foundations and applications of modern nonparametric

W e ie rstra ß -In stitu t fü r A n g e w a n d te A n a ly s is u n d S to c h a stik

Vladimir Spokoiny

Foundations and Applications of ModernNonparametric Statistics

Mohrenstr. 39, 10117 Berlin [email protected]/spokoiny October 10, 2009

LPA of Financial Time Series

Outline

1 Local Parametric Estimation of Financial Time SeriesMotivationTime-series modelingParametric EstimationLocal parametric approachAdaptive LCP procedureLocal constant caseTheoretical studyApplications to financial time seriesSimulation studyApplications

,Modern Nonparametric Statistics October 10, 2009 2 (74)

LPA of Financial Time Series Motivation

Example: Stock and log-returns of Allianz

Let St be an asset price, and Rt = log(St/St−1) , log-returns.

1974/01/02 1977/12/28 1982/01/05 1986/01/03 1990/01/05 1994/01/070

50

100

150

200

250

300

1974/01/02 1977/12/28 1982/01/05 1986/01/03 1990/01/05 1994/01/07−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

Asset prices and log-returns of Allianz between 1974/01/02 and 1997/01/07.,

Modern Nonparametric Statistics October 10, 2009 3 (74)


Objectives of the financial time series analysis

I Fast reaction on sudden structural breaks

I Stability and robustness against singular outliers

I Flexible and nonrestrictive modeling allowing for a goodinterpretability

I Simple and robust estimation procedures including an automaticchoice of tuning parameters

I Possibility for adjusting and influencing the procedure for thespecific applications

I Unified nonasymptotic theory which explains the performance ofthe proposed methods



General local parametric approach

ParametricParametric 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.


LPA of Financial Time Series Time-series modeling

Regression-like setup:

Observations Y = (Y1, . . . , YT ) .

Regression-like model:

L(Yt|Ft−1) = Pf t,

. P = (Pυ, υ ∈ U) , a given parametric family,

. Ft , the σ -field generated by Y1 . . . Yt ,

. f t , the target time varying predictable parameter process.

Filtering problem: estimate parameter process f t from Y1, . . . , Yt−1 .


LPA of Financial Time Series Time-series modeling

Conditional Heteroscedasticity Model

Rt = σtεt,

f t = σ2t = IE{R2

t |Ft−1} ∼ Ft−1,

IE(εt|Ft−1

)= 0, IE

(ε2t |Ft−1

)= 1.

. Gaussian innovations: L(εt|Ft−1

)= N(0, 1),

. Non-Gaussian innovations: L(εt|Ft−1

)∈ P , where P , a given

parametric family.

Examples of P :1. t -distribution2. Generalized Hyperbolic (GH)3. α -stable


LPA of Financial Time Series Parametric Estimation

Parametric modeling

I Y1, . . . , YT , observed data (log-returns squared).

I Parametric model:

L(Yt|Ft−1

)= Pf t

f t = fθ(Xt),

where Xt , a d -dimensional predictable “explanatory” process,Xt ∼ Ft−1 , can be partly exogenous (non-observable), and{f(·,θ),θ ∈ Θ ⊆ IRp} , a given parametric class of functions.

I The value θ completely specifies the joint distribution IPθ of thewhole data Y1, . . . , YT .



Parametric modeling. Examples

. Black-Scholes: f θ(Xt) ≡ θ , θ ∈ Θ ⊂ IR1 .

. ARCH(p): with Xt = (Yt−1, . . . , Yt−p)> ∈ IRp ,θ = (ω, α1, . . . , αp)> ∈ IRp+1 ,

fθ(Xt) = ω + α1Yt−1 + . . .+ αpYt−p

. GARCH(1,1): with Xt = (Yt−1, Yt−2, . . .) and θ = (ω, α, β) ,f t = σ2

t = fθ(Xt) follow

f t = ω + αYt−1 + βf t−1



Maximum Likelihood Estimation

Returns squared Y1, . . . , YT follow L(Yt|Ft−1

)= Pfθ(Xt) .

