Vladimir Spokoiny Foundations and Applications of Modern Nonparametric Statistics

Transcription

1 W eierstraß-institut für Angew andte Analysis und Stochastik Vladimir Spokoiny Foundations and Applications of Modern Nonparametric Statistics Mohrenstr. 39, Berlin October 10, 2009

2 Outline 1 Local Parametric Estimation of Financial Time Series Motivation Time-series modeling Parametric Estimation Local parametric approach Adaptive LCP procedure Local constant case Theoretical study Applications to financial time series Simulation study Applications Modern Nonparametric Statistics October 10, (74)

3 Motivation Example: Stock and log-returns of Allianz Let S t be an asset price, and R t = log(s t /S t 1 ), log-returns /01/ /12/ /01/ /01/ /01/ /01/ /01/ /12/ /01/ /01/ /01/ /01/07 Asset prices and log-returns of Allianz between 1974/01/02 and 1997/01/07. Modern Nonparametric Statistics October 10, (74)

4 Motivation Objectives of the financial time series analysis Fast reaction on sudden structural breaks Stability and robustness against singular outliers Flexible and nonrestrictive modeling allowing for a good interpretability Simple and robust estimation procedures including an automatic choice of tuning parameters Possibility for adjusting and influencing the procedure for the specific applications Unified nonasymptotic theory which explains the performance of the proposed methods Modern Nonparametric Statistics October 10, (74)

5 General local parametric approach Motivation Parametric Parametric risk bound Local parametric Local parametric risk bound under Small Modeling Bias condition. Oracle choice and oracle quality. Adaptive nonparametric Adaptive (LCP) procedure. Oracle bound. Modern Nonparametric Statistics October 10, (74)

6 Regression-like setup: Time-series modeling Observations Y = (Y 1,..., Y T ). Regression-like model: L(Y t F t 1 ) = P f t, P = (P υ, υ U), a given parametric family, F t, the σ -field generated by Y 1... Y t, f t, the target time varying predictable parameter process. Filtering problem: estimate parameter process f t from Y 1,..., Y t 1. Modern Nonparametric Statistics October 10, (74)

7 Conditional Heteroscedasticity Model Time-series modeling R t = σ t ε t, f t = σ 2 t = IE{R 2 t F t 1 } F t 1, IE ( ε t F t 1 ) = 0, IE ( ε 2 t F t 1 ) = 1. Gaussian innovations: L ( ε t F t 1 ) = N(0, 1), Non-Gaussian innovations: L ( ε t F t 1 ) P, where P, a given parametric family. Examples of P : 1. t -distribution 2. Generalized Hyperbolic (GH) 3. α -stable Modern Nonparametric Statistics October 10, (74)

8 Parametric modeling Parametric Estimation Y 1,..., Y T, observed data (log-returns squared). Parametric model: L ( Y t F t 1 ) = Pf t f t = f θ (X t ), where X t, a d -dimensional predictable explanatory process, X t F t 1, can be partly exogenous (non-observable), and {f(, θ), θ Θ IR p }, a given parametric class of functions. The value θ completely specifies the joint distribution IP θ of the whole data Y 1,..., Y T. Modern Nonparametric Statistics October 10, (74)

9 Parametric modeling. Examples Parametric Estimation Black-Scholes: f θ (X t ) θ, θ Θ IR 1. ARCH(p): with X t = (Y t 1,..., Y t p ) IR p, θ = (ω, α 1,..., α p ) IR p+1, f θ (X t ) = ω + α 1 Y t α p Y t p GARCH(1,1): with X t = (Y t 1, Y t 2,...) and θ = (ω, α, β), f t = σ 2 t = f θ (X t ) follow f t = ω + αy t 1 + βf t 1 Modern Nonparametric Statistics October 10, (74)

10 Maximum Likelihood Estimation Parametric Estimation Returns squared Y 1,..., Y T follow L ( Y t F t 1 ) = Pf θ (X t). Maximum likelihood estimator: θ = argmax L(θ) = log p{y t, f θ (X t )}. θ t T In words: θ is the point of maximum of the log-likelihood L(θ). We focus on the maximum likelihood: L = L( θ) = max L(θ). θ For any θ Θ, define also the maximum log-likelihood ratio L( θ, θ) = max L(θ ) L(θ). θ Modern Nonparametric Statistics October 10, (74)

