A Wavelet Regime Switching Model

Transcription

1 A Wavelet Regime Switching Model Presentation for Research Seminar with some extended material Prof. Dr. Florian Weigert Prof. Dr. Karl Frauendorfer Christian Vial St. Gallen October 10, 2017

2 Syllabus Introduction Model Results Conclusion Discussion 2

3 Introduction Research Focus 3

4 Research Focus Research Question Does a regime switching analysis based on wavelet scales provide further insights into asset price dynamics? Asset Price Processes High-frequency trader Day trader Regime 1 Asset price as a superposition of scales Pension funds Regime 2 Regime 1 4

5 Model Wavelet Regime Switching Model 5

6 Wavelet Function What Are Wavelets? 1 Morlet Wavelet Small waves Specified number of oscillations Limited life-span Scalable and moveable window Shifted over time series Properties of Wavelets Local in time and frequency domain Modeling of single deflections, abrupt changes, and finiteness of signal Orthonormal, non-orthonormal, continuous or discrete Energy conservation in time-scale space 6

7 Wavelet Definition Definition The integral of the wavelet function ψ t is න ψ t dt = 0. The wavelet is normalized such that, න ψ t 2 dt = 1. Admissibility condition ensures inverse transformation of wavelet transform Regularity condition imposes wavelet to be local in time and frequency Hence, a wavelet is a band-pass filter Admissibility Condition Let ψ (, ) be a real-valued function satisfying the admissibility condition Ψ ω 2 C Ψ = න dω <, ω where Ψ(ω) is the Fourier transform of the basic wavelet ψ(t) given by Ψ ω = න ψ t e jωt dt with exp( jωt) being the phase factor. Hence, Ψ 0 = න ψ t dt = 0. 7

8 How does it work? Morlet Wavelet Morlet Wavelet Wavelet Coefficients d j (k) x t ψj,k t dt = x, ψ j,k where j, k N + and the wavelet function is reformulated as ψ j,k t = λ 0 j/2 ψ λ0 j t kτ 0 Scaling: Dilating or compressing Translation: Advancing or delaying 8

9 Time and Frequency Spectrum Time domain Frequency domain ψ 2,τ ψ 2,τ ω ψ 4,τ ψ 4,τ ω ψ 8,τ ψ 8,τ ω Paradox of Achilles and the Tortoise Spectrum covered by scaling function 9

10 DWT Decomposition 10

11 What Is a Regime Switching Model? Unimodal - X~N(3,5), Y~N(3,1), and X =0.25 X~N(3,5) Y~N(3,1) Mixed dist. (X,Y) - Modeling fat-tailed distributions with skewness and kurtosis - Modeling non-normality Unimodality or bimodality - Useful application in many areas of research (e.g. risk management) 11

12 Multivariate Regime-Switching Model p R t+1 = μ st+1 + j=1 1Τ A j,st+1 R t+1 j + Σ 2 st+1 ε t+1 ε t+1 ~IID N 0, I N P = p 11 p 1,K p K,1 p K,K where R t+1 μ st+1 A j,st+1 1Τ2 Σ st+1 p ij s t+1 p j=1 Return vector with N assets Regime-dependent intercept Regime-dependent autoregressive N N matrices with lags j = 1,, p Regime-dependent Cholesky factorization of the variance covariance matrix Σ st+1 with Σ st+1 1Τ2 = Σ st+1 Τ Σ st = Cov[R t+1 F t, s t+1 ] Transition probability Pr{s t+1 = j s t = i} Index for the state with s t+1 = 1,, K 12

13 Graphical Representation of RS Model Return vector Mean vector VAR matrix Lagged return vector Cholesky factorized covariance matrix 1 r t+1 N r t+1 N 1 = 1 μ 1 st+1 N μ 1 st+1 1 μ 2 st+1 N μ 2 st+1 + φ 1 11,st+1 φ 1 1N,st+1 φ 1 N1,st+1 φ 1 NN,st+1 φ 2 11,st+1 φ 2 1N,st+1 φ 2 N1,st+1 φ 2 NN,st+1 r t 1 r t N N 1 + σ 1 11,st+1 σ 1 1N,st+1 σ 1 N1,st+1 σ 1 NN,st+1 σ 2 11,st+1 σ 2 1N,st+1 σ 2 N1,st+1 σ 2 NN,st ε N 1 N N N N Pr s t = j s t 1 = i 13