Maximum likelihood estimator:

θ = argmaxθ

L(θ) =∑t≤T

log p{Yt,fθ(Xt)}.

In words: θ is the point of maximum of the log-likelihood L(θ) .

We focus on the maximum likelihood:

L = L(θ) = maxθ

L(θ).

For any θ ∈ Θ , define also the maximum log-likelihood ratio

L(θ,θ) = maxθ′

L(θ′)− L(θ).



Example: Black-Scholes

Let f θ(Xt) ≡ θ , t ≤ T . Then

L(θ) = −T2

log(2πθ)−T∑i=1

Yi/(2θ) = −T2

log(2πθ)− S/(2θ),

where S = Y1 + . . .+ YT . Therefore,

θ = S/T and L(θ, θ) = −T2

log(θ/θ)− T

2(1− θ/θ) = TK(θ, θ)

where K(θ, θ′) = 0.5(θ/θ′ − 1)− 0.5 log(θ/θ′) is the Kullback-Leiblerdivergence between N(0, θ) and N(0, θ′) .



Example: GARCH(1,1) model:

Yt = f tε2t and Xt = (Yt−1, Yt−2, . . .) , where f t fulfills

f t = ω + αYt−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 + αYt−1 + βf t−1(θ).

The MLE θ (for Gaussian innovations εt ):

θ = argmaxθ

L(θ) = argminθ

∑s≤T

{ Ytf t(θ)

+ log f t(θ)}.



MLE: Exponential bound for maximum likelihood

Theorem (Golubev and S. (2009))

Let Y1, . . . , YT follow IPθ∗ . Under some regularity conditions, there isµ > 0 such that

IEθ∗ exp{µL(θ,θ∗

)}≤ Q(µ,θ∗) ≤ Q∗(µ).

Some important features:

I nonasymptotic bound, applies even for small samples

I Bound is sharp in rate.



MLE: accuracy of estimation. Basic conditions

(A1) Identifiability: θ∗ = argmaxθ

IEθ∗L(θ).

(Automatically fulfilled for log-likelihood.)

(A2) Pointwise exponential moments: for some µ > 0

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|γ|≤1

log IEθ∗ exp{

2λγ>[∇L(θ)− IEθ∗L(θ)]√

γ>V (θ)γ

}≤ κλ2 <∞.

(Easy to check for (generalized) linear models withV (θ) = nI(θ) .)



Some corollaries: confidence sets

The bound IEθ∗ exp{µ[L(θ)− L(θ∗)]

}≤ Q∗(µ) yields the

likelihood-based confidence sets (CS):

E(z) = {θ : L(θ)− L(θ) ≤ z}.

Moreover,

IPθ∗(θ∗ 6∈ E(z)

)≤ e−%µz

for some ρ ∈ (0, 1) .



Some corollaries: concentration property

Define

A(z) def= {θ : M(µ,θ,θ∗) ≤ z}

with

M(µ,θ,θ∗) = − log IEθ∗ exp{µL(θ,θ∗)

}.

Then

IPθ∗(θ 6∈ A(z)

)≤ Q∗(µ)e−%µz.

Typically A(z) is a root-T neighborhood of θ∗ .



Some corollaries: root-T consistency

From Taylor of the second order:

L(θ,θ∗) ≈ 12(θ − θ∗

)>∇2L(θ∗)(θ − θ∗

).

Under ergodicity T−1∇2L(θ∗) ≈ I(θ∗) , (Fisher IM).

Thus, the bound IEθ∗ exp{µL(θ,θ∗

)}≤ Q∗(µ) yields root-T

consistency:

T‖√I(θ∗)

(θ − θ∗

)‖2 ≤ Const.



Some corollaries: polynomial risk bound

The exponential bound IEθ∗ exp{µL(θ,θ∗

)}≤ Q(µ,θ∗) ≤ Q∗(µ)

yields the polynomial risk bound: for any r > 0

IEθ∗∣∣L(θ,θ∗)∣∣r ≤ Rr(θ∗) ≤ Rr

where Rr(θ∗) and Rr do not depend on the sample size T .



