Dr. Md. Sabiruzzaman Department of Statistics, RU Conditional Heteroscedasticity, External Events...

Dr. Md. SabiruzzamanDepartment of Statistics, RU

Conditional Heteroscedasticity, External Events and Application of Wavelet Filters in Financial Time Series

Problem Statement Financial time series possess some stylized facts like time-varying

conditional variance (volatility) and constant unconditional variance, and can be modeled by GARCH family of equations

Analyzing volatility is prior consideration of asset pricing and risk management

External events (policies and crisis) causes temporary (outlier) and permanent changes (structural break) and the unconditional variance may not constant

Identification of break points is important to determine the effect of external events and for proper modeling and forecasting

Tests for structural breaks in volatility (Kokoszka and Leipus, 2000; Andreou and Ghysels, 2002; Sanso et al., 2004; de Pooter and van Dijk, 2004 …) ignore the effect of outliers

An stochastic regime-switching model is not suitable for modeling and forecasting volatility in the presence of structural changes since permanent changes are non-stochastic

Dependency

Volatility clustering (Manderbolt, 1963)

Time varying conditional variance (Engle,1982)

Constant unconditional variance (Engle,1982)

Persistence (Bollerslev, 1986)

Excess kurtosis (Baillie and Bollerslev,1989)

Asymmetry (Zakoian, 1990)

Long memory (Baillie et al., 1996)

……

Stylized Facts of Financial Time Series

By permanent change in variance of financial time series we mean change in the unconditional variance

When a process is observed for a long period of time, permanent changes in the variance may appear as the consequences of external events like natural calamities, economic crisis or policies taken by management (Diebold, 1986; Lamoreaux and Lastraps, 1990; Morana and Beltratti, 2004; Ezaguirre et al., 2004; Rapach and Strauss, 2008 and Karoglou, 2009 )

Ignorance of structural changes in variance can result as spurious IGARCH or long memory effect (Mikosch and Starica, 2004)

From an economic point of view, structural breaks in financial markets affect fundamental financial indicators (Bates, 2000)

Permanent Change in Variance

Wavelet Filters Wavelet transform expresses possibly continuous function in

term of discontinuous wavelets. By dilating (stretching) and translating (shifting) a wavelet, we can capture features that are local both in time and frequency

The main feature of wavelet analysis is the possibility to separate out a time series into its constituent multiresolution components. The main algorithm dates back to the work of Stephane Mallat in 1988/89

The discrete wavelet transform (DWT) uses orthogonal transformations to decompose a vector X of length n=2J into vectors of wavelet coefficients D1,D2, . . . ,DJ and A1,A2, . . . ,AJ , where each set of wavelet coefficients contains n/2j data points for j = 1, . . . , J

The approximation coefficients A1,A2, . . . ,AJ contain the low-frequency content, and the detail coefficients D1,D2, . . . ,DJ contain the high-frequency content

By using a small window one looks at high frequency components and by using large window one looks at low-frequency components

Economic Process

Economic and financial systems contain variables that operate on a variety of time scales simultaneously

This implies that the relationship between variables may be different across time scales

Simple example: Securities market contains many traders operating on different time scales Long view (years), concentrate on market fundamentals Short view (months), interested in temporary deviations from

long-term growth or seasonality Really short view (hours), interested in ephemeral changes in

market behavior The most important property that wavelets possess for the

analysis of economic data is the decomposition by time-scale. Different scale components represent different features of time series. Noise is high frequency components, whereas seasonality is low frequency component.

Structural Break Detection CUSUM test for dependent process

kk

UNk 2/1supˆ

Nkk CN

kCmU 2/1

4ˆ

k

ttk xC

1

2

is an estimator of the long run fourth moment, m4

Here is a proxy volatility measure and input of the test

where and

4m̂

The asymptotic distribution of is given by where is a Brownian Bridge is a standard Brownian motion

k̂ )(sup * rWr

)1()()(* rWrWrW )(rW

k=1, 2, …., N

Kokoszka and Leipus (2000) give a consistent CUSUM statistic for detecting variance change in infinite ARCH process

2tx

Structural Break Detection CUSUM test for dependent process

The Kokoska and Leipus (KL) test together with ICSS algorithm (Inclan & Tiao, 1994) can be used to detect multiple structural breaks in volatility