14 Model Specification 14

15 Main Results 15

16 Data & Model Characteristics Data Characteristics Datastream Equity Indices Regions: North America Europe ex-uk UK Pacific ex-japan Japan Sample period: Total Return Expressed in USD and thus no currency hedging Data source: Datastream Model Characteristics Wavelet Wavelet method: dwt Wavelet function: Daubechies Filter length: 8 Scales: 1 6 Boundary condition: reflection Regime Switching Model: States: 2 Heteroscedastic Non-switching mean No autoregressive component 16

17 Descriptive Statistics Table 1 Descriptive Statistics Table 1 shows the descriptive statistics for international daily equity returns. The period under study is from July 31, 1973 to September 25, 2017 using Datastream North America, Europe ex. UK, UK, Pacific ex. Japan, and Japan total return market indices. Values are reported using continuous returns. The *, **, and *** indicate significant values at the 10%, 5%, and 1% level of confidence, respectively. N. America Europe ex. UK UK Pacific ex. Japan Japan Mean (Ann.) 9.7% 9.5% 9.8% 9.2% 6.9% Sample Std (Ann.) 16.3% 16.7% 19.8% 17.9% 20.0% Sharpe Ratio Median (Ann.) 10.9% 13.7% 11.7% 16.4% 5.9% Minimum -20.0% -10.4% -14.5% -16.6% -17.4% Maximum 10.2% 9.1% 11.8% 10.4% 11.5% Skewness Kurtosis Jarque-Bera 201'061.0*** 28'594.6*** 34'700.9*** 71'964.0*** 30'606.8*** Num. of Obs

18 Variance Contribution by Scale 18

19 Univariate Scale-Based Regimes 19

20 Correlation Table 2 Correlations Table 2 shows the unconditional sample correlation for daily Datastream North America, Europe ex. UK, UK, Pacific ex. Japan, and Japan total return market index returns from July 31, 1973 to September 25, N. America Europe ex. UK UK Pacific ex. Japan Japan N. America 1.00 Europe ex. UK UK Pacific ex. Japan Japan

21 Multivariate RS Model by Scale 21

22 Multivariate RS Model Estimation (1) Table 5 Multivariate Markov Regime-Switching Model MSH(2,0) for Original Series Table 5 shows results for the multivariate Markov regime-switching model with heteroskedasticity. The period under study is from July 31, 1973 to September 25, Correlation/ Volatilities N. America Eur. ex. UK UK Pacific ex. JP JP Transition Probabilities Regime 1 Regime 1 Regime 2 N. America Eur. ex. UK Regime UK Pacific ex. JP Duration Periods JP Regime 2 Regime Regime 1 Regime Statistics N. America Log-Likelihood 189'645 Eur. ex. UK AIC -379'215 UK BIC -379'112 Pacific ex. JP HQ 1'557 JP Number of Parameters 37 Number of Observations 57'600 Saturation-Ratio

23 Multivariate RS Model Estimation (2) Table 6 Multivariate Markov Regime-Switching Model MSH(2,0) for Scale 1 Table 6 shows results for the multivariate Markov regime-switching model with heteroskedasticity. The period under study is from July 31, 1973 to September 25, Correlation/ Volatilities N. America Eur. ex. UK UK Pacific ex. JP JP Transition Probabilities Regime 1 Regime 1 Regime 2 N. America Eur. ex. UK Regime UK Pacific ex. JP Duration Periods JP Regime 2 Regime Regime Regime Statistics N. America Log-Likelihood 95'190 Eur. ex. UK AIC -190'306 UK BIC -190'207 Pacific ex. JP HQ 778 JP Number of Parameters 37 Number of Observations 28'800 Saturation-Ratio

24 Multivariate RS Model Estimation (3) Table 7 Multivariate Markov Regime-Switching Model MSH(2,0) for Scale 4 Table 7 shows results for the multivariate Markov regime-switching model with heteroskedasticity. The period under study is from July 31, 1973 to September 25, Correlation/ Volatilities N. America Eur. ex. UK UK Pacific ex. JP JP Transition Probabilities Regime 1 Regime 1 Regime 2 N. America Eur. ex. UK Regime UK Pacific ex. JP Duration Periods JP Regime 2 Regime Regime Regime Statistics N. America Log-Likelihood 11'924 Eur. ex. UK AIC -23'774 UK BIC -23'693 Pacific ex. JP HQ 97 JP Number of Parameters 37 Number of Observations 3'600 Saturation-Ratio