Black-Scholes case

Theorem (Polzehl and S. (2006))

Let Yt i.i.d. N(0, θ∗) . Then for any z > 0

IPθ∗(L(θ, θ∗

)> z)≡ IPθ∗

(TK(θ, θ∗

)> z)≤ 2e−z,

where K(θ, θ∗) is the Kullback-Leibler divergence between N(θ) andN(θ∗) :

K(θ, θ∗) = 0.5(θ/θ∗ − 1)− 0.5 log(θ/θ∗).



Black-Scholes case: some corollaries

The bound

IPθ∗(L(θ, θ∗

)> z)≡ IPθ∗

(TK(θ, θ∗

)> z)≤ 2e−z,

yields:

I the risk bound and root-T consistency: for any r > 0

IEθ∗∣∣L(θ, θ∗)∣∣r ≡ IEθ∗∣∣TK

(θ, θ∗

)∣∣r ≤ rr

where rr depends on r only.

I confidence sets:

E(z) = {θ : K(θ, θ) ≤ z/T}.



GARCH(1,1) case

TheoremAssume Y1, . . . , YT follow GARCH(1,1) with Gaussian innovationsand the parameters θ = (ω, α, β)> . If δ ≤ α+ β ≤ 1− δ for someδ > 0 , then there is µ = µ(δ) s.t.

IEθ∗ exp{µL(θ,θ∗

)}≤ Q(µ,θ∗) ≤ Q∗(µ, δ)

and for any r > 0

IEθ∗∣∣L(θ,θ∗)∣∣r ≤ Rr(θ∗) ≤ Rr ,

I Yields root-T consistency and CS based on L(θ,θ

).

I The results can be extended to the quasi-MLE for non-Gaussianerrors under exponential moment conditions on the innovations.



Advantages of the parametric (GARCH) modeling

1. Well developed algorithms

2. Nice asymptotic and non-asymptotic theory. Root-T consistenceand asymptotic normality of the estimator θ .

3. Good in-sample properties.

4. Possibility to mimic the important stylized facts of the financialtime series (volatility clustering, leptokurtic returns and excesskurtosis).



Problems of the parametric (GARCH) modeling

1. the parameter estimates show quite high variability. Especiallyestimation of β requires about 500 observations. (consequenceof 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 isoften completely misspecified.



More references

. Mikosch and Starica (2000): Some drawbacks of GARCHmodelling

. E. Hillebrand (2004): Proved the artificial IGARCH effect in thechange 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 isdisturbed by occasional parameter shifts generates data that appearto be highly persistent when the parameter shifts are not accountedfor in global 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 structuralchanges.



Remedy of parametric modeling drawback

I Any time-homogeneous (parametric) model is wrong in long run.

I Any parametric modeling is OK if applied locally (for a sufficientlysmall time interval).

I Approach: fix your favorite (tractable, estimable, . . . ) model andapply it for a properly selected historical time intervals.


LPA of Financial Time Series Local parametric approach

Local parametric approach

Nonparametric model: L(Yt|Ft−1

)∼ Pf t

.

Local parametric assumption (LPA): for every t� there exists a timeinterval I = It� in which f t ≈ fθ(Xt) , t ∈ I .

Goal: identify the interval of homogeneity It� and estimate f t� fromthe parametric approximating model:

θI = argmaxθ

LI(θ) = argmaxθ

∑t∈I

log p(Yt,fθ(Xt)).

and apply f t� = fθI

(Xt�) .



“Small modeling bias” (SMB) condition

Nonparametric model: Yt ∼ Pf t.

Applying a parametric assumption f t = fθ(Xt) in the nonparametricsituation leads to modeling bias measured by

∆I(θ) =∑t∈I

K{f t,fθ(Xt)

}.

“SMB” ⇔ “∆I(θ) is small for some θ ”

(with a high probability).