The simulation studies of Andreou and Ghysels (2002) and Sanso et al. (2004) extend the use of KL test in more general volatile condition and for detecting multiple breaks but the size distortion problem of KL test is noted

Structural Break Detection Alternative input for KL test based on MODWT

Since a variable is a container of information due to all types of variations: both short-scale and long-scale, square of a variable is, therefore, may be contaminated by some irrelevant components

We suggest a proxy measure of volatility based on maximal overlap discrete wavelet transform (MODWT) coefficients

MODWT is a modified version of DWT given by Percival and Walden (2000)

MODWT is highly redundant but translation invariant transformation MODWT is energy preserving and properly aligned with features of

original series Unlike DWT, dyadic sample size is not necessarily required for

MODWT Like DWT, analysis of variance (ANOVA) and multiresolution analysis

(MRA) is possible with MODWT


For given any integer J01, the MODWT wavelet coefficients of {xt} form a (J0+1)N order matrix as

kth wavelet coefficients for different scales

Wavelet coefficients at level j associated with scale j

Wavelet scaling coefficients represents low frequency components

1,,1,0,

1,,1,0,

1,,1,0,

1,2,21,20,2

1,1,11,10,1

2

1

0000

0000

0

0

~~~~

~~~~

~~~~

~~~~

~~~~

~

~

~

~

~

~

NJkJJJ

NJkJJJ

Njkjjj

Nk

Nk

J

J

j

VVVV

WWWW

WWWW

WWWW

WWWW

V

W

W

W

W

W


The variance of {Xt} can be expressed in terms of multilevel wavelet coefficients as

22

1

22

0

0 ~1~1ˆ XV

NW

N J

J

jjX

Percival and Walden (2000) show that mean of is and variance

0

~JV X

22

0

~1XV

N J

As J0 , becomes much smoother and its variance tends to zero and thus for large J0

0

~JV

0

1

22 ~1

ˆJ

jjX W

N


is the contribution to the energy of {Xt} due to

changes at scale j = 2j-1

The (j,k)th wavelet periodogram, (k = 0, 1, ……, N-1)

is the kth contribution to the energy of {Xt} due to

changes at scale j We define an estimator of average variation for different

scale at point k based on wavelet periodogram

1

0

22~ N

kjkj WW

2~jkW

2222

211

2

00

~~~~kJkJkkk WWWW 1

0

1

J

jjwith

As J0 increases we will be close to the total variation of the series at point k


Since the choice of J0 depends on length of the sample, we would like

to chose js such that as J0

Fryzlewicz et al. (2003) suggest j=1/2j (j=1, 2, …, J0)

This is a good choice because and the weight j=1/2j

is quite well matching with scale j

We call (k=0, 1, …., N-1) the kth multiscale wavelet periodogram

(MSWP) that measures the variation at point k and can be used as

input for KL test

The level 1 wavelet periodogram is a special case of MSWP

which has been used by many authors to detect outlier and to

measure the volatility as well

0

1

1J

jj

2~kW

21

~kW

12

1lim

0

0 1

J

jjJ

Structural Break Detection Monte Carlo evidence: Power of the test

2000 simulations from GARCH(1,1) process with a single break (sample size=1024)

Error Distribution

Persistence Variance Change Ratio

Empirical Power SV MSWP MSWP3 WP1

0.1 0.5 2 0.9990 0.9990 0.9990 0.9960

1.5 0.9300 0.8380 0.8360 0.7860

Normal 0.2 0.6 2.5 0.9580 0.9380 0.9350 0.9380

2 0.8980 0.8460 0.8420 0.8590

0.2 0.7 3.5 0.8435 0.8290 0.8290 0.8520

3 0.7925 0.7700 0.7680 0.7900

0.1 0.5 2 1.0000 0.9985 0.9985 0.9940

1.5 0.9010 0.8350 0.8350 0.7950

GED with 2 d.f.