25 Scale Correlation 25

26 Conclusion 26

27 Conclusion Scale processes are characterized by local clusters. Most of the information i.e. energy of equity indices is contained in the lower scales. Regime switching models are capable of capturing scale specific regime processes. However, they show a high degree of persistence among scales. Correlations increase and converge for higher scales indicating that longterm economic trends become more relevant. However, the variance contribution of higher scales is relatively low. Regimes are relatively stable Scale 1: 22 Periods x 2-4 days interval = days (bull state); days (bear state) Scale 4: 2 4 years (bull state); years (bear state) Wavelet decomposition extracts features of short-term correlations and shows that they differ from long-term correlations. 27

28 Outlook Prediction model using transition matrix and synthesis of wavelet variances/covariances. Testing these predictions against other prediction models and random walk. Use higher frequency data as for example 20-min bars which show more seasonality. Usage of procedure in other academic disciplines which show clustering: Weather forecast Electricity load prediction Usage of regime switching model together with neural network. 28

29 Discussion 29

30 Appendix 30

31 Definition of Transition Probability Matrix (1) Discrete Markov Chain Finite number of regimes Only preceding state matters: Pr s t = j s t 1 τ τ=1, R t 1 τ τ=1 = Pr s t = j s t 1 = i = p ij 0,1 Time-Homogeneity P is a constant, time-independent matrix of transition probabilities. Irreducibility Let π j = Pr s t = j for j = 1,2,, N The steady-state (unconditional) probabilities π > 0, i.e. there is no absorbing state 31

32 Definition of Transition Probability Matrix (2) Ergodicity (Unconditional State Probabilities) Existence of stationary K 1 vector of steady-state probabilities π with π = P π s.t. P1 = 1. Possible to go from every state to every state (not necessarily in one step) Long-run forecast of ergodic Markov chains is independent of the current state 1 Average long-run time of occupation, i.e. lim σ T T T t=1 I st =j = π e j j = 1,, K A Markov chain may be irreducible, i.e. there is only one eigenvalue equal to unity, but not ergodic (more than one eigenvalue on the unit circle). Example: P = The eigenvalues λ 1 = 1 and λ 2 = 1 are both on the unit circle and thus the matrix P m does not converge to a fixed limit, i.e. steady-state probabilities No tendency to converge as m This leads to the definition of periodicity 32

33 Maximum Likelihood (1) Unconditional Probability State s t is generated by a distribution with unconditional probability Conditional Density The density of R t conditional on the random variable s t π j = Pr s t = j; θ f(r t s t = j; θ) for j = 1,2,, N where θ is the parameter vector Pr A B = Pr A B P B where θ is the parameter vector Joint Probability Density The joint probability density function of R t and s t is then given by p R t, s t = j; θ = f R t s t = j; θ Pr{s t = j; θ} 33

34 Maximum Likelihood (2) Problem Unconditional probability π cannot be observed Even if θ is known, we will not know with certainty which regime the process was in at a particular date Solution We can infer the value of s t based on the observations of R t given the population parameters θ We therefore introduce the conditional probability that the t-th observation was in regime j Pr{s t = j F t ; θ} for j = 1,2,, N The conditional probabilities are collected in a (N 1) vector, ξ t t where the j-th element equals Pr{s t = j F t ; θ} Similarly, ξ t t 1 represents the vector of conditional probabilities where the j-th element equals Pr s t = j F t 1 ; θ t 1 and F t 1 is at least the sigma-algebra generated by the returns series, r i i=1 34

35 Maximum Likelihood (3) Let us collect the densities f R t s t = j, F t 1 ; θ in a vector η and assume multivariate normality for each regime j, 1 η j,t = f R t s t = j, F t 1 ; θ = 2π N 1Τ2 exp 1 Σ 2 R T t μ 1 j Σj R t μ j for j = 1,2, N j Pr A B The conditional joint density of R t and s t is given by, p R t, s t = j F t 1 ; θ = Pr{s t = j F t 1 ; θ} f(r t s t = j, F t 1 ; θ) Pr{A B} = Pr{B} Pr{A B} or in vector notation ( መξ t t 1 η t ) where denotes the element-by-element (Hadamard) product The marginal density of R t is then given by the sum of these joint densities over the possible regimes, f(r t F t 1 ; θ) = 1 ( መξ t t 1 η t ) Pr{A}= Pr{A B} {B} Applying Bayes law now allows us to derive the conditional distribution of s t, መξ t t = Pr{S t R t, F t 1 } = Pr s t F t ; θ = ( መξ t t 1 η t ) 1 ( መξ t t 1 η t ) t because F t 1 F t and F t is the sigma-algebra at least generated by the return series r i i=1 Bayes Law Pr B A Pr A B Pr{B} = P A 35