Theoretic information bound

Theorem (Theoretic information bound)

Let for some θ ∈ Θ and some ∆ ≥ 0

IE∆I(θ) = IE∑t∈I

K(f t,fθ(Xt)

)≤ ∆.

Then for any ζ ∼ FI

IE log(1 + ζ) ≤ ∆+ IEθζ.

In particularly, it yields

IE log(1 +

∣∣L(θ,θ)∣∣rRr(θ)

)≤ ∆+ 1.



SMB and “Oracle” interval

LPA and SMBLPA 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−1/2I∗ .

Aim: mimic the “oracle” i.e. provide the same estimation accuracy oforder N−1/2

I∗ .


LPA of Financial Time Series Adaptive LCP procedure

Local change point procedure

Idea: select the largest possible interval of homogeneity I by testingthe parametric hypothesis ft = f(Xt,θ) , t ∈ I , against a changepoint alternative.



Why a change point alternative?

. The change point approach enables us to keep this parametricspecification just by restricting the time interval. Other testsrequire an additional estimation under alternative.

. The change point approach delivers important information aboutthe 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 alsopowerful against a smooth one.



Test of a parametric hypothesis

Problem: test the hypothesis of homogeneity H0 : f t = fθ(Xt) fort ∈ I = [t� −m, t�] .

Log-likelihood under H0 : with `(y, υ) = log p(y, υ)

LI(θ) =∑t∈I

` {Yt,fθ(Xt)} .

LR test: apply the fitted log-likelihood (FLL) as the test statistic. FLLunder H0 :

LI = LI(θ) = maxθ

LI(θ).



Test against a change-point (CP)

A CP alternative H1(τ) at τ ∈ I = [t� −m, t�] : with θ1 6= θ2

H1(τ) : f t =

{fθ1

(Xt) for t ∈ J = [τ + 1, t�],fθ2

(Xt) for t ∈ Jc = [t� −m, τ ].

Log-likelihood under H1(τ) :

LJ(θ1) + LJc(θ2) =∑t∈J

`{Yt,fθ1

(Xt)}

+∑t∈Jc

`{Yt,fθ2

(Xt)}.

LR test statistic for a given location τ :

TI,τ = maxθ1,θ2

{LJ(θ1) + LJc(θ2)} −maxθ

LI(θ) = LJ + LJc − LI

= LJ(θJ , θI) + LJc(θJc , θ).



Test against a change-point. 2

Let T be a subset of I called a tested set.

Maximum LR test: check every point of T on possible CP.

TI,T = maxτ∈T

TI,τ > zI ⇔ a CP is detected at τ = argmaxτ∈T

TI,τ .

Here zI is critical value.

� �� ' $� �t� −m t� − τ

I

JJc

t�

T



Multiscale LCP procedure

Idea: search for the largest historical interval for which the data do notcontradict the parametric assumption.

Let I0 ⊂ I1 ⊂ I2 ⊂ . . . ⊂ IK be an increasing family of historicalintervals Ik = [t−mk, t] . Define Tk = Ik \ Ik−1 = [t�−mk, t

�−mk−1] .

t� −mk t� −mk−1 t� −mk−2 t�︸︷︷︸Tk

︸︷︷︸Tk−1

︸︷︷︸Ik−2︸︷︷︸

Ik−1︸︷︷︸Ik



Multiscale LCP procedure (cont)

Each interval Tk = Ik \ Ik−1 = [t� −mk, t� −mk−1] is tested on a CP

using Ik+1 as testing interval and

Tk = TIk+1,Tk= max

τ∈Tk

TIk+1,τ

as the test statistic.

Ik is accepted ⇔ no CP is detected in T1, . . . ,Tk , that is, if

T` ≤ z` , ` ≤ k.



Multiscale LCP procedure (cont)

1. Initialization: start with k = 1

2. test Tk on a CP using Ik+1 as a testing interval: Ik is acceptedif Tk = TIk+1,Tk

≤ zk .

Set I as the largest accepted Ik , θ = θI

:

k = argmax{k : T` ≤ z` ∀` ≤ k}, I = Ik, θ = θ

I.

