Dr. Md. Sabiruzzaman Department of Statistics, RU Conditional Heteroscedasticity, External Events and Application of Wavelet Filters in Financial Time Series
Mar 29, 2015
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
Structural Break Detection Alternative input for KL test based on 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
Structural Break Detection Alternative input for KL test based on MODWT
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
Structural Break Detection Alternative input for KL test based on MODWT
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
Structural Break Detection Alternative input for KL test based on MODWT
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
Structural Break Detection Monte Carlo evidence: Power of the test
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
Structural Break Detection Monte Carlo evidence: Power of the test
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 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
Structural Break Detection Robustness
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
Structural Break Detection Robustness
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
Before volatility increase
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 Robustness
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
Structural Break Detection Winsorized KL test
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
Structural Break Detection Winsorized KL test
The following Theorem confirms that if outliers are bounded through scaled-deviation-Winsorization, the distribution of KL test is invariant
Structural Break Detection Winsorized KL test
The following theorem confirms that scaled-deviation-Winsorized KL test is consistent
Structural Break Detection Winsorized KL test
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
Before volatility increase
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 Winsorized KL test
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
Structural Break Detection Illustration
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
Structural Break Detection Illustration
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
Modeling Volatility Stochastic Breaks
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
~
Modeling Volatility Stochastic Breaks
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
Modeling Volatility Stochastic Breaks
back
back
Modeling Volatility Stochastic Breaks
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
Modeling Volatility Discrete vs Stochastic Breaks
250 500 750 1000 1250 1500 1750 20000.00
0.25
0.50
0.75
1.00
Modeling Volatility Discrete vs Stochastic Breaks
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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