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Extreme co-movements and extreme impacts in high frequency data in finance
Journal of Banking & Finance 31 (2007) 1399–1415 www.elsevier.com/locate/jbf
Extreme co-movements and extreme impacts in high frequency data in finance Zhengjun Zhang *, Kazuhiko Shinki Department of Statistics, University of Wisconsin, Madison, WI 53706-1532, USA Available online 12 January 2007
Abstract Extreme co-movement and extreme impact problems are inherently stochastic control problems, since they will influence the decision taken today and ultimately influence a decision taken in the future. Extreme co-movements among financial assets have been reported in the literature. However, extreme impacts have not been carefully studied yet. In this paper, we use the newly developed methodology to further explore extreme co-movements and extreme impacts in financial market. Particularly, two FX spot rates are studied. Based on the results of our analysis with FX returns, we conclude that there exist extreme co-movements and extreme impacts in FX returns and care has to be taken when we employ portfolio optimization models, especially models without the ability of handling extreme dependencies. Ó 2006 Elsevier B.V. All rights reserved. JEL classification: C12; C13; C4; C32; C53 Keywords: High frequency data; Exchange rate; Extreme dependence
1. Introduction In risk management, the extreme quantiles of a single risk factor, or the conditional extreme quantiles of multiple risk factors, are very important quantities. An accurate assessment of these quantities allows for effective handling of potential disasters such as stock market crashes. The difficulty of such a task varies. If the multivariate random variables used to model multiple risk factors are independent, we may only need to study *
Corresponding author. E-mail address: [email protected] (Z. Zhang).
0378-4266/$ - see front matter Ó 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.jbankfin.2006.10.019
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the extreme quantiles of all variables separately. If they are not independent however, the estimation of the conditional extreme quantiles may be very complicated, especially when the joint distribution is not known. Meanwhile, if the multivariate random variables are nearly independent or asymptotically independent in the tails, we may only need to obtain the extreme quantiles for each variable; if they are not, we hope to have some multivariate extreme value distributional functions which can adequately model the extreme observations. Yet, the most important step before building a model is to identify whether the variables of interests are extremely dependent, also known as asymptotically dependent or tail dependent. In financial studies, particularly in this study, extreme co-movement variables and extreme impact variables are extreme dependent variables. They are defined next. Extreme co-movements in financial market are characterized by the concurrence of different very large financial returns. When large observations have future impacts (for example, in the kth day or in the kth hour) – i.e., large values will be observed in the kth day or the kth hour if a large value has been observed, we say there are extreme impacts in the corresponding assets. Understanding extreme co-movements and extreme impacts among financial assets is very important in risk management and stochastic optimal control. In our data analysis, we found that there exist extreme dependencies in certain time series which were regarded as independent series from previous research results. As a result, previously reported results in the literature may be misleading, and corresponding stochastic models which do not consider extreme dependencies may be problematic. For example, as we introduce later, multivariate normal random variables can not model extreme dependent variables. There have been considerable efforts on modeling extreme co-movements – for example, Longin and Solnik (2001), Breymann et al. (2003), Dias and Embrechts (2003), Zhang (2005), Zhang and Huang (2006), among others. However, results regarding how to test and characterize the extreme dependencies in high frequency data are still very sparse, and the future impacts (extreme impacts) of very large observations at present are barely seen in the literature and in practice. In this paper, we further explore the extreme dependencies in high frequency financial data, particularly two FX spot rates (US dollar against German Mark, US dollar against Japanese Yen) which have been analyzed by Breymann et al. (2003) and Dias and Embrechts (2003). This paper aims to provide a methodology which tests extreme dependencies. The methodology can be used for various financial models in risk management and portfolio optimization. We argue that if a model does not consider extreme dependencies when variables are actually extremely dependent, it may overlook some risk factors and underestimate the risk. Hence, the methodology can help to measure the risk more accurately. Furthermore, since extreme dependencies are estimated separately for right and left tails in this paper, the methodology can be used to measure the risk under a certain condition such as financial crisis. The paper is structured as follows. Section 2 presents definitions and theories. Section 3 presents two FX spot rate data and data transformation through a generalized autoregressive conditional heteroskedasticity (GARCH) model which describes volatilities and a generalized extreme value distribution fitting. In Section 4, we test extreme dependencies for daily data and hourly data. The final section concludes. 2. Extreme dependencies, extreme co-movements, and extreme impacts For the theory of extreme independence between two random variables, there have been a series of studies in the extreme value literature. It has been shown that the bivariate
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maxima of a sequence of positively correlated normal random variables are extreme independent as long as the bivariate normal random variables are not completely correlated – i.e., their correlation is not equal to 1 – (Watson, 1954; Sibuya, 1960). It has also been shown that the sum of a sequence of random variables and the maximum of the same sequence of random variables are extreme independent under some mild conditions. Some related results are documented in Grady (2000). One can see that the study of extreme co-movements and extreme impacts is equivalent to the study of extreme dependencies between random variables. In the literature, the extreme dependency between two random variables is characterized by the so-called extreme dependence index. Sibuya (1960) introduces the asymptotic independence between two bivariate random variables with identical marginal distribution. de Haan and Resnick (1977) extend it to the case of multivariate random variables. In this paper, however, we adopt the term of extreme (in)dependence in connection to the study of extreme co-movements and extreme impacts. The definition of extreme (in)dependence between two random variables is given next. 2.1. Definitions Definition 2.1. Two identically distributed random variables X, Y are called extreme independent if k ¼ lim PrðY > ujX > uÞ ¼ 0;
ð2:1Þ
u!xF
where xF ¼ supfx 2 R : PrðX 6 xÞ < 1g. If k exists, it is called the bivariate extreme dependence index which quantifies the amount of dependence of the bivariate upper tails. If k > 0, then (X, Y) is called extreme dependent and we say there are extreme co-movements between X and Y. The extreme dependence indexes of certain bivariate random variables are presented in Embrechts et al. (2002). Zhang (submitted for publication) and Ledford and Tawn (2003) extend the definition of the extreme dependence between two random variables to the lagk extreme dependence of a sequence of random variables with identical marginal distribution. The definition of lag-k extreme dependence for a sequence of random variables is given below. Definition 2.2. A sequence of random variables {ni} is called lag-k extreme dependent if for all i kk ¼ lim Prðni > ujnkþi > uÞ > 0; u!xF
lim Prðni > ujniþkþj > uÞ ¼ 0; j > 0;
u!xF
ð2:2Þ
where xF ¼ supfx 2 R : Prðn1 6 xÞ < 1g. kk is called the lag-k extreme dependence index. When kk > 0 exists for some k > 0 and {ni}, we say there are extreme impacts within {ni}. The existence of k and kk are assumed in this paper. The problem is how to estimate k (also kk), especially when the joint distributions between any two random variables are unknown. In the following, a way to test the null hypothesis: k = 0 against the alternative: k > 0 is developed, without directly estimating k.
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2.2. Assumptions The values of extreme dependence indexes can be estimated using empirical probabilities above certain threshold values, but the estimated values may give misleading results since in general, the estimated values are greater than zero even if the random variables are actually extreme independent. A natural question then arises – i.e., whether there is extreme dependence between two random variables. This can be formulated into the following testing hypothesis problems: H 0 : X and Y are extreme independent $ H 1 : X and Y are extreme dependent; ð2:3Þ which can also be written as H 0 : k ¼ 0 vs: H 1 : k > 0;
ð2:4Þ
H 0 : kk ¼ 0 vs: H 1 : kk > 0
ð2:5Þ
or for the case of lag-k extreme dependence. We now study how to test (2.4) and (2.5) in the remaining of this section. 2.3. Transformations Theoretically, as long as the distribution function is continuous and strictly increasing, it can be transformed into any distribution function which is also continuous and strictly increasing. Moreover, the estimation of the extreme dependence index does not change depending on the distribution. This is due to the following identities: PrðX 1 > u; X 2 > uÞ PrðX 2 > uÞ 1 1 1 Pr G1 ½F 1 ðX 1 Þ > G1 1 ½F 1 ðuÞ; G2 ½F 2 ðX 2 Þ > G2 ½F 2 ðuÞ ¼ 1 Pr G1 2 ½F 2 ðX 2 Þ > G2 ½F 2 ðuÞ 1 1 1 ¼ Pr G1 1 ½F 1 ðX 1 Þ > G1 ½F 1 ðuÞjG2 ½F 2 ðX 2 Þ > G2 ½F 2 ðuÞ ;
PrðX 1 > ujX 2 > uÞ ¼
where F1(x) and F2(x) are distribution functions for continuous random variables X1 and X2, respectively; G1(y) and G2(y) are two arbitrary distribution functions. (All the distribution functions are assumed continuous and strictly increasing.) Based on these identities, without loss of any generality, we can assume that the distribution of each random variable (X1 or X2) is unit Fre´chet distribution which is G(x) = exp(1/x), x > 0. To be concrete, at first we estimate F1 and F2 from their samples of X1 and X2. After that X1 and X2 are transformed by the estimated F1 and F2 and then by the inverse of the unit Fre´chet distribution function. Note that only the distributions over a certain threshold are essential for their extreme behavior. In the literature, methods for exceedance modeling over high thresholds are widely used in the applications of extreme value theory. The theory used goes back to Pickands (1975). The only limit distribution of exceedances over threshold has been shown (Pickands, 1975) to be a generalized Pareto distribution (GPD) which has a direct connection to generalized extreme value distribution (GEV). Below we give a brief review. For full details, see Embrechts et al. (1997) and the references therein.
