Asymmetric Dependence, Tail Dependence, and the Time Interval over which the Variables Are Measured Byoung Uk Kang and Gunky Kim Preliminary version: August 30, 2013 Comments Welcome! Kang, byoung.kang@polyu.edu.hk, School of Accounting and Finance, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong; Kim, gkim@uow.edu.au, School of Mathematics and Applied Statistics, University of Wollongong, Wollongong, NSW 2522, Australia.
Asymmetric Dependence, Tail Dependence, and the Time Interval over which the Variables Are Measured Abstract The effect of time interval on the linear correlation coefficient between random variables is well documented in the literature. In this paper, we investigate the time interval effect on asymmetric dependence and tail dependence between random variables. We prove that when two random variables are characterized by asymmetric dependence (of any direction), the magnitude of asymmetry in their dependence structure decreases monotonically and approaches zero (i.e., symmetry) as the time interval increases. Also, when two random variables exhibit tail dependence, their tail dependence decreases monotonically and approaches zero (i.e., tail independence) as the time interval increases. Our results hold regardless of whether the variables are both additive, both multiplicative, or one is additive and the other is multiplicative.
1 Introduction When examining the association between variables, researchers often make an arbitrary choice of time intervals (e.g., daily, weekly, monthly). This choice may be dictated by data availability or by consideration for the balance between the number of observations and measurement errors, but not by theory. Given a considerable leeway and lack of theoretical guidance in selecting time intervals, it is important to ask whether and how the time interval selected affects the association between variables. Recent empirical research provides some guidance on this issue. Breymann et al. (2003) and Dias and Embrechts (2010) examine bivariate series of high-frequency exchange rates at several time intervals (from hourly to daily) and find that the dependence structure is best described by a t-copula with successively larger degrees of freedom as the time horizon increases. Kang et al. (2010) investigate, at a range of investment horizons (from one quarter to five years), the asymmetric dependence between hedge fund returns and market returns and find that the magnitude of asymmetry decreases as the investment horizon increases. However, all these results are empirical in nature and depend on a given set of data and time horizons considered. Extant theoretical work on the time interval effect on cross-variable association focuses mainly on measures of linear dependence. Levy and Schwarz (1997) study the linear correlation coefficient between two random variables that are both multiplicative over time (e.g., gross returns) and show that it decreases monotonically as the time interval increases, approaching zero in the limit. Levy et al. (2001) show a similar theoretical effect on the correlation coefficient when one random variable is additive (e.g., log returns) and the other is multiplicative. Of course, when two random variables are both additive, it is straightforward to show that the correlation coefficient is invariant to time interval. 1 It is important to note, however, that correlation coefficient is not a sufficient measure to describe cross-variable association other than joint normality or, more generally, ellipticality (see, e.g., Embrechts et al. 2002). Since there is no multiplicative variable that maintains normality over multiple time intervals, the time-horizon profile of correlation coefficient, documented by Levy and Schwarz (1997) and Levy et al. (2001), may not be informative about how variables intrinsic association changes as a function of the time interval. In this paper, we extend the extant theoretical work on the time interval effect by using copulas as descriptions of association. Copula-based approach is natural in situations where the association between variables is of primary interest and is valid regardless of whether the joint distribution of the variables is normal or not. 2 We are particularly interested in analyzing the time interval effect on nonlinear or asymmetric dependence (e.g., greater dependence for joint negative events than 1 Relatedly, Schneller (1975) and Levy et al. (2001) study the time interval effect on the linear regression coefficient of one random variable on another in the multiplicative-multiplicative and additive-multiplicative cases, respectively. Note that the regression coefficient is simply the correlation coefficient scaled by the ratio of two standard deviations. 