0.2 0.6 2.5 0.9415 0.9270 0.9250 0.9170

2 0.8790 0.8350 0.8350 0.8360

0.2 0.7 3.5 0.8145 0.8010 0.8010 0.8100

3 0.7750 0.7570 0.7570 0.7535


2000 simulations of GARCH(1,1) process with two breaks (sample size=1024)

Error distribution Normal

Model No. of Shifts detected SV MSWP MSWP3 WP1400GARCH(0.5,0.1,0.5) + 300GARCH(2,0.1,0.5) + 324GARCH(1,0.1,0.5)

0 0.0125 0.0290 0.0285 0.03151 0.0240 0.0380 0.0395 0.05202 0.8805 0.8835 0.8840 0.86703 0.0760 0.0475 0.0465 0.04754 0.0065 0.0020 0.0015 0.00205 0.0050 0.0000 0.0000 0.0000

400GARCH(2,0.1,0.5) + 300GARCH(0.5,0.1,0.5) + 324GARCH(1,0.1,0.5)

0 0.0000 0.0005 0.0005 0.00001 0.0265 0.0380 0.0410 0.05902 0.8705 0.8935 0.8905 0.87703 0.0940 0.0655 0.0650 0.06054 0.0085 0.0650 0.0030 0.00355 0.0005 0.000 0.0000 0.0000

400GARCH(0.5,0.1,0.5) + 300GARCH(1,0.1,0.5) + 324GARCH(2,0.1,0.5)

0 0.0000 0.0010 0.0010 0.00051 0.0580 0.1175 0.1235 0.18202 0.8480 0.8260 0.8250 0.76003 0.0870 0.0540 0.0490 0.05604 0.0065 0.0015 0.0015 0.00155 0.0005 0.0000 0.0000 0.0000

400GARCH(2,0.1,0.5) + 300GARCH(1,0.1,0.5) + 324GARCH(0.5,0.1,0.5)

0 0.0010 0.0020 0.0035 0.00351 0.0495 0.1015 0.1085 0.16502 0.8655 0.8475 0.8420 0.77653 0.0715 0.0470 0.0440 0.05204 0.0110 0.0020 0.0020 0.00255 0.0015 0.0000 0.0000 0.0005


2000 simulations of GARCH(1,1) process with two breaks (sample size=1024) Error distribution GED with 2 d.f.Model No. of Shifts detected SV MSWP MSWP3 WP1400GARCH(0.5,0.1,0.5) + 300GARCH(2,0.1,0.5) + 324GARCH(1,0.1,0.5)

0 0.0120 0.0210 0.0205 0.01251 0.0245 0.0720 0.0725 0.02402 0.8650 0.8600 0.8600 0.86403 0.0865 0.0350 0.0360 0.08604 0.0070 0.0065 0.0060 0.00855 0.0050 0.0055 0.0050 0.0050

400GARCH(2,0.1,0.5) + 300GARCH(0.5,0.1,0.5) + 324GARCH(1,0.1,0.5)

0 0.0020 0.0155 0.0145 0.00601 0.0325 0.0760 0.0760 0.02902 0.8620 0.8610 0.8610 0.86103 0.0915 0.0365 0.0360 0.09104 0.0070 0.0070 0.0090 0.00855 0.0050 0.0040 0.0035 0.0045

400GARCH(0.5,0.1,0.5) + 300GARCH(1,0.1,0.5) + 324GARCH(2,0.1,0.5)

0 0.0060 0.0140 0.0140 0.00801 0.0330 0.0820 0.0825 0.03602 0.8530 0.8520 0.8510 0.85203 0.0930 0.0460 0.0435 0.09104 0.0120 0.0060 0.0090 0.01305 0.0040 0.0000 0.0000 0.0000

400GARCH(2,0.1,0.5) + 300GARCH(1,0.1,0.5) + 324GARCH(0.5,0.1,0.5)

0 0.0040 0.0000 0.0025 0.00001 0.0365 0.0860 0.0875 0.05102 0.8360 0.8350 0.8350 0.83503 0.1055 0.0570 0.0570 0.09704 0.0180 0.0190 0.0140 0.01705 0.0000 0.0030 0.0040 0.0000

Structural Break Detection Monte Carlo evidence: Size of the test

2000 simulations from GARCH(1,1) process (sample size=1024)

Error Distribution

Persistence Constant Empirical Size SV MSWP MSWP3 WP1

0.1 0.5 1.0 0.0850 0.0340 0.0350 0.0420

0.5 0.0870 0.0380 0.0360 0.0430

Normal 0.2 0.6 1.0 0.0830 0.0400 0.0390 0.0850

0.5 0.0960 0.0540 0.0510 0.0840

0.2 0.7 1.0 0.0935 0.0465 0.0445 0.0580

0.5 0.0915 0.0390 0.0380 0.0520

0.1 0.5 1.0 0.0810 0.0470 0.0460 0.0550

0.5 0.0790 0.0510 0.0510 0.0540

GED with 2 d.f.

