Asymmetric Dependence, Tail Dependence, and the. Time Interval over which the Variables Are Measured
|
|
- Debra Watkins
- 5 years ago
- Views:
Transcription
1 Asymmetric Dependence, Tail Dependence, and the Time Interval over which the Variables Are Measured Byoung Uk Kang and Gunky Kim Preliminary version: August 30, 2013 Comments Welcome! Kang, School of Accounting and Finance, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong; Kim, School of Mathematics and Applied Statistics, University of Wollongong, Wollongong, NSW 2522, Australia.
2 Asymmetric Dependence, Tail Dependence, and the Time Interval over which the Variables Are Measured Abstract The effect of time interval on the linear correlation coefficient between random variables is well documented in the literature. In this paper, we investigate the time interval effect on asymmetric dependence and tail dependence between random variables. We prove that when two random variables are characterized by asymmetric dependence (of any direction), the magnitude of asymmetry in their dependence structure decreases monotonically and approaches zero (i.e., symmetry) as the time interval increases. Also, when two random variables exhibit tail dependence, their tail dependence decreases monotonically and approaches zero (i.e., tail independence) as the time interval increases. Our results hold regardless of whether the variables are both additive, both multiplicative, or one is additive and the other is multiplicative.
3 1 Introduction When examining the association between variables, researchers often make an arbitrary choice of time intervals (e.g., daily, weekly, monthly). This choice may be dictated by data availability or by consideration for the balance between the number of observations and measurement errors, but not by theory. Given a considerable leeway and lack of theoretical guidance in selecting time intervals, it is important to ask whether and how the time interval selected affects the association between variables. Recent empirical research provides some guidance on this issue. Breymann et al. (2003) and Dias and Embrechts (2010) examine bivariate series of high-frequency exchange rates at several time intervals (from hourly to daily) and find that the dependence structure is best described by a t-copula with successively larger degrees of freedom as the time horizon increases. Kang et al. (2010) investigate, at a range of investment horizons (from one quarter to five years), the asymmetric dependence between hedge fund returns and market returns and find that the magnitude of asymmetry decreases as the investment horizon increases. However, all these results are empirical in nature and depend on a given set of data and time horizons considered. Extant theoretical work on the time interval effect on cross-variable association focuses mainly on measures of linear dependence. Levy and Schwarz (1997) study the linear correlation coefficient between two random variables that are both multiplicative over time (e.g., gross returns) and show that it decreases monotonically as the time interval increases, approaching zero in the limit. Levy et al. (2001) show a similar theoretical effect on the correlation coefficient when one random variable is additive (e.g., log returns) and the other is multiplicative. Of course, when two random variables are both additive, it is straightforward to show that the correlation coefficient is invariant to time interval. 1 It is important to note, however, that correlation coefficient is not a sufficient measure to describe cross-variable association other than joint normality or, more generally, ellipticality (see, e.g., Embrechts et al. 2002). Since there is no multiplicative variable that maintains normality over multiple time intervals, the time-horizon profile of correlation coefficient, documented by Levy and Schwarz (1997) and Levy et al. (2001), may not be informative about how variables intrinsic association changes as a function of the time interval. In this paper, we extend the extant theoretical work on the time interval effect by using copulas as descriptions of association. Copula-based approach is natural in situations where the association between variables is of primary interest and is valid regardless of whether the joint distribution of the variables is normal or not. 2 We are particularly interested in analyzing the time interval effect on nonlinear or asymmetric dependence (e.g., greater dependence for joint negative events than 1 Relatedly, Schneller (1975) and Levy et al. (2001) study the time interval effect on the linear regression coefficient of one random variable on another in the multiplicative-multiplicative and additive-multiplicative cases, respectively. Note that the regression coefficient is simply the correlation coefficient scaled by the ratio of two standard deviations. 2 Sklar (1959) s theorem states that any joint distribution can be represented in terms of the marginals and a dependence function, termed the copula. While information concerning individual variables (e.g., mean, standard deviation, skewness, kurtosis) is entirely determined by the marginal distributions, the dependence relationship between the variables is completely described by the copula. 1
4 for joint positive events), as this type of association has been widely observed in financial data and has important implications in portfolio allocation and risk management (see, e.g., Longin and Solnik 2001; Ang and Chen 2002; Patton 2006, 2007; Hong et al. 2007). While one might suggest to conduct this analysis by means of exceedance correlation (used, e.g., in Longin and Solnik 2001, Ang and Chen 2002, Hong et al. 2007), exceedance correlation is less ideal for our purpose as it can change across time intervals due to purely marginal aspects (e.g., skewness). Copulas allow us to abstract from the time interval effect on marginal distributions and concentrate on the effect on variables intrinsic association and asymmetries therein. In addition, copulas permit an examination of joint behavior at the tails of distribution, because measures of such tail behavior, known as tail dependence, can be directly expressed in terms of the copula associated with its joint distribution (see, e.g., Joe 1997). Thus, by examining the time-horizon profile of copula, we can also address some of the suggestions of Poon et al. (2004), who propose investigating the time-aggregating properties of extreme values and the effect of investment time horizon on tail dependence, among other issues. Our theoretical results are easy to summarize. First, when two random variables are characterized by asymmetric dependence (of any direction), the magnitude of asymmetry in their dependence structure decreases monotonically and approaches zero (i.e., symmetry) as the time interval increases. Second, when two random variables exhibit tail dependence, their tail dependence decreases monotonically and approaches zero (i.e., tail independence) as the time interval increases. Finally, all these results hold regardless of whether the variables are both multiplicative, whether one is additive and the other is multiplicative, or whether both are additive. It is worth emphasizing that our finding of asymptotic symmetry and tail independence in the multiplicative-multiplicative and additive-multiplicative cases is not a mere manifestation of asymptotic independence, which one might infer from Levy and Schwarz s (1997) and Levy et al. s (2001) finding of asymptotic zero correlation. First of all, zero correlation coefficient does not necessarily mean independence (the converge is always true, however). 