Similarly Ik is the latest non-rejected interval after k steps andθk = θ

Ik.



Parameters of the procedure

The only parameters of the procedure are the “critical values” zk .

Their choice slightly depends on the given set of intervals Ik .

A proposal for intervals Ik = [t� −mk, t�] : start with some m0 , e.g.

m0 = 5 , then mk+1 = amk for a > 1 , e.g. a = 1.25 .



Choice of the critical values zk

Parametric risk bound: for every θ∗ ∈ Θ

IEθ∗∣∣LIk(θk,θ∗)

∣∣r ≤ Rr(θ∗) .

Propagation condition: given ρ > 0 , provide for all θ∗ ∈ Θ and allk ≤ K

IEθ∗∣∣LIk(θk, θk)

∣∣r ≤ ρRr(θ∗).


LPA of Financial Time Series Local constant case

Local constant case

I LR test statistic on a change-point at τ :

TI,τ = NJK(θJ , θI) +NJcK(θJc , θI)

I sup-LR test statistic for Ik+1,Tk :

Tk = TIk+1,Tk= max

τ∈Ik

TIk+1,τ .

I LCP procedure:

k = max{k : T` ≤ z` ∀` ≤ k}, I = Ik, θ = θ

I.

I Propagation condition: IEθ∗∣∣NIkK

(θIk , θk

)∣∣r ≤ ρrr .


LPA of Financial Time Series Theoretical study

“Oracle result”

Theorem (“Oracle” bound)

Assume maxk≤k◦ IE∆Ik(θ) ≤ ∆ for some k◦ , θ and ∆ . Then

IE log(

1 +

∣∣L(θk◦ , θ)∣∣rRr(θ)

)1(k ≤ k◦) ≤ ∆+ ρ

and for k > k◦ and k ≤ k

LIk◦(θk◦ , θk◦+1

)≤ zk◦ , . . . , LIk−1

(θk−1, θk

)≤ zk−1 .

In words: the adaptive estimate θ belongs with a high probability tothe CS of the “oracle” θk◦ .


LPA of Financial Time Series Theoretical study

“Oracle result” in the LC case

Theorem (“Oracle” bound)

Assume u0 ≤ Nk/Nk+1 ≤ u < 1 . Let

maxk≤k◦

IE∆Ik(θ) ≤ ∆

for some k◦ , θ and ∆ . Then

IE log(

1 +

∣∣Nk◦K(θk◦ , θ

)∣∣rrr

)≤ ∆+ ρ+ log

(1 +

∣∣cuzk◦∣∣rrr

).

Yields rate optimality of ft� = θ for a smoothly varying ft .


LPA of Financial Time Series Applications to financial time series

Typical related problem in financial engineering

I short term ahead forecasting

I risk management

I portfolio optimization

I high frequency data monitoring.



Forecast from the local model

Using the latest estimated model, one can build a forecast, at least,for a short horizon h :

fLC

t,h = f t ,

fARCHt,h = IEt(Yt+h|Ft) = ωt

h−1∑k=0

αkt + αht f t,

fGARCHt,h = IEt(Yt+h|Ft) = ωt

h−1∑k=0

(αt + βt)k + (αt + βt)hf t,

Usually h ∈ H , a forecasting horizon set, e.g. one day, H = {1} , ortwo weeks, H = {1, . . . , 10} .



Minimum forecasting error criterion

Define mean forecasting error

TFET,h =∑t≤T

∑h∈H

Λ(R2t+h, f t,h

)where Λ(·, ·) , a loss function, e.g.

Λ(v, v′) =∣∣K(v, v′)

∣∣cwith c either 1 or 1/2.

Following to Cheng, Fan and Spokoiny (2003), the data driven choiceof r, ρ can be done by minimizing the following objective function:

(r, ρ) = argminr,ρ

TFET,h


LPA of Financial Time Series Simulation study

Set-up

Consider the conditional heteroscedasticity model: Yt = f(Xt)ε2twhere L

(εt|Ft−1

)= N(0, 1) .