2 Sklar (1959) s theorem states that any joint distribution can be represented in terms of the marginals and a dependence function, termed the copula. While information concerning individual variables (e.g., mean, standard deviation, skewness, kurtosis) is entirely determined by the marginal distributions, the dependence relationship between the variables is completely described by the copula. 1
for joint positive events), as this type of association has been widely observed in financial data and has important implications in portfolio allocation and risk management (see, e.g., Longin and Solnik 2001; Ang and Chen 2002; Patton 2006, 2007; Hong et al. 2007). While one might suggest to conduct this analysis by means of exceedance correlation (used, e.g., in Longin and Solnik 2001, Ang and Chen 2002, Hong et al. 2007), exceedance correlation is less ideal for our purpose as it can change across time intervals due to purely marginal aspects (e.g., skewness). Copulas allow us to abstract from the time interval effect on marginal distributions and concentrate on the effect on variables intrinsic association and asymmetries therein. In addition, copulas permit an examination of joint behavior at the tails of distribution, because measures of such tail behavior, known as tail dependence, can be directly expressed in terms of the copula associated with its joint distribution (see, e.g., Joe 1997). Thus, by examining the time-horizon profile of copula, we can also address some of the suggestions of Poon et al. (2004), who propose investigating the time-aggregating properties of extreme values and the effect of investment time horizon on tail dependence, among other issues. Our theoretical results are easy to summarize. First, when two random variables are characterized by asymmetric dependence (of any direction), the magnitude of asymmetry in their dependence structure decreases monotonically and approaches zero (i.e., symmetry) as the time interval increases. Second, when two random variables exhibit tail dependence, their tail dependence decreases monotonically and approaches zero (i.e., tail independence) as the time interval increases. Finally, all these results hold regardless of whether the variables are both multiplicative, whether one is additive and the other is multiplicative, or whether both are additive. It is worth emphasizing that our finding of asymptotic symmetry and tail independence in the multiplicative-multiplicative and additive-multiplicative cases is not a mere manifestation of asymptotic independence, which one might infer from Levy and Schwarz s (1997) and Levy et al. s (2001) finding of asymptotic zero correlation. First of all, zero correlation coefficient does not necessarily mean independence (the converge is always true, however). 3 Furthermore, we show that the dependence structure between two random variables converges in the limit to the Gaussian copula, rather than to the independence copula. It is interesting to note that the dependence parameter of the limiting Gaussian copula is equal to the single-period correlation coefficient between the two random variables (or their logarithms). Hence, unless the single-period correlation coefficient is zero, the variables in the limit cannot be treated as independent in the cases considered by Levy and Schwarz (1997) and Levy et al. (2001). Perhaps our most striking result is that the time interval effect arises even in the additiveadditive case, which has potentially important implications for the study of term structure of optimal portfolio allocation across assets. The asset allocation literature, which typically works with log returns in modeling return dynamics (hence with additive variables), shows that predictability of asset returns can drive a wedge between the asset allocation strategies of short- and long- 3 One can easily think of two variables that are certainly related but whose correlation coefficient is zero (e.g., Y = X 2, where X N(0, 1)). 2
