Vladimir Spokoiny (joint with J.Polzehl) Varying coefficient GARCH versus local constant volatility modeling.

W e ie rstra ß -In stitu t fü r A n g e w a n d te A n a ly sis u n d S to c h a stik STATDEP 2005 Vladimir Spokoiny (joint with J.Polzehl) Varying coefficient GARCH versus local constant volatility modeling. Mohrenstr 39 10117 Berlin spokoiny@wias-berlin.de www.wias-berlin.de 29.1.2005

Outline Introduction: GARCH modeling. Varying coefficient GARCH. Estimation by AWS Semiparametric modeling. Simulated results and applications to some financial datasets Conclusion and Outlooks STATDEP 2005 29.1.2005 2

Volatility Estimation Problem Let S t an asset price process, and R t = log(s t /S t 1 ) log-returns. Conditional Heteroscedasticity Model: R t = σ t ε t, ε t N (0, 1). Typical Problem: estimate volatility θ T = σ 2 T from R 1,..., R T. STATDEP 2005 29.1.2005 3

GARCH(1,1) model Returns R 1,..., R T follow R t = σ t ε t with ε t N (0, 1). GARCH(1,1) model: X s = σs 2 fulfills X s = Ψ s θ = ω + αy s 1 + βx s 1. Here Ψ s = ( ) 1, Y s 1, X s 1 and θ = (ω, α, β). Maximum likelihood estimator: Define for a given θ = (ω, α, β) the process X s (θ) by X s (θ) = Ψ s (θ)θ = ω + αy s 1 + βx s 1 (θ). The MLE θ θ = argsup θ L(θ) = arginf θ s ( ) R 2 s X s (θ) + log X s(θ). If ε t N (0, 1), then θ a quasi MLE. STATDEP 2005 29.1.2005 4

Estimating equation The MLE θ fulfills the estimating equation dl(θ)/dθ = 0 leading to ( ) Rs 2 X s (θ)) X s (θ) 2 X s (θ) = 0. s For a given θ, the process X s (θ) is defined by the structural linear equation X s (θ) = Ψ s (θ)θ = ω + αy s 1 + βx s 1 (θ), s > t 0, X t0 = η. Here η, the initial value. Similarly for the derivatives X s (θ) = dx s (θ)/dθ and 2 X s (θ) = d 2 X s (θ)/dθ 2 : X s (θ) = Ψ s (θ) + β X s 1 (θ), s > t 0, X t0 = 0, 2 X s (θ) = Ψ s (θ) + Ψ s (θ) + β 2 X s 1 (θ), s > t 0, 2 X t0 = 0 with Ψ s (θ) = (0, 0, X s 1 (θ)). STATDEP 2005 29.1.2005 5

Numerical solution by Newton-Raphson Fix some initial value θ (0). Let, for k 1, θ (k 1) be the estimated parameter vector after step k 1. Compute X s (k) = X s (θ (k 1) ) and the derivatives X s (k) d 2 X s (θ (k 1) )/dθ 2. Define the update θ (k) as θ (k) = θ (k 1) + (B (k) ) 1 S (k) = dx s (θ (k 1) )/dθ and 2 X (k) s = with S (k) = s B (k) = s + s X (k) 2( ) Rs 2 X (k) s ( X (k) 2 X (k) s s s X (k) s X s (k), ) X (k) 2( ) ( Rs 2 X (k) 2 s s X (k) s X (k) s ( X (k) s ) ) 2 X (k) s. It is recommended to check that L(θ (k) ) < L(θ (k 1) ). Otherwise consider θ (k) = θ (k 1) + ρ(b (k) ) 1 S (k) for some ρ < 1, e.g. ρ = 1/2. STATDEP 2005 29.1.2005 6