11 Example: Black-Scholes Parametric Estimation Let f θ (X t ) θ, t T. Then L(θ) = T T 2 log(2πθ) Y i /(2θ) = T 2 i=1 log(2πθ) S/(2θ), where S = Y Y T. Therefore, θ = S/T and L( θ, θ) = T 2 log( θ/θ) T 2 (1 θ/θ) = T K( θ, θ) where K(θ, θ ) = 0.5(θ/θ 1) 0.5 log(θ/θ ) is the Kullback-Leibler divergence between N(0, θ) and N(0, θ ). Modern Nonparametric Statistics October 10, (74)

12 Example: GARCH(1,1) model: Parametric Estimation Y t = f t ε 2 t and X t = (Y t 1, Y t 2,...), where f t fulfills f t = ω + αy t 1 + βf t 1. Here θ = (ω, α, β). Define for a given θ = (ω, α, β) the process f t (θ) by f t (θ) = ω + αy t 1 + βf t 1 (θ). The MLE θ (for Gaussian innovations ε t ): θ = argmax θ L(θ) = argmin θ s T { Yt } f t (θ) + log f t(θ). Modern Nonparametric Statistics October 10, (74)

13 Parametric Estimation MLE: Exponential bound for maximum likelihood Theorem (Golubev and S. (2009)) Let Y 1,..., Y T follow IP θ. Under some regularity conditions, there is µ > 0 such that IE θ exp { µl ( θ, θ )} Q(µ, θ ) Q (µ). Some important features: nonasymptotic bound, applies even for small samples Bound is sharp in rate. Modern Nonparametric Statistics October 10, (74)

14 Parametric Estimation MLE: accuracy of estimation. Basic conditions (A1) Identifiability: θ = argmax IE θ L(θ). (Automatically fulfilled for log-likelihood.) (A2) Pointwise exponential moments: for some µ > 0 θ M(µ, θ, θ ) def = log IE exp { µl(θ, θ ) } <. (Automatically fulfilled for log-likelihood with µ 1.) (A3) Exponential moments for L(θ) : for λ λ { } sup log IE θ exp 2λ γ [ L(θ) IE θ L(θ)] κλ 2 <. γ 1 γ V (θ)γ (Easy to check for (generalized) linear models with V (θ) = ni(θ).) Modern Nonparametric Statistics October 10, (74)

15 Some corollaries: confidence sets Parametric Estimation The bound IE θ exp { µ[l( θ) L(θ )] } Q (µ) yields the likelihood-based confidence sets (CS): E(z) = {θ : L( θ) L(θ) z}. Moreover, for some ρ (0, 1). ( IP θ θ E(z) ) e ϱµz Modern Nonparametric Statistics October 10, (74)

16 Some corollaries: concentration property Parametric Estimation Define with Then A(z) def = {θ : M(µ, θ, θ ) z} M(µ, θ, θ ) = log IE θ exp { µl(θ, θ ) }. IP θ ( θ A(z) ) Q (µ)e ϱµz. Typically A(z) is a root- T neighborhood of θ. Modern Nonparametric Statistics October 10, (74)

17 Some corollaries: root- T consistency Parametric Estimation From Taylor of the second order: L( θ, θ ) 1 2( θ θ ) 2 L(θ ) ( θ θ ). Under ergodicity T 1 2 L(θ ) I(θ ), (Fisher IM). Thus, the bound IE θ exp { µl ( θ, θ )} Q (µ) yields root- T consistency: T I(θ ) ( θ θ ) 2 Const. Modern Nonparametric Statistics October 10, (74)

18 Some corollaries: polynomial risk bound Parametric Estimation The exponential bound IE θ exp { µl ( θ, θ )} Q(µ, θ ) Q (µ) yields the polynomial risk bound: for any r > 0 IE θ L ( θ, θ ) r R r (θ ) R r where R r (θ ) and R r do not depend on the sample size T. Modern Nonparametric Statistics October 10, (74)

19 Black-Scholes case Parametric Estimation Theorem (Polzehl and S. (2006)) Let Y t i.i.d. N(0, θ ). Then for any z > 0 IP θ ( L ( θ, θ ) ) ( > z IP θ T K ( θ, θ ) ) > z 2e z, where K(θ, θ ) is the Kullback-Leibler divergence between N(θ) and N(θ ) : K(θ, θ ) = 0.5(θ/θ 1) 0.5 log(θ/θ ). Modern Nonparametric Statistics October 10, (74)

