Efficient estimation of a semiparametric dynamic copula model

Transcription

1 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 Day

2 Outline Introduction Semi-parametric dynamic copula Motivation The Model Asymptotic theory Bandwidth selection Simulations and applications Simulations Empirical example Conclusions

3 Problems and Solutions Problems Modeling dependence is critical for financial time series Model volatility of returns of financial assets Different approaches are used to model dynamic correlations BUT: return distributions often reveal skewness and lower-tail dependencies Copula Allow to model nonlinear dependence Look beyond correlation (i.e., linear dependence), which is required for non-elliptical distributions

4 What are Copulas? Copulas allow us to model the dependence relationships among r.v. independently of their marginal distributions 1 Definition: Function C : [0, 1] 2 [0, 1] such that is a copula Example: Clayton copula F(x, y) = C{F 1 (x), F 2 (y)} C(u, v) = ( ) 1/θ u θ + v θ 1 where θ (0, ) - dependence parameter 1 For more details see Embrechts, Lindskog and McNeil (2001)

5 Outline Introduction Semi-parametric dynamic copula Motivation The Model Asymptotic theory Bandwidth selection Simulations and applications Simulations Empirical example Conclusions

6 Motivation Dependence may vary over time For C = {C θ, θ Θ} we allow θ to be time-varying Patton (2006): dependence parameter is a parametric function of lagged u t, v t Here: we assume θ to be a smooth function of time

7 Outline Introduction Semi-parametric dynamic copula Motivation The Model Asymptotic theory Bandwidth selection Simulations and applications Simulations Empirical example Conclusions

8 Modeling of marginal distributions Assume {X t } - a stochastic processes, e.g. X t = (X 1t, X 2t ) X i,t F t 1 N(µ it, σit 2 ), i = 1, 2 where F t = σ(x t, X t 1,..., X 0 ) µ it e.g. ARMA σit 2 e.g. GARCH Estimate the parameter vector φ = (φ 1, φ 2 ) Standardize z it = X it µ it σ it N(0, 1)

9 Estimating a copula model Joint distribution of z 1t, z 2t F(z 1t, z 2t ) = C(Φ(z 1t ; φ 1 ), Φ(z 2t ; φ 2 ); θ) The joint log-likelihood L(θ, φ) = T ln c(φ(z 1t ; φ 1 ), Φ(z 2t ; φ 2 ); θ) t=1 + T ln ϕ(z 1t ; φ 1 ) + t=1 = L C (θ, φ) + L V (φ) T ln ϕ(z 2t ; φ 2 ) t=1 (φ, θ) = [φ 1, φ 2, θ] is the parameter vector to be estimated c(u, v) = 2 C(u,v) u v

10 Estimating a copula model Two-step Maximum likelihood (ML) First step Second step φ = arg max φ Φ L V (φ) θ = arg max θ Θ L C (θ, φ) θ = arg max θ Θ L C (θ, φ) θ(τ) = arg max θ Efficient estimator: T ln c(φ 1 ( ), Φ 2 ( ); θ) K h (t/t τ) t=1 { T } 1 T

11 Outline Introduction Semi-parametric dynamic copula Motivation The Model Asymptotic theory Bandwidth selection Simulations and applications Simulations Empirical example Conclusions

12 Asymptotic theory Let l t (θ) = ln c(φ(z 1t ; φ 1 ), Φ(z 2t ; φ 2 ); θ) Define s(τ) = E [ (l t (θ))2 t/t = τ ] J(τ) = E [l t (θ) t/t = τ] g(τ) = E [l t (θ) t/t = τ] Theorem 1: Under certain assumptions Th ( θ(τ) θ(τ) h 2 b(τ)) d N(0, V θ (τ)), where b(τ) = { θ (τ) 2 + J(τ) 1 g(τ) 2 θ (τ) 2 } µ 2 (K ) V θ (τ) = J 2 (τ)s(τ) K 2 K h ( ) is a Kernel h is a bandwidth

13 Outline Introduction Semi-parametric dynamic copula Motivation The Model Asymptotic theory Bandwidth selection Simulations and applications Simulations Empirical example Conclusions

14 Local likelihood estimation Bandwidth selection We need to choose bandwidth h to balance the bias and variance. Estimator for MSE is MSE p (τ; h) = B 2 p(τ; h) + V p (τ; h) Bandwidth ĥ p = argmin h { } MSE p (x; h)w(x)dx where w(x) is a weight function

15 Outline Introduction Semi-parametric dynamic copula Motivation The Model Asymptotic theory Bandwidth selection Simulations and applications Simulations Empirical example Conclusions

16 Simulations design Step 1: Simulate {r 1,t, r 2,t } T t=1 such that: ɛ i,t GARCH(1, 1), i = 1, 2 z i,t N(0, 1) F(z 1,t, z 2,t ) = C (Φ(z 1,t ), Φ(z 2,t ); θ) θ = θ(t/t ) # replicas K = 100 # observations T = 100, 500 and 1000 Step 2: Estimate φ, θ t and φ Step 3: Test the performance of the model: MSE and dynamic quantile (DQ) test of Engle and Manganelli (2004)

17 Simulations design Functions for the dependence parameter: 1. Constant: θ(u) = 0, 1, 2 and 3 2. Step: θ(u) = 2 + 1(u > 0.5) 3. Slow sine θ(u) = 2 + sin(50u/3) 4. Fast sine θ(u) = 2 + sin(50u) 5. Slow sine, big amplitude (Slow BA) θ(u) = sin(50u/3) 6. Fast sine, big amplitude (Fast BA) θ(u) = sin(50u)

18 Simulations design One replica example 3.5 Estimated theta True theta Time-invariante theta CI K = 100, h1 = Figure: True θ (red line), time-varying θ t (black line), time-invariant θ (blue line) and 95% confidence intervals

19 Simulations results Constant models MSE*1000 MSE Model ω α β ω α β θ θ θ = 0 5.1E E θ = 1 5.9E E θ = 2 5.0E E θ = 3 5.1E E θ = 2 to 4 5.6E E Table: Mean squared error MSE for copula dependency parameter θ and for parameter vectors φ and φ.

20 Simulations results Sine models MSE*1000 MSE Model ω α β ω α β θ θ Slow sine 6.3E E Slow BA sine 5.2E E Fast sine 5.6E E Fast BA sine 5.6E E Table: Mean squared error MSE for copula dependency parameter θ and for parameter vectors φ and φ.

21 Simulations results Time increment models MSE*1000 MSE Model ω α β ω α β θ θ T= E E T= E E T= E E Table: Mean squared error MSE for copula dependency parameter θ and for parameter vectors φ and φ.

22 Outline Introduction Semi-parametric dynamic copula Motivation The Model Asymptotic theory Bandwidth selection Simulations and applications Simulations Empirical example Conclusions

23 Data set: MSCI Germany and UK Data: Morgan Stanley Capital International (MSCI) index for Germany and UK weekly quotations, 11 October October 2008 Model: rg, r UK - log-returns rg and r UK show presence of autocorrelation r G,t = 0.08r G,t 1 + ɛ G,t (0.03) r UK,t = 0.10r UK,t 1 + ɛ UK,t (0.03) for ɛg,t and ɛ UK,t build GARCH(1,1) with Student errors h G,t = 0.13E (0.08E 4) (0.03) ɛ2 G,t h G,t 1, (0.03) ν G = 8.05 (1.77) h UK,t = 0.32E (0.24E 4) (0.05) ɛ2 UK,t h UK,t 1, (0.09) ν UK = (3.30)

24 Data set: MSCI Germany and UK Volatilities estimated from univariate GARCH models with Student errors. Volatilities 0.09 GERMANY UK /12/1992 7/02/ /04/ /06/ /08/2005 Figure: Germany (blue line) and UK (green line)

25 Data set: MSCI Germany and UK Scatter plots Scatter plot of std. residuals. Kendalls tau Scatter plot of probability transforms UK 0 UK GERMANY GERMANY (a) z G vs. z UK (b) F(z G ) vs. F(z UK ) Figure: Scatter plots for the standardized returns r G versus r UK and for F (r G ; ν G ) versus F (r UK ; ν UK ), where F( ; ν) is a Student distribution with ν d.o.f.

26 Data set: MSCI Germany and UK Transform of estimated dependence to Kendalls τ for GERMANY and UK, h = 0.03 Kendalls τ: rgumbel copula Constant τ C (θ) Confidence Interval /12/1992 7/02/ /04/ /06/ /08/ Estimated correlation, DCC model (Engle) Estimated DCC /12/1992 7/02/ /04/ /06/ /08/2005 Figure: Estimated dependence θ(t) and 95% confidence intervals (upper panel) transformed to Kendall s tau and estimated corelation via Dynamic Conditional Correlation (DCC) model (lower panel)

27 Data set: MSCI Germany and UK 0.2 VaR, EW, DCM, GERMANY UK 0.2 VaR, EW, DCC, GERMANY UK 0.15 Log returns VaR, α = Log returns VaR, α = Exceedance 0.1 Exceedance /12/1992 7/02/ /04/ /06/ /08/ /12/1992 7/02/ /04/ /06/ /08/ VaR, LS, DCM, GERMANY UK 0.2 VaR, LS, DCC, GERMANY UK 0.15 Log returns VaR, α = Log returns VaR, α = Exceedance 0.1 Exceedance /12/1992 7/02/ /04/ /06/ /08/ /12/1992 7/02/ /04/ /06/ /08/2005 Figure: Value-at-Risk for Dynamic Copula model (left panels) and for Dynamic Conditional Correlation (DCC) model (right panels) for equal weighted (EW) (upper panels) and long-short (LS) (lower panels) portfolios

28 Finally What has been done: Simulation results for different copulas Theory for efficient estimator of φ Asymptotic theory for the estimator of θ In progress: Goodness-of-fit test for time-varying copulas Extend to higher dimensions Improved DCC model for higher dimensions

29 For Further Reading I J. Fan, M. Farmen, I. Gijbels Local maximum likelihood estimation and inference J. R. Statist. Soc., 60, Part 3, pp , 1998 A.J.Patton Modeling asymmetric exchange rate dependence INTERNATIONAL ECONOMIC REVIEW, Vol. 47, No. 2, May 2006 R. Dahlhaus A likelihood approximation for locally stationary processes The Annals of Statistics, 28, , 2000 F.C. Drost, C.A.J. Klaassen, and B.J.M. Werker Adaptive estimation in time-series models The Annals of Statistic, 25, , 1997