follows. In Section 2, we present definitions of safe haven and hedge currencies, specify the copula models and the estimation method, and provide criteria for identifying safe haven and hedge currencies. In Section 3, we describe the data and report the results. Section 4 concludes the paper. 2 Empirical methodology 2.1 Definitions of safe haven and hedge currencies We first define safe haven currency in the context of our approach. Baur and Lucey (2010) and Baur and McDermott (2010) define a safe haven asset in general as an asset that is uncorrelated or negatively correlated with another asset or portfolio in times of market stress or turmoil. The literature on safe haven currency applies this kind of definition by replacing an asset with a currency (see, for instance, Ranaldo and Söderlind, 2010; Hossfeld and MacDonald, 2015; Fatum and Yamamoto, 2016). In this paper, we further modify this definition in accordance with our copula-based approach and it is given as follows: Safe haven currency: a currency is a safe haven if it is positively correlated with stock 4

8 returns in times of market stress or turmoil. 9 Note that safe haven behavior is characterized not as a negative relationship but as a positive relationship between currency and stock returns. This is because some of the copula functions used here can only describe the positive dependence between variables. We can easily make that constraint suitable for the definition of safe haven currency, by using data on the exchange rate that is expressed as the number of units of a possibly safe haven currency (one of USD, GBP, EUR, CHF, JPY; hereafter referred to as foreign currency ) per unit of the currency used in one of the US, UK, and Euro area stock markets (one of USD, GBP, EUR; hereafter domestic currency ). In a similar way, we define hedge currency as follows: Hedge currency: a currency is a hedge if it is positively correlated with stock returns on average. The distinction between a safe haven and a hedge currency is that safe haven behavior in a stock market is supposed to emerge only when the market is in stress or turbulence, whereas hedge behavior can arise even in normal times. This difference is crucial to identify safe haven and hedge currencies, and copula functions presented below are able to describe both relationships with different dependence measures. 2.2 Copula method Our copula-based approach characterizes the dependence structure between stock returns and exchange rate changes and then determines safe haven and hedge currencies by criteria that are consistent with their definitions presented above. The copula method builds on Sklar s theorem which shows that a joint distribution function can be decomposed into its marginal distributions and a copula function. 10 Our discussion here is confined to a bivariate case. Let 9 This definition does not include the word uncorrelated to formally determine safe haven (and hedge) currencies with statistical tests. 10 See Joe (1997, 2015) and Nelsen (2006) for a comprehensive exposition of copulas. 5

9 X 1 and X 2 be random variables (representing stock returns and exchange rate changes in our empirical analysis) with a joint distribution F and respective marginal distributions F 1 and F 2, and let x 1 and x 2 be realizations of X 1 and X 2, respectively. Then Sklar s theorem states that under the continuity of F 1 and F 2, there exists a unique copula C : [0, 1] 2 [0, 1] such that F (x 1, x 2 ) = C(F 1 (x 1 ), F 2 (x 2 )). (1) Conversely, if C is a copula and F 1 and F 2 are distribution functions, then F defined by (1) is a joint distribution function with marginal distributions F 1 and F 2. The copula function C can be seen as a joint distribution function of U 1 = F 1 (X 1 ) and U 2 = F 2 (X 2 ), i.e. C(u 1, u 2 ) = Pr(U 1 u 1, U 2 u 2 ), where U 1 and U 2 follow a standard uniform distribution, U 1, U 2 U(0, 1), based on the probability integral transformation. An important implication of (1) is that we can construct a flexible and complex model for the joint distribution of variables of interest by combining a copula function with respective marginal distributions. In other words, a copula function captures the dependence structure between variables that is completely separated from their marginal distributions. We can derive (lower and upper) tail dependence for copulas, which measures the probability that variables of interest jointly take extreme values. More specifically, for a bivariate copula C, lower tail dependence (λ L ) and upper tail dependence (λ U ) are defined below: λ L = lim [ Pr u 0 = lim u 0 Pr ] X 1 F1 1 (u) X 2 F2 1 (u) [ ] X 2 F2 1 (u) X 1 F1 1 (u) C(u, u) = lim, (2) u 0 u [ ] λ U = lim Pr X 1 F 1 u 1 1 (u) X 2 F2 1 (u) [ ] = lim Pr X 2 F 1 u 1 2 (u) X 1 F1 1 (u) 1 2u + C(u, u) = lim, (3) u 1 1 u 6