Aim: compare varying-coefficient modeling for1. constant f θ(·) = θ ;2. ARCH(1): fθ(Xt) = ω + αYt−1 .3. GARCH(1,1): fθ(Xt) = ω + αYt−1 + βXt−1 .

Set-up: mk = m0ak , a = 1.25 , m0 = 10 , mK = 570 .

CV’s zk are computed for r = 1 and ρ = 1 (default) with the linearshape

zk = b0 + b1 log(|Ik|/|I0|) = b0 + b1k log(a).



Quantiles of the test of homogeneity. LC

85%, 90%, and 95% quantiles of the test Tk for the constant volatility.,



Quantiles of the test of homogeneity: ARCH

85%, 90%, and 95% quantiles of the test Tk for ARCH (1, 0.2, 0) .,



Finite-sample CV’s

Model (ω, α, β) z(10) Slope z(570)(0.1, 0.0, 0.0) 15.4 -0.55 5.5(0.1, 0.2, 0.0) 16.6 -0.40 9.4(0.1, 0.4, 0.0) 23.4 -0.74 10.1(0.1, 0.6, 0.0) 30.8 -1.05 11.9(0.1, 0.8, 0.0) 73.6 -3.37 16.4(0.1, 0.1, 0.8) 19.5 -0.29 14.3(0.1, 0.2, 0.7) 26.3 -0.68 14.1(0.1, 0.3, 0.6) 25.1 -0.58 14.6(0.1, 0.4, 0.5) 28.9 -0.74 15.6(0.1, 0.5, 0.4) 29.8 -0.83 14.9(0.1, 0.6, 0.3) 34.4 -1.05 15.5(0.1, 0.7, 0.2) 27.1 -0.66 15.2(0.1, 0.8, 0.1) 29.2 -0.75 15.7

(0.1, 0.05, 0.90) 16.1 -0.14 13.6(0.1, 0.10, 0.85) 19.4 -0.23 15.8(0.1, 0.20, 0.75) 36.2 -1.15 15.5



Influence r and ρ on the CV’s

Model (ω, α, β) r ρ z(10) Slope z(570)(0.1, 0.0, 0.0) 1.0 0.5 16.3 -0.50 7.3

1.0 1.0 15.4 -0.54 5.51.0 1.5 14.9 -0.58 4.50.5 0.5 10.7 -0.20 7.10.5 1.0 8.9 -0.19 5.50.5 1.5 7.7 -0.17 4.6

(0.1, 0.2, 0.0) 1.0 0.5 16.0 -0.27 11.21.0 1.0 16.5 -0.39 9.51.0 1.5 16.4 -0.45 8.30.5 0.5 11.7 -0.09 10.10.5 1.0 10.3 -0.09 8.50.5 1.5 9.3 -0.10 7.5

(0.1, 0.1, 0.8) 1.0 0.5 18.7 -0.09 17.11.0 1.0 19.4 -0.28 14.41.0 1.5 18.6 -0.29 13.40.5 0.5 11.7 -0.09 10.10.5 1.0 10.3 -0.10 8.50.5 1.5 9.3 -0.10 7.5



Low and high GARCH-effect

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, . . . , 1000 .



Series for low GARCH-effect

Simulated GARCH(1,1)-time series and their volatilities for lowGARCH-effect; t = 1, . . . , 1000 .



Series for high GARCH-effect

Simulated GARCH(1,1)-time series and their volatilities for highGARCH-effect; t = 1, . . . , 1000 .



Procedure for low GARCH-effect

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 ;



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).



One day ahead forecasting ability

Absolute prediction errors one day ahead averaged over last month.,



10 days ahead prediction error

Absolute prediction errors 10 days ahead averaged over last month.,



Discussion

. all methods are sensitive to jumps in volatility, especially the first one att = 500 .