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Consider the distribution function of a random variable X conditionally on exceeding some high threshold u, we have F ðu þ yÞ F ðuÞ ; y P 0: F u ðyÞ ¼ P ðX 6 u þ yjX > uÞ ¼ 1 F ðuÞ As u ! xF = sup{x : F(x) < 1}, Pickands (1975) shows that the GPD are the only nondegenerate distribution that approximate Fu(y) for u large. The limit distribution of Fu(y) is given by y 1=n : ð2:6Þ Gðy; r; nÞ ¼ 1 1 þ n r þ In the GPD, y+ = max(y, 0), n is the tail index, also called shape parameter, which gives a precise characterization of the shape of the tail of the GPD. For the case of n > 0, it is long-tailed, i.e., 1 G(y) decays at rate y1/n for large y. The case n < 0 corresponds to a distribution function which has a finite upper end point at r/n. The case n = 0 yields the exponential distribution with mean r: y Gðy; r; 0Þ ¼ 1 exp : r The Pareto, or GPD, and other similar distributions have long been used models for longtailed processes. After a GPD is fitted to the exceedance values of observations over a high threshold u, the data can be transformed to unit Fre´chet scales based on the fitted GPD function. An alternative way to obtain unit Fre´chet scales is to fit the generalized extreme value distribution (GEV) (2.7) to local maxima of observations: " 1=n # xl H ðxÞ ¼ exp 1 þ n ; ð2:7Þ w þ where l is a location parameter, w > 0 is a scale parameter, and n is a shape parameter which is the same parameter as in (2.6). Pickands (1975) first established the rigorous connection of the GPD with the GEV. The GEV parameter estimations can be obtained from a GPD fitting. Smith (2003) is a good reference about GEV and GPD fitting and model diagnosis. GEV parameters are estimated by the maximum likelihood method in this paper. 2.4. The theory Since Fre´chet distribution has a simple form with a positive support, it is convenient to develop a test statistic on it. The gamma test for testing extreme independencies between unit Fre´chet random variables is introduced here. Theorem 2.3. Suppose {Vi} and {Wi}, i = 1, . . . , n are two series of independent identically distributed random variables with distribution function G(x) = Pr(V 6 x) = exp{1/(u + x)}, x P 0, u > 0. Then 8 t t ð1þtÞ=u > < e ; if 0 < t < 1; uþWi t 1þt ð2:8Þ 6t ¼ 1þ Pr t 1 ð1þtÞ=u > uþVi : þ e ; if t P 1; 1þt 1þt
1=u 1 lim Pr n maxðu þ W i Þ=ðu þ V i Þ þ 1 6 x ¼ eð1e Þ=x : ð2:9Þ n!1
i6n
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Moreover,
1=u lim Pr n minðu þ W i Þ=ðu þ V i Þ 6 x ¼ 1 eð1e Þx :
n!1
i6n
ð2:10Þ
The random variables maxi6n(u + Wi)/(u + Vi) and maxi6n(u + Vi)/(u + Wi) are tail independent – i.e.,
ðu þ W i Þ ðu þ V i Þ 1 1 þ 1 6 x; n max þ1 6y lim Pr n max n!1 i6n ðu þ V i Þ i6n ðu þ W i Þ ¼ eð1e
1=u Þ=xð1e1=u Þ=y
;
ð2:11Þ
Furthermore, the random variable Cu;n ¼
maxfðu þ W i Þ=ðu þ V i Þg þ maxfðu þ V i Þ=ðu þ W i Þg 2 ; maxfðu þ W i Þ=ðu þ V i Þg maxfðu þ V i Þ=ðu þ W i Þg 1
ð2:12Þ
is asymptotically gamma distributed, i.e., L
nCu;n ! f; where f is Gamma(2, 1 e
ð2:13Þ 1/u
) distributed.
A proof of Theorem 2.3 is given in Zhang (submitted for publication). Under certain mixing conditions, Zhang (submitted for publication) also shows that (2.12) and (2.13) hold when Vis are not independent, and Wis are not independent (and hence (Vi, Wi)s are not iid). Note that (2.8) is not continuous at t = 1. This is caused by the point mass at x = 0 in G(x), and the point mass is closed to one when u is very large. However, this does not induce a problem from (2.9)–(2.12), because only t P 1 part is essential in (2.8). Actually the maximum of (u + Wi)/(u + Vi) over i = 1, . . . , n is larger than one when n is large enough for u. Cu,n is termed as tail quotient correlation coefficient for a very large threshold u in Zhang (submitted for publication). Notice that Cu,n only involves the values above the threshold, i.e., the tail values of observed sequences (or after transformed to unit Fre´chet scales). In practice, Eqs. (2.12) and (2.13) together provide a gamma test statistic which can be used to determine whether extreme dependence between two random variables is significant or not. When nCu,n > fa, H0 of (2.3) is rejected, where fa is the upper ath percentile of the gamma(2, 1 e1/u) distribution. Using simulated examples, Zhang (submitted for publication) shows that the gamma test can efficiently identify the extreme (in)dependencies between various dependence variables. Since the gamma test is simple and practically efficient, in this study, we shall apply the gamma test to the data. Suppose now the variables have been transformed to have unit Fre´chet margins. In empirical study, the limit distribution of exceedances over a threshold for the original random variables may be estimated by using GPD at first, and then it can be transformed into unit Fre´chet random variable as shown in Section 2.3. In the case of testing extreme dependence, we let V i ¼ ðX i uÞI fX i >ug , W i ¼ ðY i uÞI fY i >ug , where I(Æ) is the indicator function. Values of the threshold u are taken at 95%th (or higher level) sample percentile. In the case of testing lag-k extreme dependence, we let Yi = Xi+k, choose a bivariate subsequence fðX il ; Y il Þ; l ¼ 1; . . . ; i ¼ 1; . . .g such that il il1 > k, and compute the value of the test
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statistic for the subsequence. If the observed process is lag-j dependence, the gamma test would reject the H0 of (2.3) when k = j, but it would retain the H0 of (2.3) when k = j + 1. In the literature, Ledford and Tawn (1996), Peng (1999), Draisma et al. (2004), and others studied testing extreme dependence problems in which the null hypothesis is that the two random variables are extreme dependent. Under a class of models, various test statistics have been proposed. The derivation of a test statistic under the null hypothesis of extreme dependence is still an open problem. So far, we have established a testing procedure for determining whether there exist extreme dependencies between random variables. Once the H0 of (2.3) is rejected, we conclude there exists extreme dependence between two random variables of interest. We now turn to real data analysis. 3. The data and data transformation Our objectives are to study (1) the extreme dependencies between USD/DEM and USD/YEN returns, and (2) extreme impacts on each of the two FX rates. Hence, in this study, let X1, X2, . . . be USD/DEM returns (daily or hourly), Y1, Y2, . . . be USD/YEN returns (daily or hourly), and n1, n2, . . . be either of these two series. 3.1. The data The data was collected by Olson and Associates. The observations cover the period from 27 April 1986 until 25 October 1998. The six different time horizons considered are one, two, four, eight, twelve hour and one day periods. Breymann et al. (2003) conducted filtering and deseasonalization on the data. Extreme dependence indexes (and hence extreme co-movements, extreme impacts) are associated with identically distributed random variables, but asset return series such as a FX spot rate is considered not identically distributed due to the long memory of its volatility. Using the deseasonalized data, Dias and Embrechts (2003) conducted dynamic copula modeling with multivariate GARCH model. Since their results are rather convincing, we standardize the deseasonalized data by GARCH model. In this paper, we use the deseasonalized one hour and one day data and then fit them to univariate GARCH(1, 1) models (Bollerslev, 1986). The deseasonalized data is divided by estimated standard deviations and standardized return data (also called devolatized data, or GARCH residuals) are obtained. We then study extreme co-movements and extreme impacts within the standardized returns. Note that we have restricted our GARCH model fitting to GARCH(1, 1). Other GARCH type models, such as heavy tailed GARCH models (see Mikosch, 2003; Straumann, 2005, and others), GARCH models with dependent residual random variables (see Lee and Hansen, 1994, etc.), are potential applications. For a comparison, we plot deseasonalized data against standardized data for USD/ DEM, USD/YEN in days and hours – see Fig. 1. We also plot USD/DEM deseasonalized data against USD/YEN deseasonalized data, USD/DEM standardized data against USD/ YEN standardized data, in days and hours respectively – see Fig. 2. The plots for the (nonstandardized) deseasonalized data and the standardized data look similar but not identical. Particularly for daily data, most estimated standard deviations fall in the range between 0.9 and 1.2 which results in diagonal patterns which can clearly be seen in the left
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20
15
15
10
10
Standardized returns
Standardized returns
USD/DEM 1 Day
20
5 0 –5 –10
0 –5
–10
–15 –20 –20
5
–15
–10
0
10
20
–20 –20
Deseasonalized returns
–10
15
15
10 5 0 –5
10
20
5 0 –5
–10
–15
–15 0
20
10
–10
–10
10
USD/YEN 1 Hour
20
Standardized returns
Standardized returns
USD/YEN 1 Day
20
–20 –20
0
Deseasonalized returns
10
Deseasonalized returns
20
–20 –20
–10
0
Deseasonalized returns
Fig. 1. Comparison between deseasonalized data and standardized data. The top plots are for USD/DEM. The bottom plots are for USD/YEN. The left plots are for 1 day data. The right plots are for 1 h data.
panel in Fig. 1. Hence, using the standardized data is better to avoid misspecification in testing extreme dependencies. 3.2. Data transformation Following Section 2.3, we estimate the distribution of standardized data and transform it into unit Fre´chet scales so that the gamma test can be used to test the presences of extreme dependencies. The maximum likelihood estimates of GEV parameters are summarized in Tables 1–3. For daily data, from Table 1, we see that standardized negative returns of German Mark quoted against US dollar has a fat tail (the estimated shape parameter value is greater than 0), but the standardized positive returns are now short tailed. We also see that both returns of Japanese Yen quoted against US dollar are still fat tailed. Notice that all estimated shape parameter values are different, which in turn suggests asymmetry between negative returns and positive returns. Also notice that the estimated shape parameter values are close to zero, and hence, the standardized daily returns can be thought to have Gaussian tails. For hourly data, since the data points are very large – about 78,230 points. We split the data into 10 sub-series for the convenience of computation. From Tables 2 and 3, we see
Z. Zhang, K. Shinki / Journal of Banking & Finance 31 (2007) 1399–1415 Deseasonalized returns 1 Day
20
15
15
10
10
5
5
USD/YEN
USD/YEN
20
0 –5
–5
–10
–15
–15
–10
0
10
20
Deseasonalized returns 1 Hour
0
–10
–20 –20
–20 –20
–10
USD/DEM Standardized returns 1 Day
20
15
15
10
10
5
5
0 –5
–15
–15
0
20
10
20
Standardized returns 1 Hour
–5
–10
–10
10
0
–10
–20 –20
0
USD/DEM
USD/YEN
USD/YEN
20
1407
–20 –20
USD/DEM
–10
0
10
20
USD/DEM
Fig. 2. Comparison between USD/DEM and USD/YEN. The top plots are for deseasonalized data. The bottom plots are standardized data. The left plots are for 1 day data. The right plots are for 1 h data.
Table 1 Estimated values of parameters in GEV fitting using standardized return series and threshold value u = 1.2 for daily data Series
Nu
l (SE)
log w (SE)
n (SE)
USD/DEM negative USD/DEM positive
341 323
3.305482 (0.145446) 3.024353 (0.107579)
0.438589 (0.151050) 0.667407 (0.119296)
0.001969 (0.061733) 0.060415 (0.050207)
USD/YEN negative USD/YEN positive
328 297
3.561064 (0.168287) 2.949730 (0.132979)
0.297793 (0.154565) 0.496941 (0.145713)
0.011028 (0.063915) 0.056199 (0.060298)
Nu is the number of exceedances over u. Numbers in parentheses are estimated standard deviations.
that almost all series are fat tailed, except one USD/YEN standardized positive series having a negative estimated shape parameter value. From the split series, one can see that all estimated scale parameter values, all estimated location parameter values, and all estimated shape parameter values are different though they can be thought approximately close to each other for a particular parameter. Dias and Embrechts (2003) also find there are change points in the series. Therefore, it is reasonable to split the whole series into subseries. What we have done here is simply and equally splitting the whole sequence into 10 subintervals.