10 and one can state that the copula has lower tail dependence if λ L (0, 1] and upper tail dependence if λ U (0, 1]. In particular, the lower tail dependence (2) is used to identify safe haven currencies in our analysis as it indicates the probability that large declines in stock prices and large appreciations of currency jointly occur. In this paper we estimate five copulas (Normal, Student s t, Clayton, Gumbel, BB7) 11 whose functional forms are presented in the second column of Table 1. As seen in the last two columns, they have different structures of tail dependence: Normal copula has no tail dependence (λ L = λ U = 0); Student s t copula has the same degree of lower and upper tail dependence (λ L = λ U ); Clayton copula only has lower tail dependence (λ U = 0); Gumbel copula only has upper tail dependence (λ L = 0); and BB7 copula has both upper and lower tail dependence that are allowed to be different (λ L λ U ). As shown in the third column of the table, dependence parameters we directly estimate are linear correlations of the Normal and Student s t copulas as well as tail dependence of the Clayton, Gumbel, and BB7 copulas. The tail dependence of the Student s t copula is computed after estimating the linear correlation ρ and the degrees of freedom ν, through the formula λ L = λ U = 2T ν+1 ( ) (ν + 1)(1 ρ)/(1 + ρ), where T ν+1 ( ) denotes the cumulative distribution function of the standard Student s t with ν + 1 degrees of freedom. While we estimate the five copulas above whose dependence measures are assumed to be constant, we furthermore allow for time-varying dependence for Normal and Student s t copulas in order to capture, if any, a more flexible relationship between stock returns and exchange rate changes over time. Following Patton (2006), the evolution of the linear correlation ρ for the Normal copula is specified as the following ARMA(1, m)-type process, ρ t = Λ ω 0 + ω 1 ρ t 1 + ω 2 1 m m Φ 1 (u 1,t j ) Φ 1 (u 2,t j ), (4) j=1 11 This estimation set of copulas is the same as the one in Reboredo (2013) except that he estimates the symmetrized Joe-Clayton (SJC) copula proposed by Patton (2006) instead of the BB7 copula. The reason we estimate the BB7 copula is that it is easier to compute its concordance measure (Blomqvist s β), which is used in determining hedge currencies. As argued by Patton (2006), there is some slight asymmetry in the BB7 copula even when the lower and upper tail dependence are equal. 7

11 where Λ(x) = (1 e x )(1 + e x ) 1 is the modified logistic transformation, designed to keep ρ t in ( 1, 1) at all times. Φ 1 (x) denotes the inverse of the standard Normal distribution function and it is replaced with the inverse of the standard Student s t distribution when Student s t copula is estimated. 12 The copula method requires specifying marginal distributions F 1 (x 1 ) and F 2 (x 1 ) as well as a copula function C to build the joint distribution of variables of interest. In this study, marginal distributions are characterized by an AR(p) GJR GARCH(q,r) model with the disturbance term following the Normal, Student s t, or skewed t distribution: p i X it = α i0 + α il X i,t l + ε it, (5) l=1 ε it = σ it η it, (6) q i r i σit 2 = β i0 + β il σi,t l 2 + ψ il ε 2 i,t l + ψ il D i,t l ε 2 i,t l, (7) l=1 l=1 l=1 r i for i = 1, 2 and t = 1,..., T. The AR model (5) and the GARCH model (7) represent conditional mean and conditional variance of the marginal distribution of X it, respectively. The so-called GJR terms, suggested by Glosten et al. (1993), appear in the last of the righthand side of Eq.(7), where the dummy variable D i,t l takes the value of 1 if ε i,t l is negative and 0 otherwise. The GJR terms, therefore, take into account the leverage effect, namely the effect that a negative shock to financial returns causes higher volatility of the returns in the future than a positive shock does. It is assumed that the standardized disturbance η it in Eq.(6) follows an i.i.d. Normal, Student s t, or skewed t distribution. The degrees of freedom parameter ν of the Student s t distribution controls thickness of the tails, which enables us to describe well-known fat tails of financial return data. The skewed t distribution, proposed by Hansen (1994), characterizes asymmetry by a skewness parameter λ ( 1, 1) as well as fat tails by a degrees of freedom parameter ν. It reduces to the Student s t distribution when λ = 0, to the skewed Normal distribution when ν, and to the Normal distribution when 12 We set m equal to max(r 1, r 2), where r i for i = 1, 2 denotes the number of lags of MA terms in the GARCH(q i,r i) model specified below. 8