Advantages and problems of the parametric (GARCH) modeling 1.Well developed algorithms 2. Nice asymptotic theory. Root-n consistence and asymptotic normality of the estimator θ under mild regularity assumptions. 3.Good in-sample properties. 4.Possibility to mimic the important stylized facts of the financial time series like volatility clustering, leptokurtic returns and excess kurtosis etc. Problems: 1. the parameter estimates show quite high variability. Especially estimation of β requires about 500 observations. 2. The parametric structure and stationarity of the process is crucial. If the GARCH assumption is violated, the MLE estimator θ is often completely misspecified. STATDEP 2005 29.1.2005 7

Example: Change point GARCH Estimates of ω Estimates of α Estimates of β ω 1.2 Median estimate.05/.95 quantiles.25/.75 quantiles true parameter α 0.0 0.1 0.2 0.3 β 500 1000 1500 2000 time 500 1000 1500 2000 time 500 1000 1500 2000 time The true parameters (red) and the pointwise quantiles of the MLE s ω t, α t, β t obtained from the last 500 observations before t for the change point GARCH model. STATDEP 2005 29.1.2005 8

Models with time varying coefficients Returns R 1,..., R T follow R s = σ s ε s with ε s N (0, 1). Varying coefficient GARCH(1,1) model: X s = σ 2 s fulfills X s = Ψ s θ s = ω s + α s Y s 1 + β s X s 1. Here Y s = R 2 s and ω s, α s, β s some functions of time (predictable processes). Examples: Change point GARCH: θ s piecewise constant, Mikosch and Starica (2000). Smooth transition models: θ s, a smooth function of s, Fan, Jiang, Zhang, Zhou (2001). θ s piecewise smooth. We only assume that for every t there exists a time interval in which θ s θ t, Polzehl and Spokoiny (2000,2002,2003), Mercurio and Spokoiny (2004). STATDEP 2005 29.1.2005 9

Nonparametric modeling Define Θ = (θ t, t T ). The process X t (Θ) is described by X s (Θ) = Ψ s (Θ)θ s = ω s + α s Y s 1 + β s X s 1 (Θ) with Ψ s (Θ) = (1, Y s 1, X s 1 (Θ)). Leads to the log likelihood L(Θ) = t l(r t, X t (Θ)) with l(u, σ 2 ) = log φ(u/σ). Goal: maximize L(Θ) over the class of all admissible parameter processes: Θ = argmax Θ L(Θ). STATDEP 2005 29.1.2005 10

Local perturbation approach Problem: a local change of the parameter process Θ yields a global change of the latent process X(Θ). As a consequence, local likelihood approach does not apply directly. Idea: optimize L(Θ) by local perturbations of the process L(Θ). A local neighborhood of every time point t is described by weights w t,s [0, 1]. Important special cases: Kernel weights: w t,s = K( t s /h), where h, a bandwidth. Windowed weights: w t,s = 1(s I t ) for some time interval I t. Suppose that a preliminary estimate Θ = (θ t ) is fixed. Define for every θ, a process X t,s (θ) = X t,s (W t, θ; Θ ), s 1, by a recurrent equation X t,s (θ) = Ψ t,s (θ) (w t,s θ + (1 w t,s )θ s), s > t 0 where Ψ t,s (θ) = (1, Y s 1, X t,s 1 (θ)). This process describes the local perturbation at t of the process Θ. STATDEP 2005 29.1.2005 11

Local estimation Let Θ = (θ s) t t0 be a preliminary estimate and W t = ( w t,s )t t 0, a local model at t. Define the the local (weighted) pseudo MLE θ t of θ t as θ t = argsup L(W t, θ, Θ ) = argsup l ( Y s, X t,s (θ) ) θ θ for X t,s (θ) = Ψ t,s (θ) (w t,s θ + (1 w t,s )θ s) s > t 0. Remark: X t,s (θ) also depends on Θ. θ t solves the local estimating equation X t,s (θ) ( Y s X t,s (θ) ) X t,s (θ) 2 = 0. s Can be solved by the Newton-Raphson algorithm. s STATDEP 2005 29.1.2005 12