3 Furthermore, we show that the dependence structure between two random variables converges in the limit to the Gaussian copula, rather than to the independence copula. It is interesting to note that the dependence parameter of the limiting Gaussian copula is equal to the single-period correlation coefficient between the two random variables (or their logarithms). Hence, unless the single-period correlation coefficient is zero, the variables in the limit cannot be treated as independent in the cases considered by Levy and Schwarz (1997) and Levy et al. (2001). Perhaps our most striking result is that the time interval effect arises even in the additiveadditive case, which has potentially important implications for the study of term structure of optimal portfolio allocation across assets. The asset allocation literature, which typically works with log returns in modeling return dynamics (hence with additive variables), shows that predictability of asset returns can drive a wedge between the asset allocation strategies of short- and long- 3 One can easily think of two variables that are certainly related but whose correlation coefficient is zero (e.g., Y = X 2, where X N(0, 1)). 2
5 term investors (see, e.g., Campbell and Viceira 2002, 2005). Our results suggest that even when returns are unpredictable, asset allocation recommendations can still be a function of the investment horizon, through the effects of changing portfolio downside risk and tail risk across investment horizons. The remainder of this paper is organized as follows. Section 2 fixes notations for each of the three cases considered. Section 3 provides some simulation examples to illustrate our theoretical results. Section 3 formally presents our theoretical results. And Section 4 concludes. 2 Notations Since we consider all three different cases considered separately in the literature, we find it useful to fix some notations first. 2.1 The Additive-Additive Case Let (X 1, Y 1 ), (X 2, Y 2 ),... be a sequence of independent, identically distributed (i.i.d) pairs of additive variables. Let the expected value of X and Y, respectively, be E(X) = µ X and E(Y ) = µ Y. The variances are denoted by Var(X) = σ 2 X and Var(Y ) = σ2 Y. We define two new variables, W n and V n, to denote an n-fold increase of the differencing interval. These variables are given by W n = X 1 + X X n and V n = Y 1 + Y Y n. We denote the joint distribution of (W n, V n ) by H aa (W n, V n ) = C aa (F a (W n ), G a (V n )), (1) where F a denotes the marginal distribution of W n, G a denotes the marginal distribution of V n, and C aa denotes the copula that couples the marginals into the join distribution (Sklar, 1959). The linear correlation coefficient between W n and V n is denoted by ρ aa. 2.2 The Multiplicative-Multiplicative Case Let (X 1, Y 1 ), (X 2, Y 2 ),... be a sequence of i.i.d pairs of multiplicative variables. As a multiplicative framework is only appropriate for positive variables, assume X 1 > 0 and Y 1 > 0, as in Levy and Schwarz (1997). We define two new variables, W n and V n, to denote an n-fold increase of the differencing interval. These variables are given by W n = X 1 X 2 X n and V n = Y 1 Y 2 Y n. 3
6 We denote the joint distribution of (W n, V n) by H mm (W n, V n) = C mm (F m (W n), G m (V n)), (2) where F m denotes the marginal distribution of W n, G m denotes the marginal distribution of V n, and C mm denotes the copula that couples the marginals into the join distribution. The linear correlation coefficient between W n and V n is denoted by ρ mm. 2.3 The Additive-Multiplicative Case Let (X 1, Y 1 ), (X 2, Y 2 ),... be a sequence of i.i.d pairs of additive and multiplicative variables. We denote the joint distribution of (W n, V n) by H am (W n, V n) = C am (F a (W n ), G m (V n)), (3) where C am denotes the copula that couples the marginals F a and G m into the join distribution. The linear correlation coefficient between W n and V n is denoted by ρ am. Throughout, we drop the superscript on single-period distribution functions and dependence measures (i.e., when n = 1). Also, for notational convenience, we sometimes omit the subscripts aa, mm, and am when doing so causes no confusion. 3 Illustrative Examples Before formally presenting our results, we first provide some simulation examples to illustrate our main theoretical findings. Taking as an example the additive-additive case, our simulation exercise proceeds as follows: First, for a given n, we generate n independent draws of (X, Y ) from a bivariate data characterized by asymmetric dependence and tail dependence (see below). Compounding these n draws amounts to generating one draw of (W n, V n ) from the density function corresponding to H aa. We repeat this step 500,000 times to create the simulated joint distributions of n-period data. Based on the simulated data, we then compute various dependence measures such as the linear correlation coefficient, quantile dependence, and tail dependence. Finally, we repeat this procedure for a wide range of n to reveal the profile of the selected dependence measures across time intervals. For this illustration, we use monthly returns on the market portfolio and on the ten size portfolios from July 1963 to December We know from Hong et al. (2007) that the dependence structure between the market portfolio and the smallest size portfolio is asymmetric in that returns exhibit stronger correlations for downside moves than upside moves. Ang and Chen (2002) and Patton (2004) also document the same form of asymmetries in a similar data set. The market and size portfolios are constructed using all NYSE/AMEX/NASDAQ stocks and are taken from Professor 4