20 Black-Scholes case: some corollaries Parametric Estimation The bound yields: IP θ ( L ( θ, θ ) ) ( > z IP θ T K ( θ, θ ) ) > z 2e z, the risk bound and root- T consistency: for any r > 0 where r r depends on r only. confidence sets: IE θ L ( θ, θ ) r IE θ T K ( θ, θ ) r r r E(z) = {θ : K( θ, θ) z/t }. Modern Nonparametric Statistics October 10, (74)

21 GARCH(1,1) case Parametric Estimation Theorem Assume Y 1,..., Y T follow GARCH(1,1) with Gaussian innovations and the parameters θ = (ω, α, β). If δ α + β 1 δ for some δ > 0, then there is µ = µ(δ) s.t. IE θ exp { µl ( θ, θ )} Q(µ, θ ) Q (µ, δ) and for any r > 0 IE θ L ( θ, θ ) r R r (θ ) R r, Yields root- T consistency and CS based on L ( θ, θ ). The results can be extended to the quasi-mle for non-gaussian errors under exponential moment conditions on the innovations. Modern Nonparametric Statistics October 10, (74)

22 Parametric Estimation Advantages of the parametric (GARCH) modeling 1. Well developed algorithms 2. Nice asymptotic and non-asymptotic theory. Root- T consistence and asymptotic normality of the estimator θ. 3. Good in-sample properties. 4. Possibility to mimic the important stylized facts of the financial time series (volatility clustering, leptokurtic returns and excess kurtosis). Modern Nonparametric Statistics October 10, (74)

23 Parametric Estimation Problems of the parametric (GARCH) modeling 1. the parameter estimates show quite high variability. Especially estimation of β requires about 500 observations. (consequence of an unfortunate parametrization of the GARCH model) 2. The parametric structure and stationarity of the process is crucial. If the parametric assumption is violated, the MLE estimator θ is often completely misspecified. Modern Nonparametric Statistics October 10, (74)

24 More references Parametric Estimation Mikosch and Starica (2000): Some drawbacks of GARCH modelling E. Hillebrand (2004): Proved the artificial IGARCH effect in the change point GARCH model Bos, Franses and Ooms (1998), Andreou and Ghysels (2002): parameter changes in ARCH models. One possible interpretation: low persistence volatility process that is disturbed by occasional parameter shifts generates data that appear to be highly persistent when the parameter shifts are not accounted for in global estimation. Modern Nonparametric Statistics October 10, (74)

25 Parametric Estimation Drawback of time-homogeneous (parametric) modeling Any time-homogeneous (parametric) model is wrong in long run. Main limitation: Parametric time-homogeneous models cannot match structural changes. Modern Nonparametric Statistics October 10, (74)

26 Remedy of parametric modeling drawback Parametric Estimation Any time-homogeneous (parametric) model is wrong in long run. Any parametric modeling is OK if applied locally (for a sufficiently small time interval). Approach: fix your favorite (tractable, estimable,... ) model and apply it for a properly selected historical time intervals. Modern Nonparametric Statistics October 10, (74)

27 Local parametric approach Local parametric approach Nonparametric model: L ( Y t F t 1 ) Pf t. Local parametric assumption (LPA): for every t there exists a time interval I = I t in which f t f θ (X t ), t I. Goal: identify the interval of homogeneity I t and estimate f t from the parametric approximating model: θ I = argmax θ and apply f t = f θi (X t ). L I (θ) = argmax θ log p(y t, f θ (X t )). t I Modern Nonparametric Statistics October 10, (74)

28 Small modeling bias (SMB) condition Local parametric approach Nonparametric model: Y t P f t. Applying a parametric assumption f t = f θ (X t ) in the nonparametric situation leads to modeling bias measured by I (θ) = t I K { f t, f θ (X t ) }. SMB I (θ) is small for some θ (with a high probability). Modern Nonparametric Statistics October 10, (74)

29 Theoretic information bound Local parametric approach Theorem (Theoretic information bound) Let for some θ Θ and some 0 IE I (θ) = IE t I K ( f t, f θ (X t ) ). Then for any ζ F I IE log(1 + ζ) + IE θ ζ. In particularly, it yields ( IE log 1 + L ( θ, ) θ r ) + 1. R r (θ) Modern Nonparametric Statistics October 10, (74)