Estimation by Adaptive Weights Iterative adaptive procedure 1.Start with a local estimate θ (0) t for every t with a small bandwidth h 0. 2.Use the obtained local estimates new local models W (k) t θ (k 1) t to compute the weights w (k) t,s that describe = {w (k) t,s, s [t h k, t + h k ]} for a larger bandwidth h k 3.Recompute the local estimates θ (k) t for the models W (k) t. 4.Increase h k and repeat from step 2. STATDEP 2005 29.1.2005 13

Procedure (0) 1. Initialization: For every t, define β t = 0 and W (0) t = {w (0) t,s } with w (0) t,s θ (0) t = argmax θ L(W (0) t, θ) and k = 1. 2. Iteration: for every t = 1,..., T Calculate the adaptive weights: For every point s compute the penalty { s (k) t,s = λ 1 L(W (k 1) t,, Θ (k 1) ) L(W (k 1), Compute θ (k 1) t w (k) ( ) ( t,s = K loc t s 2 /h 2 (k)) (k) k Kst s t,s, W t t θ (k 1) s, Θ (k 1) ) = {w (k) t,s, s T }. = K l ( (t s)/h 0 2 ). Set Estimation of the parameter θ t : Define for every θ the process X (k) t,s (θ) by the recurrent formula ( ) X (k) t,s (θ) = Ψ (k) t,s (θ) w (k) t,s θ + (1 w (k) t,s ) θ (k 1) t, s > t 0, with Ψ (k) t,s (θ) = (1, Y s 1, X (k) t,s 1 θ (k) t = argsup θ Θ L (k) (W (k) t, θ, Θ (k 1) ) = arginf θ Θ (k) (θ)), and define the local MLE θ t as ( ) Rs/X 2 (k) t,s (θ) + log X (k) t,s (θ) s I t }., Θ(k) = ( θ (k) t ). 3. Stopping: Increase k by 1, set h k = ah k 1. If h k h max continue with step 2. Otherwise terminate. STATDEP 2005 29.1.2005 14

Semiparametric modeling The varying coefficient model σ 2 s = Ψ s θ s = ω s + α s Y s 1 + β s σ 2 s 1. may be too variable. Local estimation requires relatively large local neighborhoods, hence, low sensitivity to structural changes. Semiparametric approach: the parameters ω and α may depend on time, while β is a constant. Idea: iterate the steps of nonparametric estimation of ω s and α s and global estimation of β. STATDEP 2005 29.1.2005 15

Semiparametric modeling Denote γ t = (ω t, α t ) and Γ = (γ t ) t t0. For a fixed β, the process X s (Γ, β) and its derivatives are defined from X s (Γ, β) = ω s + α s R 2 s 1 + βx s 1 (Γ, β). With a fixed Γ, the estimating equation for β L(Γ, β)/ β = 0. With a fixed β, for a local model W t, an approximation Γ and the parameter vector γ, define the local perturbation X s (γ) = X s (W t, γ, Γ, β) using the recurrent formula X t,s (γ) = (1 w t,s ) ( ω s + α s R 2 s 1) + wt,s ( ω + αr 2 s 1 ) + βxt,s 1 (γ), s > t 0 and obtain the local estimate γ t by optimizing the corresponding L(γ) = L(W t, γ, Γ, β) w.r.t. γ. STATDEP 2005 29.1.2005 16

Local constant volatility modeling An important special case of the previous model with α s 0 and β s 0 : σ 2 s = ω s that is, σ 2 s coincides with the function (process) ω s. Local time homogeneity means that ω s is locally constant, see Mercurio and Spokoiny (2004). For the local constant case there exists an rigorous theory, see Mercurio and Spokoiny (2004) and Polzehl and Spokoiny (2002). Some properties: If σ 2 s σ 2, then the final estimate σ 2 s coincides with a high probability with the global MLE σ 2. The procedure delivers the optimal (in rate) sensitivity to the change points. If σ s is smooth in s, then we obtain the optimal nonparametric rate of estimation. STATDEP 2005 29.1.2005 17

