When is a copula constant? A test for changing relationships

Transcription

1 When is a copula constant? A test for changing relationships Fabio Busetti and Andrew Harvey Bank of Italy and University of Cambridge November 2007 usetti and Harvey (Bank of Italy and University of Cambridge) When is a copula constant? November / 54

2 Introduction Understanding and measuring the relationship between movements in di erent assets plays a key role in designing a portfolio. The multivariate normal distribution is not usually suitable for this task for two reasons: asset returns are not normally distributed, and, their comovements are not adequately captured by correlation coe cients. The second of these points has led to an explosion of interest in copulas. A copula models the relationships between variables independently of their marginal distributions. It does so by means of a joint distribution function with uniform marginals, obtained from the distribution functions of the original marginals. Busetti and Harvey (Bank of Italy and University of Cambridge) When is a copula constant? November / 54

3 The stock returns data used as illustrations are General Motors and IBM daily returns from April 7th, 1986, to April 7th, GM IBM Busetti and Harvey (Bank of Italy and University of Cambridge) When is a copula constant? November / 54

4 Ranks of IBM returns Figure: Domain of the empirical copula for GM and IBM Busetti and Harvey (Bank of Italy and University of Cambridge) When is a copula constant? November / 54 Empirical copula for GM and IBM Instead of considering the original variables, we consider the pairs of corresponding ranks. A plot of these ranks, re-scaled by dividing by one plus the sample size, forms the domain of the empirical copula Ranks of GM returns

5 The strength of the relationship between two variables can be measured by Kendall s-tau. Kendall s-tau is computed from ranks and is more appropriate as a measure of concordance than a correlation coe cient. The dependence parameter for some simple copulas can be shown to be a function of Kendall s-tau. usetti and Harvey (Bank of Italy and University of Cambridge) When is a copula constant? November / 54

6 The strength of the relationship between two variables can be measured by Kendall s-tau. Kendall s-tau is computed from ranks and is more appropriate as a measure of concordance than a correlation coe cient. The dependence parameter for some simple copulas can be shown to be a function of Kendall s-tau. If the relationship between two variables changes over time then Kendall s Tau will change when computed from di erent subsamples. usetti and Harvey (Bank of Italy and University of Cambridge) When is a copula constant? November / 54

7 The strength of the relationship between two variables can be measured by Kendall s-tau. Kendall s-tau is computed from ranks and is more appropriate as a measure of concordance than a correlation coe cient. The dependence parameter for some simple copulas can be shown to be a function of Kendall s-tau. If the relationship between two variables changes over time then Kendall s Tau will change when computed from di erent subsamples. Evidence that this might happen is provided by Van Der Goorbergha, Genest and Werker (2005) and Patton (2006). Cherubini et al (p.179) report time-varying correlation in a Gaussian copula. usetti and Harvey (Bank of Italy and University of Cambridge) When is a copula constant? November / 54

8 To get an idea of possible changes in the relationship we divided the rst 3000 observations up into 6 group of 500, that is 1 to 500, 501 to 1000, and so on. This leaves the nal group from 3001 to Busetti and Harvey (Bank of Italy and University of Cambridge) When is a copula constant? November / 54

9 To get an idea of possible changes in the relationship we divided the rst 3000 observations up into 6 group of 500, that is 1 to 500, 501 to 1000, and so on. This leaves the nal group from 3001 to The correlation coe cients and Kendall s tau are given in the table below. It is interesting that the correlations are higher at the beginning than at the end. There certainly appears to be a considerable degree of movement, but we need to know if it is statistically signi cant r Tau usetti and Harvey (Bank of Italy and University of Cambridge) When is a copula constant? November / 54

10 1.0 Obs: 1 500; τ = 0.37; ρ = Obs: ; τ = 0.34; ρ = Obs: ; τ = 0.26; ρ = Obs: ; τ = 0.10; ρ = Obs: ; τ = 0.15; ρ = Obs: ; τ = 0.08; ρ = Obs: ; τ = 0.22; ρ = Figure: Empirical copulas for GM and IBM usetti and Harvey (Bank of Italy and University of Cambridge) When is a copula constant? November / 54

11 The aim is to develop a suite of tests for changes in di erent parts of the copula, as well as an overall test for changing dependence. These tests do not require a model for the copula usetti and Harvey (Bank of Italy and University of Cambridge) When is a copula constant? November / 54

12 The aim is to develop a suite of tests for changes in di erent parts of the copula, as well as an overall test for changing dependence. These tests do not require a model for the copula They can be regarded as an extension of the stationarity tests for time-varying quantiles proposed in Busetti and Harvey (2007). usetti and Harvey (Bank of Italy and University of Cambridge) When is a copula constant? November / 54

13 The aim is to develop a suite of tests for changes in di erent parts of the copula, as well as an overall test for changing dependence. These tests do not require a model for the copula They can be regarded as an extension of the stationarity tests for time-varying quantiles proposed in Busetti and Harvey (2007). The time-varying quantile test are constructed from time series of indicator variables. usetti and Harvey (Bank of Italy and University of Cambridge) When is a copula constant? November / 54

14 Consider the proportion of cases in which the observations lie below a particular quantile in both series. The question is whether this proportion is constant over time. usetti and Harvey (Bank of Italy and University of Cambridge) When is a copula constant? November / 54

15 Consider the proportion of cases in which the observations lie below a particular quantile in both series. The question is whether this proportion is constant over time. Looking at di erent quantiles allows us to focus on di erent aspects of the relationship. Busetti and Harvey (Bank of Italy and University of Cambridge) When is a copula constant? November / 54

16 Consider the proportion of cases in which the observations lie below a particular quantile in both series. The question is whether this proportion is constant over time. Looking at di erent quantiles allows us to focus on di erent aspects of the relationship. Changing dependence may be of particular importance in the lower tails, because of its relevance for the value at risk (VaR) of a portfolio. Busetti and Harvey (Bank of Italy and University of Cambridge) When is a copula constant? November / 54