7 Ken French s web site. risk-free rate. Following Ang and Chen (2002), all risky returns are in excess of the Figure 1 illustrates the time interval effect on the linear correlation coefficient between two random variables, when their joint distribution is characterized by asymmetric dependence. Since Levy and Schwarz (1997) and Levy et al. (2001) do not make any distributional assumption in deriving their results, we should see that their results continue to hold here. As expected, Figure 1 shows that ρ mm and ρ am decrease monotonically as n increases, and that ρ aa is not affected by the selected time interval. Figure 1 about here Figure 2 computes, for each n, the sum of the squared differences between quantile dependences at q and those at 1 q, using q = {0.1, 0.2, 0.3, 0.4, 0.5}. 4 Quantile dependence at a threshold level q is defined as λ aa (q) Pr[F a (W n ) q G a (V n ) q], if q 0.5, Pr[F a (W n ) q G a (V n ) q], if q 0.5, C aa (q,q) q, if q 0.5, 1 2q+C aa (q,q) 1 q, if q 0.5, for the additive-additive case; the quantile dependence for the multiplicative-multiplicative and additive-multiplicative cases can be similarly defined as above, for example, by using m or mm in place of a or aa for the multiplicative-multiplicative case. Clearly, the magnitude of asymmetry in the dependence structure decreases monotonically and approaches zero as the time interval increases, illustrating our first main result. (4) Figure 2 about here Figure 3 plots the tail dependence between the simulated data against n. Tail dependence can be thought of as the limit of quantile dependence as q 0 or q 1: λ L λ U lim q 0 lim q 1 C (q, q), (5) q 1 2q + C (q, q). 1 q Here, we estimate tail dependence nonparametrically based directly on extreme observations (as in, e.g., Schmidt and Stadtmüller 2006). The threshold level used is the 1% (i.e., q = 0.01), which corresponds to 5,000 observations in each tail. Figure 3 illustrates our second main result: The tail 4 This is essentially Patton s (2012) statistic for asymmetric dependence, with a bootstrap estimate of V λ replaced by the identify matrix to facilitate direct comparison across data sets. 5
8 dependence decreases monotonically and approaches zero as the time interval increases. Finally, Figures 2 and 3 also illustrate that the observed cross-horizon patterns do not change regardless of whether the variables are both additive, both multiplicative, or one is additive and the other is multiplicative, which is our final main result. Figure 3 about here 4 The Main Results Like Levy and Schwarz (1997) and Levy et al. (2001), we proceed in two steps: First, we establish the asymptotic result (e.g., lim n λ L = 0) and then the monotonicity (e.g., λ / n 0). Unlike L the previous studies, however, our theoretical approach deals with the entire dependence structure, rather than a measure of it. 4.1 The Additive-Additive Case The Central Limit Theorem (CLT) states that, as n, ( (W n nµ X )/ nσ X (V n nµ Y )/ nσ Y ) d N [( ) ( 0, 0 1 ρ aa ρ aa 1 )]. (6) Thus, the asymptotic copula of (W n nµ X )/ nσ X and (V n nµ Y )/ nσ Y is the Gaussian copula with dependence parameter equal to ρ aa, regardless of the single-period copula of X and Y. This result can be easily extended to the asymptotic copula of W n and V n. Proposition 1. Let (X 1, Y 1 ), (X 2, Y 2 ),... be a sequence of i.i.d. bivariate random vectors that satisfies the conditions of a CLT. Then, as n, ) C aa (u, v) Φ ρaa (Φ 1 (u), Φ 1 (v), (7) where Φ ρ is the bivariate standard normal cdf with correlation ρ. Proof. Let W n = (W n nµ X )/ nσ X and Ṽn = (V n nµ Y )/ nσ Y. The cdf s of Wn and Ṽn are denoted by using tilde on the corresponding cdf s of W n and V n. Then, by a CLT, as n, H aa ( W n, Ṽn) Φ ρaa ( W n, Ṽn). (8) Note that the asymptotic copula of ( W n, Ṽn) is the Gaussian copula whose dependence parameter is ρ aa. Also note that the copula of ( W n, Ṽn) is the same as that of (W n, V n ) for all n (including the limit) because W n and Ṽn, respectively, are strictly increasing in W n and V n. Therefore, as n, ) C aa (u, v) Φ ρaa (Φ 1 (u), Φ 1 (v). 6
9 There are two important implications from this proposition: First, regardless of its direction, the asymmetry in the dependence structure will disappear as we add up the variables because their copula converges to a Gaussian copula. Second, for this reason, their tail dependence (if any) will also disappear as n. Let us assume that the copula of (W n, V n ), for a finite n, is a mixture of the single-period copula and its limiting copula; that is, ) C aa (u, v) = (1 α n ) C aa (u, v) + α n Φ ρaa (Φ 1 (u), Φ 1 (v), (9) where α n : {1, 2,..., } [0, 1] is a function of n. Under this assumption, our proof for the monotonicity boils down to showing that αn n 0. Although the true dependence structure may not be perfectly described by the mixture model, our unreported simulation study confirms that this model works well. 5 Proposition 2. Under the mixture model in (9), α n is monotonically increasing in n. Proof. By the Berry-Esseen theorem and by Proposition 1, we have sup C (u,v) [0,1] 2 aa (u, v) Φ ρaa (Φ 1 (u), Φ 1 (v)) = O(n 1/2 ), for all n, where Φ is the univariate standard normal cdf. Under the mixture model in (9), it follows that sup C (u,v) [0,1] 2 = sup (u,v) [0,1] 2 = (1 α n ) sup aa (u, v) Φ ρaa (Φ 1 (u), Φ (v)) 1 (1 α n ) C aa (u, v) + α n Φ ρaa (Φ 1 (u), Φ 1 (v) (u,v) [0,1] 2 Note that sup (u,v) [0,1] 2 C aa (u, v) Φ ρaa (Φ 1 (u), Φ (v)) 1. Therefore, α n is monotonically increasing in n. ) ( Φ ρaa Φ 1 (u), Φ (v)) 1 C aa (u, v) Φ ρaa (Φ 1 (u), Φ 1 (v)) = O(1). Hence, 1 αn = O(n 1/2 ). 4.2 The Multiplicative-Multiplicative Case Proposition 3. Let (ln X 1, ln Y 1 ), (ln X 2, ln Y 2 ),... be a sequence of i.i.d. bivariate random vectors that satisfies the conditions of a CLT. Then, as n, ( ) C mm(u, v) Φ ρln X ln Y Φ 1 (u), Φ 1 (v), (10) where ρ ln X ln Y is the linear correlation coefficient between ln X and ln Y. 5 The simulation results are readily available from the authors upon request. 7
10 Proof. From the definition, it follows that ln W n = ln X 1 + +ln X n and ln V n = ln Y 1 + +ln Y n. Then, by Proposition 1, as n, the copula of (ln W n, ln V n) becomes the Gaussian copula with dependence parameter equal to ρ ln X ln Y. Note that the copula of (W n, V n) is the same as that of (ln W n, ln V n) for all n (including the limit) because ln W n and ln V n, respectively, are strictly increasing in W n and V n. Therefore, the asymptotic copula of (W n, V n) is also the Gaussian copula with dependence parameter equal to ρ ln X ln Y. Proposition 4. Suppose that the copula of (ln W n, ln V n), for a finite n, is a mixture of the singleperiod copula and its limiting copula with a function of n, β n : {1, 2,..., } [0, 1], being the weight on the latter. Under such a model, by Proposition 2, β n is monotonically increasing in n. Since the copula of (W n, V n) is the same as that of (ln W n, ln V n) for all n, by Proposition 4, the degree of asymmetry or tail dependence in C mm will monotonically converge to zero as we compound the variables. 