Simulated results We use a set of six artificial examples to illustrate the predictive performance parametric, non- and semiparametric GARCH(1,1) models and the local constant volatility model from Polzehl and Spokoiny (2003). The sample size is set to T = 1000. Parameters are local constant and may change every 125 observations. 1.a parametric GARCH(1,1) model with common parameters ( β = 0.8 ), 2.a local constant constant volatility model ( β = 0.0 ), 3.semiparametric GARCH(1,1) models with small β, 4.entirely nonparametric GARCH with small β, 5.semiparametric GARCH(1,1) models with large β, 6.entirely nonparametric GARCH with large β. STATDEP 2005 29.1.2005 18

Parameters of simulated examples Ex 1: parametric GARCH(1,1) Ex 2: local const. volatility Ex 3: SP GARCH(1,1) parameter value 0.0 0.2 0.4 0.6 0.8 ω t α t β t parameter value 0.0 0.2 0.4 0.6 0.8 ω t α t β t parameter value 0.0 0.2 0.4 0.6 0.8 ω t α t β t 0 200 400 600 800 1000 Index 0 200 400 600 800 1000 Index 0 200 400 600 800 1000 Index Ex 4: NP GARCH(1,1) Ex 5: NP GARCH(1,1) Ex 6: SP GARCH(1,1) parameter value 0.0 0.2 0.4 0.6 0.8 ω t α t β t parameter value 0.0 0.2 0.4 0.6 0.8 ω t α t β t parameter value 0.0 0.2 0.4 0.6 0.8 ω t α t β t 0 200 400 600 800 1000 Index 0 200 400 600 800 1000 Index 0 200 400 600 800 1000 Index Parameters of simulated examples as functions of time. Procedure: All estimates are computed sequentially based on the observations from the past. For the parametric GARCH(1,1) model the last 250 observations are used. STATDEP 2005 29.1.2005 19

Criteria to compare the behaviour of the estimates: Mean estimated value of β 1 750 t>250 A predictive likelihood risk PL(k) with horizon k = 10 ( ) 1 T k k PL(k) = log (T k 250)k X t+s t + X t+s X t+s t t=251 where X t+s t - the predicted volatility at t + s from R s for s t. Mean probability P EVaR (δ, k) of exceeding VaR at level δ and time horizon k ( 1 T k t+k ) P EVaR (δ, k) = P R s < VaR t (δ, k). T k 250 t=251 β t s=1 s=t+1 where Value at Risk (VaR) at level δ and time horizon k is defined as k VaR t (δ, k) = q δ s=1 X t+s t with q δ denoting the δ -quantile of the standard Gaussian distribution. a mean VaR at level δ and time horizon k MVaR(δ, k) = 1 T k 250 T k t=251 VaR t (δ, k) STATDEP 2005 29.1.2005 20

