Forecasting Using Locally Stationary Wavelet...

Forecasting Using Locally Stationary Wavelet Processes

Yingfu Xie, Jun Yu and Bo Ranneby

Research Report Centre of Biostochastics Swedish University of Report 2007:2 Agricultural Sciences ISSN 1651-8543

Forecasting Using Locally Stationary Wavelet Processes

Yingfu Xie1, Jun Yu and Bo Ranneby

Centre of BiostochasticsSwedish University of Agricultural Sciences

SE-901 83 Umea, Sweden

Abstract

Locally stationary wavelet (LSW) processes, built on non-decimatedwavelets, can be used to analyze and forecast non-stationary time se-ries. They have been proved useful in the analysis of financial data. Inthis paper we first carry out a sensitivity analysis, then propose somepractical guidelines for choosing the wavelet bases for these processes.The existing forecasting algorithm is found vulnerable to outliers, and anew algorithm is proposed to avoid the sensitivity to extreme observa-tions. The new algorithm is shown stable and outperforms the existingalgorithm when applied to real financial data. The volatility forecastingability of LSW modeling based on our new algorithm is then discussedand shown to be competitive with traditional GARCH models.

Keywords: Locally stationary wavelet processes, non-decimated wavelets,sensitivity analysis, GARCH, volatility forecasting.

1E-mail address to the correspondence author: [email protected]

1 Introduction

Recently, many studies have been published in which time-scale or time-frequency techniques such as wavelet transform have been applied, in additionto traditional time domain analysis of stochastic processes, especially financialtime series. One advantage of using wavelet transform is that it depends lesson specifications of the dependent structure and distribution of the originalseries because of the ‘whitening’ property of wavelets, i.e., wavelet coefficientsare often less correlated than the original data, see Vidakovic (1999). Anotheradvantage is that not only stationary time series but also non-stationary onescan be treated in the same framework using wavelets (see e.g. Mallat et al.,1998).

The locally stationary wavelet (LSW) process is a relatively new tool, orig-inally proposed by Nason et al. (2000), which incorporates a class of stochas-tic processes based on non-decimated wavelets. By defining an evolutionarywavelet spectrum (EWS), which is the analogue of the usual spectrum forstationary processes, the power (variance) of LSW processes can be measuredlocally over time and scale. Estimation theory has also been developed forthe EWS and localized autocovariance. Fryzlewicz (2005) showed that theLSW model can capture most of the stylized facts of financial time series. Inaddition, Fryzlewicz et al. (2003) developed an algorithm to forecast the LSWprocesses. The predictor is simply the linear combination of previous observa-tions with the predictor coefficients obtained by minimizing the mean squareprediction error (MSPE). Hence the forecasting of non-stationary processes ispossible with this algorithm. However, it is found that this algorithm has noprotection against the occurrence of outliers, since an inverse of a covariancematrix is unavoidable. The presence of outliers, as a result of the frequentlysingular matrices, makes it very difficult to evaluate the performance of thisalgorithm (usually in terms of sample MSPE). In this paper, a new algorithmis proposed to avoid this problem, i.e., we impose suitable restrictions on thepredictor coefficients when minimizing the MSPE. Two intuitive restrictionsare examined and we focus on the algorithm with the aim of producing a pre-diction coefficient vector with unit length. Both the new algorithm and theoriginal are tested on real data. The results show that even in one-step-aheadout-of-sample prediction, outliers can appear in the original algorithm, makingit difficult to evaluate, while our new algorithm works consistently well.

The volatility forecasting ability of LSW modeling based on our new algo-rithm is now analyzed. Volatility is a central concept in capital asset pricing,portfolios investment and risk management, especially for products such asoptions. Volatility forecasting has been the subject of considerable attention

1

in the last 20 years. In a comprehensive survey summarizing 93 studies, Poonand Granger (2003) classify volatility forecasting models into four categories:HISVOL (Historical volatility), ARCH family (Engle, 1982), ISD (option im-plied standard deviation) and SV (stochastic volatility). LSW is not includedin any of Poon and Granger’s categories, so we include a new comparisonhere. In this paper, we compare the volatility forecasting abilities of the LSWmodel and GARCH models, including standard GARCH (Bollerslev, 1986),Exponential GARCH (Nelson, 1991) and Regime-Switching GARCH (e.g.,Hamilton and Susmel, 1994; Gray, 1996; Xie and Yu, 2005; Xie, 2007), basedon log-returns of the S&P500 index. LSW modeling based on our algorithmis shown to be quite competitive with GARCH models.

Our paper is organized as follows: in Section 2 we briefly introduce thedefinition of the LSW process and the estimation of its EWS. In addition, asensitivity analysis of the wavelet selection problem is described. The newforecasting algorithm and applications to real financial data are presented inSection 3. The model’s ability to forecast volatility is compared with that ofthe GARCH models in Section 4. Finally, the results and their implicationsare considered in the Discussion (Section 5).

2 Locally stationary wavelet modeling

In this section, we briefly introduce the definition of LSW processes, EWSand its estimation, mainly based on Nason et al. (2000). The sensitivityof selection of wavelet bases in LSW modeling is examined using numericalexamples.

2.1 Locally stationary wavelet processes

Definition 1 (Nason et al. (2000)) An LSW process is a sequence ofdoubly-indexed stochastic processes {Xt,T }t=0,...,T−1 having the following rep-resentation in the mean-square sense

Xt,T =−1∑

j=−J

∑

k

ωj,k;T ψj,k−tξj,k, (1)

where ξj,k is a random orthonormal increment sequence, and ψj,k is a discretenon-decimated family of wavelets for j = −1,−2, ...,−J(T ), k = 0, ..., T − 1based on a mother wavelet ψ(t) of compact support. The following propertiesare also assumed:

2

1. Eξj,k = 0 for all j, k. Hence EXt,T = 0 for all t and T .

2. cov(ξj,k, ξl,m) = δjlδkm.