17 Consider the proportion of cases in which the observations lie below a particular quantile in both series. The question is whether this proportion is constant over time. Looking at di erent quantiles allows us to focus on di erent aspects of the relationship. Changing dependence may be of particular importance in the lower tails, because of its relevance for the value at risk (VaR) of a portfolio. We might also consider the question of whether a single test, based on the quadrants formed from the two medians, is useful as an overall test of changing concordance. usetti and Harvey (Bank of Italy and University of Cambridge) When is a copula constant? November / 54

18 GM Q(05) Q(95) Q(25) Q(75) Figure: Time-varying quantiles tted to GM Busetti and Harvey (Bank of Italy and University of Cambridge) When is a copula constant? November / 54

19 Quantiles and Quantics Let ξ(τ) denote the τ th quantile. The probability that an observation is less than ξ(τ) is τ, where 0 < τ < 1. Given a set of T observations, y t, t = 1,.., T, the sample quantile, eξ(τ), can be obtained by sorting the observations in ascending order. usetti and Harvey (Bank of Italy and University of Cambridge) When is a copula constant? November / 54

20 Quantiles and Quantics Let ξ(τ) denote the τ th quantile. The probability that an observation is less than ξ(τ) is τ, where 0 < τ < 1. Given a set of T observations, y t, t = 1,.., T, the sample quantile, eξ(τ), can be obtained by sorting the observations in ascending order. The residuals associated with a quantile may be coded as indicators. The τ quantile indicator is τ 1, if yt < ξ(τ) IQ(y t ξ(τ)) =, t = 1,..., T (1) τ, if y t > ξ(τ) usetti and Harvey (Bank of Italy and University of Cambridge) When is a copula constant? November / 54

21 Quantiles and Quantics Let ξ(τ) denote the τ th quantile. The probability that an observation is less than ξ(τ) is τ, where 0 < τ < 1. Given a set of T observations, y t, t = 1,.., T, the sample quantile, eξ(τ), can be obtained by sorting the observations in ascending order. The residuals associated with a quantile may be coded as indicators. The τ quantile indicator is τ 1, if yt < ξ(τ) IQ(y t ξ(τ)) =, t = 1,..., T (1) τ, if y t > ξ(τ) Note that IQ(0) is not determined but we will constrain it to lie in the range [τ 1, τ]. usetti and Harvey (Bank of Italy and University of Cambridge) When is a copula constant? November / 54

22 Quantiles and Quantics Let ξ(τ) denote the τ th quantile. The probability that an observation is less than ξ(τ) is τ, where 0 < τ < 1. Given a set of T observations, y t, t = 1,.., T, the sample quantile, eξ(τ), can be obtained by sorting the observations in ascending order. The residuals associated with a quantile may be coded as indicators. The τ quantile indicator is τ 1, if yt < ξ(τ) IQ(y t ξ(τ)) =, t = 1,..., T (1) τ, if y t > ξ(τ) Note that IQ(0) is not determined but we will constrain it to lie in the range [τ 1, τ]. If ξ(τ) is the unique population τ quantile and y has a continuous positive density in the neighbourhood of ξ(τ), then IQ(y t ξ(τ)) has a mean of zero and a variance of τ(1 τ). usetti and Harvey (Bank of Italy and University of Cambridge) When is a copula constant? November / 54

23 Quantiles and Quantics Let ξ(τ) denote the τ th quantile. The probability that an observation is less than ξ(τ) is τ, where 0 < τ < 1. Given a set of T observations, y t, t = 1,.., T, the sample quantile, eξ(τ), can be obtained by sorting the observations in ascending order. The residuals associated with a quantile may be coded as indicators. The τ quantile indicator is τ 1, if yt < ξ(τ) IQ(y t ξ(τ)) =, t = 1,..., T (1) τ, if y t > ξ(τ) Note that IQ(0) is not determined but we will constrain it to lie in the range [τ 1, τ]. If ξ(τ) is the unique population τ quantile and y has a continuous positive density in the neighbourhood of ξ(τ), then IQ(y t ξ(τ)) has a mean of zero and a variance of τ(1 τ). The sample τ quantile indicator, or τ quantic, at time t is IQ(y t eξ(τ)), t = 1,..., T. usetti and Harvey The (Bank quantics of Italy and sum University to zero. of Cambridge) When is a copula constant? November / 54

24 Tests based on quantics A test of the null hypothesis that the τ based on the quantics. quantile is constant may be If the alternative hypothesis is that the τ quantile follows a random walk, a modi ed version of the stationarity test of Nyblom and Mäkeläinen (1983) is appropriate. η τ (Q) = 1 T 2 τ(1 τ) T t=1 t IQ(y t i=1 eξ(τ))! 2 Asymptotic distribution is Cramér-von Mises (CvM); the 5% critical value is Busetti and Harvey (Bank of Italy and University of Cambridge) When is a copula constant? November / 54

25 Nyblom and Harvey (2001) show that the test has high power against an integrated random walk, which when tted yields a curve close to a cubic spline. usetti and Harvey (Bank of Italy and University of Cambridge) When is a copula constant? November / 54

26 Nyblom and Harvey (2001) show that the test has high power against an integrated random walk, which when tted yields a curve close to a cubic spline. Harvey and Streibel (1998) show that it has an LBI interpretation against a highly persistent stationary rst-order autoregression usetti and Harvey (Bank of Italy and University of Cambridge) When is a copula constant? November / 54

