Time-varying copulas: a survey

Transcription

1 Time-varying copulas: a survey Olga Reznikova Institute of Statistics Université catholique de Louvain joint with Hans Manner Department of Quantitative Economics Maastricht University Young Researchers Day 5 February 21

2 Outline Introduction Copula estimation Time-varying copula models Parametric models Stochastic and semiparametric models LCP and RSC Simulation study and model selection Empirical example Conclusion For Further Reading

3 Outline Introduction Copula estimation Time-varying copula models Parametric models Stochastic and semiparametric models LCP and RSC Simulation study and model selection Empirical example Conclusion For Further Reading

4 Motivation example 1 Std returns Std returns GERMANY FRANCE Std returns Std returns JAPAN JAPAN FRANCE Std returns 5 5 GERMANY Std returns Std returns UK UK UK FRANCE Std returns 5 5 GERMANY Std returns 5 5 JAPAN Std returns Std returns USA USA USA USA FRANCE 5 5 GERMANY 5 5 JAPAN 5 5 UK Figure: Scatter plots of standardized returns of G5 countries, weekly observations from 11 October 1989 till 31 May 26

5 Motivation example 2 Estimated correlation, DCC model (Engle), DJ and NQ Estimated DCC /29/1998 8/1/2 9/3/22 9/2/24 1/3/26 Figure: Correlation estimated with DCC model (Engle), DJ and NQ, daily observations 17 July 1996 till 21 October 28

6 Estimating a copula model The Copula model F (X 1t, X 2t ) = C {F 1 (X 1t ), F 2 (X 2t )} The joint pdf f (X 1t, X 2t ) = c(f 1 (X 1t ; φ 1 ), F 2 (X 2t ; φ 2 ); θ) The joint log-likelihood 2 f i (X it, ; φ i ) i=1 L(θ, φ) = T ln c(f 1 (X 1t ; φ 1 ), F 2 (X 2t ; φ 2 ); θ) t=1 T T + ln f 1 (X 1t ; φ 1 ) + ln f 2 (z 2t ; φ 2 ) t=1 t=1 L(θ, φ) = L C (θ, φ) + L V (φ) (φ, θ) = [φ 1, φ 2, θ] is the parameter vector to be estimated c(u, v) = 2 C(u,v) u v

7 Estimating a copula model Two-step Maximum likelihood First step Second step φ = arg max φ Φ L V (φ) θ = arg max L C (θ, φ) Drawback loss in efficiency Solution apply Newton-Rhapson algorithm

8 Outline Introduction Copula estimation Time-varying copula models Parametric models Stochastic and semiparametric models LCP and RSC Simulation study and model selection Empirical example Conclusion For Further Reading

9 Parametric models Patton Patton (26): θ is a function of lagged past observations and autoregressive term ) ρ t = Λ 1 ( ω + βλ 1 1 (ρ t 1) + α 1 m θ t = Λ 2 ω + βθ t 1 + α 1 m m Φ 1 (U 1,t i )Φ 1 (U 2,t i ) i=1 m 1 u t j v t j j= Dynamic conditional correlation (DCC) Heinen, Valdesogo (28): The correlation is driven by the crossproduct of lagged standardized residuals and autoregressive term R t = diag{q} 1/2 Q t diag{q} 1/2 Q t = Ω(1 α β) + αy t 1 Y t 1 + βq t 1 τ t = 2 arcsin(ρt), θt = γ(τt) π where Y it = Φ 1 (U i,t ), Y t = (Y 1t, Y 2t )

10 Stochastic and semiparametric models Stochastic autoregressive copula (SCAR) Hafner, Manner (29): θ is driven by an independent stochastic process λ t = ω + βλ t 1 + σ ηη t η t iid N(, 1) θ = Λ(λ t) Semiparametric dynamic copula (SDC) Hafner, Reznikova (29): θ a smooth function of time L C (θ; h, τ) = T l(u 1t, U 2t ; θ)k h (t/t τ) t=1 θ(τ) = arg max L(θ; h, τ) θ where K ( ) is a kernel and h is a bandwidth

11 Local parametric fitting Local change point (LCP) Giacomini et al. (29): θ is approximated by a constant on a time invariant interval I t = [t m t, t[, t = 1,..., T Idea: test sequentially the nested intervals from I t on the presence of the break point.