30 SMB and Oracle interval Local parametric approach LPA and SMB LPA applies as long as SMB condition IE I (θ) holds. Oracle choice Oracle interval I is the largest one under SMB. Leads to estimation quality of order N 1/2 I. Aim: mimic the oracle i.e. provide the same estimation accuracy of order N 1/2 I. Modern Nonparametric Statistics October 10, (74)

31 Local change point procedure Adaptive LCP procedure Idea: select the largest possible interval of homogeneity I by testing the parametric hypothesis f t = f(x t, θ), t I, against a change point alternative. Modern Nonparametric Statistics October 10, (74)

32 Why a change point alternative? Adaptive LCP procedure The change point approach enables us to keep this parametric specification just by restricting the time interval. Other tests require an additional estimation under alternative. The change point approach delivers important information about the location of change points. The tools of the parametric theory (based on fitted log-likelihood) continue to apply. Test which is powerful against a change point alternative is also powerful against a smooth one. Modern Nonparametric Statistics October 10, (74)

33 Test of a parametric hypothesis Adaptive LCP procedure Problem: test the hypothesis of homogeneity H 0 : f t = f θ (X t ) for t I = [t m, t ]. Log-likelihood under H 0 : with l(y, υ) = log p(y, υ) L I (θ) = t I l {Y t, f θ (X t )}. LR test: apply the fitted log-likelihood (FLL) as the test statistic. FLL under H 0 : L I = L I ( θ) = max L I (θ). θ Modern Nonparametric Statistics October 10, (74)

34 Test against a change-point (CP) Adaptive LCP procedure A CP alternative H 1 (τ) at τ I = [t m, t ] : with θ 1 θ 2 { f H 1 (τ) : f t = θ1 (X t ) for t J = [τ + 1, t ], f θ2 (X t ) for t J c = [t m, τ]. Log-likelihood under H 1 (τ) : L J (θ 1 ) + L J c(θ 2 ) = t J l { Y t, f θ1 (X t ) } + t J c l { Y t, f θ2 (X t ) }. LR test statistic for a given location τ : T I,τ = max {L J (θ 1 ) + L J c(θ 2 )} max L I (θ) = L J + L J c L I θ 1,θ 2 θ = L J ( θ J, θ I ) + L J c( θ J c, θ). Modern Nonparametric Statistics October 10, (74)

35 Test against a change-point. 2 Adaptive LCP procedure Let T be a subset of I called a tested set. Maximum LR test: check every point of T on possible CP. T I,T = max T I,τ > z I a CP is detected at τ = argmax T I,τ. τ T τ T Here z I is critical value. J c t m I T J t τ t Modern Nonparametric Statistics October 10, (74)

36 Multiscale LCP procedure Adaptive LCP procedure Idea: search for the largest historical interval for which the data do not contradict the parametric assumption. Let I 0 I 1 I 2... I K be an increasing family of historical intervals I k = [t m k, t]. Define T k = I k \ I k 1 = [t m k, t m k 1 ]. }{{}}{{} t m k t m k 1 t m k 2 t } {{ } T k } T k 1 {{ I k 2 } } {{ I k 1 } I k Modern Nonparametric Statistics October 10, (74)

37 Multiscale LCP procedure (cont) Adaptive LCP procedure Each interval T k = I k \ I k 1 = [t m k, t m k 1 ] is tested on a CP using I k+1 as testing interval and as the test statistic. T k = T Ik+1,T k = max τ T k T Ik+1,τ I k is accepted no CP is detected in T 1,..., T k, that is, if T l z l, l k. Modern Nonparametric Statistics October 10, (74)

38 Multiscale LCP procedure (cont) Adaptive LCP procedure 1. Initialization: start with k = 1 2. test T k on a CP using I k+1 as a testing interval: I k is accepted if T k = T Ik+1,T k z k. Set Î as the largest accepted I k, θ = θî : k = argmax{k : Tl z l l k}, Î = I k, θ = θî. Similarly Îk is the latest non-rejected interval after k steps and θ k = θîk. Modern Nonparametric Statistics October 10, (74)