27 Nyblom and Harvey (2001) show that the test has high power against an integrated random walk, which when tted yields a curve close to a cubic spline. Harvey and Streibel (1998) show that it has an LBI interpretation against a highly persistent stationary rst-order autoregression Nyblom (1989) shows test has power against a break, or breaks, in an otherwise stationary time series. usetti and Harvey (Bank of Italy and University of Cambridge) When is a copula constant? November / 54

28 Nyblom and Harvey (2001) show that the test has high power against an integrated random walk, which when tted yields a curve close to a cubic spline. Harvey and Streibel (1998) show that it has an LBI interpretation against a highly persistent stationary rst-order autoregression Nyblom (1989) shows test has power against a break, or breaks, in an otherwise stationary time series. Long-run variance corrects size when quantile is stationary. usetti and Harvey (Bank of Italy and University of Cambridge) When is a copula constant? November / 54

29 Joint test for the time invariance of a distribution A joint test to see if a group of N quantiles show evidence of changing over time can be based on a generalisation of (2), namely η τ (Q; N) = T 2 " # 0 " # T t t IQ i S 1 IQ i, (2) t=1 i=1 i=1 where the j th element of the N 1 vector IQ i is IQ(y i eξ(τ j )) and the jk th element of the N N covariance matrix S is τ j (1 τ k ) for τ k > τ j. Under the null hypothesis of IID observations, the limiting distribution of η τ (Q; N) is Cramér-von Mises with N degrees of freedom, denoted CvM(N). Busetti and Harvey (Bank of Italy and University of Cambridge) When is a copula constant? November / 54

30 Bivariate series Consider a bivariate series, y 1t and y 2t, t = 1,..., T. By converting to ranks we can obtain the sample quantiles and the empirical copula. The empirical copula yields the proportion of cases in which both observations in a pair are less than, or equal to, particular quantiles, eξ(τ 1 ) and eξ(τ 2 ). This proportion will be denoted as C T (τ 1, τ 2 ). The tests in Busetti and Harvey (2007) were designed to detect movements in the quantiles of the distributions of univariate series. If there are two series and their marginal distributions are constant, we can move on to address the question of whether their copula is changing over time. As with univariate series, the tests are based on indicators, but we now have to consider combinations of quantiles from the two series. Busetti and Harvey (Bank of Italy and University of Cambridge) When is a copula constant? November / 54

31 Bivariate series To simplify matters, we will set τ 1 = τ 2 = τ, 0 < τ < 1, and explore the possibility of movements C (τ, τ), the probability that both observations lie below their respective τ quantiles. usetti and Harvey (Bank of Italy and University of Cambridge) When is a copula constant? November / 54

32 Bivariate series To simplify matters, we will set τ 1 = τ 2 = τ, 0 < τ < 1, and explore the possibility of movements C (τ, τ), the probability that both observations lie below their respective τ quantiles. Since there are four quadrants associated with a given τ, some attention needs to be paid to the other three. The probability that both observations lie above their respective τ quantiles is known as the survival function, and denoted C (τ, τ). It is equal to 1 2τ + C (τ, τ). The probabilities of being in the other two quadrants are the same, namely τ C (τ, τ). Busetti and Harvey (Bank of Italy and University of Cambridge) When is a copula constant? November / 54

33 Copulas and concordance The value of C (τ, τ) indicates the strength of dependence at τ. With perfect concordance, C (τ, τ) = τ and C (τ, τ) = 1 τ, while with perfect discordance both these probabilities are zero. For independent series, C (τ, τ) = τ 2, while C (τ, τ) = (1 τ) 2, and the probabilities of being in one of the other quadrants are both τ(1 τ). usetti and Harvey (Bank of Italy and University of Cambridge) When is a copula constant? November / 54

34 Copulas and concordance The value of C (τ, τ) indicates the strength of dependence at τ. With perfect concordance, C (τ, τ) = τ and C (τ, τ) = 1 τ, while with perfect discordance both these probabilities are zero. For independent series, C (τ, τ) = τ 2, while C (τ, τ) = (1 τ) 2, and the probabilities of being in one of the other quadrants are both τ(1 τ). Summing C (τ, τ) and C (τ, τ) gives a simple measure of what Kruskal (1958, p 818) calls quadrant association. usetti and Harvey (Bank of Italy and University of Cambridge) When is a copula constant? November / 54

35 As an illustration, the Clayton copula is de ned in terms of uniform variates, u 1 and u 2, as C (u 1, u 2 ) = (u θ 1 + u θ 2 1) 1/θ, θ 2 [ 1, ) and for a given τ, C (τ, τ)/τ = 2 τ, C (τ, τ) ' τ/2. τ θ 1/θ. For θ = 1 and a small usetti and Harvey (Bank of Italy and University of Cambridge) When is a copula constant? November / 54

36 As an illustration, the Clayton copula is de ned in terms of uniform variates, u 1 and u 2, as C (u 1, u 2 ) = (u θ 1 + u θ 2 1) 1/θ, θ 2 [ 1, ) and for a given τ, C (τ, τ)/τ = 2 τ, C (τ, τ) ' τ/2. τ θ 1/θ. For θ = 1 and a small The Clayton copula is asymmetric in that the upper tail probability for 1 τ, that is the survival copula, C (1 τ, 1 τ), is not the same as C (τ, τ). The relationship is C (1 τ, 1 τ) = 1 2(1 τ) + C (1 τ, 1 τ) = 2τ 1 + (1 τ)(2 (1 τ) θ ) 1/θ For θ = 1, we get 2τ 1 + (1 τ)(1 + τ) 1. This is.018 for τ =.1, while for τ =.05, it is These probabilities are much smaller than those for the lower tail. usetti and Harvey (Bank of Italy and University of Cambridge) When is a copula constant? November / 54

37 Clayton Copula Figure: CDF of the Clayton copula Figure: PDF of the Clayton copula usetti and Harvey (Bank of Italy and University of Cambridge) When is a copula constant? November / 54

