Definitions of dependence measures and basic properties Asymptotic variance of the tau-estimators for copulas Asymptotic variance for elliptical distributions Modelling and Estimation of Stochastic Dependence Uwe Schmock Based on joint work with Dr. Barbara Dengler Financial and Actuarial Mathematics and Christian Doppler Laboratory for Portfolio Risk Management Vienna University of Technology, Austria www.fam.tuwien.ac.at 101. Annual Meeting of the Swiss Association of Actuaries September 10./11., 2010, St. Gallen Uwe Schmock (TU Vienna) Modelling and Estimation of Stochastic Dependence
Definitions of dependence measures and basic properties Asymptotic variance of the tau-estimators for copulas Asymptotic variance for elliptical distributions Research areas for stochastic dependence (1) Modelling and estimation of dependent credit rating transitions → Ph. D. thesis of Verena Goldammer (2010) Market and credit risk aggregation: a bottom-up approach → Ph. D. thesis of Robert Schöftner (2010) Adapted dependence → Ph. D. project of Karin Hirhager Relaxing the independence of biometric and financial market risks when estimating the risk of unit-linked life insurance contracts Modelling consumer behaviour dependent on financial market development (related to American option) → Variable annuities Uwe Schmock (TU Vienna) Modelling and Estimation of Stochastic Dependence
Definitions of dependence measures and basic properties Asymptotic variance of the tau-estimators for copulas Asymptotic variance for elliptical distributions Research areas for stochastic dependence (2) Generalization of Panjer’s recursion for dependent claim numbers (collective model, CreditRisk + ) → Ph. D. project of Cordelia Rudolph Joint term-structure models for credit spreads and risk-free interest rates → Ph. D. project of Sühan Altay We aim for Non-negative interest rates and credit spreads, Negative covariation between them, Zero-coupon bond prices easy to calculate. Asymptotic variance of estimators of dependence (linear correlation, Kendall’s tau) → mainly the Ph. D. thesis of Barbara Dengler (2010) Uwe Schmock (TU Vienna) Modelling and Estimation of Stochastic Dependence
Definitions of dependence measures and basic properties Asymptotic variance of the tau-estimators for copulas Asymptotic variance for elliptical distributions Outline Definitions of dependence measures and basic properties 1 Linear correlation coefficient Kendall’s tau Applications of asymptotic variance Asymptotic variance of the tau-estimators for different 2 copulas Definitions and general formula Examples Asymptotic variance of the dependence measure for 3 elliptical distributions Elliptical distributions and measures of dependence Asymptotic variance for spherical distributions Asymptotic variance for uncorrelated t-distributions Uwe Schmock (TU Vienna) Modelling and Estimation of Stochastic Dependence
Definitions of dependence measures and basic properties Linear correlation coefficient Asymptotic variance of the tau-estimators for copulas Kendall’s tau Asymptotic variance for elliptical distributions Applications of asymptotic variance Linear correlation coefficient Definition The linear correlation coefficient for a random vector ( X , Y ) with non-zero finite variances is defined as C ov [ X , Y ] ̺ = � � . V ar [ X ] V ar [ Y ] Estimator The standard estimator for a sample ( X 1 , Y 1 ) , . . . , ( X n , Y n ) is � n i = 1 ( X i − X n )( Y i − Y n ) ̺ n = ˆ �� n �� n i = 1 ( X i − X n ) 2 i = 1 ( Y i − Y n ) 2 � n � n where X n = 1 i = 1 X i and Y n = 1 i = 1 Y i . n n Uwe Schmock (TU Vienna) Modelling and Estimation of Stochastic Dependence
Definitions of dependence measures and basic properties Linear correlation coefficient Asymptotic variance of the tau-estimators for copulas Kendall’s tau Asymptotic variance for elliptical distributions Applications of asymptotic variance Asymptotic behaviour of the standard estimator Theorem (Asymptotic normality, e.g. Witting/Müller-Funk ’95, p. 108) For an i. i. d. sequence of non-degenerate real-valued random variables ( X j , Y j ) , j ∈ N , with E [ X 4 ] < ∞ and E [ Y 4 ] < ∞ , the ̺ n , normalized with √ n , are asymptotically standard estimators ˆ normal, √ � � d � � 0 , σ 2 n ̺ n − ̺ ˆ → N , n → ∞ . ̺ The asymptotic variance is � � � σ 40 � 1 + ̺ 2 + ̺ 2 σ 22 + σ 04 − 4 σ 31 − 4 σ 13 σ 2 ̺ = , σ 2 σ 2 2 σ 20 σ 02 4 σ 11 σ 20 σ 11 σ 02 20 02 where σ kl := E [( X − µ X ) k ( Y − µ Y ) l ] , µ X := E [ X ] , µ Y := E [ Y ] . Uwe Schmock (TU Vienna) Modelling and Estimation of Stochastic Dependence