term investors (see, e.g., Campbell and Viceira 2002, 2005). Our results suggest that even when returns are unpredictable, asset allocation recommendations can still be a function of the investment horizon, through the effects of changing portfolio downside risk and tail risk across investment horizons. The remainder of this paper is organized as follows. Section 2 fixes notations for each of the three cases considered. Section 3 provides some simulation examples to illustrate our theoretical results. Section 3 formally presents our theoretical results. And Section 4 concludes. 2 Notations Since we consider all three different cases considered separately in the literature, we find it useful to fix some notations first. 2.1 The Additive-Additive Case Let (X 1, Y 1 ), (X 2, Y 2 ),... be a sequence of independent, identically distributed (i.i.d) pairs of additive variables. Let the expected value of X and Y, respectively, be E(X) = µ X and E(Y ) = µ Y. The variances are denoted by Var(X) = σ 2 X and Var(Y ) = σ2 Y. We define two new variables, W n and V n, to denote an n-fold increase of the differencing interval. These variables are given by W n = X 1 + X 2 + + X n and V n = Y 1 + Y 2 + + Y n. We denote the joint distribution of (W n, V n ) by H aa (W n, V n ) = C aa (F a (W n ), G a (V n )), (1) where F a denotes the marginal distribution of W n, G a denotes the marginal distribution of V n, and C aa denotes the copula that couples the marginals into the join distribution (Sklar, 1959). The linear correlation coefficient between W n and V n is denoted by ρ aa. 2.2 The Multiplicative-Multiplicative Case Let (X 1, Y 1 ), (X 2, Y 2 ),... be a sequence of i.i.d pairs of multiplicative variables. As a multiplicative framework is only appropriate for positive variables, assume X 1 > 0 and Y 1 > 0, as in Levy and Schwarz (1997). We define two new variables, W n and V n, to denote an n-fold increase of the differencing interval. These variables are given by W n = X 1 X 2 X n and V n = Y 1 Y 2 Y n. 3
We denote the joint distribution of (W n, V n) by H mm (W n, V n) = C mm (F m (W n), G m (V n)), (2) where F m denotes the marginal distribution of W n, G m denotes the marginal distribution of V n, and C mm denotes the copula that couples the marginals into the join distribution. The linear correlation coefficient between W n and V n is denoted by ρ mm. 2.3 The Additive-Multiplicative Case Let (X 1, Y 1 ), (X 2, Y 2 ),... be a sequence of i.i.d pairs of additive and multiplicative variables. We denote the joint distribution of (W n, V n) by H am (W n, V n) = C am (F a (W n ), G m (V n)), (3) where C am denotes the copula that couples the marginals F a and G m into the join distribution. The linear correlation coefficient between W n and V n is denoted by ρ am. Throughout, we drop the superscript on single-period distribution functions and dependence measures (i.e., when n = 1). Also, for notational convenience, we sometimes omit the subscripts aa, mm, and am when doing so causes no confusion. 3 Illustrative Examples Before formally presenting our results, we first provide some simulation examples to illustrate our main theoretical findings. Taking as an example the additive-additive case, our simulation exercise proceeds as follows: First, for a given n, we generate n independent draws of (X, Y ) from a bivariate data characterized by asymmetric dependence and tail dependence (see below). Compounding these n draws amounts to generating one draw of (W n, V n ) from the density function corresponding to H aa. We repeat this step 500,000 times to create the simulated joint distributions of n-period data. Based on the simulated data, we then compute various dependence measures such as the linear correlation coefficient, quantile dependence, and tail dependence. Finally, we repeat this procedure for a wide range of n to reveal the profile of the selected dependence measures across time intervals. For this illustration, we use monthly returns on the market portfolio and on the ten size portfolios from July 1963 to December 2012. We know from Hong et al. (2007) that the dependence structure between the market portfolio and the smallest size portfolio is asymmetric in that returns exhibit stronger correlations for downside moves than upside moves. Ang and Chen (2002) and Patton (2004) also document the same form of asymmetries in a similar data set. The market and size portfolios are constructed using all NYSE/AMEX/NASDAQ stocks and are taken from Professor 4