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Table 2 Estimated values of parameters in GEV fitting using standardized return series and threshold value u = 1.2 for USD/DEM hourly data Nu
l (SE)
Negative series 1 2 3 4 5 6 7 8 9 10
674 683 674 721 643 680 607 638 681 777
7.672220 6.804680 6.848543 6.397347 6.502903 5.710691 9.375714 6.223442 7.156392 6.964012
(0.690882) (0.574425) (0.662581) (0.512778) (0.613648) (0.449641) (0.040530) (0.540852) (0.637399) (0.650297)
0.374201 0.166911 0.168704 0.004694 0.121396 0.165534 0.915906 0.013994 0.427417 0.309103
(0.217174) (0.204156) (0.242075) (0.203877) (0.231543) (0.209283) (0.029643) (0.221815) (0.194720) (0.209685)
0.115845 0.093685 0.091014 0.065917 0.094638 0.053644 0.232811 0.065775 0.167165 0.139668
(0.041122) (0.037496) (0.045428) (0.037243) (0.042807) (0.038622) (0.014273) (0.041196) (0.035584) (0.036994)
Positive series 1 2 3 4 5 6 7 8 9 10
626 649 730 715 661 733 661 702 752 734
8.638055 7.484258 6.337594 5.867251 7.728190 6.496009 6.574191 8.179001 5.620976 5.867552
(1.004749) (0.788383) (0.471663) (0.460927) (0.776550) (0.560787) (0.644013) (0.860773) (0.336546) (0.436857)
0.723005 0.409703 0.002262 0.067402 0.543708 0.027732 0.248934 0.635715 0.217165 0.074778
(0.237559) (0.237312) (0.186881) (0.201041) (0.211443) (0.221312) (0.225150) (0.219334) (0.159370) (0.192102)
0.193144 0.138325 0.068004 0.077697 0.175673 0.068059 0.137402 0.187576 0.047628 0.076242
(0.042616) (0.043425) (0.033955) (0.036576) (0.037787) (0.040607) (0.040816) (0.039510) (0.029366) (0.035387)
log w (SE)
n (SE)
Nu is the number of exceedances over u. Numbers in parentheses are estimated standard deviations.
Notice that daily positive returns of USD/DEM are fitted with a short tailed parameter value, while hourly positive returns of USD/DEM are fitted with long-tailed parameter values, we argue that the accumulation of data causes the changing of long tail to short tail. But USD/YEN daily positive returns are still long-tailed. In other words, the price movements between two different currencies are significantly different. We explore extreme co-movements and extreme impacts for different returns next. 4. Extreme co-movements and extreme impacts In this section, we use the gamma test to detect extreme co-movements and extreme impacts in two exchange spot rates. In order to do so, the standardized data is converted into unit Fre´chet scale for the observations above certain threshold using the fitted GEV function and its connection to the generalized Pareto distribution. The gamma test is implemented for each certain length of time period, since the extreme dependence between variables may vary by period. The result is reported as the rejection rate, that is the number of rejecting null hypotheses over the total number of tests. 4.1. Extreme co-movements of daily data Usually, the notion of extreme co-movements in the stock market means that large jumps in returns from two different stocks happen at the same time (in this data analysis,
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Table 3 Estimated values of parameters in GEV fitting using standardized return series and threshold value u = 1.2 for USD/YEN hourly data Nu
l (SE)
Negative series 1 2 3 4 5 6 7 8 9 10
652 687 666 745 700 680 636 676 654 669
7.689437 7.690174 5.943126 7.035376 7.092855 7.822926 5.816896 7.047284 8.083421 7.467914
(0.750974) (0.646647) (0.473148) (0.599662) (0.622079) (0.819186) (0.434661) (0.647908) (0.887155) (0.745009)
0.392191 0.430634 0.159317 0.260564 0.296081 0.495258 0.076391 0.395617 0.585002 0.377056
(0.222850) (0.182003) (0.213052) (0.198535) (0.203666) (0.232781) (0.188880) (0.202182) (0.234861) (0.231108)
0.120125 0.136266 0.037921 0.116090 0.122993 0.152490 0.073258 0.161767 0.170735 0.128335
(0.040821) (0.032090) (0.039476) (0.036004) (0.037237) (0.042750) (0.035085) (0.036758) (0.042717) (0.042880)
Positive series 1 2 3 4 5 6 7 8 9 10
654 639 729 735 662 678 671 730 695 715
7.159015 8.821497 6.089159 5.025677 6.939650 5.917257 6.661354 6.851863 6.058247 6.156229
(0.649723) (1.327930) (0.489057) (0.313351) (0.593644) (0.495130) (0.566859) (0.620021) (0.528339) (0.529878)
0.315680 0.858021 0.084831 0.577958 0.543708 0.125039 0.177595 0.233014 0.058887 0.024695
(0.211308) (0.299076) (0.206855) (0.190994) (0.195076) (0.223383) (0.206427) (0.211337) (0.227645) (0.219840)
0.123436 0.239423 0.056796 0.022408 0.137517 0.052073 0.105988 0.116898 0.065642 0.071737
(0.039201) (0.053912) (0.037509) (0.035185) (0.036059) (0.041514) (0.038571) (0.038044) (0.042113) (0.040185)
log w (SE)
n (SE)
Nu is the number of exceedances over u. Numbers in parentheses are estimated standard deviations.
we use the same day and the same hour). When jumps in returns have an impact in the following trading days or hours, it is better to consider cross-sectional extreme dependencies or to extend the meaning of extreme co-movements to that large jumps in returns from two different stocks happen in a short time period of several consecutive days or hours. We use the extended meaning and apply the gamma test to all transformed data. In each test, we use threshold values of 95th percentiles of the data and test level of a = .05. The total number of tests equals the sample size subtracting the maximal length of test windows (in this study we choose 500 consecutive points in each test window). The testing results are summarized in Table 4. In the table, N. USD/DEM stands for the negative returns of USD/DEM. P. USD/ DEM stands for the positive returns of USD/DEM. N. USD/YEN stands for negative returns of USD/YEN. P. USD/YEN stands for positive returns of USD/YEN. In the table, the cell of Row (A, B) and Column (Day 0, Day i) means that the return value of asset A on Day 0 and the return value of asset B on Day i are formed in pairs and used in the test. Table 4 This table summarizes the test results by using the gamma test and subset scheme of daily data Pairs
(Day 0, Day 0)
(Day 0, Day 1)
(Day 0, Day 2)
(N. USD/DEM, N. USD/YEN) (N. USD/DEM, P. USD/YEN) (P. USD/DEM, N. USD/YEN) (P. USD/DEM, P. USD/YEN)
1.0000 0.00 0.00 0.4858
0.00 0.00 0.00 0.00
0.00 0.00 0.00 0.00
The number in each cell is the rejection rate of the null hypothesis of extreme independence.
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From Table 4, we see that there are clearly extreme co-movements between negative USD/DEM returns and negative USD/YEN returns since the rejection rate of the null hypothesis of extreme independence is 100%. The rejection rate of the null hypothesis of extreme independence between positive returns is 49%. This number (49%) suggests there were extreme co-movement between positive USD/DEM returns and positive USD/YEN returns, but extreme co-movements did not appear in all subintervals – i.e., there were change points that price movements behaved differently. A change point analysis conducted by Dias and Embrechts (2003) also suggests such phenomena. 4.2. Extreme co-movements of hourly data Like in the previous section, we use the extended meaning and perform the gamma test to all transformed data. There are totally 2615 tests. In each test, we use threshold values of 95th percentiles of the data and test level of a = .05. The testing results are summarized in Table 5. In the table, the cell of Row (A, B) and Column (Hour 0, Hour i) means that the return value of asset A on Hour 0 and the return value of asset B on Hour i are formed in pairs and used in the test. From Table 5, we see that there are clearly extreme co-movements between USD/DEM hourly price change and USD/YEN hourly price change. Notice that there are more time periods of positive extreme co-movements than time periods of negative extreme co-movements which show an asymmetric price movements in hourly return data. We also see the difference between hourly data and daily data. During market recession, the chance of extreme co-movements will more likely be increasing (from a low rejection rate in hourly data to a high rejection rate in daily data of null hypothesis of extreme independence). During market expansion, however, the chance of extreme co-movements is likely decreasing from hourly movements to daily movements. We also see that extreme co-movements only occur at the same day or the same hour because all numbers in columns (Day 0, Day 1) and (Day 0, Day 2) are zeros. This phenomenon suggests that a large price change of one spot rate at a particular time does not have an extreme impact on another spot rate at the time points after 1 h or 1 day. This is interesting. It is suggesting that the cross-sectional extreme impacts only occur in a very short time period, probably a ‘tick by tick’ time period. 4.3. Extreme impacts of daily data From the previous two sections, we have seen that extreme co-movements between two different exchange currencies were found using the gamma test. We now turn to study extreme impacts within each exchange spot rate sequence. In order to test extreme impacts, Table 5 This table summarizes the test results by using the gamma test and subset scheme of hourly data Pairs
(Hour 0, Hour 0)
(Hour 0, Hour 1)
(Hour 0, Hour 2)
(N. USD/DEM, N. USD/YEN) (N. USD/DEM, P. USD/YEN) (P. USD/DEM, N. USD/YEN) (P. USD/DEM, P. USD/YEN)
0.7908 0.00 0.00 0.8563
0.00 0.00 0.00 0.00
0.00 0.00 0.00 0.00
The number in each cell is the rejection rate of the null hypothesis of extreme independence.
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we need to project a univariate sequence into a bivariate sequence. A simple way to accomplish this is to let X 0i ¼ X i , Y 0i ¼ X iþk , i = 1, . . . , n k for a particular lag-k. For each return sequence (negative or positive), the rejection rate of null hypothesis of extreme independence varies from case to case. One can see that identifying extreme impacts is more difficult than identifying extreme co-movements between two different risk factors. Sometimes, the gamma test alone may not be sufficient to suggest a lag-k extreme dependence. Other statistical quantities together with the test results may help to identify a best lag-k. A natural choice would be the empirical estimation of the extreme dependence index. The empirical formula used in this study is Pm i¼1 I ðX i >u;Y i >uÞ ^ kemp ¼ P ; m i¼1 I ðX i >uÞ where m (m = 500 in this study) is the number of points used in each test window, {Xi} is one time series in the data set, {Yi} is another time series in the data set. u is a threshold value which is the minimum of the two 95th sample percentiles. The tested results are summarized in Tables 6 and 7. In the tables, R.R. denotes the rejection rate of rejecting the null hypothesis of extreme independence. There are totally 671 tests in which each uses 500 data points. For each data set of 500 data points, we compute the empirical estimates of extreme dependence index using the above empirical formula. (M.E.D.I.) stands for the average of estimated extreme dependence indexes. [S. std] stands for the sample standard deviation of estimated extreme dependence indexes. Table 6 Lag-k tail independence test for transformed return data Lag
1 2 3 4
USD/DEM negative
USD/DEM positive
R.R.