Definitions of dependence measures and basic properties Linear correlation coefficient Asymptotic variance of the tau-estimators for copulas Kendall’s tau Asymptotic variance for elliptical distributions Applications of asymptotic variance Kendall’s tau Definition Kendall’s tau for a random vector ( X , Y ) is defined as τ = P [ ( X − � X )( Y − � ] − P [ ( X − � X )( Y − � Y ) > 0 Y ) < 0 ] � �� � � �� � concordance discordance = E [ sgn ( X − � X ) sgn ( Y − � Y ) ] , where ( � X , � Y ) is an independent copy of ( X , Y ) . Estimator (Representation as U-statistic) The tau-estimator for a sample ( X 1 , Y 1 ) , . . . , ( X n , Y n ) is � n � − 1 � τ n = ˆ sgn ( X i − X j ) sgn ( Y i − Y j ) . 2 1 ≤ i < j ≤ n Uwe Schmock (TU Vienna) Modelling and Estimation of Stochastic Dependence
Definitions of dependence measures and basic properties Linear correlation coefficient Asymptotic variance of the tau-estimators for copulas Kendall’s tau Asymptotic variance for elliptical distributions Applications of asymptotic variance U-statistics Definition Fix m ∈ N . For n ≥ m let Z 1 , . . . , Z n be random variables taking values in the measurable space ( Z , Z ) and let κ : Z m → R be a symmetric measurable function. The U-statistic ˆ U n ( κ ) belonging to the kernel κ of degree m is defined as � n � − 1 � ˆ U n ( κ ) := κ ( Z i 1 , . . . , Z i m ) . m 1 ≤ i 1 < ··· < i m ≤ n The tau-estimator is a U-statistic with kernel κ τ of degree 2: κ τ : R 2 × R 2 → R , � � ( x , y ) , ( x ′ , y ′ ) = sgn ( x − x ′ ) sgn ( y − y ′ ) . κ τ Uwe Schmock (TU Vienna) Modelling and Estimation of Stochastic Dependence
Definitions of dependence measures and basic properties Linear correlation coefficient Asymptotic variance of the tau-estimators for copulas Kendall’s tau Asymptotic variance for elliptical distributions Applications of asymptotic variance Properties of the tau-estimator If the observations are i. i. d., then ˆ τ n is an unbiased estimate of τ . Theorem (Asymptotic normality, e.g. Borovskikh ’96) For an i. i. d. sequence of R 2 -valued random vectors, the τ n , normalized with √ n , are asymptotically tau-estimators ˆ normal, √ � � d � � 0 , σ 2 τ n − τ → N n → ∞ . n ˆ , τ The asymptotic variance is � � E [ sgn ( X − � X ) sgn ( Y − � σ 2 τ = 4 V ar Y ) | X , Y ] , where ( � X , � Y ) is an independent copy of ( X , Y ) . Uwe Schmock (TU Vienna) Modelling and Estimation of Stochastic Dependence
Definitions of dependence measures and basic properties Linear correlation coefficient Asymptotic variance of the tau-estimators for copulas Kendall’s tau Asymptotic variance for elliptical distributions Applications of asymptotic variance Applications of asymptotic variance Asymptotic normality leads to asymptotic confidence intervals of the form � � τ n − σ τ τ n + σ τ √ n u 1 + α √ n u 1 + α ˆ 2 , ˆ 2 for given confidence level α ∈ ( 0 , 1 ) , where u 1 + α is the 2 corresponding quantile of the standard normal distribution. This allows in particular to test for dependence. Estimators can be evaluated by their asymptotic variance and different ways of estimation can be compared, e.g. for elliptical distributions. Uwe Schmock (TU Vienna) Modelling and Estimation of Stochastic Dependence
Definitions of dependence measures and basic properties Definitions and general formula Asymptotic variance of the tau-estimators for copulas Examples Asymptotic variance for elliptical distributions Definition of a copula and Sklar’s theorem Definition A two-dimensional copula C is a distribution function on [ 0 , 1 ] 2 with uniform marginal distributions. Let ( X , Y ) be an R 2 -valued random vector with marginal distribution functions F and G . Then, by Sklar’s theorem, there exists a copula C such that � � P [ X ≤ x , Y ≤ y ] = C F ( x ) , G ( y ) , x , y ∈ R . If the marginal distribution functions F and G are continuous, then Sklar’s theorem also gives uniqueness of the copula C . Uwe Schmock (TU Vienna) Modelling and Estimation of Stochastic Dependence
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