Ken French s web site. risk-free rate. Following Ang and Chen (2002), all risky returns are in excess of the Figure 1 illustrates the time interval effect on the linear correlation coefficient between two random variables, when their joint distribution is characterized by asymmetric dependence. Since Levy and Schwarz (1997) and Levy et al. (2001) do not make any distributional assumption in deriving their results, we should see that their results continue to hold here. As expected, Figure 1 shows that ρ mm and ρ am decrease monotonically as n increases, and that ρ aa is not affected by the selected time interval. Figure 1 about here Figure 2 computes, for each n, the sum of the squared differences between quantile dependences at q and those at 1 q, using q = {0.1, 0.2, 0.3, 0.4, 0.5}. 4 Quantile dependence at a threshold level q is defined as λ aa (q) Pr[F a (W n ) q G a (V n ) q], if q 0.5, Pr[F a (W n ) q G a (V n ) q], if q 0.5, C aa (q,q) q, if q 0.5, 1 2q+C aa (q,q) 1 q, if q 0.5, for the additive-additive case; the quantile dependence for the multiplicative-multiplicative and additive-multiplicative cases can be similarly defined as above, for example, by using m or mm in place of a or aa for the multiplicative-multiplicative case. Clearly, the magnitude of asymmetry in the dependence structure decreases monotonically and approaches zero as the time interval increases, illustrating our first main result. (4) Figure 2 about here Figure 3 plots the tail dependence between the simulated data against n. Tail dependence can be thought of as the limit of quantile dependence as q 0 or q 1: λ L λ U lim q 0 lim q 1 C (q, q), (5) q 1 2q + C (q, q). 1 q Here, we estimate tail dependence nonparametrically based directly on extreme observations (as in, e.g., Schmidt and Stadtmüller 2006). The threshold level used is the 1% (i.e., q = 0.01), which corresponds to 5,000 observations in each tail. Figure 3 illustrates our second main result: The tail 4 This is essentially Patton s (2012) statistic for asymmetric dependence, with a bootstrap estimate of V λ replaced by the identify matrix to facilitate direct comparison across data sets. 5
dependence decreases monotonically and approaches zero as the time interval increases. Finally, Figures 2 and 3 also illustrate that the observed cross-horizon patterns do not change regardless of whether the variables are both additive, both multiplicative, or one is additive and the other is multiplicative, which is our final main result. Figure 3 about here 4 The Main Results Like Levy and Schwarz (1997) and Levy et al. (2001), we proceed in two steps: First, we establish the asymptotic result (e.g., lim n λ L = 0) and then the monotonicity (e.g., λ / n 0). Unlike L the previous studies, however, our theoretical approach deals with the entire dependence structure, rather than a measure of it. 4.1 The Additive-Additive Case The Central Limit Theorem (CLT) states that, as n, ( (W n nµ X )/ nσ X (V n nµ Y )/ nσ Y ) d N [( ) ( 0, 0 1 ρ aa ρ aa 1 )]. (6) Thus, the asymptotic copula of (W n nµ X )/ nσ X and (V n nµ Y )/ nσ Y is the Gaussian copula with dependence parameter equal to ρ aa, regardless of the single-period copula of X and Y. This result can be easily extended to the asymptotic copula of W n and V n. Proposition 1. Let (X 1, Y 1 ), (X 2, Y 2 ),... be a sequence of i.i.d. bivariate random vectors that satisfies the conditions of a CLT. Then, as n, ) C aa (u, v) Φ ρaa (Φ 1 (u), Φ 1 (v), (7) where Φ ρ is the bivariate standard normal cdf with correlation ρ. Proof. Let W n = (W n nµ X )/ nσ X and Ṽn = (V n nµ Y )/ nσ Y. The cdf s of Wn and Ṽn are denoted by using tilde on the corresponding cdf s of W n and V n. Then, by a CLT, as n, H aa ( W n, Ṽn) Φ ρaa ( W n, Ṽn). (8) Note that the asymptotic copula of ( W n, Ṽn) is the Gaussian copula whose dependence parameter is ρ aa. Also note that the copula of ( W n, Ṽn) is the same as that of (W n, V n ) for all n (including the limit) because W n and Ṽn, respectively, are strictly increasing in W n and V n. Therefore, as n, ) C aa (u, v) Φ ρaa (Φ 1 (u), Φ 1 (v). 6