Results for the simulated examples Ex 1 Ex 2 Ex 3 Ex 4 Ex 5 Ex 6 Mean β 0.8 0.0 0.2 0.181 0.65 0.8 Mean β Seq. GARCH 0.536 0.777 0.782 0.803 0.809 0.810 Mean β Seq. NP-GARCH 0.561 0.567 0.475 0.519 0.576 0.627 Mean β Seq. SP-GARCH 0.329 0.218 0.192 0.186 0.286 0.367 PL(10) Seq. GARCH -10.86 14.56 13.16 13.93 0.756-8.814 PL(10) Seq. SP-GARCH -10.96 14.72 14.00 14.51 1.564-7.806 PL(10) Seq. NP-GARCH -10.83 14.75 14.45 15.02 2.084-7.150 PL(10) Seq. loc.volat. -10.82 15.14 14.99 15.63 2.591-6.918 100 P EVaR (0.01, 10) Seq. GARCH 1.11 1.51 2.49 2.37 2.47 2.47 100 P EVaR (0.01, 10) Seq. NP-GARCH 1.41 1.32 2.08 1.83 2.08 2.07 100 P EVaR (0.01, 10) Seq. SP-GARCH 1.24 1.31 2.01 1.89 1.95 1.85 100 P EVaR (0.01, 10) Seq. loc. volat. 1.23 1.19 1.97 1.72 1.97 1.97 MVaR(0.01, 10) Seq. GARCH 7.13 2.05 2.09 2.00 3.72 5.95 MVaR(0.01, 10) Seq. NP-GARCH 7.08 2.12 2.20 2.10 3.88 6.14 MVaR(0.01, 10) Seq. SP-GARCH 7.22 2.12 2.20 2.10 3.92 6.30 MVaR(0.01, 10) Seq. loc. volat. 7.17 2.09 2.16 2.07 3.84 6.16 Simulation results for artificial examples 1-6. Simulation size 10. Mean estimated values of beta, mean predictive likelihood, probability of exceeding the Value at Risk and mean Value at Risk obtained for the sequential GARCH(1,1) model (last 250 observations), AWS for nonparametric GARCH(1,1), AWS for semiparametric GARCH(1,1) and using the local constant volatility AWS procedure. STATDEP 2005 29.1.2005 21

Applications to financial time series We now apply our methodology to two time series, the German DAX index (August 1991 to July 2003) and the US$-GBP exchange rate (January 1990 to December 2000). Criteria: 1.the predictive empirical likelihood risk PEL(k) at different time horizons k ranging from 2 weeks to half a year. ( ) 1 T k k PEL(k) = log (T k 250)k X t+s t + R2 t+s X t+s t t=251 where X t+s t means the predicted volatility at t + s from R s for s t. 2.the excess probability P EVaR 3.the mean Value at Risk MVaR 4.tail index estimate of the standardized returns s=1 STATDEP 2005 29.1.2005 22

Analysis of German DAX DAX : Logarithmic returns Volatility 0.00 0.02 0.04 rdax 0.05 0.05 SQRT of volatility process Parametric GARCH(1,1) SQRT of volatility process AWS for Nonparametric GARCH(1,1) Volatility 0.01 0.03 0.05 Volatility 0.00 0.02 0.04 0.06 SQRT of volatility process AWS for Semiparametric GARCH(1,1) SQRT of volatility process local constant AWS (Volatility) Volatility 0.005 0.020 0.035 1992 1994 1996 1998 2000 2002 DAX: Logarithmic returns (top) and estimated volatility processes. Given are global estimates (red) and sequential estimates (obtained from the last 250 observations, green) by parametric GARCH(1,1), AWS for nonparametric GARCH(1,1), AWS for semiparametric GARCH(1,1) and the local constant volatility model (from top to bottom) STATDEP 2005 29.1.2005 23

Cointegration in DAX: fact or artifact? Mean value of the estimate β t of the parameter β t using the parametric GARCH(1,1) model and its non- and semiparametric generalizations: GARCH(1,1) NP-GARCH(1,1) SP-GARCH(1,1) Global 0.942 0.569 0.105 Sequential 0.855 0.589 0.228 STATDEP 2005 29.1.2005 24

DAX Applications: Analysis of standardized returns Model assumption R t = σ t ε t with ε t N (0, 1). Define ε t = R t / σ t. DAX: of R t 2 DAX: of R 2 ^ t X t (Seq. GARCH(1,1)) DAX: of R 2 ^ t X t (Seq. NP GARCH(1,1)) DAX: of R 2 ^ t X t (Seq. SP GARCH(1,1)) DAX: of R 2 ^ t X t (Seq. local const. AWS) 2 DAX: of R t DAX: of R 2 ^ t X t (GARCH(1,1)) DAX: of R 2 ^ t X t (NP GARCH(1,1)) DAX: of R 2 ^ t X t (SP GARCH(1,1)) DAX: of R 2 ^ t X t (local const. AWS) of squared log returns and squared standardized residuals obtained for the four methods using sequential (top) and global (bottom) volatility estimates, respectively. Dotted straight line - the 95% significant level. STATDEP 2005 29.1.2005 25