39 Parameters of the procedure Adaptive LCP procedure The only parameters of the procedure are the critical values z k. Their choice slightly depends on the given set of intervals I k. A proposal for intervals I k = [t m k, t ] : start with some m 0, e.g. m 0 = 5, then m k+1 = am k for a > 1, e.g. a = Modern Nonparametric Statistics October 10, (74)

40 Adaptive LCP procedure Choice of the critical values z k Parametric risk bound: for every θ Θ IE θ LIk ( θ k, θ ) r R r (θ ). Propagation condition: given ρ > 0, provide for all θ Θ and all k K IE θ L Ik ( θ k, θ k ) r ρr r (θ ). Modern Nonparametric Statistics October 10, (74)

41 Local constant case Local constant case LR test statistic on a change-point at τ : T I,τ = N J K( θ J, θ I ) + N J ck( θ J c, θ I ) sup-lr test statistic for I k+1, T k : LCP procedure: T k = T Ik+1,T k = max τ I k T Ik+1,τ. k = max{k : Tl z l l k}, Î = I k, θ = θî. Propagation condition: IE θ N Ik K ( θik, θ ) k r ρrr. Modern Nonparametric Statistics October 10, (74)

42 Oracle result Theoretical study Theorem ( Oracle bound) Assume max k k IE Ik (θ) for some k, θ and. Then ( IE log 1 + and for k > k and k k L ( θk, θ ) r R r (θ) ) 1( k k ) + ρ L Ik ( θk, θ k +1) zk,..., L Ik 1 ( θk 1, θ k ) zk 1. In words: the adaptive estimate θ belongs with a high probability to the CS of the oracle θ k. Modern Nonparametric Statistics October 10, (74)

43 Oracle result in the LC case Theoretical study Theorem ( Oracle bound) Assume u 0 N k /N k+1 u < 1. Let for some k, θ and. Then ( Nk K ( θk, θ ) r ) IE log 1 + r r max IE k k I k (θ) ( + ρ + log 1 + cu z k r ). r r Yields rate optimality of ft = θ for a smoothly varying f t. Modern Nonparametric Statistics October 10, (74)

44 Typical related problem in financial engineering Applications to financial time series short term ahead forecasting risk management portfolio optimization high frequency data monitoring. Modern Nonparametric Statistics October 10, (74)

45 Forecast from the local model Applications to financial time series Using the latest estimated model, one can build a forecast, at least, for a short horizon h : f LC t,h = f t, f ARCH t,h f GARCH t,h h 1 = IE t (Y t+h F t ) = ω t α t k + α t h f t, k=0 h 1 = IE t (Y t+h F t ) = ω t ( α t + β t ) k + ( α t + β t ) h f t, k=0 Usually h H, a forecasting horizon set, e.g. one day, H = {1}, or two weeks, H = {1,..., 10}. Modern Nonparametric Statistics October 10, (74)

46 Minimum forecasting error criterion Define mean forecasting error TFE T,h = t T where Λ(, ), a loss function, e.g. with c either 1 or 1/2. Applications to financial time series Λ ( Rt+h 2, f ) t,h h H Λ(v, v ) = K(v, v ) c Following to Cheng, Fan and Spokoiny (2003), the data driven choice of r, ρ can be done by minimizing the following objective function: ( r, ρ) = argmin TFE T,h r,ρ Modern Nonparametric Statistics October 10, (74)

47 Set-up Simulation study Consider the conditional heteroscedasticity model: Y t = f(x t )ε 2 t where L ( ε t F t 1 ) = N(0, 1). Aim: compare varying-coefficient modeling for 1. constant f θ ( ) = θ ; 2. ARCH(1): f θ (X t ) = ω + αy t GARCH(1,1): f θ (X t ) = ω + αy t 1 + βx t 1. Set-up: m k = m 0 a k, a = 1.25, m 0 = 10, m K = 570. CV s z k are computed for r = 1 and ρ = 1 (default) with the linear shape z k = b 0 + b 1 log( I k / I 0 ) = b 0 + b 1 k log(a). Modern Nonparametric Statistics October 10, (74)

48 Quantiles of the test of homogeneity. LC Simulation study 85%, 90%, and 95% quantiles of the test T k for the constant volatility. Modern Nonparametric Statistics October 10, (74)