M.E.D.I.
S. std
R.R.
M.E.D.I.
S. std
1.0000 1.0000 1.0000 1.0000
0.0242 0.0182 0.0182 0.0121
0.0415 0.0647 0.1313 0.0419
0.1853 0.7844 0.0000 0.0000
0.0062 0.0123 0.0062 0.0062
0.0261 0.0437 0.0092 0.0113
R.R. denotes the rejection rate of rejecting the null hypothesis of extreme independence. M.E.D.I. means empirical estimation of extreme dependence index. [S. std] stands for the sample standard deviation of estimated extreme dependence indexes.
Table 7 Lag-k tail independence test for transformed return data Lag
1 2 3 4
USD/YEN negative
USD/YEN positive
R.R.
M.E.D.I.
S. std
R.R.
M.E.D.I.
S. std
1.0000 1.0000 1.0000 1.0000
0.0189 0.0252 0.0126 0.0126
0.0421 0.0975 0.0445 0.0333
1.0000 1.0000 0.0000 0.0000
0.006 0.0119 0.0238 0.0179
0.076 0.1017 0.0133 0.0105
R.R. denotes the rejection rate of rejecting the null hypothesis of extreme independence. M.E.D.I. means empirical estimation of extreme dependence index. [S. std] stands for the sample standard deviation of estimated extreme dependence indexes.
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We see from these two tables, up to lag-4, negative returns have extreme impacts all the time. Extreme impacts within positive returns were found to last about three consecutive days. 4.4. Extreme impacts of hourly data Like the previous section, we project each univariate sequence into a bivariate sequence. The testing results are summarized in Tables 8 and 9 for lag-k = 1, 2, . . . , 10, 96, . . . , 100. Table 8 Lag-k tail independence test for transformed return data (USD/DEM) Lag
1 2 3 4 5 6 7 8 9 10 .. . 96 97 98 99 100
Negative
Positive
R.R.
M.E.D.I.
S. std
R.R.
M.E.D.I.
S. std
0.9409 0.9485 0.8498 0.7268 0.866 0.7855 0.8356 0.9547 0.8104 0.7791 .. . 0.8653 1.0000 1.0000 0.0000 1.0000
0.0298 0.0277 0.0154 0.0157 0.0146 0.011 0.0085 0.0077 0.0108 0.0087 .. . 0.0013 0.0015 0.0005 0.0015 0.0008
0.0664 0.0604 0.0636 0.0522 0.065 0.0486 0.0575 0.063 0.0594 0.0528 .. . 0.0569 0.0866 0.0455 0.0131 0.0773
0.8625 0.8729 0.8601 0.6782 0.7295 0.8159 0.7273 0.8699 0.7425 0.755 .. . 1.0000 1.0000 1.0000 1.0000 1.0000
0.0259 0.0187 0.0218 0.0138 0.0144 0.0108 0.0103 0.011 0.0054 0.01 .. . 0.0003 0.001 0.0013 0.0005 0.0008
0.0624 0.0537 0.0624 0.0564 0.0557 0.0439 0.0592 0.0655 0.0464 0.0548 .. . 0.135 0.1088 0.0699 0.0533 0.1099
M.E.D.I. means empirical estimation of extreme dependence index. Table 9 Lag-k tail independence test for transformed return data Lag
1 2 3 4 5 6 7 8 9 10 .. . 96 97 98 99 100
Negative
Positive
R.R.
M.E.D.I.
S. std
R.R.
M.E.D.I.
S. std
0.8911 0.8645 0.8537 0.8871 0.8406 0.9538 0.7673 0.847 0.9058 0.9071 .. . 1.0000 1.0000 1.0000 1.0000 0.609
0.0247 0.0263 0.0173 0.0162 0.0142 0.0108 0.0111 0.007 0.0085 0.0057 .. . 0.0003 0.0008 0.0021 0.0013 0.0008
0.0699 0.0552 0.0524 0.0672 0.0649 0.063 0.0574 0.062 0.0713 0.0669 .. . 0.0403 0.1107 0.1519 0.0455 0.02
0.9195 0.8587 0.8584 0.9011 0.747 0.8392 0.745 0.9389 0.8759 0.6969 .. . 1.0000 0.9652 1.0000 0.6199 1.0000
0.0212 0.0189 0.0146 0.0156 0.0141 0.0084 0.0097 0.0097 0.0077 0.0069 .. . 0.0015 0.0003 0.001 0.0008 0.0008
0.0598 0.0587 0.0464 0.0546 0.0482 0.0428 0.0519 0.0645 0.0661 0.0476 .. . 0.0652 0.0459 0.0272 0.0229 0.0677
M.E.D.I. means empirical estimation of extreme dependence index (USD/YEN).
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Plot for hourly data lag–k test
1
0.9
0.8
Rejection rates
0.7
0.6
0.5
0.4
0.3 N. USD/DEM P. USD/DEM N. USD/YEN P. USD/YEN
0.2
0.1
0
0
10
20
30
40
50 lag–k
60
70
80
90
100
Fig. 3. Rejection rates of lag-1 to lag-100 extreme dependence tests for hourly data.
We see there are extreme impacts of hourly returns (negative or positive) for very large lags. The rejection rates are also plotted in Fig. 3. From the figure, we see that the rejection rates within the first 30 lags are around 80%. Then the rejection rates are varying in a wide range when lag-k is increasing. From the previous section, we have seen that extreme impacts of positive daily data appear to last within the following two consecutive days. From results in Tables 8 and 9, we see that extreme impacts of positive hourly data appear to last more than four days (will be higher if larger lags are used in test). This phenomenon suggests that extreme impacts of high frequency data are stronger than extreme impacts of low frequency data. It also suggests great care must be taken when modeling high frequency data. 5. Conclusion This paper has presented ways of analyzing extreme co-movements and extreme impacts among financial market variables. We focused on two FX spot rates. We found that there exist extreme co-movements and extreme impacts in financial market data. The methodology proposed in this paper of testing extreme co-movements and extreme impacts among two FX spot rates can be used to assessing extreme co-movements and extreme impacts among other financial returns, such as stock returns. We have proposed procedures of: (1) fitting GARCH(1, 1) (can be GARCH(p, q)) model to price returns; (2)
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obtaining devolatized return data; (3) fitting GPD to exceedance values over a typical high threshold value; (4) transforming the observed values to unit Fre´chet scales by fitted GPD (or GEV) functions; (5) performing the gamma test to the transformed data; (6) identifying extreme co-movements and extreme impacts among financial returns. Based on the results of our analysis with FX spot rates, we argue that extreme co-movement and extreme impact problems are inherently stochastic control problems, since they will influence the decision taken today and ultimately influence a decision taken in the future. For the daily data, there are existing models which are able to model extreme co-movements and extreme impacts among the interest variables. Those models are studied, for example, by Deheuvels (1983), Davis and Resnick (1989), Smith and Weissman (1996), Hall et al. (2002), Zhang and Smith (2004, submitted for publication), Zhang (2005), Chamu´ Morales (2005), Zhang and Huang (2006), Heffernan et al. (submitted for publication), among others. The models have been used in stochastic optimal control and portfolio optimizations. For the high frequency data (for example, hourly data), efforts of constructing reasonable models are still needed. Our ultimate goal is to construct portfolio optimization models for high frequency data, especially models which are applicable during the time of market recession or market expansion. Acknowledgements The work by the first author was supported in part by NSF Grants DMS-0505528, and DMS-0630210, Wisconsin Alumni Foundation, and Institute for Mathematical Research, ETH Zu¨rich. The authors thank Dr. Dias for sharing the data with us. References Bollerslev, T., 1986. Generalized autoregressive conditional heteroscedasticity. J. Econom. 31, 307–327. Breymann, W., Dias, A., Embrechts, P., 2003. Dependence structures for multivariate high-frequency data in finance. Quant. Fin. 3, 1–14. Chamu´ Morales, F., 2005. Estimation of max-stable processes using Monte Carlo methods with applications to financial risk assessment. PhD dissertation, Department of Statistics at the University of North Carolina. Davis, R.A., Resnick, S.I., 1989. Basic properties and prediction of Max-ARMA processes. Adv. Appl. Prob. 21, 781–803. de Haan, L., Resnick, S.I., 1977. Limit Theory for Multivariate Sample Extremes. Z. Wahrscheinlichkeitstheorie verw Gebiete 40, 317–337. Deheuvels, P., 1983. Point processes and multivariate extreme values. J. Multivariate Anal. 13, 257–272. Dias, A. Embrechts, P., 2003. Dynamic copula models for multivariate high-frequency data in finance, web reference, ETHZ. Draisma, G., Drees, H., Ferreira, A., de Haan, L., 2004. Bivariate tail estimation: Dependence in asymptotic independence. Bernoulli 10, 251–280. Embrechts, P., Klu¨ppelberg, C., Mikosch, T., 1997. Modelling Extremal Events for Insurance and Finance. Springer. Embrechts, P., McNeil, A., Straumann, D., 2002. Correlation and dependence in risk management: Properties and pitfalls. In: Dempster, M.A.H. (Ed.), Risk Management: Value at Risk and Beyond. Cambridge University Press, Cambridge, pp. 176–223. Grady, A., 2000. A higher order expansion for the joint density of the sum and the maximum with applications to the estimation of the climatological trends. PhD Dissertation, Department of Statistics, University of North Carolina. Hall, P., Peng, L., Yao, Q., 2002. Moving-maximum models for extrema of time series. J. Statist. Plann. Infer. 103, 51–63.