3. The amplitudes ωj,k;T are real constants and for each j ≤ −1 there existsa Lipschitz-continuous function Wj(z) for z ∈ (0, 1) which satisfies

−1∑

j=−∞W 2

j (z) < ∞ uniformly in z ∈ (0, 1)

with Lipschitz constants Lj which are uniformly bounded in j and

−1∑

j=−∞2−jLj < ∞.

In addition, there exists a sequence of constants Cj fulfilling∑

j Cj < ∞such that for each T

supk=0,...,T−1

|ωj,k;T −Wj(k/T )| ≤ Cj/T.

It is well-known that a stationary stochastic process Xt, t ∈ Z, can bewritten as

Xt =∫ π

−πA(δ) exp(iδt)dζ(δ), (2)

where dζ(δ) is an orthonormal increment process (Priestley, 1981). Now, theidea behind the LSW process is to replace the set of harmonics {exp(iδt)|δ ∈[−π, π]} in (2) with a set of non-decimated wavelets ψj,k and the spectrumA(δ) by the time-varying ωj,k;T . In fact, LSW processes include all stationaryprocesses with absolutely summable autocovariance as special cases (Nasonet al., 2000, Corollary 2). Assumption 3 demands that a smooth Wj(z), as afunction of rescaled time, z, controls the variation of ωj,k, as a function of k, sothat it cannot change too quickly, allowing effective estimates to be obtainedfor the model. In this assumption, a rescaled time z = k/T is used whichimplies that as T →∞, instead of more and more future data, more detailedlocal information of Wj(z) are collected. For a more detailed description ofthe model and of rescaled time, we refer the reader to Nason et al. (2000) andDahlhaus (1997).

From the definition of LSW processes, direct calculation gives the covari-ance structure with lag τ as

Cov(Xt,T , Xt+τ,T ) =∑

j

∑

k

ω2j,k;T ψj,k−tψj,k−t−τ . (3)

3

Considering of τ = 0 leads to Definition 2.

Definition 2 (Nason et al. (2000)) The Evolutionary Wavelet Spectrum(EWS) of sequence {Xt,T }t=0,...,T−1 for infinite sequence T ≥ 1 is definedas

Sj(z) = W 2j (z), for j = −1, ...,−J(T ), z ∈ (0, 1).

Under Assumption 3 of Definition 1, Sj(z) = limT→∞ |ωj,[zT ];T |2 and∑−1−∞ Sj(z) < ∞ uniformly in z ∈ (0, 1).

The EWS measures the local power (variance) at a particular time z andscale j, which is the analogue of usual spectrum for stationary processes. Inthe stationary case, however, it is independent of time, i.e., Sj = ω2

j,k;T = W 2j .

The autocovariance is defined as follows:

cT (z, τ) = Cov(X[zT ],T , X[zT ]+τ,T )

and the local autocovariance with EWS Sj(z) as

c(z, τ) =−1∑

j=−∞Sj(z)Ψj(τ),

where the Ψj(τ) =∑∞−∞ ψj,kψj,k−τ is defined as the autocorrelation wavelets.

From (3) it can be seen that ‖ cT − c ‖L∞= O(T−1) (Nason et al., 2000),which implies that the local autocovariance is the “autocorrelation wavelet”transform of the EWS. We know that for a stationary process, the au-tocovariance and spectrum are Fourier transforms of each other. This isthe analogous result for LSW processes. In particular, the local varianceσ2(z) := c(z, 0) =

∑−1j=−∞ Sj(z) since Ψj(0) = 1 for all values of j.

Definition 3 (Nason et al. (2000)) The empirical wavelet coefficients ofan LSW process Xt,T are given by

dj,k;T =T−1∑

t=0

Xt,T ψj,k−t,

where ψj,k is the same wavelet basis used to build Xt,T in Definition 1. Thewavelet periodogram of Xt,T is defined as Ij,k = |dj,k;T |2.

The wavelet periodogram is the “building block” for the estimation ofEWS. Defining AJ as the inner product of the autocorrelation wavelet, whoseelement Aj,l =< Ψj , Ψl >=

∑τ Ψj(τ)Ψl(τ), led Nason et al. (2000) to the

following proposition:

4

Proposition 1 Assuming that the innovations ξj,k in Definition 1 are Gaus-sian, we have, for the LSW process Xt,T ,

EIj,k =∑

l

AjlSl(k/T ) + O(T−1). (4)

Hence, for the vector of periodogram I(k) := {Il,k}l=−1,...−J and the correctedperiodogram vector L(k) = A−1

J I(k),

EL(k) = EA−1J I(k) = S(k) + O(T−1), (5)

where S(k) := {Sj(k/T )}j=−1,...,−J . In addition,

V arIj,k = 2{∑

l

AjlSl(k/T )}2 + O(2−j/T ). (6)

Proposition 1 enables us to use the corrected wavelet periodogram asan unbiased estimate of EWS. However, equation (6) implies that the (cor-rected) wavelet periodogram is an inconsistent estimate of EWS and needs tobe smoothed. Nason et al. (2000) used translation-invariant linear wavelet(TILW) smoothing (Coifman and Donoho, 1995) to do this. Fryzlewicz (2005)compared TILW and cubic B-splines smoothing and showed that they werealmost equally powerful. In this paper, we used spline smoothing. For amore detailed description of estimating EWS and local variance, see Fryzlewicz(2005).

2.2 Sensitivity of wavelet selection in LSW modeling

It should be observed that the wavelet periodogram used to estimate the EWSof process {Xt,T } has to be constructed using the true wavelet basis of {Xt,T }(Definition 3). In practice this is unrealistic, prompting Nason et al. (2000) toask the following questions. How can we choose an appropriate wavelet basison which to build the model, and what happens if we choose an inappropriatebasis? Since these questions were posed they have not previously been an-swered in the literature. In this subsection, we describe a sensitivity analysis,based on numerical examples, that was conducted to demonstrate the effectof selecting the wrong wavelet on the estimate of EWS.