49 Simulation study Quantiles of the test of homogeneity: ARCH 85%, 90%, and 95% quantiles of the test T k for ARCH (1, 0.2, 0). Modern Nonparametric Statistics October 10, (74)

50 Finite-sample CV s Simulation study Model (ω, α, β) z(10) Slope z(570) (0.1, 0.0, 0.0) (0.1, 0.2, 0.0) (0.1, 0.4, 0.0) (0.1, 0.6, 0.0) (0.1, 0.8, 0.0) (0.1, 0.1, 0.8) (0.1, 0.2, 0.7) (0.1, 0.3, 0.6) (0.1, 0.4, 0.5) (0.1, 0.5, 0.4) (0.1, 0.6, 0.3) (0.1, 0.7, 0.2) (0.1, 0.8, 0.1) (0.1, 0.05, 0.90) (0.1, 0.10, 0.85) (0.1, 0.20, 0.75) Modern Nonparametric Statistics October 10, (74)

51 Influence r and ρ on the CV s Simulation study Model (ω, α, β) r ρ z(10) Slope z(570) (0.1, 0.0, 0.0) (0.1, 0.2, 0.0) (0.1, 0.1, 0.8) Modern Nonparametric Statistics October 10, (74)

52 Low and high GARCH-effect Simulation study Low GARCH-effect: α + β 1, high GARCH: α + β 1. GARCH(1,1) parameters of low (upper panel) and high (lower panel) GARCH-effect simulations for t = 1,..., Modern Nonparametric Statistics October 10, (74)

53 Series for low GARCH-effect Simulation study Simulated GARCH(1,1)-time series and their volatilities for low GARCH-effect; t = 1,..., Modern Nonparametric Statistics October 10, (74)

54 Series for high GARCH-effect Simulation study Simulated GARCH(1,1)-time series and their volatilities for high GARCH-effect; t = 1,..., Modern Nonparametric Statistics October 10, (74)

55 Procedure for low GARCH-effect Simulation study Here α + β 0.3. Choice of ρ and r by minimizing TFE : LC: ρ = 0.5, r = 0.5 ; local ARCH: ρ = 1.0, r = 0.5 ; local GARCH: ρ = 1.5, r = 0.5 ; Modern Nonparametric Statistics October 10, (74)

56 Simulation study Parameter estimates for low GARCH-effect The parametric (upper row) and locally adaptive GARCH (lower row) estimate, t = 250,..., 1000 : true parameter (thick dashed), pointwise mean (solid line), 10% and 90% quantiles (dotted lines). Modern Nonparametric Statistics October 10, (74)

57 One day ahead forecasting ability Simulation study Absolute prediction errors one day ahead averaged over last month. Modern Nonparametric Statistics October 10, (74)

58 10 days ahead prediction error Simulation study Absolute prediction errors 10 days ahead averaged over last month. Modern Nonparametric Statistics October 10, (74)

59 Discussion Simulation study all methods are sensitive to jumps in volatility, especially the first one at t = 500. the local GARCH performs rather similarly to the parametric GARCH in general: they are equal before the breaks, t < 500 ; the local GARCH outperforms the parametric one after the first break, 550 < t < 750, and also at the end of the period, 900 < t < 1000 ; the local ARCH performs as well as the GARCH methods and even outperforms them after several structural breaks, 550 < t < 750 and 900 < t < the local constant method is lacking behind the other two adaptive methods whenever there is a longer time period without a structural break, but keeps up with them in periods with more frequent volatility changes, 500 < t < 750. Modern Nonparametric Statistics October 10, (74)

60 High GARCH-effect Simulation study α + β 0.9. The optimal choice of the parameters ρ and r by minimizing TFE leads to: LC: r = 0.5, ρ = 1.5 ; ARCH: r = 0.5, ρ = 1.5 ; GARCH: r = 1.0, ρ = 0.5. Modern Nonparametric Statistics October 10, (74)

61 1 day ahead prediction Simulation study Absolute prediction errors 1 day ahead averaged over last month. Modern Nonparametric Statistics October 10, (74)

62 10 days ahead prediction Simulation study Absolute prediction errors 10 days ahead averaged over last month. Modern Nonparametric Statistics October 10, (74)

63 DAX log-returns. 1/1990 to 12/2002 Applications Log-returns of DAX series from January 1, 1990 till December 31, Modern Nonparametric Statistics October 10, (74)