0.2 0.6 1.0 0.0820 0.0450 0.0437 0.0466

0.5 0.0850 0.0490 0.0490 0.0560

0.2 0.7 1.0 0.0920 0.0410 0.0410 0.0520

0.5 0.0960 0.0395 0.0310 0.0410

Structural Break Detection Monte Carlo evidence: Spread of detection

Structural Break Detection Robustness

KL test is robust to short lived jumps or outliers of moderate size (Andreou and Ghysels, 2002)

Rodrigues and Rubia (2011) show that in presence of bounded outliers KL test is consistent and the asymptotic distribution is invariant

We investigate the extent of robustness of KL test and found that it is heavily sensitive to outliers of large size

It’s a crucial decision which one is to address first- the breaks or the outliers when both are present, because presence of outliers may influence break detection and vice versa

We propose KL test for detecting breaks controlling the effect of outliers by Winsorization. Winsorization replaces extreme points with cutoff values

Standardized sensitivity curve (SV) (Maronna et al., 2006)

Influence function (IF) (Hampel, 1974)

200

11))(( x

N

lxSSC l

ll xN

lFSxIF

2

00 1),;(

dFxN

lIxFS tlttll

22)(

-5 -4 -3 -2 -1 0 1 2 3 4 50

5

10

15

20 SC of KL test

outlier

sen

sit

ivit

y


1000 observations from a GARCH(1,1) process with a single break is simulated and the break detected by KL test is assumed to be the true break point. An outlier is then set before (after) the break, and KL test is applied to the data containing outlier each time. Frequencies of correct detection in 1000 replications for different positions and magnitudes of outliers are reported

Outlier location Persistence Outlier size

3MAD 4MAD 5MAD 6MAD 7MAD

Before volatility increase

0.1 0.5 950 917 901 840 788

0.2 0.6 945 914 844 809 736

0.2 0.7 958 920 891 854 792

After volatility increase 0.1 0.5 946 913 899 836 817

0.2 0.6 948 899 845 805 755

0.2 0.7 950 899 853 802 758


1000 observations from a GARCH(1,1) process with a single break is simulated and the break detected by KL test is assumed to be the true break point. A pair of outliers is then set before (after) the break, and KL test is applied to the data containing outlier each time. Frequencies of correct detection in 1000 replications for different positions and magnitudes of outliers are reported

Outlier location Persistence Outlier size

3MAD 4MAD 5MAD 6MAD 7MAD


0.1 0.5 917 867 788 512 259

0.2 0.6 897 821 727 564 377

0.2 0.7 919 866 802 735 649

After volatility increase 0.1 0.5 924 881 818 718 641

0.2 0.6 893 835 757 680 628

0.2 0.7 894 833 761 691 674


Structural Break Detection Winsorized KL test

Scaled-deviation-Winsorization (Wu and Zuo, 2008):

(*)

Winsorized KL test:

(**)

)()()}({ )( UxLxxDtWt tttIUILIxx

~

~)(

t

t

xxD ~~ L ~~ U

Wll

W UNl 2/1supˆ

WlWWNWkWWl SmCN

lCmU 2/1

42/1

4 ˆˆ

WNWlWl CN

lCS

l

tWtWl NlxC

1

2 ...,2,1,

The advantage of scaled-deviation-Winsorization is that by choosing η appropriately it is possible to construct an interval that includes all the good observations. Mathematically, for a small positive quantity ε, we can chose η such that

The chance of mistreating in scaled-deviation-Winsorization technique is low and it can produce the same set of observations as in the original data if no observation is too far from the centre

The fraction of Winsorized points is not fixed but data-dependent

1}~~~~Pr{ tr


Standardized sensitivity curve (SV)

Influence Function (IF)