36 Maximum Likelihood (4) Iteration Given the initial state probabilities መξ 1 0 and the parameter set θ the following formulas can be iterated over t = 1,2,, T መξ t t = ( መξ t t 1 η t ) 1 ( መξ t t 1 η t ) መξ t+1 t = P መξ t t Optimization The ML sums probability densities over the space መξ 1 1 መξ 2 2 መξ T T for K T possible evolutions where P denotes the transition probability matrix. If we optimize the log-likelihood function where T L θ = log f(r t F t 1 ; θ) t f R t F t 1 ; θ = 1 ( መξ t t 1 η t ) (unconditional density w.r.t. s t ) we can derive parameter estimates θ. Conditions P1 = 1 መξ t t 1 = 1 p ij 0 መξ t t 0 Σ K e k is positive (semi-) definite k = 1,2,, K 36

37 Implied Conditional Moments (1) Expected mean E t R t+1 = = s t+1 E t R t+1 s t+1 Pr{s t+1 F t } s t+1 E t R t+1 s t+1 Pr s t+1 s t, F t Pr s t F t = ξ t t PμƸ The τ period ahead forecast is then given by since E t R t+τ = ξ t t P τ μƹ መξ t+τ t = መξ t t P τ 37

38 Implied Conditional Moments (2) Expected variance Var t R t+τ = s t+1 E t R t+τ E t R t+τ 2 s t+τ Pr{s t+1 F t } where = ξ t t P τ ψ since and ψ = μ 1 E t R t+τ 2 + σ 1 2 μ K E t R t+τ 2 + σ K 2 መξ t+τ t = መξ t t P τ R t = μ + Σ 1Τ2 ε with ε~n(0, I N ) 38

39 DWT vs. Maximal-Overlap DWT (MODWT) Drawbacks of Discrete Wavelet Transform Dyadic length requirement Non-shift invariant Sensitivity due to down-sampling Maximal-Overlap Discrete Wavelet Transform (MODWT) (Percival & Walden, 2000) Zero-phasing property Non-dyadic treatment of sample sizes Complete time resolution Asymptotically more efficient wavelet variance estimator But Highly redundant Non-orthogonal 39

40 Discrete Wavelet Transform Pyramid Algorithm s 0 Low-Pass Filter 2 s 1 Low-Pass Filter 2 s 2 High-Pass Filter 2 d 1 High-Pass Filter 2 d 2 IDWT (MRA) Algorithm s 2 2 Low-Pass Filter* Low-Pass + s Filter* s 0 d 2 2 High-Pass Filter* d 1 2 High-Pass Filter* Pioneered by Mallat (1989) 40

41 MODWT Pyramid Algorithm s 0 Low-Pass Filter 2 2 s 1 Low-Pass Filter 2 2 s 2 High-Pass Filter 2 2 d 1 High-Pass Filter 2 2 d 2 IDWT (MRA) Algorithm s Low-Pass Filter* Low-Pass + s Filter* 2 s 0 d High-Pass Filter* d High-Pass Filter* Pioneered by Mallat (1989) 41

42 Scaling and Translation Property Mother wavelet Wavelet function ψ λ,τ = 1 λ ψ t τ λ Father wavelet Scaling function φ λ,τ = 1 λ φ t τ λ where λ and τ are the scale and translation variable, respectively. Frequency Spectrum Ψ λ,τ ω = 1 λ ψ t λ λ = λψ λω e jωτ e jωt dt Wavelet transform filter: λψ λω Impulse response function: 1 λ ψ t λ A dilation t/τ (i.e. λ > 1) of the wavelet function in the time domain is equivalent to a contraction by λω and a translation by λ of its Fourier transform in the frequency domain 42

43 Energy & Additive Decomposition of MRA Energy Decomposition Additive Decomposition J x 2 = j=1 d j 2 + sj 2 x = D j + S J j=1 D j := wavelet detail, S J :=wavelet smooth J The equality J x 2 = d j 2 + sj 2 = j=1 J D j 2 + SJ 2 j=1 does no longer hold for the MODWT as it is no orthonormal system. 43