The analysis was conducted as follows. We predetermined a set of waveletsfor the comparison and constructed true LSW processes with known EWS andlocal variance based on these wavelets. For each process, 50 realizations weregenerated. These wavelets were then applied to all realizations to constructthe wavelet periodogram and estimate the EWS. The estimates were then

5

compared with the true values and the averages (over 50 realizations) of meansquare errors (MSE) were summarized.

The wavelets we selected are compactly supported orthogonal waveletsfrom Daubechies (1992) with different filter lengths, including: Haar, Daubletsd4, d8, d12 and d20, Coiflets c6, c12, c18 and c30 and least-asymmetricwavelets s8, s12 and s20. These wavelets are generally representative, withrespect to their symmetry and smoothness, of orthogonal wavelets. Descrip-tions and properties of these wavelets can be found in Daubechies (1992),Bruce and Gao (1996) and Percival and Walden (2000).

Three processes with different wavelets were considered.

• First, a non-stationary process defined by:

Xt,T =∑

k

(√S(t/T )ψ−1,k−tεk

), T = 2000, (7)

where S(t/T ) = 0.1 + cos2(3πt/T + 0.25π), ψ−1,. are the non-decimatedwavelet filters at scale j = −1 and εk is standard Gaussian. By definition,the EWS of this process is simply S(t/T ). The results of MSE for thisprocess are presented in Table 1.

• The second process was:

Xrt,T =

∑

k

ψ−r,k−tεk, (8)

where ψ−r,. are the non-decimated wavelet filters at scale j = −r. Forthese processes ωj,k;T and Wj equal to 1 when j = −r and otherwiseequal to 0. So, the EWS Sj and local variance of Xr

t,T are also equal to1 when j = −r and otherwise equal to 0. In this example, we chooser = 2 and T = 2000. This sequence is stationary, and the results arepresented in Table 2.

• The third process was the concatenation of X1t,T , X2

t,T , X3t,T and X4

t,T asdefined in (8), each with length 1024 and standard Gaussian noise. Thetrue EWS should be S−1(k/T ) = 1 for k = 1, ..., 1024 and 0 for othervalues of k; S−2(k/T ) = 1 for k = 1025, ..., 2048, and so on. The resultsare shown in Table 3.

From Table 1 it can be seen that the wavelet selection was not so sensitivefor the example based on a non-stationary process. In fact, wavelet s8 isquite robust. It has the smallest MSE for eight out of 12 cases (plus another

6

Tab

le1:

Ave

rage

(ove

r50

real

izat

ions

)M

SE(×

104)of

LSW

mod

elin

gba

sed

ondi

ffere

ntw

avel

ets.

The

real

izat

ions

are

gene

rate

dfr

omth

eno

n-st

atio

nary

proc

ess

(7)

wit

hth

esa

me

wav

elet

s.T

hebo

ldnu

mbe

rsar

eth

esm

alle

stin

thei

rre

spec

tive

colu

mns

.

Use

d\Tru

eH

aar

d4c6

d8s8

c12

d12

s12

c18

d20

s20

c30

Haa

r26

.08

34.3

128

.72

37.5

637

.83

40.3

637

.72

42.6

437

.86

47.6

341

.67

45.8

2d4

24.4

523

.17

19.4

424

.08

23.7

726

.40

22.9

524

.89

25.1

729

.95

27.9

429

.21

c627

.32

25.4

522

.04

26.4

826

.60

28.8

525

.36

27.0

426

.35

32.0

730

.27

32.0

2d8

30.8

022

.47

20.5

220

.73

20.9

723

.09

18.9

020

.10

20.0

323

.20

23.6

224

.07

s830

.49

22.3

920

.10

20.9

520

.58

22.9

918

.79

19.7

021

.28

22.6

623

.14

23.6

4c1

235

.23

25.4

623

.22

23.6

723

.49

25.7

620

.98

22.0

522

.98

25.1

825

.64

26.3

6d1

239

.28

27.2

124

.74

24.1

823

.69

25.8

420

.54

21.5

423

.73

24.5

024

.18

25.2

1s1

238

.24

26.7

924

.20

23.1

722

.78

25.3

120

.31

21.0

222

.89

23.5

124

.60

24.7

6c1

845

.23

31.1

428

.56

27.1

026

.60

29.4

723

.32

24.0

925

.88

26.9

528

.32

28.2

0d2

052

.68

36.3

133

.80

30.4

832

.31

32.3

226

.32

27.1

129

.18

29.6

130

.47

30.3

9s2

049

.96

33.8

431

.39

27.7

427

.94

30.5

923

.98

24.5

527

.47

26.2

728

.03

27.8

7c3

050

.21

34.5

131

.58

27.8

928

.40

30.6

623

.98

24.5

226

.47

26.1

627

.80

27.3

3

7

Tab

le2:

Ave

rage

(ove

r50

real

izat

ions

)M

SE(×

104)of

LSW

mod

elin

gba

sed

ondi

ffere

ntw

avel

ets.

The

real

izat

ions

are

gene

rate

dfr

omth

est

atio

nary

proc

ess

(8)

wit

hth

esa

me

wav

elet

s.T

hebo

ldnu

mbe

rsar

eth

esm

alle

stin

thei

rre

spec

tive

colu

mns

.

Use

d\Tru

eH

aar

d4c6

d8s8

c12

d12

s12

c18

d20

s20

c30

Haa

r96

.43

154.

8914

6.65

251.

4425

3.72

273.

7933

9.07

318.

8932

2.46

394.

5341

0.97

421.

18d4

106.

4876

.65

70.4

011

1.61

112.

7412

3.96

168.

2615

4.54

155.

9420

6.42

218.

3222

2.24

c612

4.14

83.5

378

.17

120.

9412

1.41

132.

5818

0.32

165.

7316

4.04

221.

9123

5.36

236.

33d8

196.

1192

.62

84.7

660

.25

62.2

863

.74

80.9

971

.91

71.7

992

.94

105.

0210

2.97

s819

5.54

91.8

684

.21

59.6

963

.59

64.4

780

.81

71.5

671

.33

92.2

210

4.84

103.

14c1

223

2.57

108.