38 Bivariate series The coe cients of tail dependence are measures of pairwise dependence that depend on the copula. The coe cient of lower tail dependence is lim τ!0 C (τ, τ)/τ, while the coe cient of upper tail dependence is lim τ!1 C (τ, τ)/(1 τ). usetti and Harvey (Bank of Italy and University of Cambridge) When is a copula constant? November / 54

39 Bivariate series The coe cients of tail dependence are measures of pairwise dependence that depend on the copula. The coe cient of lower tail dependence is lim τ!0 C (τ, τ)/τ, while the coe cient of upper tail dependence is lim τ!1 C (τ, τ)/(1 τ). If two variables have a bivariate normal distribution, they are asymptotically independent in the tails as the coe cients of tail dependence are both zero. On the other hand, a t-copula does exhibit tail dependence. For θ > 0, the Clayton copula exhibits lower tail dependence, that is lim τ!0 C (τ, τ)/τ = 2 1/θ, and as θ!, this coe cient tends to one. usetti and Harvey (Bank of Italy and University of Cambridge) When is a copula constant? November / 54

40 Bivariate quantics We will de ne the bivariate τ quantic - or τ bi-quantic - as BIQ(y 1t eξ 1 (τ), y 2t eξ 2 (τ)) (3) = C T (τ, τ) I (y 1t eξ 1 (τ), y 2t eξ 2 (τ)), t = 1,..., T where I (.) is the indicator function. usetti and Harvey (Bank of Italy and University of Cambridge) When is a copula constant? November / 54

41 Bivariate quantics We will de ne the bivariate τ quantic - or τ bi-quantic - as BIQ(y 1t eξ 1 (τ), y 2t eξ 2 (τ)) (3) = C T (τ, τ) I (y 1t eξ 1 (τ), y 2t eξ 2 (τ)), t = 1,..., T where I (.) is the indicator function. By construction this has a mean of zero and a variance of C T (τ, τ)(1 C T (τ, τ)). When there is no ambiguity we will abbreviate BIQ(y 1t eξ 1 (τ), y 2t eξ 2 (τ) to BIQ(τ). usetti and Harvey (Bank of Italy and University of Cambridge) When is a copula constant? November / 54

42 Bivariate quantics We will de ne the bivariate τ quantic - or τ bi-quantic - as BIQ(y 1t eξ 1 (τ), y 2t eξ 2 (τ)) (3) = C T (τ, τ) I (y 1t eξ 1 (τ), y 2t eξ 2 (τ)), t = 1,..., T where I (.) is the indicator function. By construction this has a mean of zero and a variance of C T (τ, τ)(1 C T (τ, τ)). When there is no ambiguity we will abbreviate BIQ(y 1t eξ 1 (τ), y 2t eξ 2 (τ) to BIQ(τ). Associated with BIQ(τ) are three complementary bivariate τ quantics. The rst is BIQ(y 1t > eξ 1 (τ), y 2t > eξ 2 (τ)) (4) = C T (τ, τ) I (y 1t > eξ 1 (τ), y 2t > eξ 2 (τ)) (5) usetti and Harvey (Bank of Italy and University of Cambridge) When is a copula constant? November / 54

43 We call this the survival τ BIQ(τ). quantic and use the shorthand notation usetti and Harvey (Bank of Italy and University of Cambridge) When is a copula constant? November / 54

44 We call this the survival τ quantic and use the shorthand notation BIQ(τ). Remember that C T (τ, τ) = 1 2τ + C T (τ, τ). The variance of BIQ(τ) is C T (τ, τ)(1 C T (τ, τ)). usetti and Harvey (Bank of Italy and University of Cambridge) When is a copula constant? November / 54

45 We call this the survival τ quantic and use the shorthand notation BIQ(τ). Remember that C T (τ, τ) = 1 2τ + C T (τ, τ). The variance of BIQ(τ) is C T (τ, τ)(1 C T (τ, τ)). The other two complementary τ bi-quantics are BIQ(y 1t eξ 1 (τ), y 2t > eξ 2 (τ)) (6) = C T (τ, τ) I (y 1t eξ 1 (τ), y 2t > eξ 2 (τ)) (7) and BIQ(y 1t > eξ 1 (τ), y 2t eξ 2 (τ)) (8) = C T (τ, τ) I (y 1t > eξ 1 (τ), y 2t eξ 2 (τ)) (9) where C T (τ, τ) = τ C T (τ, τ). usetti and Harvey (Bank of Italy and University of Cambridge) When is a copula constant? November / 54

46 Tests We now consider tests of the null hypothesis that C (τ, τ) is constant. Test statistics can be formed from the bi-quantics in the same way as for the quantics. Thus, for a given τ, η τ (BQ; BB) = 1 T 2 C T (τ, τ)(1 C T (τ, τ)) T t=1! 2 t BIQ(τ) i=1 (10) Busetti and Harvey (Bank of Italy and University of Cambridge) When is a copula constant? November / 54

47 Tests We now consider tests of the null hypothesis that C (τ, τ) is constant. Test statistics can be formed from the bi-quantics in the same way as for the quantics. Thus, for a given τ, η τ (BQ; BB) = 1 T 2 C T (τ, τ)(1 C T (τ, τ)) T t=1! 2 t BIQ(τ) i=1 (10) The asymptotic distribution of this stationarity test statistic when the series are jointly stationary is Cramér von Mises. Since there are limits on the range of C (τ, τ) the alternative the alternative hypothesis is best thought of as one in which C (τ, τ) is slowly changing, or subject to breaks, rather than being a random walk. usetti and Harvey (Bank of Italy and University of Cambridge) When is a copula constant? November / 54