12 λ = 0 and ν. The parameters of marginal distributions and a copula are estimated through maximum likelihood (ML) estimation. We can derive the log-likelihood function at time t by modifying Eq.(1) for the analysis of time series data, 13 differentiating both sides of the modified equation with respect to x 1t and x 2t, and taking the logarithm. Then the log-likelihood at time t is given by log f(x 1t, x 2t Ω t 1 ; θ) = log f 1 (x 1t Ω t 1 ; θ 1 ) + log f 2 (x 2t Ω t 1 ; θ 2 ) + log c(u 1t, u 2t Ω t 1 ; θ c ), (8) where Ω t 1 denotes the information set available at the end of time t 1, f the conditional joint density of X 1t and X 2t, f i the conditional marginal density of X it for i = 1, 2, c the conditional copula density, and u it = F i (x it Ω t 1 ; θ i ) for i = 1, 2. We define a vector of all parameters as θ = [θ 1, θ 2, θ c] where θ 1 and θ 2 are parameter vectors for the conditional marginal models of X 1t and X 2t, and θ c is a parameter vector for the copula model. Since θ 1, θ 2, and θ c separately appear in each term of Eq.(8), we can use a two-stage estimation procedure. First, parameters of conditional marginal distributions are estimated by ML: ˆθ i arg max θ i T log f i (x it Ω t 1 ; θ i ), (9) t=1 for i = 1, 2. Second, given ˆθ 1 and ˆθ 2, parameters of a conditional copula are estimated by ML: ˆθ c arg max θ c T log c(û 1t, û 2t Ω t 1 ; θ c ), (10) t=1 where û it = F i (x it Ω t 1 ; ˆθ i ) for i = 1, 2. This procedure has the benefit of being computationally tractable, at the cost of a loss of full efficiency (Patton, 2009). In the first stage, the best 13 Patton (2006) extends Sklar s theorem (1) to the one applicable to time series data. 9

13 marginal model for each series is selected according to the Akaike information criteria (AIC). After that, in the second stage we estimate several copulas and select the best copula based on the AIC Criteria for determining safe haven and hedge currencies We next explain how to identify safe haven and hedge currencies after selecting the best copula model for each pair of stock returns and exchange rate changes. Table 2 summarizes these criteria, which are designed to be consistent with definitions of safe haven and hedge currencies given in Section 2.1. We use different frameworks for the identification, depending on whether the best copula is in the class of constant dependence copulas or the class of time-varying dependence copulas. When the constant dependence Student s t, Clayton, or BB7 is selected as the best copula and furthermore their lower tail dependence is statistically significant, we classify a currency under study into a safe haven currency because the lower tail dependence indicates, in our analysis, the probability that an extremely large decline in stock prices and an extremely large appreciation in the currency jointly occur. When the constant dependence Normal or Gumbel copula is selected, on the other hand, there is no evidence of safe haven since these two copulas have no lower tail dependence. With regard to identification of hedge currency, we need to see the dependence structure on average rather than the dependence structure in times of extreme market movements, according to the definition of hedge currency provided in Section 2.1. In particular, we resort to the linear correlation or the concordance measure (Kendall s τ or Blomqvist s β) of the best copula, since they represent the average relationship between stock returns and exchange rate changes over the sample period. A currency is categorized as a hedge currency when the constant dependence Normal or Student s t copula is selected as the best copula, and 14 Reboredo (2013) evaluates the performance of copula models using the AIC adjusted for small-sample bias. We used it instead of the simple AIC, but the same copula function was selected as the best copula for all of the equity currency pairs. 10