3299

.03

67.2

170

.70

67.7

986

.02

78.5

276

.80

99.6

811

1.81

109.

19d1

227

6.36

133.

7712

5.52

68.0

6469

.77

67.3

169

.36

61.7

062

.39

69.2

681

.03

76.4

6s1

227

7.19

133.

3512

4.28

67.2

870

.21

66.5

870

.41

64.5

862

.70

68.6

679

.83

75.9

2c1

833

3.59

163.

6515

1.73

81.1

184

.66

78.1

578

.36

72.3

872

.21

77.8

291

.76

83.0

3d2

038

5.80

201.

7318

9.58

95.5

696

.99

89.2

779

.81

70.0

170

.93

62.8

273

.14

65.4

4s2

038

5.28

202.

4419

0.19

95.2

197

.42

90.5

980

.23

71.0

472

.19

59.7

972

.42

64.6

7c3

039

4.37

207.

9919

5.82

98.0

499

.44

91.2

579

.92

70.8

971

.44

59.6

470

.54

61.6

6

8

Tab

le3:

Ave

rage

(ove

r50

real

izat

ions

)M

SE(×

104)of

LSW

mod

elin

gba

sed

ondi

ffere

ntw

avel

ets.

The

real

izat

ions

are

gene

rate

dfr

omth

eco

ncat

enat

edpr

oces

sde

scri

bed

inse

ctio

n2.

2,w

ith

the

sam

ew

avel

ets.

The

bold

num

bers

are

the

smal

lest

inth

eir

resp

ecti

veco

lum

ns.

Use

d\Tru

eH

aar

d4c6

d8s8

c12

d12

s12

c18

d20

s20

c30

Haa

r12

3.9

197.

818

4.6

300.

330

1.2

290.

334

5.5

355.

935

7.9

410.

445

8.5

392.

1d4

160.

315

0.0

134.

318

0.0

179.

917

6.9

202.

420

9.7

203.

225

3.2

286.

624

5.6

c618

5.5

171.

415

4.5

205.

421

2.1

233.

322

8.2

234.

624

3.9

290.

330

4.2

272.

0d8

232.

416

0.2

154.

513

9.5

159.

914

7.6

150.

615

9.4

157.

218

0.1

179.

317

7.3

s823

1.3

155.

814

8.5

143.

515

0.5

148.

214

6.6

158.

115

5.4

180.

617

5.7

169.

9d1

230

7.9

194.

420

6.0

167.

816

6.7

160.

116

7.8

176.

016

8.3

178.

719

2.0

184.

5s1

230

7.9

190.

519

7.9

165.

216

7.2

156.

015

8.8

166.

516

2.7

174.

118

3.3

179.

9c1

837

5.9

238.

124

4.9

211.

921

3.0

209.

222

5.9

219.

021

1.9

222.

124

7.0

231.

5d2

041

0.4

275.

526

7.7

242.

524

1.3

230.

323

3.1

229.

423

1.9

228.

125

8.8

237.

6s2

039

6.6

266.

826

3.8

229.

823

3.4

225.

922

2.3

220.

621

6.4

217.

924

1.7

220.

4c3

042

0.5

315.

330

0.5

290.

629

1.6

264.

827

6.1

266.

126

8.3

270.

729

1.0

285.

5

9

two cases for d8, which has properties very similar to s8 except symmetry).Wavelets with longer filters are almost all dominated by s8 (except one whichis dominated by d8). In addition, the results for the concatenated process(Table 3) exhibit a similar dominance of s8 with respect to averages of MSE(for seven out of 12 cases it is the smallest, plus another two for d8). Notethat the concatenated process is also non-stationary.

However, the selection of wavelet bases is more sensitive for the stationaryprocess examined here. For instance, the MSEs increase in the first column ofTable 2 nearly monotonically, demonstrating that the more the filter length ofa wavelet basis differs from the true one (Haar here) the worse the performance.The first row of MSE in Table 2 shows that using the Haar wavelet in theestimation, the longer filter the true wavelet basis has, the larger MSE theLSW model produces. This even happens with the concatenated process.Other columns and rows exhibit similar trends. In addition, the Daublets, ‘d’,and Least-asymmetric wavelets, ‘s’, have similar smoothness characteristics,such as vanishing moments (VM), number of derivatives and Holder exponent(HE), while Coiflet c12 (VM 3, HE 1.45 ) is close to s8 and d8 (VM 3, HE1.62 and 1.4), c18 (VM 5, HE 2.21) to s12 and d12 (VM 5, HE 2.19 and2.12) and c30 (VM 9, HE 3.47) to s20 and d20 (VM 9, HE 3.38 and 3.31)(see Bruce and Gao, 1996, for a description of the Holder exponent and theseproperties). Taking the above into consideration, it is always the true waveletor wavelets with similar smoothness characteristics that produce the best fit.In general, for stationary processes, the selection of wavelet bases is sensitive,and is mainly determined by the smoothness of the wavelet. Care should betaken when choosing the wavelet basis in LSW modeling. In contrast, as canbe seen from equation (3), for a wavelet with filter length L the covarianceCov(Xt,T , Xt+τ,T ) = 0 for τ > (2−J−1)(L−1)+1. For a stationary series, thisproperty may be useful when choosing the wavelet filter length, by inspectingthe sample covariance and determining the minimum non-trivial scale.

3 Forecasting

In this section we first introduce and assess the original forecasting algorithmfor LSW processes published by Fryzlewicz et al. (2003). We found that thisalgorithm cannot prevent outliers from occurring. This decreases its useful-ness, especially when it is evaluated using, e.g., MSPE. Therefore we proposea new algorithm, by imposing restrictions on the predictor coefficients. Here,the original and new algorithms are applied to real financial data and theirperformance is compared.