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Heffernan, J.E., Tawn, J.A., Zhang, Z., submitted for publication. Asymptotically (in)dependent multivariate maxima of moving maxima processes. Ledford, A.W., Tawn, J.A., 1996. Statistics for near independence in multivariate extreme values. Biometrika 83, 169–187. Ledford, A.W., Tawn, J.A., 2003. Diagnostics for dependence within time series extremes. J. Royal Stat. Soc. B. 65, 521–543. Lee, S.-W., Hansen, B.E., 1994. Asymptotic theory for the GARCH(1, 1) quasi-maximum likelihood estimator. Econom. Theory 10, 29–52. Longin, F., Solnik, B., 2001. Extreme correlation of international equity markets. J. Fin. LVI (2), 649–675. Mikosch, T., 2003. Modeling dependence and tails of financial time series. In: Finkenstadt, B., Rootze´n, H. (Eds.), Extreme Value in Finance, Telecommunications and the Environment. Chapman and Hall/CRC. Peng, L., 1999. Estimation of the coefficient of tail dependence in bivariate extremes. Statist. Probab. Lett. 43, 399–409. Pickands III, J., 1975. Statistical inference using extreme order statistics. Ann. Statist. 3 (1), 119–131. Sibuya, M., 1960. Bivariate extreme statistics, I. Ann. Inst. Statist. Math. 11, 195–210. Smith, R.L., 2003. Statistics of extremes, with applications in the environment, insurance and finance. In: Finkenstadt, B., Rootze´n, H. (Eds.), Extreme Value in Finance Telecommunications and the Environment. Chapman and Hall/CRC. Smith, R.L., Weissman, I., 1996. Characterization and estimation of the multivariate extremal index. Technical report, the University of North Carolina. Straumann, D., 2005. Estimation in Conditionally Heteroscedastic Time Series Models, Lecture Notes in Statistics, vol. 181. Springer. Watson, G.S., 1954. Extreme values in samples from m-dependent stationary stochastic processes. Ann. Math. Statist. 25, 798–800. Zhang, Z., 2005. A new class of tail-dependent time series models and its applications in financial time series. Adv. Econom. 20 (B), 323–358. Zhang, Z., Huang, J., 2006. Extremal financial risk model and portfolio evaluation. Comput. Statist. Data Anal. 51, 2313–2338. Zhang, Z., Smith, R.L., 2004. The behavior of multivariate maxima of moving maxima processes. J. Appl. Probab. 41, 1113–1123. Zhang, Z., Smith, R.L., submitted for publication. On the estimation and application of Max-stable processes. Zhang, Z., submitted for publication. Quotient correlation: A sample based alternative to Pearson’s correlation.
Extreme co-movements and extreme impacts in high frequency data in finance Zhengjun Zhang *, Kazuhiko Shinki Department of Statistics, University of Wisconsin, Madison, WI 53706-1532, USA Available online 12 January 2007
Abstract Extreme co-movement and extreme impact problems are inherently stochastic control problems, since they will influence the decision taken today and ultimately influence a decision taken in the future. Extreme co-movements among financial assets have been reported in the literature. However, extreme impacts have not been carefully studied yet. In this paper, we use the newly developed methodology to further explore extreme co-movements and extreme impacts in financial market. Particularly, two FX spot rates are studied. Based on the results of our analysis with FX returns, we conclude that there exist extreme co-movements and extreme impacts in FX returns and care has to be taken when we employ portfolio optimization models, especially models without the ability of handling extreme dependencies. Ó 2006 Elsevier B.V. All rights reserved. JEL classification: C12; C13; C4; C32; C53 Keywords: High frequency data; Exchange rate; Extreme dependence
1. Introduction In risk management, the extreme quantiles of a single risk factor, or the conditional extreme quantiles of multiple risk factors, are very important quantities. An accurate assessment of these quantities allows for effective handling of potential disasters such as stock market crashes. The difficulty of such a task varies. If the multivariate random variables used to model multiple risk factors are independent, we may only need to study *
Corresponding author. E-mail address: [email protected] (Z. Zhang).
0378-4266/$ - see front matter Ó 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.jbankfin.2006.10.019
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the extreme quantiles of all variables separately. If they are not independent however, the estimation of the conditional extreme quantiles may be very complicated, especially when the joint distribution is not known. Meanwhile, if the multivariate random variables are nearly independent or asymptotically independent in the tails, we may only need to obtain the extreme quantiles for each variable; if they are not, we hope to have some multivariate extreme value distributional functions which can adequately model the extreme observations. Yet, the most important step before building a model is to identify whether the variables of interests are extremely dependent, also known as asymptotically dependent or tail dependent. In financial studies, particularly in this study, extreme co-movement variables and extreme impact variables are extreme dependent variables. They are defined next. Extreme co-movements in financial market are characterized by the concurrence of different very large financial returns. When large observations have future impacts (for example, in the kth day or in the kth hour) – i.e., large values will be observed in the kth day or the kth hour if a large value has been observed, we say there are extreme impacts in the corresponding assets. Understanding extreme co-movements and extreme impacts among financial assets is very important in risk management and stochastic optimal control. In our data analysis, we found that there exist extreme dependencies in certain time series which were regarded as independent series from previous research results. As a result, previously reported results in the literature may be misleading, and corresponding stochastic models which do not consider extreme dependencies may be problematic. For example, as we introduce later, multivariate normal random variables can not model extreme dependent variables. There have been considerable efforts on modeling extreme co-movements – for example, Longin and Solnik (2001), Breymann et al. (2003), Dias and Embrechts (2003), Zhang (2005), Zhang and Huang (2006), among others. However, results regarding how to test and characterize the extreme dependencies in high frequency data are still very sparse, and the future impacts (extreme impacts) of very large observations at present are barely seen in the literature and in practice. In this paper, we further explore the extreme dependencies in high frequency financial data, particularly two FX spot rates (US dollar against German Mark, US dollar against Japanese Yen) which have been analyzed by Breymann et al. (2003) and Dias and Embrechts (2003). This paper aims to provide a methodology which tests extreme dependencies. The methodology can be used for various financial models in risk management and portfolio optimization. We argue that if a model does not consider extreme dependencies when variables are actually extremely dependent, it may overlook some risk factors and underestimate the risk. Hence, the methodology can help to measure the risk more accurately. Furthermore, since extreme dependencies are estimated separately for right and left tails in this paper, the methodology can be used to measure the risk under a certain condition such as financial crisis. The paper is structured as follows. Section 2 presents definitions and theories. Section 3 presents two FX spot rate data and data transformation through a generalized autoregressive conditional heteroskedasticity (GARCH) model which describes volatilities and a generalized extreme value distribution fitting. In Section 4, we test extreme dependencies for daily data and hourly data. The final section concludes. 2. Extreme dependencies, extreme co-movements, and extreme impacts For the theory of extreme independence between two random variables, there have been a series of studies in the extreme value literature. It has been shown that the bivariate
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maxima of a sequence of positively correlated normal random variables are extreme independent as long as the bivariate normal random variables are not completely correlated – i.e., their correlation is not equal to 1 – (Watson, 1954; Sibuya, 1960). It has also been shown that the sum of a sequence of random variables and the maximum of the same sequence of random variables are extreme independent under some mild conditions. Some related results are documented in Grady (2000). One can see that the study of extreme co-movements and extreme impacts is equivalent to the study of extreme dependencies between random variables. In the literature, the extreme dependency between two random variables is characterized by the so-called extreme dependence index. Sibuya (1960) introduces the asymptotic independence between two bivariate random variables with identical marginal distribution. de Haan and Resnick (1977) extend it to the case of multivariate random variables. In this paper, however, we adopt the term of extreme (in)dependence in connection to the study of extreme co-movements and extreme impacts. The definition of extreme (in)dependence between two random variables is given next. 2.1. Definitions Definition 2.1. Two identically distributed random variables X, Y are called extreme independent if k ¼ lim PrðY > ujX > uÞ ¼ 0;
ð2:1Þ
u!xF
where xF ¼ supfx 2 R : PrðX 6 xÞ < 1g. If k exists, it is called the bivariate extreme dependence index which quantifies the amount of dependence of the bivariate upper tails. If k > 0, then (X, Y) is called extreme dependent and we say there are extreme co-movements between X and Y. The extreme dependence indexes of certain bivariate random variables are presented in Embrechts et al. (2002). Zhang (submitted for publication) and Ledford and Tawn (2003) extend the definition of the extreme dependence between two random variables to the lagk extreme dependence of a sequence of random variables with identical marginal distribution. The definition of lag-k extreme dependence for a sequence of random variables is given below. Definition 2.2. A sequence of random variables {ni} is called lag-k extreme dependent if for all i kk ¼ lim Prðni > ujnkþi > uÞ > 0; u!xF
lim Prðni > ujniþkþj > uÞ ¼ 0; j > 0;
u!xF
ð2:2Þ
where xF ¼ supfx 2 R : Prðn1 6 xÞ < 1g. kk is called the lag-k extreme dependence index. When kk > 0 exists for some k > 0 and {ni}, we say there are extreme impacts within {ni}. The existence of k and kk are assumed in this paper. The problem is how to estimate k (also kk), especially when the joint distributions between any two random variables are unknown. In the following, a way to test the null hypothesis: k = 0 against the alternative: k > 0 is developed, without directly estimating k.
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2.2. Assumptions The values of extreme dependence indexes can be estimated using empirical probabilities above certain threshold values, but the estimated values may give misleading results since in general, the estimated values are greater than zero even if the random variables are actually extreme independent. A natural question then arises – i.e., whether there is extreme dependence between two random variables. This can be formulated into the following testing hypothesis problems: H 0 : X and Y are extreme independent $ H 1 : X and Y are extreme dependent; ð2:3Þ which can also be written as H 0 : k ¼ 0 vs: H 1 : k > 0;
ð2:4Þ
H 0 : kk ¼ 0 vs: H 1 : kk > 0
ð2:5Þ
or for the case of lag-k extreme dependence. We now study how to test (2.4) and (2.5) in the remaining of this section. 2.3. Transformations Theoretically, as long as the distribution function is continuous and strictly increasing, it can be transformed into any distribution function which is also continuous and strictly increasing. Moreover, the estimation of the extreme dependence index does not change depending on the distribution. This is due to the following identities: PrðX 1 > u; X 2 > uÞ PrðX 2 > uÞ 1 1 1 Pr G1 ½F 1 ðX 1 Þ > G1 1 ½F 1 ðuÞ; G2 ½F 2 ðX 2 Þ > G2 ½F 2 ðuÞ ¼ 1 Pr G1 2 ½F 2 ðX 2 Þ > G2 ½F 2 ðuÞ 1 1 1 ¼ Pr G1 1 ½F 1 ðX 1 Þ > G1 ½F 1 ðuÞjG2 ½F 2 ðX 2 Þ > G2 ½F 2 ðuÞ ;
PrðX 1 > ujX 2 > uÞ ¼
where F1(x) and F2(x) are distribution functions for continuous random variables X1 and X2, respectively; G1(y) and G2(y) are two arbitrary distribution functions. (All the distribution functions are assumed continuous and strictly increasing.) Based on these identities, without loss of any generality, we can assume that the distribution of each random variable (X1 or X2) is unit Fre´chet distribution which is G(x) = exp(1/x), x > 0. To be concrete, at first we estimate F1 and F2 from their samples of X1 and X2. After that X1 and X2 are transformed by the estimated F1 and F2 and then by the inverse of the unit Fre´chet distribution function. Note that only the distributions over a certain threshold are essential for their extreme behavior. In the literature, methods for exceedance modeling over high thresholds are widely used in the applications of extreme value theory. The theory used goes back to Pickands (1975). The only limit distribution of exceedances over threshold has been shown (Pickands, 1975) to be a generalized Pareto distribution (GPD) which has a direct connection to generalized extreme value distribution (GEV). Below we give a brief review. For full details, see Embrechts et al. (1997) and the references therein.