14 moreover their linear correlation is significantly greater than zero. Meanwhile, we rely on Kendall s τ for Clayton and Gumbel copulas and on Blomqvist s β for BB7 copula, the latter is because we can more easily estimate it and its standard error than Kendall s τ. 15 Hence, we classify a currency into a hedge currency when the constant dependence Clayton, Gumbel, or BB7 is selected as the best copula and their corresponding concordance measure takes a significant positive value. When the time-varying dependence Normal or Student-t copula is chosen as the best copula, we develop an alternative way to identify safe haven and hedge currencies. It is based on a threshold model and its linear version, the former testing whether a currency is a safe haven and the latter a hedge. Our threshold model is given by φ 0,L + φ 1,L ˆρ t 1 + ξ t, if ˆσ 1t γ, ˆρ t = φ 0,H + φ 1,H ˆρ t 1 + ξ t, if ˆσ 1t > γ, (11) where ˆρ t is the estimated time-varying linear correlation of the Normal or Student-t copula; ˆσ 1t, used as a threshold variable in our threshold model, is the estimated time-varying volatility of a stock market (one of the US, UK, and Euro area stock markets) which is obtained from estimation of the stock returns marginal model; and γ is a threshold value. The threshold model (11) specifies ˆρ t as an AR(1) model to capture its high persistence, but the coefficients are allowed to change depending on the level of stock market volatility. This model seems to overlap the evolution equation of linear correlation (4). However, it should be emphasized that, while Eq.(4) is a part of the copula model that specifies the data generating process of stock returns and exchange rate changes, the threshold model (11) is just an equation for the test of safe haven currency. As discussed in the Introduction, using a threshold model in this 15 Kendall s τ for Clayton is defined as τ = δ/(δ+2), Kendall s τ for Gumbel as τ = (δ 1)/δ, and Blomqvist s β for BB7 as β = 4β 1 where β = 1 (1 [2{1 2 κ } δ 1] 1/δ ) 1/κ (see Joe, 2015). We can obtain Kendall s τ and its standard error for the first two copulas by re-estimating the corresponding best copula in which the copula parameter δ is specified as a function of Kendall s τ instead of tail dependence. We can compute Blomqvist s β and its standard error for BB7 by using estimates of the copula parameters δ and κ, which are then obtained by re-estimating BB7 without imposing tail dependence equations. 11

15 way can preserve advantages of the copula model as the data generating process. As presented in Table 2, we classify a currency into a safe haven currency if the following two conditions are satisfied: (1) the mean of the process in times of high stock market volatility ρ H = φ 0,H /(1 φ 1,H ) is positive; (2) ρ H is greater than the mean of the process in times of low stock market volatility ρ L = φ 0,L /(1 φ 1,L ). Note that we use entirely distinct measures of dependence to identify safe haven currency, depending on which class of copula is selected as the best copula; lower tail dependence for constant dependence copulas and (time-varying) linear correlation for time-varying dependence copulas. However, both criteria are consistent with the definition of safe haven currency given in Section 2.1 and thus we can rely on these two criteria without inconsistencies. The following linear variant of the threshold model (11) is used to identify hedge currency: ˆρ t = φ 0 + φ 1 ˆρ t 1 + ξ t. (12) We regard a currency as a hedge currency if the mean of the process ρ = φ 0 /(1 φ 1 ) is positive, since it measures the average relationship between stock returns and exchange rate changes, regardless of the level of stock market uncertainty. Estimation of the threshold model (11) is carried out by conditional least squares (see for example, Hansen, 2000). That is, conditional on a threshold value γ, the model is linear and thus can be estimated by least squares. Repeating this for a number of γ, 16 we estimate γ so as to minimize the sum of squared errors. A problem with regard to inference of the threshold model is that γ is not identified under the null hypothesis of no threshold effect (i.e., when φ 0,L = φ 0,H and φ 1,L = φ 1,H ), implying that the asymptotic distribution of the test is nonstandard. Hansen (1996) shows that the simulation-based Lagrange multiplier (LM) tests have excellent size and good power. We follow this kind of procedure and employ a heteroskedasticity-consistent bootstrap LM test, computing the p-value with 1,000 bootstrap samples. 16 We consider values of γ ranging between the 0.15th and 0.85th quantile of ˆσ 1t. 12