AxN

lFSxIF lWWl

2

00 1),;(

U U

lt

L L

lt dFUN

dFIUdFLN

dFILA 2222 11

UxLxUxLWWl IUILIxN

lx

N

lxSSC

000

2220

200 1

11

1

11))((

-5 -4 -3 -2 -1 0 1 2 3 4 50

2

4

6

8

10 SC of Winsorized KL test

outliersen

sit

ivit

y


The following Theorem confirms that if outliers are bounded through scaled-deviation-Winsorization, the distribution of KL test is invariant


The following theorem confirms that scaled-deviation-Winsorized KL test is consistent


1000 observations from a GARCH(1,1) process with a single break is simulated and the break detected by KL test is assumed to be the true break point. A pair of outliers (6MAD) is then set before (after) the break, and KL test is again applied to Winsorized data each time. Frequencies of correct detection in 10000 replications for different location of outliers are reported

Outlier location Persistence Cutoff Value Ordinary Scaled-deviation

99% Quantile

η=3 η=4 η=5

No outlier 0.1 0.5 7933 8871 9887 99900.2 0.6 6854 7642 9345 96810.2 0.7 6981 7462 9018 9626


0.1 0.5 8001 9017 8626 78420.2 0.6 7215 7881 8342 72710.2 0.7 7444 7643 8778 7971

After volatility increase

0.1 0.5 8191 8743 8855 82000.2 0.6 6928 7616 8107 76880.2 0.7 6890 6990 8016 7493


Structural Break Detection Illustration

S&P500 daily index and return series over the period January 02, 1980 to September 10, 2010. The vertical lines in the lower panel indicate breaks in long-run variance. Shift 1(17/05/1991): Increase in the index of leading economic indicators, Shift 2(20/07/1997): Asian market crisis, Shift 3(29/04/2003): Iraq invasion and drop in oil prices, Shift 4(23/07/2007): Weakening US housing market


KLSE daily index and return series over the period January 1, 1998 to December 31, 2008. The vertical lines in the lower panel indicate breaks in long-run variance. Shift 1(19/08/1999): Recovery of stock market and real economy, Shift 2(11/10/2001): Tax exemption to encourage investment, Shift 3(17/06/2004): Booming Asian economy, Shift 4(23/02/2007): High global liquidity and increase in fuel price


Index SV MSWP MSWP3 WP1

S&P500May 16, 1991 May 17, 1991 May 17, 1991 May 16, 1991

Mar 26, 1997 Jul 20, 1997 Jul 20, 1997 Jul 07, 1997

Apr 28, 2003 Mar 29, 2003 Mar 29, 2003 Mar 28, 2003

Jul 23, 2007 Jul 23, 2007 Jul 23, 2007 Jul 19, 2007

KLSEAug 13, 1999 Aug 19, 1999 Aug 19, 1999 Aug 18, 1999

Sep 21, 2001 Oct 11, 2001 Oct 11, 2001 Nov 08, 2001

Jul 06, 2004 Jun 17, 2004 Jun 17, 2004 -

Feb 26, 2007 Feb 23, 2007 Feb23, 2007 Feb 20, 2007

Modeling Volatility Discrete Breaks

GARCH(1,1) model with k dummies (Lamoureux and Lastrapes, 1990)

where zt N(0,1) and Laumoreux and Lastrapes (1990) only allow the drift to be

changed over periods. A straightforward extension can be adding dummies for other parameters as well (see Kim et al., 2010, for example)

To determine the timing of structural breaks, there are model free methods available for detecting break points in volatile series (see Kokoszka and Leipus, 2000, for instance)

ttt hz

1111 ttktktt vhDDh

12

1 ttt hv

Modeling Volatility Stochastic Breaks

Switching ARCH (Cai, 1994)

The latent variable St is assumed to follow a first order Markov process with transition probabilities

ttt hu

ttt zSccr 10

tktktt zbzbz 11

)1,0(~.. Niiut

0,0)( 10110 ttt SSS

g

iititt Sh

1

21 0,)(

qSSpqSSp

pSSppSSp

tttt

tttt

1]1|0[]1|1[

1]0|1[]0|0[

11

11


SWARCH model (Hamilton and Susmel, 1994)