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Consider the distribution function of a random variable X conditionally on exceeding some high threshold u, we have F ðu þ yÞ F ðuÞ ; y P 0: F u ðyÞ ¼ P ðX 6 u þ yjX > uÞ ¼ 1 F ðuÞ As u ! xF = sup{x : F(x) < 1}, Pickands (1975) shows that the GPD are the only nondegenerate distribution that approximate Fu(y) for u large. The limit distribution of Fu(y) is given by y 1=n : ð2:6Þ Gðy; r; nÞ ¼ 1 1 þ n r þ In the GPD, y+ = max(y, 0), n is the tail index, also called shape parameter, which gives a precise characterization of the shape of the tail of the GPD. For the case of n > 0, it is long-tailed, i.e., 1 G(y) decays at rate y1/n for large y. The case n < 0 corresponds to a distribution function which has a finite upper end point at r/n. The case n = 0 yields the exponential distribution with mean r: y Gðy; r; 0Þ ¼ 1 exp : r The Pareto, or GPD, and other similar distributions have long been used models for longtailed processes. After a GPD is fitted to the exceedance values of observations over a high threshold u, the data can be transformed to unit Fre´chet scales based on the fitted GPD function. An alternative way to obtain unit Fre´chet scales is to fit the generalized extreme value distribution (GEV) (2.7) to local maxima of observations: " 1=n # xl H ðxÞ ¼ exp 1 þ n ; ð2:7Þ w þ where l is a location parameter, w > 0 is a scale parameter, and n is a shape parameter which is the same parameter as in (2.6). Pickands (1975) first established the rigorous connection of the GPD with the GEV. The GEV parameter estimations can be obtained from a GPD fitting. Smith (2003) is a good reference about GEV and GPD fitting and model diagnosis. GEV parameters are estimated by the maximum likelihood method in this paper. 2.4. The theory Since Fre´chet distribution has a simple form with a positive support, it is convenient to develop a test statistic on it. The gamma test for testing extreme independencies between unit Fre´chet random variables is introduced here. Theorem 2.3. Suppose {Vi} and {Wi}, i = 1, . . . , n are two series of independent identically distributed random variables with distribution function G(x) = Pr(V 6 x) = exp{1/(u + x)}, x P 0, u > 0. Then 8 t t ð1þtÞ=u > < e ; if 0 < t < 1; uþWi t 1þt ð2:8Þ 6t ¼ 1þ Pr t 1 ð1þtÞ=u > uþVi : þ e ; if t P 1; 1þt 1þt
1=u 1 lim Pr n maxðu þ W i Þ=ðu þ V i Þ þ 1 6 x ¼ eð1e Þ=x : ð2:9Þ n!1
i6n
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Moreover,
1=u lim Pr n minðu þ W i Þ=ðu þ V i Þ 6 x ¼ 1 eð1e Þx :
n!1
i6n
ð2:10Þ
The random variables maxi6n(u + Wi)/(u + Vi) and maxi6n(u + Vi)/(u + Wi) are tail independent – i.e.,
ðu þ W i Þ ðu þ V i Þ 1 1 þ 1 6 x; n max þ1 6y lim Pr n max n!1 i6n ðu þ V i Þ i6n ðu þ W i Þ ¼ eð1e
1=u Þ=xð1e1=u Þ=y
;
ð2:11Þ
Furthermore, the random variable Cu;n ¼
maxfðu þ W i Þ=ðu þ V i Þg þ maxfðu þ V i Þ=ðu þ W i Þg 2 ; maxfðu þ W i Þ=ðu þ V i Þg maxfðu þ V i Þ=ðu þ W i Þg 1
ð2:12Þ
is asymptotically gamma distributed, i.e., L
nCu;n ! f; where f is Gamma(2, 1 e
ð2:13Þ 1/u
) distributed.
A proof of Theorem 2.3 is given in Zhang (submitted for publication). Under certain mixing conditions, Zhang (submitted for publication) also shows that (2.12) and (2.13) hold when Vis are not independent, and Wis are not independent (and hence (Vi, Wi)s are not iid). Note that (2.8) is not continuous at t = 1. This is caused by the point mass at x = 0 in G(x), and the point mass is closed to one when u is very large. However, this does not induce a problem from (2.9)–(2.12), because only t P 1 part is essential in (2.8). Actually the maximum of (u + Wi)/(u + Vi) over i = 1, . . . , n is larger than one when n is large enough for u. Cu,n is termed as tail quotient correlation coefficient for a very large threshold u in Zhang (submitted for publication). Notice that Cu,n only involves the values above the threshold, i.e., the tail values of observed sequences (or after transformed to unit Fre´chet scales). In practice, Eqs. (2.12) and (2.13) together provide a gamma test statistic which can be used to determine whether extreme dependence between two random variables is significant or not. When nCu,n > fa, H0 of (2.3) is rejected, where fa is the upper ath percentile of the gamma(2, 1 e1/u) distribution. Using simulated examples, Zhang (submitted for publication) shows that the gamma test can efficiently identify the extreme (in)dependencies between various dependence variables. Since the gamma test is simple and practically efficient, in this study, we shall apply the gamma test to the data. Suppose now the variables have been transformed to have unit Fre´chet margins. In empirical study, the limit distribution of exceedances over a threshold for the original random variables may be estimated by using GPD at first, and then it can be transformed into unit Fre´chet random variable as shown in Section 2.3. In the case of testing extreme dependence, we let V i ¼ ðX i uÞI fX i >ug , W i ¼ ðY i uÞI fY i >ug , where I(Æ) is the indicator function. Values of the threshold u are taken at 95%th (or higher level) sample percentile. In the case of testing lag-k extreme dependence, we let Yi = Xi+k, choose a bivariate subsequence fðX il ; Y il Þ; l ¼ 1; . . . ; i ¼ 1; . . .g such that il il1 > k, and compute the value of the test
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statistic for the subsequence. If the observed process is lag-j dependence, the gamma test would reject the H0 of (2.3) when k = j, but it would retain the H0 of (2.3) when k = j + 1. In the literature, Ledford and Tawn (1996), Peng (1999), Draisma et al. (2004), and others studied testing extreme dependence problems in which the null hypothesis is that the two random variables are extreme dependent. Under a class of models, various test statistics have been proposed. The derivation of a test statistic under the null hypothesis of extreme dependence is still an open problem. So far, we have established a testing procedure for determining whether there exist extreme dependencies between random variables. Once the H0 of (2.3) is rejected, we conclude there exists extreme dependence between two random variables of interest. We now turn to real data analysis. 3. The data and data transformation Our objectives are to study (1) the extreme dependencies between USD/DEM and USD/YEN returns, and (2) extreme impacts on each of the two FX rates. Hence, in this study, let X1, X2, . . . be USD/DEM returns (daily or hourly), Y1, Y2, . . . be USD/YEN returns (daily or hourly), and n1, n2, . . . be either of these two series. 3.1. The data The data was collected by Olson and Associates. The observations cover the period from 27 April 1986 until 25 October 1998. The six different time horizons considered are one, two, four, eight, twelve hour and one day periods. Breymann et al. (2003) conducted filtering and deseasonalization on the data. Extreme dependence indexes (and hence extreme co-movements, extreme impacts) are associated with identically distributed random variables, but asset return series such as a FX spot rate is considered not identically distributed due to the long memory of its volatility. Using the deseasonalized data, Dias and Embrechts (2003) conducted dynamic copula modeling with multivariate GARCH model. Since their results are rather convincing, we standardize the deseasonalized data by GARCH model. In this paper, we use the deseasonalized one hour and one day data and then fit them to univariate GARCH(1, 1) models (Bollerslev, 1986). The deseasonalized data is divided by estimated standard deviations and standardized return data (also called devolatized data, or GARCH residuals) are obtained. We then study extreme co-movements and extreme impacts within the standardized returns. Note that we have restricted our GARCH model fitting to GARCH(1, 1). Other GARCH type models, such as heavy tailed GARCH models (see Mikosch, 2003; Straumann, 2005, and others), GARCH models with dependent residual random variables (see Lee and Hansen, 1994, etc.), are potential applications. For a comparison, we plot deseasonalized data against standardized data for USD/ DEM, USD/YEN in days and hours – see Fig. 1. We also plot USD/DEM deseasonalized data against USD/YEN deseasonalized data, USD/DEM standardized data against USD/ YEN standardized data, in days and hours respectively – see Fig. 2. The plots for the (nonstandardized) deseasonalized data and the standardized data look similar but not identical. Particularly for daily data, most estimated standard deviations fall in the range between 0.9 and 1.2 which results in diagonal patterns which can clearly be seen in the left
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20
15
15
10
10
Standardized returns
Standardized returns
USD/DEM 1 Day
20
5 0 –5 –10
0 –5
–10
–15 –20 –20
5
–15
–10
0
10
20
–20 –20
Deseasonalized returns
–10
15
15
10 5 0 –5
10
20
5 0 –5
–10
–15
–15 0
20
10
–10
–10
10
USD/YEN 1 Hour
20
Standardized returns
Standardized returns
USD/YEN 1 Day
20
–20 –20
0
Deseasonalized returns
10
Deseasonalized returns
20
–20 –20
–10
0
Deseasonalized returns
Fig. 1. Comparison between deseasonalized data and standardized data. The top plots are for USD/DEM. The bottom plots are for USD/YEN. The left plots are for 1 day data. The right plots are for 1 h data.
panel in Fig. 1. Hence, using the standardized data is better to avoid misspecification in testing extreme dependencies. 3.2. Data transformation Following Section 2.3, we estimate the distribution of standardized data and transform it into unit Fre´chet scales so that the gamma test can be used to test the presences of extreme dependencies. The maximum likelihood estimates of GEV parameters are summarized in Tables 1–3. For daily data, from Table 1, we see that standardized negative returns of German Mark quoted against US dollar has a fat tail (the estimated shape parameter value is greater than 0), but the standardized positive returns are now short tailed. We also see that both returns of Japanese Yen quoted against US dollar are still fat tailed. Notice that all estimated shape parameter values are different, which in turn suggests asymmetry between negative returns and positive returns. Also notice that the estimated shape parameter values are close to zero, and hence, the standardized daily returns can be thought to have Gaussian tails. For hourly data, since the data points are very large – about 78,230 points. We split the data into 10 sub-series for the convenience of computation. From Tables 2 and 3, we see
Z. Zhang, K. Shinki / Journal of Banking & Finance 31 (2007) 1399–1415 Deseasonalized returns 1 Day
20
15
15
10
10
5
5
USD/YEN
USD/YEN
20
0 –5
–5
–10
–15
–15
–10
0
10
20
Deseasonalized returns 1 Hour
0
–10
–20 –20
–20 –20
–10
USD/DEM Standardized returns 1 Day
20
15
15
10
10
5
5
0 –5
–15
–15
0
20
10
20
Standardized returns 1 Hour
–5
–10
–10
10
0
–10
–20 –20
0
USD/DEM
USD/YEN
USD/YEN
20
1407
–20 –20
USD/DEM
–10
0
10
20
USD/DEM
Fig. 2. Comparison between USD/DEM and USD/YEN. The top plots are for deseasonalized data. The bottom plots are standardized data. The left plots are for 1 day data. The right plots are for 1 h data.