Heavy tail behaviour The estimated tail index for the returns and standardized returns ε t : log residuals residuals residuals local const. returns GARCH NP-GARCH SP-GARCH Volatility Global 0.315 0.261 0.210 0.158 0.202 Sequential 0.315 0.278 0.169 0.151 0.173 Tail index of absolute logarithmic returns and standardized residuals. Critical values for Gaussian distributions with same sample size: 0.193 (.95), 0.202 (.99). STATDEP 2005 29.1.2005 26

Out-of-sample performance Predictive empirical likelihood PEL(k) for four different time horizons ranging from two weeks to half a year ( ) 1 T k k PEL(k) = log (T k 250)k X t+s t + R2 t+s. X t+s t t=251 Method two weeks one month three months six months GARCH(1,1) 7.568 7.541 7.439 7.337 NP-GARCH(1,1) 7.548 7.501 7.345 7.225 SP-GARCH(1,1) 7.284 7.335 7.213 7.133 local const. 7.562 7.519 7.394 7.305 Mean predictive likelihood for different forecast horizons. The best result for each time horizon in blue. s=1 STATDEP 2005 29.1.2005 27

DAX: Value-at-Risk performance 1. The excess probability P EVaR Level GARCH(1,1) NP-GARCH(1,1) SP-GARCH(1,1) local const. volatility 0.01 0.0133 0.0152 0.0152 0.0118 0.05 0.0575 0.0598 0.0636 0.0499 Probability to exceed the Value at Risk at 10 trading days. The best result in blue. 2. Mean VaR at level δ = 0.01 and δ = 0.05 and time horizon k = 10 MVaR(δ, k) = 1 T k 250 T k t=251 VaR t (δ, k) Level GARCH(1,1) NP-GARCH(1,1) SP-GARCH(1,1) local const. volatility 0.01 0.1029 0.1054 0.1053 0.1053 0.05 0.0728 0.0746 0.0745 0.0744 Value at Risk at 10 trading days. The best result in blue. STATDEP 2005 29.1.2005 28

Analysis of the US$-GBP exchange rate US$ GBP Exchange rate : Logarithmic returns rusdgbp 0.03 0.00 0.02 SQRT of volatility process Parametric GARCH(1,1) Volatility 0.000 0.010 Volatility 0.000 0.010 0.020 Volatility 0.000 0.010 0.020 estimated sequential (n=250) estimated sequential (n=250) estimated sequential (n=250) SQRT of volatility process AWS for Nonparametric GARCH(1,1) SQRT of volatility process AWS for Semiparametric GARCH(1,1) SQRT of volatility process local constant AWS (Volatility) Volatility 0.000 0.010 estimated sequential (n=250) 1990 1992 1994 1996 1998 2000 DAX: Logarithmic returns (top) and estimated volatility processes. Given are global estimates (red) and sequential estimates (obtained from the last 250 observations, green) by parametric GARCH(1,1), AWS for nonparametric GARCH(1,1), AWS for semiparametric GARCH(1,1) and the local constant volatility model (from top to bottom) STATDEP 2005 29.1.2005 29

Cointegration in US$-GBP exchange rate: fact or artifact? Mean value of the estimate β t of the parameter β t using the parametric GARCH(1,1) model and its non- and semiparametric generalizations: GARCH(1,1) NP-GARCH(1,1) SP-GARCH(1,1) Global 0.945 0.357 0.065 Sequential 0.843 0.364 0.268 STATDEP 2005 29.1.2005 30