10

3.1 Fryzlewicz’s algorithm

Fryzlewicz et al. (2003) developed a forecasting algorithm for LSW processes.Observing that LSW processes have a linear form, a convenient option toconsider is a linear predictor for h steps ahead forecast of Xt−1+h,T , givenobservations X0,T , X1,T , ..., Xt−1,T , as

Xt−1+h,T =t−1∑

s=0

bt−1−s;T Xs,T . (9)

The coefficients bj,T , j = 0, ..., t− 1, are chosen to minimize the MSPE definedas E(Xt−1+h,T − Xt−1+h,T )2. That is, the vector bt = (b0,T , ..., bt−1,T )′ ( ′

denoting transposition) is such that

bt = arg minb′t

[(b′t,−1)Σt+h−1;T (b′t,−1)′

], (10)

where Σt+h−1;T is the covariance matrix of X0,T , ..., Xt−1,T and Xt−1+h,T . Di-rectly taking the derivative over the quadratic form in (10) then equating itto zero leads to a linear equation system for solving bt

Σt−1;Tbt = Ct−1+h ,Ct−1,h + C′

h,t−1

2, (11)

where Σt−1;T is the covariance matrix of X0,T , ..., Xt−1,T , Ct−1,h is thecolumn vector of covariances between X0,T , ..., Xt−1,T and Xt−1+h,T andCh,t−1 the vector of covariances between Xt−1+h,T and X0,T , ..., Xt−1,T .These (co)variances can be estimated by estimating the local autocovariance.Fryzlewicz et al. (2003, Remark 8) provided an estimate for these, which wasinconsistent and suggested to be smoothed using, for instance, standard kernelsmoothing.

In practice, there are two compromises to be made with respect to theabove algorithm. First, Σt;T in (10) depends on the amplitudes ωj,k;T , whichare not uniquely defined due to the redundancy of non-decimated wavelet fam-ilies. Based on technical considerations, Fryzlewicz et al. (2003) approximated(b′t,−1)Σt+h−1;T (b′t,−1)′ by (b′t,−1)Bt+h−1;T (b′t,−1)′, where Bt+h−1;T is a(t + 1)× (t + 1) matrix whose (m, n)-th element is

−1∑

j=−J

Sj(n + m

2T)Ψj(n−m)

and can be estimated by estimating the EWS Sj . Second, considering thenon-stationary nature and local smoothness of the process, it is recommended

11

that only the most recent p observations in (9) should be used, rather thanthe entire sequence, i.e.,

X(p)t−1+h,T =

t−1∑s=t−p

bt−1−s;T Xs,T . (12)

The parameter p, as well as g, the bandwidth of a kernel used to smooth theinconsistent estimator of local autocovariance, can be selected automaticallyby so-called Adaptive Forecasting (see Fryzlewicz et al., 2003, for detail):Suppose we observe the sequence until Xt−1,T and want to predict Xt−1+h,T .But move first, say, s + h steps backwards and start to predict Xt−s,T usingX0,T , ..., Xt−h−s with initial parameters (ps, gs). With some predeterminedcriterion (usually minimum distance criteria or relative absolute predictionerror) and parameter space of (p, g), we obtain the optimal pair of (ps∗, gs∗)and use it as the start value in the next prediction of Xt−s+1,T , and so on.After this training process, an updated pair (p1∗, g1∗) is finally obtained for theactual forecasting. The number s can be chosen to be the length of the largestsegment at the end of sequence containing no apparent visually observablebreakpoints. When feasible, we can run the algorithm several times using(p1∗, g1∗) as the start value in the next iteration until it performs reasonablywell.

3.2 The new algorithm

Fryzlewicz’s algorithm may work well for short forecasting horizons (usuallysmall p’s) and carefully chosen parameters (see Fryzlewicz et al., 2003, andFryzlewicz, 2005, for examples). However, we find that in this algorithm ex-traordinary high value of bt is often obtained when solving (11) because thecovariance matrix often become singular, even for moderately large value t in(9) (or p in (12)). A similar problem is well known in linear regression whenthere are many regressors. Consequently, the forecasts predict abnormallylarge values (outliers). After some investigation we found that this problemwas difficult to circumvent without artificially intervening in a case-by-casemanner. There are a number of potential remedies for this, for example,avoiding the use of the quadratic form of prediction error during the mini-mization, and instead selecting an alternative yet robust criterion. Further,it is also possible to avoid the use of MSPE when evaluating the performanceof this forecasting algorithm, thus reducing the effect of outliers. However, itmay be of more interest to prevent outliers from occurring in the first place.Here a new forecasting algorithm is proposed to achieve this.

12

We suggest imposing some restriction on the predictor coefficients bt whenminimizing the quadratic form of (10). Particular forms of restrictions shouldbe specified for different problems to obtain a solution. An obvious constraintin this is to require the sum of bt to be one, or

b′t1 = 1, (13)

where 1 is the unit vector with same length as bt. This actually works as aweighted average predictor with data-driven coefficients. The solution of (10)with constraint (13) is quite simple using the Lagrangian Multiplier (LM)method. bt is the solution of the following equation system

(Σt−1;T 1

1′ 0

)(bt

−12λ

)=

(Ct−1+h

1

), (14)

where λ is the Lagrangian multiplier. However, imposing constraint (13) can-not prevent the excessively predictor coefficients from occurring. Hence, itdoes not fit our purpose and so we will mainly focus on another convenientchoice, namely the requirement for a unit length of vector bt, i.e.,

b′tbt = 1. (15)

It should be mentioned that, by imposing a restriction to bt, the parameterspace of bt is reduced and we may obtain only local maxima. In addition, thesolving of bt under condition (36) is more complicated. Similarly using theLM method, we obtain

bt = (Σt−1;T − λI)−1 Ct−1+h, (16)

where I is the identity matrix of same size as Σ, λ is again the Lagrangianmultiplier and satisfies

C′t−1+h(Σt−1;T − λI)−1(Σt−1;T − λI)−1Ct−1+h = 1 (17)

andΣt−1;T − λI > 0 (positive definite). (18)

It is difficult, if not impossible at all, to obtain an analytic solution ofλ from equation (17). Instead we use numerical computing. The numeri-cal experiments show that in general this equation is very unsmooth aroundzero and there are usually multiple roots near zero for λ depending on thecovariance matrix. Alternatively, we can try to solve this by minimizing the

13

Table 4: MSPE (×105) of the two algorithms with different wavelets bases inone-step-ahead forecasting of FTSE 100 data F1659, ..., F1758.