. the local GARCH performs rather similarly to the parametric GARCH ingeneral: they are equal before the breaks, t < 500 ; the local GARCHoutperforms the parametric one after the first break, 550 < t < 750 , andalso at the end of the period, 900 < t < 1000 ;

. the local ARCH performs as well as the GARCH methods and evenoutperforms them after several structural breaks, 550 < t < 750 and900 < t < 1000 .

. the local constant method is lacking behind the other two adaptivemethods whenever there is a longer time period without a structuralbreak, but keeps up with them in periods with more frequent volatilitychanges, 500 < t < 750 .



High GARCH-effect

α+ β ≤ 0.9 .

The “optimal” choice of the parameters ρ and r by minimizing TFEleads to:

. LC: r = 0.5, ρ = 1.5 ;

. ARCH: r = 0.5, ρ = 1.5 ;

. GARCH: r = 1.0, ρ = 0.5 .



1 day ahead prediction

Absolute prediction errors 1 day ahead averaged over last month.,



10 days ahead prediction

Absolute prediction errors 10 days ahead averaged over last month.,


LPA of Financial Time Series Applications

DAX log-returns. 1/1990 to 12/2002

Log-returns of DAX series from January 1, 1990 till December 31, 2002.,



DAX analysis

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) forthe adaptive methods with respect to the global (parametric) GARCH:

RPE t,h =21∑m=1

∑h∈H

Λ(R2t−m+h, f t−m,h

)/ 21∑m=1

∑h∈H

Λ(R2t−m+h, f

GARCH

t−m,h).



Results for DAX, 1/1991 to 3/1997

The ratio RPE t,h for h = 1 for 3 adaptive methods. DAX from January,1991 to March, 1997.



DAX Analysis. 1/1991 to 3/1997

. 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 1994.

. the local constant and local ARCH methods are optimal at thebeginning of the period, where we have less than 500 observations.

. A similar behavior can be observed after the break detected in October1994;

. In the other parts of the data, the performance of all methods is almostthe 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).



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 toJune, 2001.



DAX Analysis. Period 7/1999 to 6/2001

. Structural change of the market in 1999 (stabilization after thebreaks 1997(Asian) and 1998 (Russian) crashes) detected bylocal constant and ARCH but not by GARCH.

. Local GARCH performs almost as the global one

. GARCH models are preferable for the stable market (middle2000).

. In terms of the global prediction error, the local ARCH is best.

. Generally, LC and local ARCH provide almost the same results.



S&P 500. 1/1994 to 12/1996

The ratio RPE t,h for h = 1 for 3 adaptive methods. S&P 500 from January,1994 to December, 1996.

Stable market ⇔ low GARCH-effect, all methods are equally well.,



S&P 500 from 1/1990 to 12/2004

The log-returns of S&P 500 from January 1, 2000 till December 31, 2004.



Results for S&P 500. 1/2000 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 fromJanuary, 2003 to December, 2004.



Conclusion and Outlook

The new approach

. can be applied and studied in a unified way for a wide class ofdifferent 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 faras the in sample properties or short time ahead forecasting isconcerned.



Extensions and Applications

The local parametric approach has been successfully applied to thefollowing problems:

Mercurio, D. & Spokoiny, V. (2004) Annals of StatisticsLocal constant volatility estimation.

Spokoiny, V. (2008) Annals of StatisticsMultiscale local constant change point procedure.

Polzehl, J. and Spokoiny, V. (2006). Probab. T. Rel. FieldsVarying coefficient GARCH modeling

Cizek, P., Härdle, W. and Spokoiny, V. (2005)Varying coefficient GARCH modeling

Grama, I. and Spokoiny, V. (2008). Annals of StatisticsTail-index estimation.



Giacomini, E., Härdle, W., and Spokoiny (2008).Estimation of time varying copulaeJournal of Business and Economic Statistics.

Härdle, W., Herwartz, H. and Spokoiny, V. (2003). J. of FinancialEconometricsMultivariate volatility estimation.

Chen, Y., Härdle, W. and Spokoiny, V. (2008).Multivariate volatility estimation.

Chen, Y., and Spokoiny, V. (2007)Robust risk management.


foundations and applications of modern nonparametric

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