Dr. Md . Sabiruzzaman Department of Statistics, RU

1. Conditional Heteroscedasticity, External Events and Application of Wavelet Filters in Financial Time Series Dr. Md. Sabiruzzaman Department of Statistics, RU

2. Problem Statement • Financial time series possess some stylized facts like time-varying conditional variance (volatility) and constantunconditional 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) causestemporary (outlier) and permanentchanges (structural break) and the unconditional variance may not constant • Identification of break pointsis 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

3. Stylized Facts of Financial Time Series • 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) • ……

4. Permanent Change in Variance • 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)

5. 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 StephaneMallatin 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

6. 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.

7. k=1, 2, …., N where and is an estimator of the long run fourth moment, m4 Here is a proxy volatility measureandinputof the test The asymptotic distribution of is given by where is a Brownian Bridge is a standard Brownian motion Structural Break Detection CUSUM test for dependent process Kokoszka and Leipus (2000) give a consistent CUSUM statistic for detecting variance change in infinite ARCH process

8. 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

9. 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

10. Wavelet coefficients at level j associated with scale j Wavelet scaling coefficients represents low frequency components kth wavelet coefficients for different scales Structural Break Detection Alternative input for KL test based on MODWT For given any integer J01, the MODWT wavelet coefficients of {xt} form a (J0+1)N order matrix as

11. 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 Percival and Walden (2000) show that mean of is and variance As J0 , becomes much smoother and its variance tends to zero and thus for large J0

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

13. 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 kthmultiscale 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

14. Structural Break Detection Monte Carlo evidence: Power of the test 2000 simulations from GARCH(1,1) process with a single break (sample size=1024)

15. Structural Break Detection Monte Carlo evidence: Power of the test 2000 simulations of GARCH(1,1) process with two breaks (sample size=1024)

16. Structural Break Detection Monte Carlo evidence: Power of the test 2000 simulations of GARCH(1,1) process with two breaks (sample size=1024)

17. Structural Break Detection Monte Carlo evidence: Size of the test 2000 simulations from GARCH(1,1) process (sample size=1024)

18. Structural Break Detection Monte Carlo evidence: Spread of detection

19. Structural Break Detection Monte Carlo evidence: Spread of detection

20. 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. Winsorizationreplaces extreme points with cutoff values

21. Structural Break Detection Robustness • Standardized sensitivity curve (SV) (Maronna et al., 2006) • Influence function (IF) (Hampel, 1974)

22. 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

23. 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

24. Structural Break Detection Winsorized KL test Scaled-deviation-Winsorization (Wu and Zuo, 2008): (*) Winsorized KL test: (**)

25. Structural Break Detection Winsorized KL test • 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

26. Structural Break Detection Winsorized KL test • Standardized sensitivity curve (SV) • Influence Function (IF)

27. 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

28. Structural Break Detection Winsorized KL test The following theorem confirms that scaled-deviation-Winsorized KL test is consistent

29. 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

30. 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

31. 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

32. Structural Break Detection Illustration

33. 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)

34. Modeling Volatility Stochastic Breaks Switching ARCH (Cai, 1994) The latent variable Stis assumed to follow a first order Markov process with transition probabilities

35. 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

36. 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

37. Modeling Volatility Stochastic Breaks back back

38. 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

39. 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

40. Modeling Volatility Discrete vs Stochastic Breaks

41. Modeling Volatility Discrete vs Stochastic Breaks

42. Modeling Volatility Forecasting

43. Identification and verification of break points Fitting a proper break model and test significance of breaks Interested in studying breaks Breaks identified No break identified Not interested in studying breaks Selection of Sample period 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 Steps for Analyzing Volatility in Presence of Breaks

44. 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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