There are two important implications from this proposition: First, regardless of its direction, the asymmetry in the dependence structure will disappear as we add up the variables because their copula converges to a Gaussian copula. Second, for this reason, their tail dependence (if any) will also disappear as n. Let us assume that the copula of (W n, V n ), for a finite n, is a mixture of the single-period copula and its limiting copula; that is, ) C aa (u, v) = (1 α n ) C aa (u, v) + α n Φ ρaa (Φ 1 (u), Φ 1 (v), (9) where α n : {1, 2,..., } [0, 1] is a function of n. Under this assumption, our proof for the monotonicity boils down to showing that αn n 0. Although the true dependence structure may not be perfectly described by the mixture model, our unreported simulation study confirms that this model works well. 5 Proposition 2. Under the mixture model in (9), α n is monotonically increasing in n. Proof. By the Berry-Esseen theorem and by Proposition 1, we have sup C (u,v) [0,1] 2 aa (u, v) Φ ρaa (Φ 1 (u), Φ 1 (v)) = O(n 1/2 ), for all n, where Φ is the univariate standard normal cdf. Under the mixture model in (9), it follows that sup C (u,v) [0,1] 2 = sup (u,v) [0,1] 2 = (1 α n ) sup aa (u, v) Φ ρaa (Φ 1 (u), Φ (v)) 1 (1 α n ) C aa (u, v) + α n Φ ρaa (Φ 1 (u), Φ 1 (v) (u,v) [0,1] 2 Note that sup (u,v) [0,1] 2 C aa (u, v) Φ ρaa (Φ 1 (u), Φ (v)) 1. Therefore, α n is monotonically increasing in n. ) ( Φ ρaa Φ 1 (u), Φ (v)) 1 C aa (u, v) Φ ρaa (Φ 1 (u), Φ 1 (v)) = O(1). Hence, 1 αn = O(n 1/2 ). 4.2 The Multiplicative-Multiplicative Case Proposition 3. Let (ln X 1, ln Y 1 ), (ln X 2, ln Y 2 ),... be a sequence of i.i.d. bivariate random vectors that satisfies the conditions of a CLT. Then, as n, ( ) C mm(u, v) Φ ρln X ln Y Φ 1 (u), Φ 1 (v), (10) where ρ ln X ln Y is the linear correlation coefficient between ln X and ln Y. 5 The simulation results are readily available from the authors upon request. 7
Proof. From the definition, it follows that ln W n = ln X 1 + +ln X n and ln V n = ln Y 1 + +ln Y n. Then, by Proposition 1, as n, the copula of (ln W n, ln V n) becomes the Gaussian copula with dependence parameter equal to ρ ln X ln Y. Note that the copula of (W n, V n) is the same as that of (ln W n, ln V n) for all n (including the limit) because ln W n and ln V n, respectively, are strictly increasing in W n and V n. Therefore, the asymptotic copula of (W n, V n) is also the Gaussian copula with dependence parameter equal to ρ ln X ln Y. Proposition 4. Suppose that the copula of (ln W n, ln V n), for a finite n, is a mixture of the singleperiod copula and its limiting copula with a function of n, β n : {1, 2,..., } [0, 1], being the weight on the latter. Under such a model, by Proposition 2, β n is monotonically increasing in n. Since the copula of (W n, V n) is the same as that of (ln W n, ln V n) for all n, by Proposition 4, the degree of asymmetry or tail dependence in C mm will monotonically converge to zero as we compound the variables. 4.3 The Additive-Multiplicative Case Proposition 5. Let (X 1, ln Y 1 ), (X 2, ln Y 2 ),... be a sequence of i.i.d. bivariate random vectors that satisfies the conditions of a CLT. Then, as n, ( ) C am(u, v) Φ ρx ln Y Φ 1 (u), Φ 1 (v), (11) where ρ X ln Y is the linear correlation coefficient between X and ln Y. Proof. From the definition, it follows that ln V n = ln Y 1 + + ln Y n. Then, by Proposition 1, as n, the copula of (W n, ln V n) becomes the Gaussian copula with dependence parameter equal to ρ X ln Y. Note that the copula of (W n, V n) is the same as that of (W n, ln V n) for all n (including the limit) because ln V n is strictly increasing in V n. Therefore, the asymptotic copula of (W n, V n) is also the Gaussian copula with dependence parameter equal to ρ X ln Y. Proposition 6. Suppose that the copula of (W n, ln V n), for a finite n, is a mixture of the singleperiod copula and its limiting copula with a function of n, γ n : {1, 2,..., } [0, 1], being the weight on the latter. Under such a model, γ n is monotonically increasing in n. Since the copula of (W n, V n) is the same as that of (W n, ln V n) for all n, by Proposition 6, the degree of asymmetry or tail dependence in C am will monotonically converge to zero as we compound the variables. 5 Concluding Remarks The effect of time interval on the linear correlation coefficient between two random variables is well documented in the literature. In this paper, we investigate the time interval effect on asymmetric dependence and tail dependence between random variables. We prove that when two random 8
variables are characterized by asymmetric dependence (of any direction), the magnitude of asymmetry in their dependence structure decreases monotonically and approaches zero (i.e., symmetry) as the time interval increases. Also, when two random variables exhibit tail dependence, their tail dependence decreases monotonically and approaches zero (i.e., tail independence) as the time interval increases. Our results hold regardless of whether the variables are both additive, both multiplicative, or one is additive and the other is multiplicative. 9