4.3 The Additive-Multiplicative Case Proposition 5. Let (X 1, ln Y 1 ), (X 2, ln Y 2 ),... be a sequence of i.i.d. bivariate random vectors that satisfies the conditions of a CLT. Then, as n, ( ) C am(u, v) Φ ρx ln Y Φ 1 (u), Φ 1 (v), (11) where ρ X ln Y is the linear correlation coefficient between X and ln Y. Proof. From the definition, it follows that ln V n = ln Y ln Y n. Then, by Proposition 1, as n, the copula of (W n, ln V n) becomes the Gaussian copula with dependence parameter equal to ρ X ln Y. Note that the copula of (W n, V n) is the same as that of (W n, ln V n) for all n (including the limit) because ln V n is strictly increasing in V n. Therefore, the asymptotic copula of (W n, V n) is also the Gaussian copula with dependence parameter equal to ρ X ln Y. Proposition 6. Suppose that the copula of (W n, ln V n), for a finite n, is a mixture of the singleperiod copula and its limiting copula with a function of n, γ n : {1, 2,..., } [0, 1], being the weight on the latter. Under such a model, γ n is monotonically increasing in n. Since the copula of (W n, V n) is the same as that of (W n, ln V n) for all n, by Proposition 6, the degree of asymmetry or tail dependence in C am will monotonically converge to zero as we compound the variables. 5 Concluding Remarks The effect of time interval on the linear correlation coefficient between two random variables is well documented in the literature. In this paper, we investigate the time interval effect on asymmetric dependence and tail dependence between random variables. We prove that when two random 8
11 variables are characterized by asymmetric dependence (of any direction), the magnitude of asymmetry in their dependence structure decreases monotonically and approaches zero (i.e., symmetry) as the time interval increases. Also, when two random variables exhibit tail dependence, their tail dependence decreases monotonically and approaches zero (i.e., tail independence) as the time interval increases. Our results hold regardless of whether the variables are both additive, both multiplicative, or one is additive and the other is multiplicative. 9
12 References Ang, A., J. Chen Asymmetric correlations of equity portfolios. J. Financial Econom Breymann, W., A. Dias, P. Embrechts Dependence structures for multivariate high-frequency data in finance. Quant. Finance 3(1) Campbell, J. Y., L. M. Viceira Strategic Asset Allocation: Portfolio Choice for Long-Term Investors. Oxford University Press, Oxford, UK. Campbell, J. Y., L. M. Viceira The term structure of the risk-return trade-off. Financial Anal. J. 61(1) Dias, A., P. Embrechts Modeling exchange rate dependence dynamics at different time horizons. J. Internat. Money Finance Embrechts, P., A. J. McNeil, D. Straumann Correlation and dependence in risk management: Properties and pitfalls. In Risk Management: Value at Risk and Beyond, M. Dempster, ed. Cambridge University Press, Cambridge, UK. Hong, Y., J. Tu, G. Zhou Asymmetries in stock returns: Statistical tests and economic evaluation. Rev. Financial Stud. 20(5) Joe, H Multivariate Models and Dependence Concepts. Chapman & Hall, London, UK. Levy, H., I. Guttman, I. Tkatch Regression, correlation, and the time interval: Additive-multiplicative framework. Management Sci. 47(8) Levy, H., G. Schwarz Correlation and the time interval over which the variables are measured. J. Econometrics Longin, F., B. Solnik Extreme correlation of international equity markets. J. Finance 56(2) Kang, B. U., F. In, G. Kim, T. S. Kim A longer look at the asymmetric dependence between hedge funds and the equity market. J. Financial Quant. Anal. 45(3) Patton, A. J On the out-of-sample importance of skewness and asymmetric dependence for asset allocation. J. Financial Econometrics 2(1) Patton, A. J Modelling asymmetric exchange rate dependence. Internat. Econom. Rev. 47(2)
13 Patton, A. J Copula methods for forecasting multivariate time series. In Handbook of Economic Forecasting, G. Elliott, A. Timmermann, eds. Elsevier, Oxford. Poon, S.-H., M. Rockinger, J. Tawn Extreme value dependence in financial markets: Diagnostics, models, and financial implications. Rev. Financial Stud. 17(2) Schmidt, R., U. Stadtmüller Non-parametric estimation of tail dependence. Scand. J. Statist Schneller, M. I Regression analysis for multiplicative phenomena and its implication for the measurement of investment risk. Management Sci. 22(4) Sklar, A Fonctions de répartition à n dimensions et leurs marges. Publications de l Institut Statistique de l Université de Paris
14 FIGURE 1 Time Interval Effect on the Linear Correlation Coefficient Correlation ρ aa 0.55 ρ mm ρ am Time Interval Figure 1 shows the linear correlation coefficients between the returns on the market portfolio and the smallest size decile portfolio, at various time intervals. The data used in this figure is generated using historical simulation based on monthly returns from July 1963 to December
15 FIGURE 2 Sum of the Squared Differences between Quantile Dependences at q and those at 1 q, using q = {0.1, 0.2, 0.3, 0.4, 0.5}. Sum of Squared Differences between Quantile Dependences SSD aa SSD mm SSD am Time Interval Figure 2 shows, at various time intervals, the sum of the squared differences between quantile dependences at q and those at 1 q, using q = {0.1, 0.2, 0.3, 0.4, 0.5}. The data used in this figure is generated using historical simulation based on monthly returns on the market portfolio and the smallest size decile portfolio from July 1963 to December
16 FIGURE 3 Time Interval Effect on the Tail Dependence Lower Tail Dependence λ Laa λ Lmm λ Lam Time Interval Upper Tail Dependence λ Uaa λ Umm λ Uam Time Interval Figure 3 shows the 1% and 99% quintile dependences between the market portfolio and the smallest size decile portfolio, at various time intervals. The data used in this figure is generated using historical simulation based on monthly returns from July 1963 to December
A Goodness-of-fit Test for Copulas
A Goodness-of-fit Test for Copulas Artem Prokhorov August 2008 Abstract A new goodness-of-fit test for copulas is proposed. It is based on restrictions on certain elements of the information matrix and
More informationDependence Patterns across Financial Markets: a Mixed Copula Approach
Dependence Patterns across Financial Markets: a Mixed Copula Approach Ling Hu This Draft: October 23 Abstract Using the concept of a copula, this paper shows how to estimate association across financial
More informationThe Instability of Correlations: Measurement and the Implications for Market Risk
The Instability of Correlations: Measurement and the Implications for Market Risk Prof. Massimo Guidolin 20254 Advanced Quantitative Methods for Asset Pricing and Structuring Winter/Spring 2018 Threshold
More informationA simple graphical method to explore tail-dependence in stock-return pairs