48 Tests We now consider tests of the null hypothesis that C (τ, τ) is constant. Test statistics can be formed from the bi-quantics in the same way as for the quantics. Thus, for a given τ, η τ (BQ; BB) = 1 T 2 C T (τ, τ)(1 C T (τ, τ)) T t=1! 2 t BIQ(τ) i=1 (10) The asymptotic distribution of this stationarity test statistic when the series are jointly stationary is Cramér von Mises. Since there are limits on the range of C (τ, τ) the alternative the alternative hypothesis is best thought of as one in which C (τ, τ) is slowly changing, or subject to breaks, rather than being a random walk. In the Monte Carlo experiments, the test based on (10) is denoted BB to signify that the indicator is unity when both observations are below their respective quantiles. The test constructed using the survival bi-quantics is denoted AA. usetti and Harvey (Bank of Italy and University of Cambridge) When is a copula constant? November / 54

49 A combined test can be constructed from any three bi-quantics, provided that C T (τ, τ) < τ. The statistic is η τ (BQ; 3) = T 2 T t=1 " # 0 " # t t BIQ i S 1 BIQ i, (11) i=1 i=1 where the elements of the 3 1 vector BIQ i are chosen from (3), (4), (6) and (8), and S is their sample covariance matrix. Under the null hypothesis of serially independent observations with a constant copula, the limiting distribution of η τ (BQ; 3) is Cramér-von Mises with 3 degrees of freedom. We can add BIQ(τ)and BIQ(τ) to give BIQ(τ) + BIQ(τ) = 1 2τ + 2C T (τ, τ) I (y 1t eξ 1 (τ), y 2t eξ 2 (τ) or y 1t > eξ 1 (τ), y 2t > A test statistic formed from this series will be denoted η τ (BQ + BQ). Since it is based on the sum of C T (τ, τ) and C T (τ, τ), we will call it the quadrant association test. Its limiting distribution under the null hypothesis is Cramér-von Mises with one degree of freedom. usetti and Harvey (Bank of Italy and University of Cambridge) When is a copula constant? November / 54

50 Single comprehensive tests of changing dependence If a single overall test is required, it may be based on the combined or quadrant association tests for τ = 0.5. usetti and Harvey (Bank of Italy and University of Cambridge) When is a copula constant? November / 54

51 Single comprehensive tests of changing dependence If a single overall test is required, it may be based on the combined or quadrant association tests for τ = 0.5. We might compare the power of these tests with a stationarity based on the series (y 1t y 1 )(y 2t y 2 ), t = 1,..., T We call this the changing correlation test. Its performance will depend on the whole joint distribution, not just the copula. usetti and Harvey (Bank of Italy and University of Cambridge) When is a copula constant? November / 54

52 Single comprehensive tests of changing dependence If a single overall test is required, it may be based on the combined or quadrant association tests for τ = 0.5. We might compare the power of these tests with a stationarity based on the series (y 1t y 1 )(y 2t y 2 ), t = 1,..., T We call this the changing correlation test. Its performance will depend on the whole joint distribution, not just the copula. The tests based on τ = 0.5 could be extended to joint tests based on a range of τ 0 s. Busetti and Harvey (Bank of Italy and University of Cambridge) When is a copula constant? November / 54

53 Bi-quantilogram The pattern of serial correlation can be captured by computing the correlograms of the bivariate τ quantics. These might be called bi-quantilograms. They are not the same as the cross-quantilograms of Linton and Whang (2007), though these may also be useful. Box-Ljung Q-statistics may be formed from the bi-quantilograms and used as an alternative to stationarity tests for assessing change in the copula. Busetti and Harvey (Bank of Italy and University of Cambridge) When is a copula constant? November / 54

54 Monte Carlo We consider three cases of data generating processes, denoted by A,B,C; the rst two are based on a time-varying Gaussian copula while the third is characterized by a Clayton copula, with lower tail dependence. In general the two degree of freedom test was dominated by the others and so we do not report the results for it. usetti and Harvey (Bank of Italy and University of Cambridge) When is a copula constant? November / 54

55 Monte Carlo We consider three cases of data generating processes, denoted by A,B,C; the rst two are based on a time-varying Gaussian copula while the third is characterized by a Clayton copula, with lower tail dependence. In general the two degree of freedom test was dominated by the others and so we do not report the results for it. The speed of convergence to the asymptotic distribution may be slow when τ is close to zero or one. The simulation experiments therefore do not include τ = 0.10 or For larger sample sizes the relative performances of the tests should be similar to what we report here for for τ =.25, 0.5 and usetti and Harvey (Bank of Italy and University of Cambridge) When is a copula constant? November / 54

56 Monte Carlo We consider three cases of data generating processes, denoted by A,B,C; the rst two are based on a time-varying Gaussian copula while the third is characterized by a Clayton copula, with lower tail dependence. In general the two degree of freedom test was dominated by the others and so we do not report the results for it. The speed of convergence to the asymptotic distribution may be slow when τ is close to zero or one. The simulation experiments therefore do not include τ = 0.10 or For larger sample sizes the relative performances of the tests should be similar to what we report here for for τ =.25, 0.5 and All tests are at the 5% signi cance level. usetti and Harvey (Bank of Italy and University of Cambridge) When is a copula constant? November / 54

57 Monte Carlo A The observations are generated from a bivariate Gaussian distribution with time varying correlation ρ t = 1 eµ t 1 + e µ, t where µ t is a random walk process obtained from Gaussian innovations with mean zero and variance equal to q 2. Logistic - so ρ t 2 ( 1, 1). usetti and Harvey (Bank of Italy and University of Cambridge) When is a copula constant? November / 54