References Ang, A., J. Chen. 2002. Asymmetric correlations of equity portfolios. J. Financial Econom. 63 443 494. Breymann, W., A. Dias, P. Embrechts. 2003. Dependence structures for multivariate high-frequency data in finance. Quant. Finance 3(1) 1 14. Campbell, J. Y., L. M. Viceira. 2002. Strategic Asset Allocation: Portfolio Choice for Long-Term Investors. Oxford University Press, Oxford, UK. Campbell, J. Y., L. M. Viceira. 2005. The term structure of the risk-return trade-off. Financial Anal. J. 61(1) 34 44. Dias, A., P. Embrechts. 2010. Modeling exchange rate dependence dynamics at different time horizons. J. Internat. Money Finance 29 1687 1705. Embrechts, P., A. J. McNeil, D. Straumann. 2002. Correlation and dependence in risk management: Properties and pitfalls. In Risk Management: Value at Risk and Beyond, M. Dempster, ed. Cambridge University Press, Cambridge, UK. Hong, Y., J. Tu, G. Zhou. 2007. Asymmetries in stock returns: Statistical tests and economic evaluation. Rev. Financial Stud. 20(5) 1547 1581. Joe, H. 1997. Multivariate Models and Dependence Concepts. Chapman & Hall, London, UK. Levy, H., I. Guttman, I. Tkatch. 2001. Regression, correlation, and the time interval: Additive-multiplicative framework. Management Sci. 47(8) 1150 1159. Levy, H., G. Schwarz. 1997. Correlation and the time interval over which the variables are measured. J. Econometrics 76 341 350. Longin, F., B. Solnik. 2001. Extreme correlation of international equity markets. J. Finance 56(2) 649 676. Kang, B. U., F. In, G. Kim, T. S. Kim. 2010. A longer look at the asymmetric dependence between hedge funds and the equity market. J. Financial Quant. Anal. 45(3) 763 789. Patton, A. J. 2004. On the out-of-sample importance of skewness and asymmetric dependence for asset allocation. J. Financial Econometrics 2(1) 130 168. Patton, A. J. 2006. Modelling asymmetric exchange rate dependence. Internat. Econom. Rev. 47(2) 527 556. 10
Patton, A. J. 2012. Copula methods for forecasting multivariate time series. In Handbook of Economic Forecasting, G. Elliott, A. Timmermann, eds. Elsevier, Oxford. Poon, S.-H., M. Rockinger, J. Tawn. 2004. Extreme value dependence in financial markets: Diagnostics, models, and financial implications. Rev. Financial Stud. 17(2) 581 610. Schmidt, R., U. Stadtmüller. 2006. Non-parametric estimation of tail dependence. Scand. J. Statist. 33 307 335. Schneller, M. I. 1975. Regression analysis for multiplicative phenomena and its implication for the measurement of investment risk. Management Sci. 22(4) 422 426. Sklar, A. 1959. Fonctions de répartition à n dimensions et leurs marges. Publications de l Institut Statistique de l Université de Paris 8 229 231. 11
FIGURE 1 Time Interval Effect on the Linear Correlation Coefficient 0.85 0.8 0.75 Correlation 0.7 0.65 0.6 ρ aa 0.55 ρ mm ρ am 0.5 0 50 100 150 200 250 300 Time Interval Figure 1 shows the linear correlation coefficients between the returns on the market portfolio and the smallest size decile portfolio, at various time intervals. The data used in this figure is generated using historical simulation based on monthly returns from July 1963 to December 2012. 12
FIGURE 2 Sum of the Squared Differences between Quantile Dependences at q and those at 1 q, using q = {0.1, 0.2, 0.3, 0.4, 0.5}. Sum of Squared Differences between Quantile Dependences 0.12 0.1 0.08 0.06 0.04 0.02 SSD aa SSD mm SSD am 0 0 50 100 150 200 250 300 350 400 450 500 Time Interval Figure 2 shows, at various time intervals, the sum of the squared differences between quantile dependences at q and those at 1 q, using q = {0.1, 0.2, 0.3, 0.4, 0.5}. The data used in this figure is generated using historical simulation based on monthly returns on the market portfolio and the smallest size decile portfolio from July 1963 to December 2012. 13
FIGURE 3 Time Interval Effect on the Tail Dependence Lower Tail Dependence 1 0.8 0.6 0.4 0.2 λ Laa λ Lmm λ Lam 0 0 50 100 150 200 250 300 Time Interval Upper Tail Dependence 1 0.8 0.6 0.4 0.2 λ Uaa λ Umm λ Uam 0 0 50 100 150 200 250 300 Time Interval Figure 3 shows the 1% and 99% quintile dependences between the market portfolio and the smallest size decile portfolio, at various time intervals. The data used in this figure is generated using historical simulation based on monthly returns from July 1963 to December 2012. 14