Haar d4 c6 s8 d10Fryzlewicz’s algorithm 168.17 43.66 753.84 16.78 24.27Unit-length algorithm 26.05 23.79 27.07 24.55 24.01

function f(λ) , (C′t−1+h(Σt−1;T −λI)−1(Σt−1;T −λI)−1Ct−1+h−1)2. In prac-

tice, selecting one of the minima of f(λ) over the interval (−1, 1) satisfying(18) gives quite satisfactory result in our forecasting applications, using theoptimization routine in R (R Development Core Team, 2005) or S-plusr. Thetwo practical settings, approximating the MSPE by (b′t,−1)Bt+h−1;T (b′t,−1)′

and using the ’clipped’ series (12) for prediction, are also used in our proce-dure. The adaptive forecasting procedure to obtain the nuisance parametersis also applied.

3.3 Applications and comparison of the algorithms

In this section of study, we compare these algorithms and demonstrate theusefulness of the new algorithm and the need to avoid the influence of outliersin the evaluation.

In the first experiment, our algorithm with bt obtained from (16), denotedas the unit-length algorithm, and Fryzlewicz’s algorithm (11) are applied tothe log-returns of daily FTSE 100 index from 22/23 October 1992 to 10/11May 2001, Ft. We start from F1658 and forecast one step ahead for everystep, resulting in a total of 100 steps. (This sequence was also investigated byFryzlewicz, 2005). The parameters are automatically chosen by the adaptiveforecasting procedure for given starting setups, for example, start values of pand g and so on. The MSPE is presented in Table 4. From the sensitivityanalysis, we may use wavelet s8 as an example. Wavelets Haar, d4, c6 andd10 are also included here for comparison.

Table 5 shows the results for SPt, the S&P500 index daily log-returns from2 Jan 1990 to 29 Dec 2000, from the Center for Research in Securities Prices(CRSP) database. The one-step-ahead forecasting starts from a half sampleSP1390 and runs for 100 steps. Another example with a longer forecastinghorizon is also considered. Table 6 shows the MSPE of these two algorithmsfor a ten-step-ahead forecast of the same S&P500 return series.

Clearly, the performance of the original algorithm of Fryzlewicz et al.(2003) is severely affected by the outliers. Even in the one-step-ahead fore-

14

Table 5: MSPE (×106) of the two algorithms with different wavelets bases inone-step-ahead forecasting of S&P500 data SP1391, ..., SP1490


Table 6: MSPE (×106) of the two algorithms with different wavelets bases inten-step-ahead forecasting of S&P500 data SP1400, ..., SP1499.


casts, where smaller p’s are usually adapted, the algorithm still causes thisproblem. For example, a single outlier (0.8442) dominates near 85% MSPE inthe FTSE experiment with the c6 wavelet basis. The outlier (−0.3404) givesa 96.36% MSPE in the S&P500 experiment based on the Haar wavelet. More-over, most results in the ten-step-ahead experiment (Table 6) for Fryzlewicz’salgorithm are unsatisfactory due to outliers. Once again, such outliers aredifficult to predict and avoid without artificially intervening on a case-by-casebasis. Note that in the real calculation Fryzlewicz et al. (2003) deliberately settheir forecasts to zeros when the parameter p, the number of observations tobe used in (12), obtained is less than or equal to the corresponding forecastinghorizon h. We do not see the rationale behind this and do not follow this prac-tice. For instance, in the one-step-ahead cases, whenever p = 1, Fryzlewiczet al. (2003) forecast zero values, while the forecasts for our algorithm arethe previous observations from (15). Otherwise, the setup for the two algo-rithms was identical for each experiment. For different setups, the results ofFryzlewicz’s algorithm can change dramatically for different wavelets, depend-ing on the occurrences of outliers, while results for the unit-length algorithmvary only slightly. However, the overall performance of Fryzlewicz’s algorithmis similar to (if not worse than) that reported here. As mentioned previously,the outlier problem persists in the ‘average algorithm’ with constraint (13)and its performance is not significantly better than Fryzlewicz’s algorithm.We, therefore, did not include it in this study. From these experiments theunit-length algorithm shows promise; it works consistently and outperformsFryzlewicz’s algorithm in most cases.

15

4 Application to volatility forecast

Our aim in this section is to illustrate the volatility forecasting ability of LSWmodeling. After obtaining the forecasts described in the previous section, thenext step is to obtain volatility forecasts by estimating the EWS of the se-quence together with the forecasts. Fryzlewicz’s algorithm is not suitable forthis due to the occurrence of outliers. This problem becomes more apparentin volatility forecasts since through wavelet transform more of them will beaffected by a single outlier derived from the sequence forecasting. The perfor-mance of Fryzlewicz’s algorithm can be very poor in terms of MSPE. Therefore,only the unit-length algorithm was used in this study. From the sensitivityanalysis presented in Section 2.2, wavelet s8 is chosen as a representative andsome other wavelets are also discussed.

Here, GARCH models are compared to our LSW model. Besides the stan-dard GARCH(1,1) model, GARCH(1,1) with student-t as a conditional distri-bution (GARCH-t(1,1)) and exponential GARCH (EGARCH(1,1)) model arealso considered. The GARCH-t model may capture the occurrence of fat tailsthat are often observed in financial sequences. For the EGARCH model, theprocess may be written as:

rt = σtηt, (19)

log(σ2t ) = α0 +

p∑

i=1

αi|rt−i|+ γirt−i

σt−i+

q∑

j=1

βj log(σ2t−j), (20)

where, σt is the conditional variance of rt conditioning on information up totime t−1, ηt is usually assumed to be standard Gaussian and αi, i = 0, 1, ..., p,and βj , j = 1, ..., q, are parameters to be estimated. From equation (20), notonly is the positive parameter constraint of the GARCH model unnecessary,the leverage effect exerted by bad news (negative shocks) tending to havea greater impact on the volatility than good news (positive shocks), is alsoincorporated, by introducing (usually negative) parameters γj . See Nelson(1991) for detail. Finally, we will consider the Regime-Switching GARCH(RS-GARCH) model too, which is defined by:

rt = σtηt,

σ2t = α0(Yt) +

q∑

i=1

αi(Yt)r2t−i +

p∑

j=1

βj(Yt)σ2t−j . (21)

In (21), a regular (stationary, ergodic), finite state (assuming two states inour experiment) Markov chain {Yt} is incorporated into the conditional vari-ance equation. This model can remedy the volatility persistence problem of

16

the GARCH model and has been shown to be superior to GARCH in modelexplanation and volatility forecasting in some case (e.g., Klaassen, 2002).