58 Monte Carlo A The observations are generated from a bivariate Gaussian distribution with time varying correlation ρ t = 1 eµ t 1 + e µ, t where µ t is a random walk process obtained from Gaussian innovations with mean zero and variance equal to q 2. Logistic - so ρ t 2 ( 1, 1). Table (1) - Power is a maximum at τ =.5. The lower quadrant test, BB, is more powerful for τ < 0.5, while the upper quadrant test, AA, is more powerful for τ > 0.5. Furthermore the power for BB at τ is the same as the power for AA at 1 τ; this is a re ection of the symmetry properties of the Gaussian distribution. usetti and Harvey (Bank of Italy and University of Cambridge) When is a copula constant? November / 54

59 Monte Carlo A The observations are generated from a bivariate Gaussian distribution with time varying correlation ρ t = 1 eµ t 1 + e µ, t where µ t is a random walk process obtained from Gaussian innovations with mean zero and variance equal to q 2. Logistic - so ρ t 2 ( 1, 1). Table (1) - Power is a maximum at τ =.5. The lower quadrant test, BB, is more powerful for τ < 0.5, while the upper quadrant test, AA, is more powerful for τ > 0.5. Furthermore the power for BB at τ is the same as the power for AA at 1 τ; this is a re ection of the symmetry properties of the Gaussian distribution. The table shows that the bi-quantic tests are signi cantly more powerful than the Box-Ljung bi-quantilogram Q(m)-tests. usetti and Harvey (Bank of Italy and University of Cambridge) When is a copula constant? November / 54

60 The lower part of the table contains the results for the combined test, η τ (BQ; 3), the quadrant association test, η τ (BQ + BQ), and the test of changing correlation. All of them appear to be more powerful than the single quadrant bi-quantic tests. usetti and Harvey (Bank of Italy and University of Cambridge) When is a copula constant? November / 54

61 The lower part of the table contains the results for the combined test, η τ (BQ; 3), the quadrant association test, η τ (BQ + BQ), and the test of changing correlation. All of them appear to be more powerful than the single quadrant bi-quantic tests. The quadrant association test appears to be more powerful than the combined test for τ = For small values of q the maximum power is attained by the test of changing correlation, but the quadrant association test is better for q = 0.25 and above. Busetti and Harvey (Bank of Italy and University of Cambridge) When is a copula constant? November / 54

62 Figure: Table 1 Busetti and Harvey (Bank of Italy and University of Cambridge) When is a copula constant? November / 54

63 Monte Carlo B The observations are generated from a bivariate Gaussian distribution with a structural break in the correlation coe cient. We set ρ t =.75 in the rst half of the sample and ρ t =.75,.50,.25, 0 in the second half. T = 200. usetti and Harvey (Bank of Italy and University of Cambridge) When is a copula constant? November / 54

64 Monte Carlo B The observations are generated from a bivariate Gaussian distribution with a structural break in the correlation coe cient. We set ρ t =.75 in the rst half of the sample and ρ t =.75,.50,.25, 0 in the second half. T = 200. Table 2 - Again the copula bi-quantic tests are signi cantly more powerful than those based on the bi-quantilogram, but the di erences appear to be even greater. usetti and Harvey (Bank of Italy and University of Cambridge) When is a copula constant? November / 54

65 Monte Carlo B The observations are generated from a bivariate Gaussian distribution with a structural break in the correlation coe cient. We set ρ t =.75 in the rst half of the sample and ρ t =.75,.50,.25, 0 in the second half. T = 200. Table 2 - Again the copula bi-quantic tests are signi cantly more powerful than those based on the bi-quantilogram, but the di erences appear to be even greater. Broadly speaking, these results are qualitatively the same as those obtained (A). Busetti and Harvey (Bank of Italy and University of Cambridge) When is a copula constant? November / 54

66 Figure: Table 2 Busetti and Harvey (Bank of Italy and University of Cambridge) When is a copula constant? November / 54

67 Monte Carlo C Table (3) - Clayton copula with structural break in the dependence parameter. usetti and Harvey (Bank of Italy and University of Cambridge) When is a copula constant? November / 54

68 Monte Carlo C Table (3) - Clayton copula with structural break in the dependence parameter. No longer symmetry in the BB and AA tests, though the BA and AB tests do, as before, have similar power. In fact these tests are now more powerful than the the BB and AA tests. usetti and Harvey (Bank of Italy and University of Cambridge) When is a copula constant? November / 54

69 Monte Carlo C Table (3) - Clayton copula with structural break in the dependence parameter. No longer symmetry in the BB and AA tests, though the BA and AB tests do, as before, have similar power. In fact these tests are now more powerful than the the BB and AA tests. As before the bi-quantilogram tests are signi cantly less powerful. usetti and Harvey (Bank of Italy and University of Cambridge) When is a copula constant? November / 54

70 Monte Carlo C Table (3) - Clayton copula with structural break in the dependence parameter. No longer symmetry in the BB and AA tests, though the BA and AB tests do, as before, have similar power. In fact these tests are now more powerful than the the BB and AA tests. As before the bi-quantilogram tests are signi cantly less powerful. The quadrant association test is the most powerful test overall - better than the combined bi-quantic tests. usetti and Harvey (Bank of Italy and University of Cambridge) When is a copula constant? November / 54

71 Monte Carlo C Table (3) - Clayton copula with structural break in the dependence parameter. No longer symmetry in the BB and AA tests, though the BA and AB tests do, as before, have similar power. In fact these tests are now more powerful than the the BB and AA tests. As before the bi-quantilogram tests are signi cantly less powerful. The quadrant association test is the most powerful test overall - better than the combined bi-quantic tests. The test of changing correlation has very low power. Busetti and Harvey (Bank of Italy and University of Cambridge) When is a copula constant? November / 54

72 Di erent marginals would give di erent results for the changing correlation test, but unless the joint distribution is bivariate normal, it seems that a good performance cannot be guaranteed. usetti and Harvey (Bank of Italy and University of Cambridge) When is a copula constant? November / 54