16 3 Data and results 3.1 Data Our data set consists of weekly data of stock market indices in the three major countries/area (US, UK, Euro area) and nominal exchange rates of five possible safe haven and hedge currencies (USD, GBP, EUR, CHF, JPY), all collected from Datastream over the period 1 January 1999 to 30 December The starting sample period is determined by the event that the euro was introduced to financial markets in that period. The use of weekly frequency data is appropriate for our copula analysis. Reboredo (2013), who investigates the relationship between gold and the USD using copulas, argues that daily or high-frequency data may be affected by drifts and noise that could mask the dependence relationship and complicate modeling of the marginal distributions through non-stationary variances, sudden jumps or long memory. This concern is relevant to our study as well. Stock price indices used here are S&P 500 for the US, FTSE 100 for the UK, and EURO STOXX 50 for the Euro area. As mentioned in Section 2.1, exchange rates are expressed as the number of units of foreign currency (one of USD, GBP, EUR, CHF, JPY) per domestic currency (one of USD, GBP, EUR), the latter determined by the stock market we consider in the copula model. For example, when analyzing the relationship between the USD and the UK stock returns, we use USD/GBP as the exchange rate data. Thus a decrease in exchange rates means an appreciation of foreign currency against domestic currency, and a positive relationship between stock returns and exchange rate changes is relevant to the identification of safe haven and hedge currencies. Two time series data, stock returns (X 1t ) and exchange rate changes (X 2t ), are calculated as 100 times the log-difference of stock price index and exchange rate, respectively. Fig.1 displays time series movements of stock price indices of the US, the UK, and the Euro area (left-upper panel), exchange rates of three pairs between USD, GBP and EUR (right-upper panel), exchange rates of the CHF against USD, GBP, and EUR (left-lower 13

17 panel), exchange rates of the JPY against USD, GBP, and EUR (right-lower panel). Three stock market indices move in a highly correlated way over the whole period. In particular, they decline sharply from 2000 to 2002 (corresponding to the crash of the dot-com bubble and the 9/11 attacks) and from 2007 to 2008 (corresponding to the global financial crisis). For exchange rate dynamics, sudden and substantial appreciations in the second half of 2008 are observed for the USD (against GBP and EUR), the CHF (against GBP), and the JPY (against GBP and EUR). The large appreciations of USD, CHF, and JPY at the time are also pointed out by many studies (e.g., Fratzscher, 2009; McCauly and McGuire, 2009; Kohler, 2010; De Bock and de Carvalho Filho, 2015) and the phenomena suggest that these three currencies might have played safe haven roles during the global financial crisis. Table 3 presents descriptive statistics of the time series data used in our analysis. Standard deviations of three stock returns are larger than those of exchange rate changes, showing higher risk in the stock markets. Skewness, Kurtosis, and Jarque-Bera test statistics imply that the use of Normal distribution might not be appropriate for modeling conditional marginal distributions, supporting the use of Student s t and skewed t distributions in addition to the Normal distribution. 3.2 Results for the best marginal models We first estimate several variants of marginal models and select the best model for each of our time series data. The following procedure is taken to select the best marginal model from the set of the AR(p) GJR GARCH(q,r) models (5) (7). First, we estimate several AR(p) GJR GARCH(q,r) models that vary in three aspects: (1) those models with different sets of lag length (p, q, r), each of the lag length parameters taking the value of 0 or 1 17 ; (2) those models with and without the GJR term; (3) those models with three distinct distributions for the standardized disturbance (standard Normal, Student s t, and skewed t distributions). As a result, we estimate a total of 36 marginal models for each of our time series data. Second, a 17 The models with r = 0 are omitted from our estimations since they are neither ARCH nor GARCH model. 14

18 candidate for the best marginal model is chosen according to the AIC. Third, we implement four diagnostic tests for that candidate: the Ljung-Box serial correlation test applied to the standardized residual ˆη it, the same test applied to the squared standardized residual ˆη 2 it,18 and the Kolmogorov-Smirnov (KS) and Anderson-Darling (AD) goodness-of-fit tests applied to the estimated probability integral transformation û it = F i (x it Ω t 1 ; ˆθ i ). If the candidate model is well specified, ˆη it and ˆη 2 it would exhibit no serial correlation and û it would be generated from an i.i.d. uniform (0, 1) distribution. 19 Therefore, if it passes all of the four tests at the 5% significance level, we regard it as the best marginal model. If it fails any of the four tests, we return to and repeat the second and third steps until obtaining the best marginal model. Table 4 reports results on the best marginal models for all of the time series data. 20 Panel A shows that the best marginal models of the three stock returns are all specified as the AR(1) GJR GARCH(1,1) model with the skewed t distribution. The coefficient on the GJR term is significantly positive for the three models, suggesting the presence of leverage effects in their conditional variance. The fat tail feature of stock returns is confirmed from their significantly estimated degrees of freedom parameters. The skewness is significantly negative for the three stock markets, meaning that the modes of their conditional probability density functions are positive and the density functions are skewed to the left. As for exchange rate changes (Panels B and C), their best marginal models have different characteristics except between the marginal models of exchange rates of two same currencies (e.g., GBP/USD and USD/GBP). Only one feature common to all the exchange rate series is that the GARCH(1,1) specification is the best marginal model for their conditional variance. Finally, we can confirm from p-values of four diagnostic tests reported in the last four columns, that all the marginal models are correctly specified, although it is obvious from our procedure of selecting the best marginal model. Fig.2 plots the movements of stock market volatility, ˆσ 1t, in the US, the UK, and the Euro 18 We set the lag length for the two serial correlation tests equal to 12 weeks. 19 Specifying marginal models correctly is necessary for copulas to be correctly specified. 20 To save space, we do not report estimates of the AR and GARCH coefficients, which are likely to be less informative. One feature common to all of the time series is that there is high persistence in the volatility. 15