12 Regime switching copula (RSC) Pelletier(26), Garcia, Tsafack (28), Chollete et al.(28): allow for two regimes, characterized by different levels of dependence L(θ) = η t = T log(1 ( ξ t t 1 η t)) t=1 ( c1 (U 1t, U 2t ; θ 1 ) c 2 (U 1t, U 2t ; θ 2 ) ) where ξ t t 1 is the vector of estimated transition probabilities using information until (t 1) is the Hadamard product

13 Outline Introduction Copula estimation Time-varying copula models Parametric models Stochastic and semiparametric models LCP and RSC Simulation study and model selection Empirical example Conclusion For Further Reading

14 Simulation study and model selection Simulation design: Simulate 1 observations from Gaussian copula with ρ t Step: ρ t =.2 +.6I t>5 Sine: ρ t = cos(2πt/4) AR(1): ρ t = exp(2λ t) 1 exp(2λ t ) + 1 λ t = λ t 1 +.1ɛ t Measures: MSE, Log-likelihood, Anderson-Darling test

15 Simulation study: MSE MSE = 1 K K k=1 1 T T ( ) 2 ρ k t ρ k t t=1 MSE Const DCC PATT SDC LCP SCAR RSC Step Sine AR(1)

16 Model selection by log-likelihood The fraction of times each copula is selected as the best fitting. Sine Const DCC PATT SDC LCP SCAR RSC Gaussian Clayton Frank Gumbel

17 Model selection by Anderson-Darling test Anderson-Darling test: Is the data generated by a C i? H : C i (u t, v t; ˆθ it ) = C (u t, v t; θ t ) ẑ t = C i (u t v t; ˆθ it ) = C i(u t, v t; ˆθ it ) v t U(, 1) The size and power for the AD test at 5% nominal level (the fraction of times the H is rejected) Sine Const DCC PATT SDC LCP SCAR RSC Gaussian Clayton Frank Gumbel

18 Outline Introduction Copula estimation Time-varying copula models Parametric models Stochastic and semiparametric models LCP and RSC Simulation study and model selection Empirical example Conclusion For Further Reading

19 Empirical example Data set: exchange rates Euro-USD and Yen-USD from 31 December 1999 till 3 December 25 daily returns, T = 1564 Data is corrected for autocorrelation X E t = 9.7E 5.6 X E t 1 + (1.7E 4) (.3) ɛe t X Y t = 9.8E (1.5E 4) (.3) ɛy t and conditional heteroscedasticity h E t = 3.5E (1.3E 7) (.1) ɛe t (.1) he t 1, νe = (12.3) h Y t = 5.3E (1.5E 7) (.1) ɛy t (.1) he t 1, νy = 7.11 (1.15)

20 Empirical example: Log-likelihood (a) Log-likelihood Const DCC PATT SDC LCP SCAR RSC Gaussian Gumbel Clayton Frank rot Gumbel rot Clayton

21 Empirical example: Anderson-Darling test H : C i (u t, v t ; ˆθ it ) = C (u t, v t ; θ t ) (b) AD test (Pvalues) Const DCC PATT SDC LCP SCAR RSC Gaussian Gumbel Clayton Frank rot Gumbel rot Clayton

22 Empirical example: Dynamic Quantile (DQ) test DQ test Engle and Manganelli (24): is the model correctly specified? VaR t (α) = F 1 hit α t t+1 (α) = I(X t VaR t (α)) hit α t α = δ + δ 1 hit α t δ 5hit α t 5 + δ 6 VaR t (α) + ν t H : δ =... = δ 6 = (c) DQ test (Pvalues) Const DCC PATT SDC LCP SCAR RSC Gaussian Gumbel Clayton Frank rot Gumbel rot Clayton

23 Empirical example: estimated dependence.6 Kendalls τ, Euro Yen, Frank copula Dec/ Dec/1 Dec/2 Dec/3 Dec/4 SCAR SDC DCC Constant Dec/ Dec/1 Dec/2 Dec/3 Dec/4 LCP RSC Patton Constant

24 Outline Introduction Copula estimation Time-varying copula models Parametric models Stochastic and semiparametric models LCP and RSC Simulation study and model selection Empirical example Conclusion For Further Reading

25 Conclusion Results log-likelihood is a strong model selection criterion, when variation of the dependence parameter is taken into account Anderson-Darling test has acceptable size and power properties DQ test of Engle and Manganelli (24) only shows if the model fits the data Recommendations RSC model showed good performance in the simulation study, is easy to program and is not computationally tedious

26 Outline Introduction Copula estimation Time-varying copula models Parametric models Stochastic and semiparametric models LCP and RSC Simulation study and model selection Empirical example Conclusion For Further Reading

27 For Further Reading Chollete, Heinen, Valdesogo (29) Modeling International Financial Returns with a Multivariate Regime-switching Copula. Journal of Financial Econometrics, 7(4): Garcia and Tsafack (28) Dependence structure and extreme comovements in international equity and bond markets with portfolio diversification effects. Working paper, EDHEC Risk Asset Management Research Centre Giacomini, Härdle, Spokoiny (29) Inhomogeneous dependency modelling with time varying copulae. Journal of Business and Economic Statistics, 27: Hafner and Manner (28) Dynamic stochastic copula models: Estimation, inference and applications. METEOR Research Memorandum RM/8/43, Maastricht University Hafner and Reznikova (29) Efficient estimation of a semiparametric dynamic copula model. Manuscript, Institute of Statistics, UCL Heinen and Valdesogo (28) Asymmetric CAPM Dependence for Large Dimensions: The Canonical Vine Autoregressive Copula Model. Available at SSRN: CORE Patton (26) Modelling asymmetric exchange rate dependence.international Economic Review, 47: