Recovering Copulae from Conditional Quantiles

Transcription

1 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

2 Motivation 1-1 Copulae-Based Regression Semiparametric regression based on parametric copulae. Curse of dimensionality can be mitigated. Noh et al. (2013) Misspecication problem? Nonlinear quantile regression based on parametric copulae. Chen et al. (2009) and Bouyé and Salmon (2009) High-dimensional framework?

3 Motivation 1-2 Comments by Dette et al. (2014) One-dimensional quadratic regression model Semiparametric estimation using parametric copulae family Figure 1: What if the parametric copulae are misspecied?

4 Motivation 1-3 Regression-Based Copulae Recover copulae from regression machine. Copulae cannot but regression can cover basic relations. Most of the dependence is linear. No assumptions concerning moments and distributions. Real world applications require high-dimensional copulae. Penalized quantile regression (using LASSO to select variables).

5 Outline 1. Motivation 2. Conditional Quantile and Copulae 3. Estimation Procedure 4. Sampling and Simulation 5. Further Research

6 Conditional Quantile and Copulae 2-1 Conditional and Unconditional Copulae def Let u j = F j (x j ), the conditional copulae are dened by (u 2 ) = C U2 U1=u1 (u 2) (1) From conditional copulae to unconditional copulae C(u 1, u 2 ) = F 1 (x 1 )C U2 U1=u1 (u 2) = u 1 (u 2 ) (2)

7 Conditional Quantile and Copulae 2-2 Conditional Quantile and Copulae The τ-th conditional quantile function of X 2 given X 1 = x 1, τ (0, 1], is Q(τ x 1 ) = inf{x 2 : F (x 2 X 1 = x 1 ) τ} (3) Inverting (u 2 ) = τ gives equivalence with (3) F 1 2 { 1 (τ)} = Q(τ x 1 ) (4)

8 Conditional Quantile and Copulae 2-3 Conditional Quantile and Copulae Start from a linear additive quantile regression model ˆQ(τ x 1 ) = ˆα(τ) + ˆβ(τ)x 1 (5) If the specication is correct, F 2 1{ ˆQ(τ x1 )} = τ; otherwise, min F ( ) τ {F (τ) ˆQ(τ x1 )} 2 (6) Estimate the monotonically increasing conditional quantile function F 1 ( ) by PAV (Pool-Adjacent-Violators) algorithm PAV 2 1 PAVAlgo

9 Conditional Quantile and Copulae 2-4 Moving to Higher Dimensions For random vector (X 1,..., X d ) R d, (1) becomes (u k ) = C Uk l k:ul=ul (u k) (7) Note: conditional copulae are the ratio of its partial derivatives. More details Similarly, taking inverse yields F 1 k { 1 (τ)} = Q(τ l k{x l = x l }) (8)

10 Estimation Procedure 3-1 Estimation Procedure Fit values from quantile regression. Estimate the quantile curve from isotonic regression (PAV). Take inverse to obtain the conditional copulae. Repeat and multiply all conditional copulae to get unconditional copulae (the order is not unique). C(u 1,..., u d ) = F 1 (x 1 ) d C Ui j i:u j =u j (u i ) (9) i=2

11 Estimation Procedure 3-2 Stepwise Recursion Figure 2: Stepwise recursion plot

12 Sampling and Simulation 4-1 Random Sampling Taking two-dimensional as example, Generate iid v 1, v 2 U[0, 1] Generate x as the v 1 1-th quantile from the marginal distribution x = F 1 (v ) Generate x as the v 2 2-th quantile from the conditional quantile function x = Q(v 2 2 x ) or x = F 1 {C 1 (v )} U2 U1=v1 Then (x 1, x 2 ) are sampled from the joint distribution F (x 1, x 2 ). Wei (2008).

13 Sampling and Simulation 4-2 Estimated Conditional Quantile Curves Sample from a bivariate normal copula with ρ = 0.5 and t 10 distributed margins (sample size n = 1000) By PAV algorithm, the conditional quantile function is estimated as an interpolated step function Figure 3: Estimated conditional quantile curves and cdf for dierent x1

14 Sampling and Simulation 4-3 Estimated Bivariate Joint Copula Sample from a bivariate Clayton copula with θ = 1.5 Figure 4: Estimated bivariate joint copula from Clayton copula sample

15 Sampling and Simulation 4-4 Estimated Bivariate Marginal Copulae Sample from a 3-dimensional t-copula with ρ = Figure 5: Estimated bivariate marginal copulae from 3-dimensional t-copula sample

16 Further Research 5-1 Further Research Which order in conditional decomposition? Can we derive tail-dependence coecients? Applications in asset portfolio risk measurements or network spill-over eect analysis.

17 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

18 References 6-1 References Bouyé, E. and Salmon, M. (2009) Dynamic Copula Quantile Regressions and Tail Area Dynamic Dependence in Forex Markets The European Journal of Finance, 15(7-8), Chen, X., Koenker R. and Xiao, Z. (2009) Copula-based Nonlinear Quantile Autoregression The Econometrics Journal, 12(s1), S50-S67 Dette, H., Hecke R. and Volgushev, S. (2014) Some Comments on Copula-Based Regression Journal of the American Statistical Association, 109(507),

19 References 6-2 References Härdle, W. and Okhrin O. (2010) De copulis non est disputandum - Copulae: An Overview AStA Advances in Statistical Analysis, 94(1), 1-31 Noh, H., El Ghouch, A. and Bouezmarni, T. (2013) Copula-Based Regression Estimation and Inference Journal of the American Statistical Association, 108(502), Wei, Y. (2008) An Approach to Multivariate Covariate-Dependent Quantile Contours with Application to Bivariate Conditional Growth Charts Journal of the American Statistical Association, 103(481),

20 Appendix 7-1 Pool-Adjacent-Violators Algorithm Need for it arises from monotonic smoothing of bivariate data An iterative tool for isotonic regression/monotone smoothing Results in an interpolated step function

21 Appendix 7-2 Monotonic smoothing on {(X i, Y i )} n i=1 can be formalized as: 1. Sort {(X i, Y i )} n by X into {(X i=1 (i), Y (i))} n i=1 2. Find { ˆm(X (i))} n minimizing n {Y i=1 i=1 (i) ˆm(X (i))} 2 subject to the monotonicity restriction ˆm(X (1)) ˆm(X (2)) ˆm(X (n))

22 Appendix 7-3 The PAV (from the left) can be formalized as follows: Algorithm Step 1: Start with Y (1), move to the right and stop if (Y (i), Y (i+1)) violates the monotonicity constraint, i.e., Y (i) > Y (i+1) Pool Y (i) and the adjacent Y (i+1), by replacing them both by Y(i) = Y (i+1) = (Y (i) + Y (i+1))/2 Step 2: If Y (i 1) > Y (i), pool {Y (i 1), Y (i), Y (i+1)} into one average. Continue to the left until Y (i 1) Y (i). Proceed to the right. The nal solutions are ˆm(X (i)). Return

23 Appendix 7-4 Partial Derivatives and Conditional Copulae Taking d = 3 as example, let C 2 (u 1, u 2 ) def = C(u 1, u 2, 1) and c 2(u 1 1, u 2 ) def as = C 2 (u1,u2) u1, then P(U 2 u 2, U 1 = u 1 ) can be written C 2 (u 1 + u 1, u 2 ) lim = c 2 (u 1 1, u 2 ) u1 0 u 1 The conditional distribution (u 2 ) (given xed u 1 ) is the ratio of derivatives: P(U 2 u 2 U 1 = u 1 ) = c 2 1 (u 1, u 2 ) c 1 1 (u 1) Return