A simple graphical method to explore tail-dependence in stock-return pairs Klaus Abberger, University of Konstanz, Germany Abstract: For a bivariate data set the dependence structure can not only be measured
More informationAsymmetry in Tail Dependence of Equity Portfolios
Asymmetry in Tail Dependence of Equity Portfolios Eric Jondeau This draft: August 1 Abstract In this paper, we investigate the asymmetry in the tail dependence between US equity portfolios and the aggregate
More informationTrivariate copulas for characterisation of droughts
ANZIAM J. 49 (EMAC2007) pp.c306 C323, 2008 C306 Trivariate copulas for characterisation of droughts G. Wong 1 M. F. Lambert 2 A. V. Metcalfe 3 (Received 3 August 2007; revised 4 January 2008) Abstract
More informationX
Correlation: Pitfalls and Alternatives Paul Embrechts, Alexander McNeil & Daniel Straumann Departement Mathematik, ETH Zentrum, CH-8092 Zürich Tel: +41 1 632 61 62, Fax: +41 1 632 15 23 embrechts/mcneil/strauman@math.ethz.ch
More informationFinancial Econometrics and Volatility Models Copulas
Financial Econometrics and Volatility Models Copulas Eric Zivot Updated: May 10, 2010 Reading MFTS, chapter 19 FMUND, chapters 6 and 7 Introduction Capturing co-movement between financial asset returns
More informationA TEST OF GENERAL ASYMMETRIC DEPENDENCE
A TEST OF GENERAL ASYMMETRIC DEPENDENCE LEI JIANG, ESFANDIAR MAASOUMI, JIENING PAN AND KE WU First draft: March 2015, This draft: December 9, 2015. Abstract. We extend the asymmetric correlation test in
More informationFramework for Analyzing Spatial Contagion between Financial Markets
Finance Letters, 2004, 2 (6), 8-15 Framework for Analyzing Spatial Contagion between Financial Markets Brendan O. Bradley a and Murad S. Taqqu b, a Acadian Asset Management Inc., USA b Boston University,
More informationCONTAGION VERSUS FLIGHT TO QUALITY IN FINANCIAL MARKETS
EVA IV, CONTAGION VERSUS FLIGHT TO QUALITY IN FINANCIAL MARKETS Jose Olmo Department of Economics City University, London (joint work with Jesús Gonzalo, Universidad Carlos III de Madrid) 4th Conference
More informationA simple nonparametric test for structural change in joint tail probabilities SFB 823. Discussion Paper. Walter Krämer, Maarten van Kampen
SFB 823 A simple nonparametric test for structural change in joint tail probabilities Discussion Paper Walter Krämer, Maarten van Kampen Nr. 4/2009 A simple nonparametric test for structural change in
More informationTail Dependence of Multivariate Pareto Distributions
!#"%$ & ' ") * +!-,#. /10 243537698:6 ;=@?A BCDBFEHGIBJEHKLB MONQP RS?UTV=XW>YZ=eda gihjlknmcoqprj stmfovuxw yy z {} ~ ƒ }ˆŠ ~Œ~Ž f ˆ ` š œžÿ~ ~Ÿ œ } ƒ œ ˆŠ~ œ
More informationarxiv: v2 [q-fin.st] 9 Jul 2015
Dependence structure of market states arxiv:153.94v [q-fin.st] 9 Jul 15 Desislava Chetalova, Marcel Wollschläger and Rudi Schäfer Fakultät für Physik, Universität Duisburg Essen, D 4748 Duisburg, Germany
More informationCopulas and dependence measurement
Copulas and dependence measurement Thorsten Schmidt. Chemnitz University of Technology, Mathematical Institute, Reichenhainer Str. 41, Chemnitz. thorsten.schmidt@mathematik.tu-chemnitz.de Keywords: copulas,
More informationTail dependence in bivariate skew-normal and skew-t distributions
Tail dependence in bivariate skew-normal and skew-t distributions Paola Bortot Department of Statistical Sciences - University of Bologna paola.bortot@unibo.it Abstract: Quantifying dependence between
More informationEVANESCE Implementation in S-PLUS FinMetrics Module. July 2, Insightful Corp
EVANESCE Implementation in S-PLUS FinMetrics Module July 2, 2002 Insightful Corp The Extreme Value Analysis Employing Statistical Copula Estimation (EVANESCE) library for S-PLUS FinMetrics module provides
More informationRisk Measures with Generalized Secant Hyperbolic Dependence. Paola Palmitesta. Working Paper n. 76, April 2008
Risk Measures with Generalized Secant Hyperbolic Dependence Paola Palmitesta Working Paper n. 76, April 2008 Risk Measures with Generalized Secant Hyperbolic Dependence Paola Palmitesta University of
More informationOverview of Extreme Value Theory. Dr. Sawsan Hilal space
Overview of Extreme Value Theory Dr. Sawsan Hilal space Maths Department - University of Bahrain space November 2010 Outline Part-1: Univariate Extremes Motivation Threshold Exceedances Part-2: Bivariate
More informationSimulation of Tail Dependence in Cot-copula
Int Statistical Inst: Proc 58th World Statistical Congress, 0, Dublin (Session CPS08) p477 Simulation of Tail Dependence in Cot-copula Pirmoradian, Azam Institute of Mathematical Sciences, Faculty of Science,
More informationMultivariate Distributions
IEOR E4602: Quantitative Risk Management Spring 2016 c 2016 by Martin Haugh Multivariate Distributions We will study multivariate distributions in these notes, focusing 1 in particular on multivariate
More informationDependence modelling of the joint extremes in a portfolio using Archimedean copulas : application to MSCI indices
Dependence modelling of the joint extremes in a portfolio using Archimedean copulas : application to MSI indices Dominique Guegan, Sophie A. Ladoucette To cite this version: Dominique Guegan, Sophie A.
More informationMarginal Specifications and a Gaussian Copula Estimation
Marginal Specifications and a Gaussian Copula Estimation Kazim Azam Abstract Multivariate analysis involving random variables of different type like count, continuous or mixture of both is frequently required
More informationDependence and VaR Estimation:An Empirical Study of Chinese Stock Markets using Copula. Baoliang Li WISE, XMU Sep. 2009
Dependence and VaR Estimation:An Empirical Study of Chinese Stock Markets using Copula Baoliang Li WISE, XMU Sep. 2009 Outline Question: Dependence between Assets Correlation and Dependence Copula:Basics
More informationSpeculation and the Bond Market: An Empirical No-arbitrage Framework
Online Appendix to the paper Speculation and the Bond Market: An Empirical No-arbitrage Framework October 5, 2015 Part I: Maturity specific shocks in affine and equilibrium models This Appendix present
More informationBehaviour of multivariate tail dependence coefficients
ACTA ET COMMENTATIONES UNIVERSITATIS TARTUENSIS DE MATHEMATICA Volume 22, Number 2, December 2018 Available online at http://acutm.math.ut.ee Behaviour of multivariate tail dependence coefficients Gaida
More informationMultivariate Distribution Models
Multivariate Distribution Models Model Description While the probability distribution for an individual random variable is called marginal, the probability distribution for multiple random variables is
More informationAn Econometric Study of Vine Copulas
An Econometric Study of Vine Copulas Pierre-André Maugis (Corresponding author) PSE, Université Paris 1 Panthéon-Sorbonne, 106 boulevard de l Hopital 75647 Paris Cedex 13, France E-mail: pierre-andre.maugis@malix.univ-paris1.fr.