73 Di erent marginals would give di erent results for the changing correlation test, but unless the joint distribution is bivariate normal, it seems that a good performance cannot be guaranteed. Taking all three sets of results together, the quadrant association test with τ = 0.5 is clearly the one to be recommended for detecting overall changes in dependence. It also seems to be the preferred test for other values of τ. usetti and Harvey (Bank of Italy and University of Cambridge) When is a copula constant? November / 54

74 Figure: Table 3 usetti and Harvey (Bank of Italy and University of Cambridge) When is a copula constant? November / 54

75 Correcting for time variation in the individual series The bi-quantic tests will tend to reject if the individual quantiles are time-varying. Even if stationary will still reject with non-parametric correction if persistent. usetti and Harvey (Bank of Italy and University of Cambridge) When is a copula constant? November / 54

76 Correcting for time variation in the individual series The bi-quantic tests will tend to reject if the individual quantiles are time-varying. Even if stationary will still reject with non-parametric correction if persistent. If the movements in quantiles are due solely to changes in volatility, the series may be adjusted by dividing by a measure of dispersion. usetti and Harvey (Bank of Italy and University of Cambridge) When is a copula constant? November / 54

77 Correcting for time variation in the individual series The bi-quantic tests will tend to reject if the individual quantiles are time-varying. Even if stationary will still reject with non-parametric correction if persistent. If the movements in quantiles are due solely to changes in volatility, the series may be adjusted by dividing by a measure of dispersion. A stochastic volatility (SV) model can be tted to each of the series and the resulting (two sided) smoothed estimates of the standard deviations used to rescale the observations. usetti and Harvey (Bank of Italy and University of Cambridge) When is a copula constant? November / 54

78 Correcting for time variation in the individual series The bi-quantic tests will tend to reject if the individual quantiles are time-varying. Even if stationary will still reject with non-parametric correction if persistent. If the movements in quantiles are due solely to changes in volatility, the series may be adjusted by dividing by a measure of dispersion. A stochastic volatility (SV) model can be tted to each of the series and the resulting (two sided) smoothed estimates of the standard deviations used to rescale the observations. Alternatively could rescale the observations by a time-varying estimate of the interquartile range, using the approach of De Rossi and Harvey (2006). usetti and Harvey (Bank of Italy and University of Cambridge) When is a copula constant? November / 54

79 Correcting for time variation in the individual series The bi-quantic tests will tend to reject if the individual quantiles are time-varying. Even if stationary will still reject with non-parametric correction if persistent. If the movements in quantiles are due solely to changes in volatility, the series may be adjusted by dividing by a measure of dispersion. A stochastic volatility (SV) model can be tted to each of the series and the resulting (two sided) smoothed estimates of the standard deviations used to rescale the observations. Alternatively could rescale the observations by a time-varying estimate of the interquartile range, using the approach of De Rossi and Harvey (2006). More generally time-varying τ quantiles could be tted, as in De Rossi and Harvey (2006), and the bi-quantics formed from them. usetti and Harvey (Bank of Italy and University of Cambridge) When is a copula constant? November / 54

80 Correcting for time variation in the individual series The bi-quantic tests will tend to reject if the individual quantiles are time-varying. Even if stationary will still reject with non-parametric correction if persistent. If the movements in quantiles are due solely to changes in volatility, the series may be adjusted by dividing by a measure of dispersion. A stochastic volatility (SV) model can be tted to each of the series and the resulting (two sided) smoothed estimates of the standard deviations used to rescale the observations. Alternatively could rescale the observations by a time-varying estimate of the interquartile range, using the approach of De Rossi and Harvey (2006). More generally time-varying τ quantiles could be tted, as in De Rossi and Harvey (2006), and the bi-quantics formed from them. For an overall quadrant association test based on τ = 0.5, there is no need to make any adjustments if the medians can be assumed constant. usetti and Harvey (Bank of Italy and University of Cambridge) When is a copula constant? November / 54

81 Monte Carlo Bivariate SV data generating process As with table 2, we consider a structural break in the correlation coe cient, setting ρ t =.75 in the rst half of the sample and ρ t =.75,.50,.25, 0 in the second half. usetti and Harvey (Bank of Italy and University of Cambridge) When is a copula constant? November / 54

82 Monte Carlo Bivariate SV data generating process As with table 2, we consider a structural break in the correlation coe cient, setting ρ t =.75 in the rst half of the sample and ρ t =.75,.50,.25, 0 in the second half. Then multiply each series by (independent) SV processes. usetti and Harvey (Bank of Italy and University of Cambridge) When is a copula constant? November / 54

83 Monte Carlo Bivariate SV data generating process As with table 2, we consider a structural break in the correlation coe cient, setting ρ t =.75 in the rst half of the sample and ρ t =.75,.50,.25, 0 in the second half. Then multiply each series by (independent) SV processes. Estimated by QML and divide by smoothed estimates of the standard deviation. Busetti and Harvey (Bank of Italy and University of Cambridge) When is a copula constant? November / 54

84 Table 4 reports the simulated rejection frequencies for the copula based tests and the changing correlation test, run at 5% signi cance level, for a sample size of T = 400 observations. usetti and Harvey (Bank of Italy and University of Cambridge) When is a copula constant? November / 54

85 Table 4 reports the simulated rejection frequencies for the copula based tests and the changing correlation test, run at 5% signi cance level, for a sample size of T = 400 observations. For τ 6= 0.5, ignoring time-varying volatility makes the tests oversized; the least a ected appears to be the quadrant association test. The volatility correction (pre- ltering) brings the empirical size close to the nominal 5%. Busetti and Harvey (Bank of Italy and University of Cambridge) When is a copula constant? November / 54

86 Figure: Table 4 Busetti and Harvey (Bank of Italy and University of Cambridge) When is a copula constant? November / 54