Table 1 Estimated values of parameters in GEV fitting using standardized return series and threshold value u = 1.2 for daily data Series
Nu
l (SE)
log w (SE)
n (SE)
USD/DEM negative USD/DEM positive
341 323
3.305482 (0.145446) 3.024353 (0.107579)
0.438589 (0.151050) 0.667407 (0.119296)
0.001969 (0.061733) 0.060415 (0.050207)
USD/YEN negative USD/YEN positive
328 297
3.561064 (0.168287) 2.949730 (0.132979)
0.297793 (0.154565) 0.496941 (0.145713)
0.011028 (0.063915) 0.056199 (0.060298)
Nu is the number of exceedances over u. Numbers in parentheses are estimated standard deviations.
that almost all series are fat tailed, except one USD/YEN standardized positive series having a negative estimated shape parameter value. From the split series, one can see that all estimated scale parameter values, all estimated location parameter values, and all estimated shape parameter values are different though they can be thought approximately close to each other for a particular parameter. Dias and Embrechts (2003) also find there are change points in the series. Therefore, it is reasonable to split the whole series into subseries. What we have done here is simply and equally splitting the whole sequence into 10 subintervals.
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Table 2 Estimated values of parameters in GEV fitting using standardized return series and threshold value u = 1.2 for USD/DEM hourly data Nu
l (SE)
Negative series 1 2 3 4 5 6 7 8 9 10
674 683 674 721 643 680 607 638 681 777
7.672220 6.804680 6.848543 6.397347 6.502903 5.710691 9.375714 6.223442 7.156392 6.964012
(0.690882) (0.574425) (0.662581) (0.512778) (0.613648) (0.449641) (0.040530) (0.540852) (0.637399) (0.650297)
0.374201 0.166911 0.168704 0.004694 0.121396 0.165534 0.915906 0.013994 0.427417 0.309103
(0.217174) (0.204156) (0.242075) (0.203877) (0.231543) (0.209283) (0.029643) (0.221815) (0.194720) (0.209685)
0.115845 0.093685 0.091014 0.065917 0.094638 0.053644 0.232811 0.065775 0.167165 0.139668
(0.041122) (0.037496) (0.045428) (0.037243) (0.042807) (0.038622) (0.014273) (0.041196) (0.035584) (0.036994)
Positive series 1 2 3 4 5 6 7 8 9 10
626 649 730 715 661 733 661 702 752 734
8.638055 7.484258 6.337594 5.867251 7.728190 6.496009 6.574191 8.179001 5.620976 5.867552
(1.004749) (0.788383) (0.471663) (0.460927) (0.776550) (0.560787) (0.644013) (0.860773) (0.336546) (0.436857)
0.723005 0.409703 0.002262 0.067402 0.543708 0.027732 0.248934 0.635715 0.217165 0.074778
(0.237559) (0.237312) (0.186881) (0.201041) (0.211443) (0.221312) (0.225150) (0.219334) (0.159370) (0.192102)
0.193144 0.138325 0.068004 0.077697 0.175673 0.068059 0.137402 0.187576 0.047628 0.076242
(0.042616) (0.043425) (0.033955) (0.036576) (0.037787) (0.040607) (0.040816) (0.039510) (0.029366) (0.035387)
log w (SE)
n (SE)
Nu is the number of exceedances over u. Numbers in parentheses are estimated standard deviations.
Notice that daily positive returns of USD/DEM are fitted with a short tailed parameter value, while hourly positive returns of USD/DEM are fitted with long-tailed parameter values, we argue that the accumulation of data causes the changing of long tail to short tail. But USD/YEN daily positive returns are still long-tailed. In other words, the price movements between two different currencies are significantly different. We explore extreme co-movements and extreme impacts for different returns next. 4. Extreme co-movements and extreme impacts In this section, we use the gamma test to detect extreme co-movements and extreme impacts in two exchange spot rates. In order to do so, the standardized data is converted into unit Fre´chet scale for the observations above certain threshold using the fitted GEV function and its connection to the generalized Pareto distribution. The gamma test is implemented for each certain length of time period, since the extreme dependence between variables may vary by period. The result is reported as the rejection rate, that is the number of rejecting null hypotheses over the total number of tests. 4.1. Extreme co-movements of daily data Usually, the notion of extreme co-movements in the stock market means that large jumps in returns from two different stocks happen at the same time (in this data analysis,
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Table 3 Estimated values of parameters in GEV fitting using standardized return series and threshold value u = 1.2 for USD/YEN hourly data Nu
l (SE)
Negative series 1 2 3 4 5 6 7 8 9 10
652 687 666 745 700 680 636 676 654 669
7.689437 7.690174 5.943126 7.035376 7.092855 7.822926 5.816896 7.047284 8.083421 7.467914
(0.750974) (0.646647) (0.473148) (0.599662) (0.622079) (0.819186) (0.434661) (0.647908) (0.887155) (0.745009)
0.392191 0.430634 0.159317 0.260564 0.296081 0.495258 0.076391 0.395617 0.585002 0.377056
(0.222850) (0.182003) (0.213052) (0.198535) (0.203666) (0.232781) (0.188880) (0.202182) (0.234861) (0.231108)
0.120125 0.136266 0.037921 0.116090 0.122993 0.152490 0.073258 0.161767 0.170735 0.128335
(0.040821) (0.032090) (0.039476) (0.036004) (0.037237) (0.042750) (0.035085) (0.036758) (0.042717) (0.042880)
Positive series 1 2 3 4 5 6 7 8 9 10
654 639 729 735 662 678 671 730 695 715
7.159015 8.821497 6.089159 5.025677 6.939650 5.917257 6.661354 6.851863 6.058247 6.156229
(0.649723) (1.327930) (0.489057) (0.313351) (0.593644) (0.495130) (0.566859) (0.620021) (0.528339) (0.529878)
0.315680 0.858021 0.084831 0.577958 0.543708 0.125039 0.177595 0.233014 0.058887 0.024695
(0.211308) (0.299076) (0.206855) (0.190994) (0.195076) (0.223383) (0.206427) (0.211337) (0.227645) (0.219840)
0.123436 0.239423 0.056796 0.022408 0.137517 0.052073 0.105988 0.116898 0.065642 0.071737
(0.039201) (0.053912) (0.037509) (0.035185) (0.036059) (0.041514) (0.038571) (0.038044) (0.042113) (0.040185)
log w (SE)
n (SE)
Nu is the number of exceedances over u. Numbers in parentheses are estimated standard deviations.
we use the same day and the same hour). When jumps in returns have an impact in the following trading days or hours, it is better to consider cross-sectional extreme dependencies or to extend the meaning of extreme co-movements to that large jumps in returns from two different stocks happen in a short time period of several consecutive days or hours. We use the extended meaning and apply the gamma test to all transformed data. In each test, we use threshold values of 95th percentiles of the data and test level of a = .05. The total number of tests equals the sample size subtracting the maximal length of test windows (in this study we choose 500 consecutive points in each test window). The testing results are summarized in Table 4. In the table, N. USD/DEM stands for the negative returns of USD/DEM. P. USD/ DEM stands for the positive returns of USD/DEM. N. USD/YEN stands for negative returns of USD/YEN. P. USD/YEN stands for positive returns of USD/YEN. In the table, the cell of Row (A, B) and Column (Day 0, Day i) means that the return value of asset A on Day 0 and the return value of asset B on Day i are formed in pairs and used in the test. Table 4 This table summarizes the test results by using the gamma test and subset scheme of daily data Pairs
(Day 0, Day 0)
(Day 0, Day 1)
(Day 0, Day 2)
(N. USD/DEM, N. USD/YEN) (N. USD/DEM, P. USD/YEN) (P. USD/DEM, N. USD/YEN) (P. USD/DEM, P. USD/YEN)
1.0000 0.00 0.00 0.4858
0.00 0.00 0.00 0.00
0.00 0.00 0.00 0.00
The number in each cell is the rejection rate of the null hypothesis of extreme independence.
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From Table 4, we see that there are clearly extreme co-movements between negative USD/DEM returns and negative USD/YEN returns since the rejection rate of the null hypothesis of extreme independence is 100%. The rejection rate of the null hypothesis of extreme independence between positive returns is 49%. This number (49%) suggests there were extreme co-movement between positive USD/DEM returns and positive USD/YEN returns, but extreme co-movements did not appear in all subintervals – i.e., there were change points that price movements behaved differently. A change point analysis conducted by Dias and Embrechts (2003) also suggests such phenomena. 4.2. Extreme co-movements of hourly data Like in the previous section, we use the extended meaning and perform the gamma test to all transformed data. There are totally 2615 tests. In each test, we use threshold values of 95th percentiles of the data and test level of a = .05. The testing results are summarized in Table 5. In the table, the cell of Row (A, B) and Column (Hour 0, Hour i) means that the return value of asset A on Hour 0 and the return value of asset B on Hour i are formed in pairs and used in the test. From Table 5, we see that there are clearly extreme co-movements between USD/DEM hourly price change and USD/YEN hourly price change. Notice that there are more time periods of positive extreme co-movements than time periods of negative extreme co-movements which show an asymmetric price movements in hourly return data. We also see the difference between hourly data and daily data. During market recession, the chance of extreme co-movements will more likely be increasing (from a low rejection rate in hourly data to a high rejection rate in daily data of null hypothesis of extreme independence). During market expansion, however, the chance of extreme co-movements is likely decreasing from hourly movements to daily movements. We also see that extreme co-movements only occur at the same day or the same hour because all numbers in columns (Day 0, Day 1) and (Day 0, Day 2) are zeros. This phenomenon suggests that a large price change of one spot rate at a particular time does not have an extreme impact on another spot rate at the time points after 1 h or 1 day. This is interesting. It is suggesting that the cross-sectional extreme impacts only occur in a very short time period, probably a ‘tick by tick’ time period. 4.3. Extreme impacts of daily data From the previous two sections, we have seen that extreme co-movements between two different exchange currencies were found using the gamma test. We now turn to study extreme impacts within each exchange spot rate sequence. In order to test extreme impacts, Table 5 This table summarizes the test results by using the gamma test and subset scheme of hourly data Pairs
(Hour 0, Hour 0)
(Hour 0, Hour 1)
(Hour 0, Hour 2)
(N. USD/DEM, N. USD/YEN) (N. USD/DEM, P. USD/YEN) (P. USD/DEM, N. USD/YEN) (P. USD/DEM, P. USD/YEN)
0.7908 0.00 0.00 0.8563
0.00 0.00 0.00 0.00
0.00 0.00 0.00 0.00
The number in each cell is the rejection rate of the null hypothesis of extreme independence.