More informationGaussian Slug Simple Nonlinearity Enhancement to the 1-Factor and Gaussian Copula Models in Finance, with Parametric Estimation and Goodness-of-Fit
Gaussian Slug Simple Nonlinearity Enhancement to the 1-Factor and Gaussian Copula Models in Finance, with Parametric Estimation and Goodness-of-Fit Tests on US and Thai Equity Data 22 nd Australasian Finance
More informationFinancial Econometrics and Quantitative Risk Managenent Return Properties
Financial Econometrics and Quantitative Risk Managenent Return Properties Eric Zivot Updated: April 1, 2013 Lecture Outline Course introduction Return definitions Empirical properties of returns Reading
More informationEfficient estimation of a semiparametric dynamic copula model
Efficient estimation of a semiparametric dynamic copula model Christian Hafner Olga Reznikova Institute of Statistics Université catholique de Louvain Louvain-la-Neuve, Blgium 30 January 2009 Young Researchers
More informationEstimation of direction of increase of gold mineralisation using pair-copulas
22nd International Congress on Modelling and Simulation, Hobart, Tasmania, Australia, 3 to 8 December 2017 mssanz.org.au/modsim2017 Estimation of direction of increase of gold mineralisation using pair-copulas
More informationSolutions of the Financial Risk Management Examination
Solutions of the Financial Risk Management Examination Thierry Roncalli January 9 th 03 Remark The first five questions are corrected in TR-GDR and in the document of exercise solutions, which is available
More informationRecovering Copulae from Conditional Quantiles
Wolfgang K. Härdle Chen Huang Alexander Ristig Ladislaus von Bortkiewicz Chair of Statistics C.A.S.E. Center for Applied Statistics and Economics HumboldtUniversität zu Berlin http://lvb.wiwi.hu-berlin.de
More informationIntroduction to Algorithmic Trading Strategies Lecture 10
Introduction to Algorithmic Trading Strategies Lecture 10 Risk Management Haksun Li haksun.li@numericalmethod.com www.numericalmethod.com Outline Value at Risk (VaR) Extreme Value Theory (EVT) References
More informationQuaderni di Dipartimento. Small Sample Properties of Copula-GARCH Modelling: A Monte Carlo Study. Carluccio Bianchi (Università di Pavia)
Quaderni di Dipartimento Small Sample Properties of Copula-GARCH Modelling: A Monte Carlo Study Carluccio Bianchi (Università di Pavia) Maria Elena De Giuli (Università di Pavia) Dean Fantazzini (Moscow
More informationLecture Quantitative Finance Spring Term 2015
on bivariate Lecture Quantitative Finance Spring Term 2015 Prof. Dr. Erich Walter Farkas Lecture 07: April 2, 2015 1 / 54 Outline on bivariate 1 2 bivariate 3 Distribution 4 5 6 7 8 Comments and conclusions
More informationOn the Estimation and Application of Max-Stable Processes
On the Estimation and Application of Max-Stable Processes Zhengjun Zhang Department of Statistics University of Wisconsin Madison, WI 53706, USA Co-author: Richard Smith EVA 2009, Fort Collins, CO Z. Zhang
More informationProbabilities & Statistics Revision
Probabilities & Statistics Revision Christopher Ting Christopher Ting http://www.mysmu.edu/faculty/christophert/ : christopherting@smu.edu.sg : 6828 0364 : LKCSB 5036 January 6, 2017 Christopher Ting QF
More informationProbability Distributions and Estimation of Ali-Mikhail-Haq Copula
Applied Mathematical Sciences, Vol. 4, 2010, no. 14, 657-666 Probability Distributions and Estimation of Ali-Mikhail-Haq Copula Pranesh Kumar Mathematics Department University of Northern British Columbia
More informationarxiv:physics/ v1 [physics.soc-ph] 18 Aug 2006
arxiv:physics/6819v1 [physics.soc-ph] 18 Aug 26 On Value at Risk for foreign exchange rates - the copula approach Piotr Jaworski Institute of Mathematics, Warsaw University ul. Banacha 2, 2-97 Warszawa,
More informationChain Plot: A Tool for Exploiting Bivariate Temporal Structures
Chain Plot: A Tool for Exploiting Bivariate Temporal Structures C.C. Taylor Dept. of Statistics, University of Leeds, Leeds LS2 9JT, UK A. Zempléni Dept. of Probability Theory & Statistics, Eötvös Loránd
More informationRobustness of a semiparametric estimator of a copula
Robustness of a semiparametric estimator of a copula Gunky Kim a, Mervyn J. Silvapulle b and Paramsothy Silvapulle c a Department of Econometrics and Business Statistics, Monash University, c Caulfield
More informationWhen is a copula constant? A test for changing relationships
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)
More informationNormal Probability Plot Probability Probability
Modelling multivariate returns Stefano Herzel Department ofeconomics, University of Perugia 1 Catalin Starica Department of Mathematical Statistics, Chalmers University of Technology Reha Tutuncu Department
More informationVine copulas with asymmetric tail dependence and applications to financial return data 1. Abstract
*Manuscript Vine copulas with asymmetric tail dependence and applications to financial return data 1 Aristidis K. Nikoloulopoulos 2, Harry Joe 3 and Haijun Li 4 Abstract In Aas et al. (2009) and Aas and
More information1 Description of variables
1 Description of variables We have three possible instruments/state variables: dividend yield d t+1, default spread y t+1, and realized market volatility v t+1 d t is the continuously compounded 12 month
More informationNon-parametric Estimation of Elliptical Copulae With Application to Credit Risk
Non-parametric Estimation of Elliptical Copulae With Application to Credit Risk Krassimir Kostadinov Abstract This paper develops a method for statistical estimation of the dependence structure of financial
More informationA measure of radial asymmetry for bivariate copulas based on Sobolev norm
A measure of radial asymmetry for bivariate copulas based on Sobolev norm Ahmad Alikhani-Vafa Ali Dolati Abstract The modified Sobolev norm is used to construct an index for measuring the degree of radial
More informationSelection of Copulas with Applications in Finance
Selection of Copulas with Applications in Finance Zongwu Cai a,b, Xiaohong Chen c, Yanqin Fan d, and Xian Wang a a Department of Mathematics & Statistics, University of North Carolina at Charlotte, Charlotte,
More informationIntroduction to Computational Finance and Financial Econometrics Probability Theory Review: Part 2
Introduction to Computational Finance and Financial Econometrics Probability Theory Review: Part 2 Eric Zivot July 7, 2014 Bivariate Probability Distribution Example - Two discrete rv s and Bivariate pdf
More informationUsing copulas to model time dependence in stochastic frontier models
Using copulas to model time dependence in stochastic frontier models Christine Amsler Michigan State University Artem Prokhorov Concordia University November 2008 Peter Schmidt Michigan State University
More informationDependence. Practitioner Course: Portfolio Optimization. John Dodson. September 10, Dependence. John Dodson. Outline.