87 Application to stock returns GM Q(05) Q(95) Q(25) Q(75) Figure: Quantiles tted to GM returns Busetti and Harvey (Bank of Italy and University of Cambridge) When is a copula constant? November / 54

88 3 5% 50% 95% 25% 75% Figure: IBM - IRWs tted to quantiles Busetti and Harvey (Bank of Italy and University of Cambridge) When is a copula constant? November / 54

89 Table (5) shows empirical results for the bi-variate series of GM and IBM stock returns. The upper part of the table reports the values of the statistics when the tests are applied over the full sample (3392 observations); in the lower part we consider the rst and second half-sample separately. The 5% and 1% critical values are and respectively, except for the combined test where they are and The IQ tests reported in Busetti and Harvey (2007) indicate that the quantiles for General Motors and IBM daily returns are changing over time. The bivariate statistics were computed by rescaling the observations by estimated standard deviations, from a basic SV model, or by interquartile ranges. Busetti and Harvey (Bank of Italy and University of Cambridge) When is a copula constant? November / 54

90 Figure: Table 5 Busetti and Harvey (Bank of Italy and University of Cambridge) When is a copula constant? November / 54

91 We also correct for weak dependence in the bivariate quantics, setting the bandwidth parameter as m = int 4(T /100) 1/4 ; however the results are not very di erent with m = 0. usetti and Harvey (Bank of Italy and University of Cambridge) When is a copula constant? November / 54

92 We also correct for weak dependence in the bivariate quantics, setting the bandwidth parameter as m = int 4(T /100) 1/4 ; however the results are not very di erent with m = 0. The table also shows the univariate IQ tests for the standardized data and these indicate that the quantiles have been satisfactorarily corrected for time-variation. usetti and Harvey (Bank of Italy and University of Cambridge) When is a copula constant? November / 54

93 We also correct for weak dependence in the bivariate quantics, setting the bandwidth parameter as m = int 4(T /100) 1/4 ; however the results are not very di erent with m = 0. The table also shows the univariate IQ tests for the standardized data and these indicate that the quantiles have been satisfactorarily corrected for time-variation. The null hypothesis of constant association is soundly rejected in most cases with full sample data. The sample split shows that time variation in the bivariate relationship largely occurred in the rst half of the sample. Busetti and Harvey (Bank of Italy and University of Cambridge) When is a copula constant? November / 54

94 Multivariate test A joint test may be developed to determine whether there is a changing pattern of dependence among N time series, y jt, j = 1,.., N, t = 1,..., T. The test statistic is constructed from the bi-quantic series for every possible pair. The number of pairs is N!/ (N 2)!2 = N (N 1) /2 = M. This is the number of degrees of freedom in the CvM distribution. As M!, the CvM distribution tends to N (M/6, M/45). Busetti and Harvey (Bank of Italy and University of Cambridge) When is a copula constant? November / 54

95 Tracking changes in the copula If time variation is found in the relationship between variables we might try to model it. For example Patton (2006) models changing relationship between exchange rates using a GARCH type model for the conditional correlation. Requires a parametric speci cation for the copula, a model as to how the parameter of the copula evolves as a function of past observation (or their ranks) and pre ltering by tting GARCH models to the individual series. The approach here is completely di erent in that no speci c form is assumed for the copula, changing measures of dependence can be obtained for di erent quantiles and a single measure of dependence, based on tting medians, requires no pre- ltering. Busetti and Harvey (Bank of Italy and University of Cambridge) When is a copula constant? November / 54

96 The quadrant association gives a measure of dependence in the range [0,1]. However, since C (τ, τ) + C (τ, τ) = 1 2τ + 2C (τ, τ) the same information is in C (τ, τ). This measure lies in the range [0, τ].a plot of C (τ, τ)/τ, which is between 0 and 1 may be more revealing. usetti and Harvey (Bank of Italy and University of Cambridge) When is a copula constant? November / 54

97 The quadrant association gives a measure of dependence in the range [0,1]. However, since C (τ, τ) + C (τ, τ) = 1 2τ + 2C (τ, τ) the same information is in C (τ, τ). This measure lies in the range [0, τ].a plot of C (τ, τ)/τ, which is between 0 and 1 may be more revealing. For copulas based on a single parameter, there is usually a relationship between this parameter and C (τ, τ). e.g. Clayton copula C (τ, τ) = (2τ θ 1) 1/θ. usetti and Harvey (Bank of Italy and University of Cambridge) When is a copula constant? November / 54

98 The quadrant association gives a measure of dependence in the range [0,1]. However, since C (τ, τ) + C (τ, τ) = 1 2τ + 2C (τ, τ) the same information is in C (τ, τ). This measure lies in the range [0, τ].a plot of C (τ, τ)/τ, which is between 0 and 1 may be more revealing. For copulas based on a single parameter, there is usually a relationship between this parameter and C (τ, τ). e.g. Clayton copula C (τ, τ) = (2τ θ 1) 1/θ. Single measure of dependence - C (0.5, 0.5) is appropriate. usetti and Harvey (Bank of Italy and University of Cambridge) When is a copula constant? November / 54

99 The quadrant association gives a measure of dependence in the range [0,1]. However, since C (τ, τ) + C (τ, τ) = 1 2τ + 2C (τ, τ) the same information is in C (τ, τ). This measure lies in the range [0, τ].a plot of C (τ, τ)/τ, which is between 0 and 1 may be more revealing. For copulas based on a single parameter, there is usually a relationship between this parameter and C (τ, τ). e.g. Clayton copula C (τ, τ) = (2τ θ 1) 1/θ. Single measure of dependence - C (0.5, 0.5) is appropriate. Blomqvist s beta, 4C (0.5, 0.5) 1, is in the range [ 1, 1]. usetti and Harvey (Bank of Italy and University of Cambridge) When is a copula constant? November / 54