19 area, obtained from estimations of their best marginal models. 21 First, there is considerable fluctuation in the three stock markets from October 2008 (a month after the bankruptcy of Lehman Brothers) to the beginning of Second, there is somewhat large uncertainty in the early 2000s (presumably because of the crash of the dot-com bubble and the 9/11 attacks) and from 2010 to 2012 (presumably because of the European debt crisis). Third, the volatility in the UK stock market is smaller than in the US and Euro area stock markets, implying that the UK was less affected by those crisis episodes. 3.3 Results for the best copula models and determinations of safe haven and hedge currencies As described in Section 2.2, the set of copula functions we estimate includes five constant dependence copulas (Normal, Student s t, Clayton, Gumbel, and BB7) and two time-varying dependence copulas (Normal and Student s t). In addition to these seven copulas, we further estimate constant dependence copulas but allowing for one structural break in 3 August 2007 as well as those allowing for two structural breaks in 3 August 2007 and 4 January 2013, using appropriate dummy variables. Many papers assume that the global financial crisis began in August 2007, from which safe haven behavior might have changed in the US, UK, and Euro area stock markets due to huge impacts of the crisis on the markets. In fact, Ranaldo and Söderlind (2010) and Fatum and Yamamoto (2016) provide evidence that during the crisis period from August 2007, the JPY had stronger safe haven properties but that the CHF, on the other hand, had weaker safe haven properties compared with in the pre-crisis period. The other break, January 2013, is also considered in this paper because the global financial crisis and the subsequent European debt crisis are likely to have settled down until then. This is confirmed from Fig.2 showing that movements of the volatility in the US, UK, and Euro area stock markets became stable from the beginning of The constant dependence with one or two breaks is considered for five copula functions, leading to a total of 17 copulas to be 21 Recall that we use this measure as a threshold variable of Eq.(11). 16

20 estimated for each pair of stock returns and exchange rate changes. Then we select the best copula among them based on the AIC Results for pairs whose best copulas are constant dependence copulas Table 5 provides results on the best copulas, but only for six equity-currency pairs for which constant dependence copulas (with no, one, and two structural breaks in dependence) are selected as the best one. Each column reports estimates of the copula parameters (mostly dependence parameters) for an equity-currency pair followed by estimates of alternative dependence measures in the rows of indirect est., which are obtained by re-estimating the best copula (Kendall s τ for Clayton copula in the first and third columns of the table; see also footnote 15) or from the function of estimated copula parameters (tail dependence for Student s t copula in the second, fourth, and last columns; see also Section 2.2). In the bottom rows, we give classification of currencies into safe haven and hedge currencies for three subsample periods. The JPY has become an outstanding currency both as a safe haven and as a hedge for investors holding US equities since the 2007 global financial crisis. The best copula for the US equity JPY pair (first column) is the Clayton with one break in dependence in August The lower tail dependence is statistically significant at the 1% level for the period , which means that the JPY qualifies as a safe haven for this period. With regard to the hedge role, the Kendall s τ for the period is positive but only marginally significant at the 10% level, suggesting weak evidence on the hedge role of the JPY for US stock returns in the pre-crisis period. Meanwhile, it is significantly positive at the 1% level for the period , showing strong evidence on the hedge role during the crisis and post-crisis periods. For investors holding UK and Euro area equities, the JPY has been a hedge rather than a safe haven currency since The best copula for the UK equity JPY pair (fourth column) is the Student s t copula with one break in dependence in August The linear correlation that is positive and highly significant during the period suggests strong evidence 17