We use the module FinMetrics in S-plusr to generate estimates and fore-casts for the ordinary GARCH models. With respect to the RS-GARCHmodel, based on the suggestions of Klaassen (2002) and Xie and Yu (2005),suppose we want to predict the volatility at time t based on information up tot − 1, It−1. By letting λit = Pr(Yt = i|It−1) and making use of the Bayesianrule, it is relatively straight-forward to obtain the predicted regime

λit =d∑

j=1

p(j, i)fj(r1, ..., rt−1)λjt−1∑d

k=1 fk(r1, ..., rt−1)λkt−1

, (22)

where p(j, i) is the Markov transition probability, d is the number of states(regimes), fj(r1, ..., rt) is the density (assumed to be Gaussian when MLEis calculated) of rt given all previous observations r1, ..., rt−1 and regime j.This procedure continues recursively. After obtaining the regime forecast,the volatility forecasting of the RS-GARCH model is similar to that of theGARCH model. Further details of the RS-GARCH model are presented inGray (1996), Francq et al. (2001), Xie and Yu (2005), and Xie (2007).

The data are the same S&P500 return series that we considered in Section3.2, with a total of 2780 samples. To perform the out-of-sample forecast, start-ing from a half sample (t = 1390, 29 June 1995), we estimate the parametersusing all previous observations and forecast 1 to 50 steps ahead using all thesemodels. After every 50 steps, we update the data, re-estimate the parametersand forecast again. The sample MSPEs with respect to the true volatilities(σ2,∗

t , see equation (23) below) for all forecasting horizons were summarizedas an evaluation criterion.

The definition of true volatility is a very important issue in volatility fore-casting. The standard way is to use the square of returns (or returns minusthe sample mean if it is not zero) as an approximation (see, inter alia, Poonand Granger, 2003, and Gokcan, 2000). Andersen and Bollerslev (1998), how-ever, argued that using this definition generally leads to a model with poorgoodness-of-fit, because this measurement typically displayed a large degreeof idiosyncratic, observation-by-observation variation (see particularly theirFigure 1). They instead proposed the use of cumulative squared intraday re-turns and showed strikingly improvements. Along the same line McMillanand Speight (2004) reassessed the performance of GARCH models and ar-gued that with this measure of true volatility GARCH models outperformedsmoothing and moving average models in a data set of 17 daily exchangerate series. However, the high frequency data are not always available to use

17

this technique. Alternatively, Starica (2003) and Starica and Granger (2005)used grouped realized volatility,

∑hi=1 r2

t+i as the true volatility for forecastsof (σ2

t+1+ ...+σ2t+h), in the hope that the averaging (summation) would cancel

out some of the idiosyncratic noise in the daily squared returns. It is worthynoting that Starica and Granger (2005) gave up the stationarity assumptionof whole S&P500 returns (from 1928 to 2000) and argued for the use of sta-tionary, or even independent, identically distributed (i.i.d) sequences to modelthe return series piece by piece. The i.i.d sequences have a mean of zero andconstant variances.

Recall that the S&P500 return series is non-stationary, but with a locallystationary structure. We propose that it is perhaps more appropriate to definethe true volatility as a local mean of squared observations over a symmetricinterval (t−m, t+m) around the observation at time t for some positive integerm, i.e.,

σ2,∗t =

12m + 1

m∑

i=−m

r2t+i − (

∑mi=−m rt+i

2m + 1)2. (23)

This volatility definition can also be applied to stationary processes. However,it is particularly suitable for processes with a locally stationary structure,smooth evolution of the variance or even variances which have a linear trend.Let l = 2m + 1 be the interval length. A fairly large l should be used forstationary processes, and a smaller one for non-stationary processes. In ourapplication, l = 5, 11, 19 and 31 were used, and their differences are discussedbelow.

The ratios of the MSPE of different GARCH models, over that of theLSW modeling with the s8 wavelet, are presented in Figure 1. The ratio forthe RS-GARCH model is not included in the figure to facilitate interpretationof the graph. The ratios are usually over five for most forecasting horizons.Perhaps surprisingly, in-sample estimation of the data shows only one regimeis visible. Regime prediction also always adheres to a single regime. In thiscase, the model is too complicated to produce an accurate forecast. Clearlyfrom Figure 1, the volatility forecasting for the LSW model is promising. Forwavelet s8 and l = 5, it gives more accurate forecasts than GARCH(1,1) formost of the forecast horizons. It is more than twice as good around h = 22.It is also competitive for small h’s, where it is well-known that GARCH cangive good forecasts. The GARCH model only outperforms it for long horizonsand large l. Recall that during the adaptive forecasting to obtain the nuisanceparameters p in (10) and bandwidth g for kernel smoothing, we move h stepsbackwards. In a sense, we use parameters obtained from quite ‘old’ informationfor the real forecasting when h is large. This seems to be justifiable since non-

18

stationary processes are unsuitable for long-term forecasts.For different lengths of the intervals defining the volatility, the compari-

son differs slightly. With respect to the size of the ratios, the performanceof GARCH(1,1) is, to some extent, improved with a ‘smoother’ volatilitydefinition (large l), while LSW modeling favors a ‘sharp’ definition, due tothe fact that wavelet methods are capable of preserving more detailed infor-mation about processes. The GARCH-t model behaves in a similar way tothe standard GARCH model, but is usually less accurate. In contrast, theEGARCH(1,1) model generally performs better than GARCH(1,1) except forl = 31. With respect to other wavelets, similar figures for c6, s12, d12 andc30 are also obtained (not reported here for sake of space). In general, shortwavelet (in the sense of filter length, e.g. c6, s8) preserves details of infor-mation and performs better with ‘sharp’ true volatility, while long wavelets(like s12) performs relatively better with ‘smooth’ volatility. The result forwavelet d12 was not so good, perhaps because it is a little too long and we usedthe symmetric volatility definition in (23) although wavelet d12 is very asym-metric. Over all, it is clear that wavelet c30 is not recommended for LSWforecasting for S&P500 data. It is too ‘long’ for a non-stationary process.Once again, wavelet s8, which is smooth, near-symmetric and has a supportof medium length, tends to be a good choice.