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we need to project a univariate sequence into a bivariate sequence. A simple way to accomplish this is to let X 0i ¼ X i , Y 0i ¼ X iþk , i = 1, . . . , n k for a particular lag-k. For each return sequence (negative or positive), the rejection rate of null hypothesis of extreme independence varies from case to case. One can see that identifying extreme impacts is more difficult than identifying extreme co-movements between two different risk factors. Sometimes, the gamma test alone may not be sufficient to suggest a lag-k extreme dependence. Other statistical quantities together with the test results may help to identify a best lag-k. A natural choice would be the empirical estimation of the extreme dependence index. The empirical formula used in this study is Pm i¼1 I ðX i >u;Y i >uÞ ^ kemp ¼ P ; m i¼1 I ðX i >uÞ where m (m = 500 in this study) is the number of points used in each test window, {Xi} is one time series in the data set, {Yi} is another time series in the data set. u is a threshold value which is the minimum of the two 95th sample percentiles. The tested results are summarized in Tables 6 and 7. In the tables, R.R. denotes the rejection rate of rejecting the null hypothesis of extreme independence. There are totally 671 tests in which each uses 500 data points. For each data set of 500 data points, we compute the empirical estimates of extreme dependence index using the above empirical formula. (M.E.D.I.) stands for the average of estimated extreme dependence indexes. [S. std] stands for the sample standard deviation of estimated extreme dependence indexes. Table 6 Lag-k tail independence test for transformed return data Lag
1 2 3 4
USD/DEM negative
USD/DEM positive
R.R.
M.E.D.I.
S. std
R.R.
M.E.D.I.
S. std
1.0000 1.0000 1.0000 1.0000
0.0242 0.0182 0.0182 0.0121
0.0415 0.0647 0.1313 0.0419
0.1853 0.7844 0.0000 0.0000
0.0062 0.0123 0.0062 0.0062
0.0261 0.0437 0.0092 0.0113
R.R. denotes the rejection rate of rejecting the null hypothesis of extreme independence. M.E.D.I. means empirical estimation of extreme dependence index. [S. std] stands for the sample standard deviation of estimated extreme dependence indexes.
Table 7 Lag-k tail independence test for transformed return data Lag
1 2 3 4
USD/YEN negative
USD/YEN positive
R.R.
M.E.D.I.
S. std
R.R.
M.E.D.I.
S. std
1.0000 1.0000 1.0000 1.0000
0.0189 0.0252 0.0126 0.0126
0.0421 0.0975 0.0445 0.0333
1.0000 1.0000 0.0000 0.0000
0.006 0.0119 0.0238 0.0179
0.076 0.1017 0.0133 0.0105
R.R. denotes the rejection rate of rejecting the null hypothesis of extreme independence. M.E.D.I. means empirical estimation of extreme dependence index. [S. std] stands for the sample standard deviation of estimated extreme dependence indexes.
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We see from these two tables, up to lag-4, negative returns have extreme impacts all the time. Extreme impacts within positive returns were found to last about three consecutive days. 4.4. Extreme impacts of hourly data Like the previous section, we project each univariate sequence into a bivariate sequence. The testing results are summarized in Tables 8 and 9 for lag-k = 1, 2, . . . , 10, 96, . . . , 100. Table 8 Lag-k tail independence test for transformed return data (USD/DEM) Lag
1 2 3 4 5 6 7 8 9 10 .. . 96 97 98 99 100
Negative
Positive
R.R.
M.E.D.I.
S. std
R.R.
M.E.D.I.
S. std
0.9409 0.9485 0.8498 0.7268 0.866 0.7855 0.8356 0.9547 0.8104 0.7791 .. . 0.8653 1.0000 1.0000 0.0000 1.0000
0.0298 0.0277 0.0154 0.0157 0.0146 0.011 0.0085 0.0077 0.0108 0.0087 .. . 0.0013 0.0015 0.0005 0.0015 0.0008
0.0664 0.0604 0.0636 0.0522 0.065 0.0486 0.0575 0.063 0.0594 0.0528 .. . 0.0569 0.0866 0.0455 0.0131 0.0773
0.8625 0.8729 0.8601 0.6782 0.7295 0.8159 0.7273 0.8699 0.7425 0.755 .. . 1.0000 1.0000 1.0000 1.0000 1.0000
0.0259 0.0187 0.0218 0.0138 0.0144 0.0108 0.0103 0.011 0.0054 0.01 .. . 0.0003 0.001 0.0013 0.0005 0.0008
0.0624 0.0537 0.0624 0.0564 0.0557 0.0439 0.0592 0.0655 0.0464 0.0548 .. . 0.135 0.1088 0.0699 0.0533 0.1099
M.E.D.I. means empirical estimation of extreme dependence index. Table 9 Lag-k tail independence test for transformed return data Lag
1 2 3 4 5 6 7 8 9 10 .. . 96 97 98 99 100
Negative
Positive
R.R.
M.E.D.I.
S. std
R.R.
M.E.D.I.
S. std
0.8911 0.8645 0.8537 0.8871 0.8406 0.9538 0.7673 0.847 0.9058 0.9071 .. . 1.0000 1.0000 1.0000 1.0000 0.609
0.0247 0.0263 0.0173 0.0162 0.0142 0.0108 0.0111 0.007 0.0085 0.0057 .. . 0.0003 0.0008 0.0021 0.0013 0.0008
0.0699 0.0552 0.0524 0.0672 0.0649 0.063 0.0574 0.062 0.0713 0.0669 .. . 0.0403 0.1107 0.1519 0.0455 0.02
0.9195 0.8587 0.8584 0.9011 0.747 0.8392 0.745 0.9389 0.8759 0.6969 .. . 1.0000 0.9652 1.0000 0.6199 1.0000
0.0212 0.0189 0.0146 0.0156 0.0141 0.0084 0.0097 0.0097 0.0077 0.0069 .. . 0.0015 0.0003 0.001 0.0008 0.0008
0.0598 0.0587 0.0464 0.0546 0.0482 0.0428 0.0519 0.0645 0.0661 0.0476 .. . 0.0652 0.0459 0.0272 0.0229 0.0677
M.E.D.I. means empirical estimation of extreme dependence index (USD/YEN).
Z. Zhang, K. Shinki / Journal of Banking & Finance 31 (2007) 1399–1415
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Plot for hourly data lag–k test
1
0.9
0.8
Rejection rates
0.7
0.6
0.5
0.4
0.3 N. USD/DEM P. USD/DEM N. USD/YEN P. USD/YEN
0.2
0.1
0
0
10
20
30
40
50 lag–k
60
70
80
90
100
Fig. 3. Rejection rates of lag-1 to lag-100 extreme dependence tests for hourly data.
We see there are extreme impacts of hourly returns (negative or positive) for very large lags. The rejection rates are also plotted in Fig. 3. From the figure, we see that the rejection rates within the first 30 lags are around 80%. Then the rejection rates are varying in a wide range when lag-k is increasing. From the previous section, we have seen that extreme impacts of positive daily data appear to last within the following two consecutive days. From results in Tables 8 and 9, we see that extreme impacts of positive hourly data appear to last more than four days (will be higher if larger lags are used in test). This phenomenon suggests that extreme impacts of high frequency data are stronger than extreme impacts of low frequency data. It also suggests great care must be taken when modeling high frequency data. 5. Conclusion This paper has presented ways of analyzing extreme co-movements and extreme impacts among financial market variables. We focused on two FX spot rates. We found that there exist extreme co-movements and extreme impacts in financial market data. The methodology proposed in this paper of testing extreme co-movements and extreme impacts among two FX spot rates can be used to assessing extreme co-movements and extreme impacts among other financial returns, such as stock returns. We have proposed procedures of: (1) fitting GARCH(1, 1) (can be GARCH(p, q)) model to price returns; (2)
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obtaining devolatized return data; (3) fitting GPD to exceedance values over a typical high threshold value; (4) transforming the observed values to unit Fre´chet scales by fitted GPD (or GEV) functions; (5) performing the gamma test to the transformed data; (6) identifying extreme co-movements and extreme impacts among financial returns. Based on the results of our analysis with FX spot rates, we argue that extreme co-movement and extreme impact problems are inherently stochastic control problems, since they will influence the decision taken today and ultimately influence a decision taken in the future. For the daily data, there are existing models which are able to model extreme co-movements and extreme impacts among the interest variables. Those models are studied, for example, by Deheuvels (1983), Davis and Resnick (1989), Smith and Weissman (1996), Hall et al. (2002), Zhang and Smith (2004, submitted for publication), Zhang (2005), Chamu´ Morales (2005), Zhang and Huang (2006), Heffernan et al. (submitted for publication), among others. The models have been used in stochastic optimal control and portfolio optimizations. For the high frequency data (for example, hourly data), efforts of constructing reasonable models are still needed. Our ultimate goal is to construct portfolio optimization models for high frequency data, especially models which are applicable during the time of market recession or market expansion. Acknowledgements The work by the first author was supported in part by NSF Grants DMS-0505528, and DMS-0630210, Wisconsin Alumni Foundation, and Institute for Mathematical Research, ETH Zu¨rich. The authors thank Dr. Dias for sharing the data with us. References Bollerslev, T., 1986. Generalized autoregressive conditional heteroscedasticity. J. Econom. 31, 307–327. Breymann, W., Dias, A., Embrechts, P., 2003. Dependence structures for multivariate high-frequency data in finance. Quant. Fin. 3, 1–14. Chamu´ Morales, F., 2005. Estimation of max-stable processes using Monte Carlo methods with applications to financial risk assessment. PhD dissertation, Department of Statistics at the University of North Carolina. Davis, R.A., Resnick, S.I., 1989. Basic properties and prediction of Max-ARMA processes. Adv. Appl. Prob. 21, 781–803. de Haan, L., Resnick, S.I., 1977. Limit Theory for Multivariate Sample Extremes. Z. Wahrscheinlichkeitstheorie verw Gebiete 40, 317–337. Deheuvels, P., 1983. Point processes and multivariate extreme values. J. Multivariate Anal. 13, 257–272. Dias, A. Embrechts, P., 2003. Dynamic copula models for multivariate high-frequency data in finance, web reference, ETHZ. Draisma, G., Drees, H., Ferreira, A., de Haan, L., 2004. Bivariate tail estimation: Dependence in asymptotic independence. Bernoulli 10, 251–280. Embrechts, P., Klu¨ppelberg, C., Mikosch, T., 1997. Modelling Extremal Events for Insurance and Finance. Springer. Embrechts, P., McNeil, A., Straumann, D., 2002. Correlation and dependence in risk management: Properties and pitfalls. In: Dempster, M.A.H. (Ed.), Risk Management: Value at Risk and Beyond. Cambridge University Press, Cambridge, pp. 176–223. Grady, A., 2000. A higher order expansion for the joint density of the sum and the maximum with applications to the estimation of the climatological trends. PhD Dissertation, Department of Statistics, University of North Carolina. Hall, P., Peng, L., Yao, Q., 2002. Moving-maximum models for extrema of time series. J. Statist. Plann. Infer. 103, 51–63.
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