Practitioner Course: Portfolio Optimization September 10, 2008 Before we define dependence, it is useful to define Random variables X and Y are independent iff For all x, y. In particular, F (X,Y ) (x,
More informationTail dependence coefficient of generalized hyperbolic distribution
Tail dependence coefficient of generalized hyperbolic distribution Mohalilou Aleiyouka Laboratoire de mathématiques appliquées du Havre Université du Havre Normandie Le Havre France mouhaliloune@gmail.com
More informationGeneralized quantiles as risk measures
Generalized quantiles as risk measures Bellini, Klar, Muller, Rosazza Gianin December 1, 2014 Vorisek Jan Introduction Quantiles q α of a random variable X can be defined as the minimizers of a piecewise
More informationAsymptotic distribution of the sample average value-at-risk
Asymptotic distribution of the sample average value-at-risk Stoyan V. Stoyanov Svetlozar T. Rachev September 3, 7 Abstract In this paper, we prove a result for the asymptotic distribution of the sample
More informationLecture 2: Review of Probability
Lecture 2: Review of Probability Zheng Tian Contents 1 Random Variables and Probability Distributions 2 1.1 Defining probabilities and random variables..................... 2 1.2 Probability distributions................................
More informationECONOMICS 7200 MODERN TIME SERIES ANALYSIS Econometric Theory and Applications
ECONOMICS 7200 MODERN TIME SERIES ANALYSIS Econometric Theory and Applications Yongmiao Hong Department of Economics & Department of Statistical Sciences Cornell University Spring 2019 Time and uncertainty
More informationChallenges in implementing worst-case analysis
Challenges in implementing worst-case analysis Jon Danielsson Systemic Risk Centre, lse,houghton Street, London WC2A 2AE, UK Lerby M. Ergun Systemic Risk Centre, lse,houghton Street, London WC2A 2AE, UK
More information2 (U 2 ), (0.1) 1 + u θ. where θ > 0 on the right. You can easily convince yourself that (0.3) is valid for both.
Introducing copulas Introduction Let U 1 and U 2 be uniform, dependent random variables and introduce X 1 = F 1 1 (U 1 ) and X 2 = F 1 2 (U 2 ), (.1) where F1 1 (u 1 ) and F2 1 (u 2 ) are the percentiles
More informationUniversity of Pretoria Department of Economics Working Paper Series
University of Pretoria Department of Economics Working Paper Series Predicting Stock Returns and Volatility Using Consumption-Aggregate Wealth Ratios: A Nonlinear Approach Stelios Bekiros IPAG Business
More informationMAXIMUM ENTROPIES COPULAS
MAXIMUM ENTROPIES COPULAS Doriano-Boris Pougaza & Ali Mohammad-Djafari Groupe Problèmes Inverses Laboratoire des Signaux et Systèmes (UMR 8506 CNRS - SUPELEC - UNIV PARIS SUD) Supélec, Plateau de Moulon,
More informationA Non-Parametric Approach of Heteroskedasticity Robust Estimation of Vector-Autoregressive (VAR) Models
Journal of Finance and Investment Analysis, vol.1, no.1, 2012, 55-67 ISSN: 2241-0988 (print version), 2241-0996 (online) International Scientific Press, 2012 A Non-Parametric Approach of Heteroskedasticity
More informationMultivariate Asset Return Prediction with Mixture Models
Multivariate Asset Return Prediction with Mixture Models Swiss Banking Institute, University of Zürich Introduction The leptokurtic nature of asset returns has spawned an enormous amount of research into
More informationGeneralized Hypothesis Testing and Maximizing the Success Probability in Financial Markets
Generalized Hypothesis Testing and Maximizing the Success Probability in Financial Markets Tim Leung 1, Qingshuo Song 2, and Jie Yang 3 1 Columbia University, New York, USA; leung@ieor.columbia.edu 2 City
More informationRisk Aggregation with Dependence Uncertainty
Introduction Extreme Scenarios Asymptotic Behavior Challenges Risk Aggregation with Dependence Uncertainty Department of Statistics and Actuarial Science University of Waterloo, Canada Seminar at ETH Zurich
More informationFall 2017 STAT 532 Homework Peter Hoff. 1. Let P be a probability measure on a collection of sets A.
1. Let P be a probability measure on a collection of sets A. (a) For each n N, let H n be a set in A such that H n H n+1. Show that P (H n ) monotonically converges to P ( k=1 H k) as n. (b) For each n
More informationStat 710: Mathematical Statistics Lecture 31
Stat 710: Mathematical Statistics Lecture 31 Jun Shao Department of Statistics University of Wisconsin Madison, WI 53706, USA Jun Shao (UW-Madison) Stat 710, Lecture 31 April 13, 2009 1 / 13 Lecture 31:
More informationImputation Algorithm Using Copulas
Metodološki zvezki, Vol. 3, No. 1, 2006, 109-120 Imputation Algorithm Using Copulas Ene Käärik 1 Abstract In this paper the author demonstrates how the copulas approach can be used to find algorithms for
More informationCopulas, a novel approach to model spatial and spatio-temporal dependence
Copulas, a novel approach to model spatial and spatio-temporal dependence Benedikt Gräler 1, Hannes Kazianka 2, Giovana Mira de Espindola 3 1 Institute for Geoinformatics, University of Münster, Germany
More informationMultivariate Skew Distributions Based on the GT-Copula *
Multivariate Skew Distributions Based on the GT-Copula * Beatriz Vaz de Melo Mendes ** Olcay Arslan *** Abstract In this paper we obtain the expression for the copula pertaining to the multivariate generalized
More informationDetecting Structural Differences in Tail Dependence of Financial Time Series
Detecting Structural Differences in Tail Dependence of Financial Time Series ABSTRACT Carsten Bormann Karlsruhe Institute of Technology and Melanie Schienle Karlsruhe Institute of Technology An accurate
More informationConvolution Based Unit Root Processes: a Simulation Approach
International Journal of Statistics and Probability; Vol., No. 6; November 26 ISSN 927-732 E-ISSN 927-74 Published by Canadian Center of Science and Education Convolution Based Unit Root Processes: a Simulation
More informationVolatility. Gerald P. Dwyer. February Clemson University
Volatility Gerald P. Dwyer Clemson University February 2016 Outline 1 Volatility Characteristics of Time Series Heteroskedasticity Simpler Estimation Strategies Exponentially Weighted Moving Average Use
More informationNonlinear Bivariate Comovements of Asset Prices: Theory and Tests
Nonlinear Bivariate Comovements of Asset Prices: Theory and Tests M. Corazza, A.G. Malliaris, E. Scalco Department of Applied Mathematics University Ca Foscari of Venice (Italy) Department of Economics
More informationFrequent Turbulence? A Dynamic Copula Approach
Frequent Turbulence? A Dynamic Copula Approach Lorán Chollete November 3, 2005 Abstract How common and how persistent are turbulent periods? I address these questions by developing and applying a dynamic
More informationSummary statistics. G.S. Questa, L. Trapani. MSc Induction - Summary statistics 1
Summary statistics 1. Visualize data 2. Mean, median, mode and percentiles, variance, standard deviation 3. Frequency distribution. Skewness 4. Covariance and correlation 5. Autocorrelation MSc Induction
More informationTesting Downside-Risk Efficiency Under Distress
Testing Downside-Risk Efficiency Under Distress Jesus Gonzalo Universidad Carlos III de Madrid Jose Olmo City University of London XXXIII Simposio Analisis Economico 1 Some key lines Risk vs Uncertainty.