21 that the JPY is a hedge currency for the UK stock market in this period. On the other hand, unlike the US equity JPY pair, there is only weak evidence on the safe haven role of the JPY for UK stock returns during the period , as seen from the estimate of the lower tail dependence that is significant only at the 10% level. We find similar results for the Euro equity JPY pair (last column), although the differences are that the selected best copula is the Student s t with two breaks in dependence and that evidence on the hedge role of the JPY is stronger in the crisis period than in the post-crisis period. Ranaldo and Söderlind (2010) and Fatum and Yamamoto (2016) find that the role of the JPY as a safe haven currency has increased since the 2007 global financial crisis. Our results imply that this is mainly attributed to the behavior of investors holding US equities rather than investors holding UK and Euro area equities. The CHF has been a hedge currency for UK equity investors over the full sample period. This is found from the result on the UK equity CHF pair (third column), showing that the Clayton copula with no break in dependence is selected as the best copula and that its Kendall s τ is positive and statistically significant at the 1% level. Note, however, that the lower tail dependence is not statistically significant, implying that the CHF has played a hedge role rather than a safe haven role for the UK stock market. The same result is found for the Euro are stock market, which will be provided below. In addition to the CHF and the JPY, the USD qualifies as a hedge currency for the UK stock market, but the role is transient relative to the other two currencies. The linear correlation of the Student s t copula (second column) is significantly positive only for the period As will be shown below, we find the same tendency in the Euro area stock market Results for pairs whose best copulas are time-varying dependence copulas It is found that the best copulas for alternative six equity-currency pairs belong to the class of time-varying dependence copulas. Table 6 presents estimated coefficients in Eq.(4) for these 18

22 pairs. Time-varying linear correlations of the Normal and Student s t copulas are all highly persistent, which give the justification to include one lag of the linear correlation in the test equations of safe haven and hedge currencies (11) and (12). We design these tests to be consistent with estimations of constant dependence copulas by dividing our full sample period into three subsamples, i.e. January 1999 to July 2007 (precrisis period), August 2007 to December 2012 (crisis period), and January 2013 to December 2016 (post-crisis period). Tables 7 9 provide results for these three subsamples, respectively. The first row reports LM test statistics followed by results on the threshold model in Panel A and by results on the linear model in Panel B. However, the results on the threshold model are only reported for the equity-currency pairs that support this model by the LM test. 22 In the last row, we provide classification of currencies into safe haven and hedge currencies. The LM tests support the threshold model (11) only for four equity-currency pairs over three subsample periods, and two pairs of them show evidence on safe haven currency: the Euro equity CHF pair in the period (last column, Panel A in Table 7) and the Euro equity USD pair in the period (fifth column, Panel A in Table 8). The linear correlation of the former pair in times of high volatility ρ H is significantly positive and greater than the one in times of low volatility ρ L, satisfying conditions of safe haven currency provided in Table 2. Therefore, we can state that the CHF had been a safe haven currency for investors holding Euro area equities until the 2007 global financial crisis occurred. During the period , however, the safe haven role of the CHF was substituted by the USD as shown in Table 8 (and by the JPY but with weak evidence as reported in Table 5). This implies that Euro area equity investors perceived that two crisis episodes (the 2007 global financial crisis and the subsequent European debt crisis) would have more adverse effects on the Swiss economy than the US economy. The safe haven role of the USD during the crisis period is consistent with a phenomenon that the USD substantially appreciated in the second half of 22 The linear model is also estimated for these pairs, even though their LM tests reject the null hypothesis of the linear model. By doing so, we can gain information on the average relationship between stock returns and exchange rate changes over the period, which is needed for identifying hedge currency. 19