US$-GBP exchange rate applications: Analysis of standardized returns Model assumption Y t = σ t ε t with ε t N (0, 1). Define ε t = Y t / σ t. DAX: of R t 2 USD GBP: of R 2 ^ t X t (Seq. GARCH(1,1)) USD GBP: of R 2 ^ t X t (Seq. NP GARCH(1,1 USD GBP: of R 2 ^ t X t (Seq. SP GARCH(1,1 USD GBP: of R 2 ^ t X t (Seq. local const. AWS 2 DAX: of R t USD GBP: of R 2 ^ t X t (GARCH(1,1)) USD GBP: of R 2 ^ t X t (NP GARCH(1,1)) USD GBP: of R 2 ^ t X t (SP GARCH(1,1)) USD GBP: of R 2 ^ t X t (local const. AWS) of squared log returns and squared standardized residuals obtained for the four methods using sequential (top) and global (bottom) volatility estimates, respectively. Dotted straight line - the 95% significant level. STATDEP 2005 29.1.2005 31

US$-GBP exchange rate: Heavy tail behaviour The estimated tail index for the returns and standardized returns ε t : log residuals residuals residuals local const. returns GARCH NP-GARCH SP-GARCH Volatility Global 0.311 0.362 0.177 0.152 0.163 Sequential 0.311 0.219 0.153 0.153 0.164 Tail index of absolute logarithmic returns and standardized residuals. Critical values for Gaussian distributions with same sample size: 0.193 (.95), 0.202 (.99). STATDEP 2005 29.1.2005 32

US$-GBP exchange rate: Out-of-sample performance Predictive empirical likelihood PEL(k) for four different time horizons ranging from two weeks to half a year ( ) 1 T k k PEL(k) = log (T k 250)k X t+s t + R2 t+s. X t+s t t=251 Method two weeks one month three months six months GARCH(1,1) 9.347 9.342 9.33 9.318 NP-GARCH(1,1) 9.357 9.353 9.273 9.214 SP-GARCH(1,1) 9.269 9.224 9.169 9.12 local const. Volatility 9.463 9.457 9.399 9.352 Mean predictive likelihood for different forecast horizons. The best result for each time horizon in blue. s=1 STATDEP 2005 29.1.2005 33

US$-GBP exchange rate: Value-at-Risk performance 1. The excess probability P EVaR Level GARCH(1,1) NP-GARCH(1,1) SP-GARCH(1,1) local const. volatility 0.01 0.0206 0.0197 0.0216 0.0154 0.05 0.0590 0.0619 0.0586 0.0533 Probability to exceed the Value at Risk at 10 trading days. The best result in blue. 2. Mean VaR at level δ = 0.01 and δ = 0.05 and time horizon k = 10 MVaR(δ, k) = 1 T k 250 T k t=251 VaR t (δ, k) Level GARCH(1,1) NP-GARCH(1,1) SP-GARCH(1,1) local const. volatility 0.01 0.0391 0.0405 0.0401 0.0401 0.05 0.0277 0.0286 0.0284 0.0284 Value at Risk at 10 trading days. The best result in blue. STATDEP 2005 29.1.2005 34

Conclusion and Outlook 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. STATDEP 2005 29.1.2005 35

Extension: Parametric generalized linear modeling Model: R t = σ t ε t and the process X t = g(σt 2 ) satisfies the linear equation: X t = ω + α 1 Y t 1 +... + α p Y t p + β 1 X t 1 +... + β q X t q where Y t = h(r t ) for some fixed functions g, h. Example 1. ARCH(p) X t = σt 2 and Y t = Rt 2 : σt 2 = ω + α 1 Rt 1 2 +... + α p Rt p. 2 Example 2. GARCH(p,q) X t = σ 2 t and Y t = R 2 t X t = ω + α 1 R 2 t 1 +... + α p R 2 t p + β 1 σ 2 t 1 +... + β q σ 2 t q Example 3. Exponential (G)ARCH(p) X t = log σ 2 t and Y t = h(r t ) where h(u) = log(u 2 + ɛ) or h(u) = log log(e u + e + ɛ) for some ɛ 0 : log σ 2 t = ω + α 1 h(r t 1 ) +... + α p h(y t p ) + β 1 log σ 2 t 1 +... + β q log σ 2 t q STATDEP 2005 29.1.2005 36