While Cai (1994) allows the drift of the ARCH process to be varied depending on the latent variable St, Hamilton and Susmel (1994) propose for multiplying by different constant as St indicates changes of regime

To avoid the difficulties of estimation due to path dependence Cai (1994) and Hamilton and Susmel (1994) confined themselves to ARCH models

tSt t ~ ttt hu~ )1,0(~.. Niiut

g

iitith

1

20

~


Gray (1996) propose that the conditional variance of εt-

1, given information at t-2, can be calculated by

where is the variance of εt-1, given St-1 = j. Each regime variance is

Klaassen (2002) proposes the use of pt-1(St-1 = j), j = 1, …, k, instead of pt-2(St-1 = j), that is, use of information up to time t-1 instead of t-2

k

jjtttt hjSph

11121 )(

kjhh tjtjjjt ,....,1,112

110

1jth


back

back


Hass et al. (2004) noted that these models is analytically intractable and, therefore, conditions for covariance stationarity have yet not established. They propose for

{St} is a Markov chain with a transition matrix

P=[pij]=[P(St=j|St-1=i], i, j = 1, …, k

The variance equation is defined as

where

)2(1

2110

)2( ttt

]...,,,[ 222

21

)2( ktttt

)...,,,( 21 kdiag

0,,0 10

tStt t ,

;1,0,]...,,,[ 21 iikiii

Modeling Volatility Discrete vs Stochastic Breaks

Nguyen and Bellahah (2008) find that emerging market experienced multiple breaks in volatility dynamics. The shifts in volatility are associated with events closely related to stock market liberalization, market expansions, and some major economic and political events

Sensier and van Dijk (2004) analyze 214 US macroeconomic time series and find most of the series had an experience of a break within the study period leading to a conclusion that increased stability of economic fluctuations is a widespread phenomenon. They noted that reduced output volatility is primarily accounted for by a reduction in the variance of exogenous shocks hitting the economy. They demonstrate that volatility changes are more appropriately characterized as instantaneous breaks rather than as gradual changes

McConnel and Quiros (2000) documented a structural break in the volatility of US GDP growth. They argue that changes in inventory management techniques have served to stabilize output fluctuations. They also illustrate that widely used regime switching framework is no longer a useful characterization of business cycle movement; the GDP growth is better characterized by a process with structural break in the variance


250 500 750 1000 1250 1500 1750 20000.00

0.25

0.50

0.75

1.00


200 400 600 800 1000 12000.0

0.2

0.4

0.6

0.8

1.0

Modeling Volatility Forecasting

Standard GARCH(1,1) GARCH(1,1) with dummy

In-sample Out-of-sample In-sample Out-of-sample

S&P500 Sample period Apr 29, 2003 – Nov 30, 2006 Jan 05, 1998 – Nov 30, 2006

RMSE 0.4253 0.1766 0.5008 0.1772

MAE 0.3005 0.1466 0.3653 0.1416

KLSE Sample period Jun 17 2004 – Oct 23, 2006 Oct 10 2001 – Oct 23, 2006

RMSE 0.6865 0.5390 0.9215 0.5423

MAE 0.5688 0.4515 0.7499 0.4546

Steps for Analyzing Volatility in Presence of Breaks

Identification and verification of break points

Verification of Normality, Asymmetry, Memory property and Persistence

Segmentation of sample period for in-sample and out-of-sample forecast evaluation

Selection of appropriate model based on both in-sample and out-of-sample performance

Fitting final model and making inference

Selection of Sample period

No break identified

Breaks identified

Not interested in studying breaks

Interested in studying breaks Fitting a proper break model and

test significance of breaks

Conclusion Wavelet filtering facilitate us to decompose a time series

variable into its component variations at different time scales and thus, provides an deep insight about the features of the process; use of proposed wavelet-based input improves the size property for the KL test with an assurance that the power is still reasonable

Proposed scaled-deviation-Winsorized KL is robust to detect structural breaks in variance

Stochastic switching volatility models may overlook the structural changes and so, should be utilized with care

Inclusion of observations from the period before a break do not improve forecasting for post break period

Break detection is prior to volatility modeling

Investigate on the properties of MSWP as a volatility measure, like it’s asymptotic distribution, is an interesting subject of future studies

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