23 2008, even though the US was the origin of the global financial crisis (see, e.g., Fratzscher, 2009; and McCauly and McGuire, 2009). Hedge currencies are identified from results on the linear model reported in panel B of each of the tables. It is found that the CHF has been a hedge currency for Euro area stock returns throughout the full sample period: the Euro equity CHF pair has mean values of the linear correlation, ρ, that are positive and statistically significant at the 1% level in the three subsamples (last columns, Panel B in Tables 7 9). Therefore, the CHF is an indispensable currency as a hedge rather than a safe haven for the Euro area stock market over the whole sample period, which is the same conclusion as the one for the UK stock market as shown in Table 5. On the other hand, the USD qualifies as a hedge currency in the Euro area stock market only for the short period (fifth column, Panel B in Table 8). This result again is the same as the UK stock market as reported in Table 5. The CHF serves as a hedge currency for US stock returns in the post-crisis period , as seen from a positive and significant mean value of the linear correlation (third column, Panel B in Table 9). However, such evidence is weak in the period (see Table 7), and there is no evidence in the period (see Table 8). This is a different aspect compared with the hedge role of the CHF over the whole sample period in the UK and Euro area stock markets. Table 10 summarizes all of the results regarding the classification of safe haven and hedge currencies. 23 First, UK and Euro area stock markets have the same set of hedge currencies over the whole period, although they vary across three subsample periods. In particular, the CHF has served as a hedge currency throughout the full sample period. Second, the difference between the UK and the Euro area stock markets is that there is no safe haven currency in the UK market, while the Euro area stock market had safe haven currencies until 2013 (CHF in the pre-crisis period and USD in the crisis period). This might be explained by higher volatility in the Euro area stock market relative to the volatility in the UK market in the 23 In the table, we only show the results with significance levels of 1% and 5%, excluding the results with a significance level of 10% as weak evidence. 20

24 corresponding periods (see Fig.2). Third, the hedge and safe haven behavior in the US stock market appear to be different from UK and Euro stock markets. The most remarkable feature in the US market is that the JPY has become an essential currency since 2007, serving as both a safe haven and a hedge currencies. This would be partly because direct influences of the subprime loan problem and the European debt crisis were smaller on the Japanese economy than the US, the UK, and the Euro area. 4 Conclusion In this paper, we identify safe haven and hedge currencies for the US, UK, and Euro area stock markets over the period To this end, we employ a copula-based approach in which the dependence structure between stock returns and exchange rate changes is characterized. Our identification criteria differ depending on which class of copula functions, the constant dependence copula or the time-varying dependence copula, is selected as the best copula. When a constant dependence copula is selected as the best copula, we rely on the lower tail dependence for identification of safe haven currency and on the linear correlation or concordance measure for identification of hedge currency. On the other hand, when a timevarying dependence copula is selected, we utilize a threshold model and its linear version as test equations to determine safe haven and hedge currencies. Our results reveal similarities and differences in the determination of safe haven and hedge currencies across the three stock markets. First, UK and Euro area stock markets have the same set of hedge currencies. In particular, the CHF has been the most crucial currency for the two markets over the whole sample period. Second, while there is no safe haven currency in the UK stock market, the Euro area stock market had safe haven currencies until 2012 (CHF in the period and USD in the period). This might be explained by evidence that volatility of the Euro area stock market is larger than that of the UK stock market. Third, in the US stock market, the JPY has served as both a safe haven and a hedge 21

25 since the 2007 global financial crisis. This is probably because the Japanese economy was less affected directly by this crisis and the subsequent European debt crisis than the three economies. Acknowledgements Financial support from JSPS KAKENHI Grant Number 16K03746 is gratefully acknowledged. References [1] Baur, D.G., Lucey, B.M., Is Gold a Hedge or a Safe Haven? An Analysis of Stocks, Bonds and Gold. Financial Review, 45(2), [2] Baur, D.G., McDermott, T.K., Is Gold a Safe Haven? International Evidence. Journal of Banking and Finance, 34(8), [3] Botman, D., de Carvalho Filho, I., Lam, W.R., The Curious Case of the Yen as a Safe Haven Currency: A Forensic Analysis. IMF Working Papers, 13/228. [4] Coudert, V., Guillaumin, C., Raymond, H., Looking at the Other Side of Carry Trades: Are there any Safe Haven Currencies? CEPII Working Papers. [5] De Bock, R., de Carvalho Filho, I., The Behavior of Currencies during Risk-Off Episodes. Journal of International Money and Finance, 53, [6] Fatum, R., Yamamoto, Y., Intra-safe Haven Currency Behavior during the Global Financial Crisis. Journal of International Money and Finance, 66, [7] Fratzscher, M., What Explains Global Exchange Rate Movements during the Financial Crisis? Journal of International Money and Finance, 28(8),