More information6. The econometrics of Financial Markets: Empirical Analysis of Financial Time Series. MA6622, Ernesto Mordecki, CityU, HK, 2006.
6. The econometrics of Financial Markets: Empirical Analysis of Financial Time Series MA6622, Ernesto Mordecki, CityU, HK, 2006. References for Lecture 5: Quantitative Risk Management. A. McNeil, R. Frey,
More informationRearrangement Algorithm and Maximum Entropy
University of Illinois at Chicago Joint with Carole Bernard Vrije Universiteit Brussel and Steven Vanduffel Vrije Universiteit Brussel R/Finance, May 19-20, 2017 Introduction An important problem in Finance:
More informationA PRACTICAL WAY FOR ESTIMATING TAIL DEPENDENCE FUNCTIONS
Statistica Sinica 20 2010, 365-378 A PRACTICAL WAY FOR ESTIMATING TAIL DEPENDENCE FUNCTIONS Liang Peng Georgia Institute of Technology Abstract: Estimating tail dependence functions is important for applications
More informationAsymptotic behaviour of multivariate default probabilities and default correlations under stress
Asymptotic behaviour of multivariate default probabilities and default correlations under stress 7th General AMaMeF and Swissquote Conference EPFL, Lausanne Natalie Packham joint with Michael Kalkbrener
More informationMultivariate Normal-Laplace Distribution and Processes
CHAPTER 4 Multivariate Normal-Laplace Distribution and Processes The normal-laplace distribution, which results from the convolution of independent normal and Laplace random variables is introduced by
More informationCopula Methods for Forecasting Multivariate Time Series
Copula Methods for Forecasting Multivariate Time Series Andrew J. Patton Duke University 29 May 2012 Forthcoming in the Handbook of Economic Forecasting, Volume 2. Abstract Copula-based models provide
More informationOn Backtesting Risk Measurement Models
On Backtesting Risk Measurement Models Hideatsu Tsukahara Department of Economics, Seijo University e-mail address: tsukahar@seijo.ac.jp 1 Introduction In general, the purpose of backtesting is twofold:
More informationMultivariate GARCH models.
Multivariate GARCH models. Financial market volatility moves together over time across assets and markets. Recognizing this commonality through a multivariate modeling framework leads to obvious gains
More informationHow to select a good vine
Universitetet i Oslo ingrihaf@math.uio.no International FocuStat Workshop on Focused Information Criteria and Related Themes, May 9-11, 2016 Copulae Regular vines Model selection and reduction Limitations
More informationJackknife Euclidean Likelihood-Based Inference for Spearman s Rho
Jackknife Euclidean Likelihood-Based Inference for Spearman s Rho M. de Carvalho and F. J. Marques Abstract We discuss jackknife Euclidean likelihood-based inference methods, with a special focus on the
More informationAssessing financial model risk
Assessing financial model risk and an application to electricity prices Giacomo Scandolo University of Florence giacomo.scandolo@unifi.it joint works with Pauline Barrieu (LSE) and Angelica Gianfreda (LBS)
More informationThe Slow Convergence of OLS Estimators of α, β and Portfolio. β and Portfolio Weights under Long Memory Stochastic Volatility
The Slow Convergence of OLS Estimators of α, β and Portfolio Weights under Long Memory Stochastic Volatility New York University Stern School of Business June 21, 2018 Introduction Bivariate long memory
More informationElicitability and backtesting
Elicitability and backtesting Johanna F. Ziegel University of Bern joint work with Natalia Nolde, UBC 17 November 2017 Research Seminar at the Institute for Statistics and Mathematics, WU Vienna 1 / 32
More informationCopulas. Mathematisches Seminar (Prof. Dr. D. Filipovic) Di Uhr in E
Copulas Mathematisches Seminar (Prof. Dr. D. Filipovic) Di. 14-16 Uhr in E41 A Short Introduction 1 0.8 0.6 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 The above picture shows a scatterplot (500 points) from a pair
More informationMultivariate Non-Normally Distributed Random Variables
Multivariate Non-Normally Distributed Random Variables An Introduction to the Copula Approach Workgroup seminar on climate dynamics Meteorological Institute at the University of Bonn 18 January 2008, Bonn
More informationProbability. Table of contents
Probability Table of contents 1. Important definitions 2. Distributions 3. Discrete distributions 4. Continuous distributions 5. The Normal distribution 6. Multivariate random variables 7. Other continuous
More informationA Measure of Monotonicity of Two Random Variables
Journal of Mathematics and Statistics 8 (): -8, 0 ISSN 549-3644 0 Science Publications A Measure of Monotonicity of Two Random Variables Farida Kachapova and Ilias Kachapov School of Computing and Mathematical
More informationPARSIMONIOUS MULTIVARIATE COPULA MODEL FOR DENSITY ESTIMATION. Alireza Bayestehtashk and Izhak Shafran
PARSIMONIOUS MULTIVARIATE COPULA MODEL FOR DENSITY ESTIMATION Alireza Bayestehtashk and Izhak Shafran Center for Spoken Language Understanding, Oregon Health & Science University, Portland, Oregon, USA
More informationThe Econometric Analysis of Mixed Frequency Data with Macro/Finance Applications
The Econometric Analysis of Mixed Frequency Data with Macro/Finance Applications Instructor: Eric Ghysels Structure of Course It is easy to collect and store large data sets, particularly of financial
More informationThe Use of Copula Functions in Pricing Weather Contracts for the California Wine Industry
The Use of Copula Functions in Pricing Weather Contracts for the California Wine Industry Don Cyr (contact author) Dean, Goodman School of Business Brock University St. Catharines ON, Canada dcyr@brocku,ca
More informationRobust Backtesting Tests for Value-at-Risk Models
Robust Backtesting Tests for Value-at-Risk Models Jose Olmo City University London (joint work with Juan Carlos Escanciano, Indiana University) Far East and South Asia Meeting of the Econometric Society
More informationParameter estimation of a Lévy copula of a discretely observed bivariate compound Poisson process with an application to operational risk modelling
Parameter estimation of a Lévy copula of a discretely observed bivariate compound Poisson process with an application to operational risk modelling J. L. van Velsen 1,2 arxiv:1212.0092v1 [q-fin.rm] 1 Dec
More information