64 DAX analysis Applications The parameters r and ρ were LC: r = 0.5, ρ = 1.5 ; local ARCH: r = 1.0, ρ = 1.0 ; local GARCH: r = 0.5, ρ = 1.5. We show the relative prediction error (averaged over one month) for the adaptive methods with respect to the global (parametric) GARCH: RPE t,h = 21 Λ ( Rt m+h 2, f ) / 21 t m,h Λ ( Rt m+h 2 GARCH), f t m,h. m=1 h H m=1 h H Modern Nonparametric Statistics October 10, (74)

65 Results for DAX, 1/1991 to 3/1997 Applications The ratio RPE t,h for h = 1 for 3 adaptive methods. DAX from January, 1991 to March, Modern Nonparametric Statistics October 10, (74)

66 DAX Analysis. 1/1991 to 3/1997 Applications Structural breaks on July, 1991 and June, 1992 (cf. Stapf and Werner, 2003) has been detected by all adaptive methods. One additional break detected by all methods occurs in October the local constant and local ARCH methods are optimal at the beginning of the period, where we have less than 500 observations. A similar behavior can be observed after the break detected in October 1994; In the other parts of the data, the performance of all methods is almost the same; In terms of the global prediction error, the local constant is best (0.829), followed by the local ARCH (0.844) and local GARCH (0.869). Modern Nonparametric Statistics October 10, (74)

67 Applications Results for DAX. Period 7/1999 to 6/2001 The ratio RPE t,h for h = 1 for 3 adaptive methods. DAX from July, 1999 to June, Modern Nonparametric Statistics October 10, (74)

68 DAX Analysis. Period 7/1999 to 6/2001 Applications Structural change of the market in 1999 (stabilization after the breaks 1997(Asian) and 1998 (Russian) crashes) detected by local constant and ARCH but not by GARCH. Local GARCH performs almost as the global one GARCH models are preferable for the stable market (middle 2000). In terms of the global prediction error, the local ARCH is best. Generally, LC and local ARCH provide almost the same results. Modern Nonparametric Statistics October 10, (74)

69 S&P /1994 to 12/1996 Applications The ratio RPE t,h for h = 1 for 3 adaptive methods. S&P 500 from January, 1994 to December, Stable market low GARCH-effect, all methods are equally well. Modern Nonparametric Statistics October 10, (74)

70 S&P 500 from 1/1990 to 12/2004 Applications The log-returns of S&P 500 from January 1, 2000 till December 31, Modern Nonparametric Statistics October 10, (74)

71 Applications Results for S&P /2000 to 12/2004 r = 0.5 and ρ = 1.5 for all methods. The ratio RPE t,h for h = 1 for 3 adaptive methods. The S&P 500 from January, 2003 to December, Modern Nonparametric Statistics October 10, (74)

72 Conclusion and Outlook Applications The new approach can be applied and studied in a unified way for a wide class of different models presents a consistent way of selecting the tuning parameters demonstrates a very reasonable numerical performance the simplest local constant modeling is slightly preferable as far as the in sample properties or short time ahead forecasting is concerned. Modern Nonparametric Statistics October 10, (74)

73 Extensions and Applications Applications The local parametric approach has been successfully applied to the following problems: Mercurio, D. & Spokoiny, V. (2004) Annals of Statistics Local constant volatility estimation. Spokoiny, V. (2008) Annals of Statistics Multiscale local constant change point procedure. Polzehl, J. and Spokoiny, V. (2006). Probab. T. Rel. Fields Varying coefficient GARCH modeling Cizek, P., Härdle, W. and Spokoiny, V. (2005) Varying coefficient GARCH modeling Grama, I. and Spokoiny, V. (2008). Annals of Statistics Tail-index estimation. Modern Nonparametric Statistics October 10, (74)

74 Applications Giacomini, E., Härdle, W., and Spokoiny (2008). Estimation of time varying copulae Journal of Business and Economic Statistics. Härdle, W., Herwartz, H. and Spokoiny, V. (2003). J. of Financial Econometrics Multivariate volatility estimation. Chen, Y., Härdle, W. and Spokoiny, V. (2008). Multivariate volatility estimation. Chen, Y., and Spokoiny, V. (2007) Robust risk management. Modern Nonparametric Statistics October 10, (74)