5 Discussions

As Nason et al. (2000) and Fryzlewicz (2005) pointed out, LSW modeling wasdeveloped not because it is superior to other approaches, but as an attractivealternative, because of properties such as its linearity, local stationary natureand the availability of estimation and forecasting methods. It is particularlyuseful when time and scale must be jointly considered and/or local patternsare of interest.

However, the first problem with LSW modeling for real data is choosing asuitable wavelet basis. Theoretical results concerning the estimation of EWSand local covariance (Nason et al., 2000) are all based on the assumptionthat we know the true wavelet basis. It is unclear what will happen if awrong wavelet is used. We conducted a sensitivity analysis of the selection ofdifferent wavelet bases, based on simulated data. The criterion was the meanof MSE (over 50 realizations) for estimating the EWS. For non-stationaryprocesses, the selection of wavelets was found to be not sensitive, and wavelets8 outperformed the others in most of cases. Together with its outstandingperformance in the volatility forecasting of S&P500 returns in Section 4, s8

19

0 10 20 30 40 50

0.51.0

1.52.0

2.5

MSPE

ratio

s (ov

er s8

)l = 5

0 10 20 30 40 50

1.01.5

2.02.5

MSPE

ratio

s (ov

er s8

)

l = 11

0 10 20 30 40 50

0.81.2

1.6

MSPE

ratio

s (ov

er s8

)

l = 19

0 10 20 30 40 50

0.81.0

1.21.4

MSPE

ratio

s (ov

er s8

)

l = 31

Figure 1: Ratios of GARCH(1,1) (solid), EGARCH(1,1) (dashed) and GARCH-t(1,1)(dotted) MSPEs divided by corresponding MSPE from the LSW modeling withs8 wavelet basis in forecasting S&P500 data, and unit line (dash-dotted), against theforecasting horizons. The lengths of intervals l, defining the true volatility (23), are5, 11, 19, and 31, respectively.

could be a good candidate for any non-stationary process with respect to LSWmodeling. For the stationary process, the wavelet selection is more sensitive.There is no dominating wavelet family and the true wavelet or a wavelet witha similar smoothness characteristic always provides the best fit. However, weobserved a ‘cutting’ property which indicates that for a wavelet whose filterlength is L, the covariance Cov(Xt,T , Xt+τ,T ) = 0 for τ > (2−J −1)(L−1)+1.For a stationary series, it may help to identify the right wavelet filter lengthby inspecting the sample autocorrelation and determining the minimum non-trivial scale.

We propose a new forecasting procedure with LSW processes in order toavoid the outliers that are produced by the original algorithm developed byFryzlewicz et al. (2003). We suggest that some restrictions should be imposedon the choice of predictor coefficients bt in (9) or (12) when minimizing theMSPE, while focusing on the unit-length algorithm with constraint b′tbt = 1.Applications to real data show that outliers in Fryzlewicz’s algorithm makethe evaluation of this algorithm very difficult. In contrast, the unit-length al-gorithm works consistently and outperforms the original in most of cases. Ofcourse, constraints other than (13) or (15) are other possible considerations.

20

Note also that instead of imposing constraints on bt to prevent outliers occur-ring, one can also try to use more robust minimizing and evaluation criteriainstead of the quadratic form.

Based on our unit-length algorithm, the volatility forecasting ability ofLSW modeling was investigated and compared with that of the GARCH mod-els. For the example of S&P500 return data, we introduced a new definitionof true volatility (23) by realizing the possible non-stationary nature of thissequence. Comparison using MSPE shows that the volatility forecasting withthe LSW model is promising. With wavelet s8 (among others), it outper-forms GARCH models for a large range of forecasting horizons. In general,a fair comparison is not easy to achieve because even for the same pair ofmodels, their relative merits can vary with differences in data frequency, datasize, forecast horizon, true volatility definition, evaluation criterion and otherfactors. The overall ranking from Poon and Granger (2003) suggests thatISD performs best, followed by HISVOL and GARCH models (which per-form roughly equally well, although other studies cited in their review cometo different conclusions regarding GARCH and HISVOL models). ISD entailsoption prices and is not generally available for other assets. From our experi-ment, it seems that LSW forecasting can be a valuable alternative, especiallyin a non-stationary or locally stationary situation. Note that other methodsfor non-stationary processes have also been developed. For instance, Staricaand Granger (2005) divided a long, non-stationary process into homogeneityintervals, to each of which a stationary, even i.i.d sequence was modeled. Theyalso showed the superiority of forecasting based on this methodology over theGARCH (1, 1) model. Fryzlewicz et al. (2006) used (inter alia) waveletshrinkage to study processes with stepwise variance. We conjecture that sucha technique could also be applied to stochastic processes with smoothly evolv-ing variances, such as LSW processes, and suggest that this possibility shouldbe explored in future work.

Acknowledgements

The authors thank the High-Performance Computing Center North (HPC2N)at Umea University, Sweden for providing computational assistance and Dr.Magnus Ekstrom for help with this. The computation in this paper is benefitedfrom the open S-plusr code (for the Haar wavelet only) accompanying thepaper by Fryzlewicz et al. (2003). The codes for other wavelets are availablefrom the authors upon request.

21

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