44 Multiresolution Decomposition & Analysis Multiresolution Decomposition x t = s J (k)φ J,k (t) + d j k ψ j,k (t) k j=1 k where x is the process under analysis and J the maximum scale. J Multiresolution Analysis s m = s J,k φ J,k (t) + d J,k ψ J,k (t) + d J 1,k ψ J 1,k (t) k k + + d m,k ψ m,k (t) Retains most important part De-nosing of signal Inverse discrete wavelet transform application k k 44

45 References (1) Ang, A. & Beakert, G. (2002). International asset allocation with regime shifts, Review of Financial Studies, 15, (4), Aussem, A., Campbell, J. & Murtagh, F. (1998). Wavelet-Based Feature Extraction and Decomposition Strategies for Financial Forecasting. Journal of Computational Intelligence in Finance, 6, Bultheel, A. (1995). Learning to swim in a sea of wavelets. Bull. Belg. Math. Soc. Simon Stevin, 2 (1), Capobianco, E. (2003). Empirical volatility analysis. Feature detection and signal extraction with function dictionaries. Physica A. Statistical Mechanics and its Applications, 319, Capobianco, E. (2004). Multiscale Analysis of Stock Index Return Volatility. Computational Economics, 23, Crowley, P. M. (2007). A Guide to Wavelets for Economists. Journal of Economic Surveys, 21 (2), Fay, D. & Ringwood, J. (2007). A Wavelet Transfer Model for Time Series Forecasting. International Journal of Bifurcation and Chaos, 17 (10), Gençay, R., Selçuk, F. & Whitcher, B. (2001a). Scaling properties of foreign exchange volatility. Physica A. Statistical Mechanics and its Applications, 289, Gençay, R., Selçuk, F. & Whitcher, B. (2001b). Differentiating intraday seasonalities through wavelet multi-scaling. Physica A, 289, Gençay, R., Selçuk, F. & Whitcher, B. (2003). Systematic risk and timescales. Quantitative Finance, 3, Gençay, R. & Selçuk, F. (2004). Volatility-return dynamics across different timescales. Working paper. Retrieved May 2, 2015, from tp%3a%2f%2fwww.sfu.ca%2f~rgencay%2fjarticles%2fdowlev.pdf&ei=8624vanagie_ywp0irmwcg&usg=afqjcnfw3vf3uhqzrbmyew0l kwo9rrxnea&bvm=bv ,d.bgq Gençay, R., Selçuk, F. & Whitcher, B. (2005). Multiscale systematic risk. Journal of International Money and Finance, 24, Gençay, R., Gradojevic, N., Selçuk, F. & Whitcher, B. (2010). Asymmetry of information flow between volatilities across time scales. Quantitative Finance, 10, Guidolin, M. & Ria, F. (2011). Regime shifts in mean-variance efficient frontiers. Some international evidence, Journal of Asset Management, 12 (5), Guidolin, M. & Timmermann, A. (2007). Asset allocation under multivariate regime switching, Journal of Economic Dynamics & Control, 31, In, F. & Kim, S. (2013). An Introduction to Wavelet Theory in Finance. Singapore: World Scientific Publishing Co. Pte. Ltd. 45

46 References (2) Kim, C., Yu, I. & Song, Y.H. (2002). Kohonen neural network and wavelet transform based approach to short-term load forecasting, Electric Power Systems Research, 63, Masset, P. (2008). Analysis of Financial Time-Series using Fourier and Wavelet Methods. Working Paper. Retrieved April 24, 2015, from Mallat, S. G. (1989). A theory for multiresolution signal decomposition. The wavelet representation. IEEE Transaction on Pattern Analysis and Machine Intelligence, 11, Morger, F. (2006). International asset allocation and hidden regime switching. Disseratation, University of Zurich. Zurich: Institut für Schweizerisches Bankwesen. Percival, D. B. & Mofjeld, H. O. (1995). Analysis of Subtidal Coastal Sea Level Fluctuations Using Wavelets. Journal of the American Statistical Association, 92 (439), Percival, D. B. & Walden, A. T. (2000). Wavelet Methods for Time Series Analysis. New York, NY: Cambridge University Press. Perlin, M. (2012). MS_Regress. The MATLAB package for markov regime switching models. Working Paper. Retrieved March 4, 2014 from Ranta, M. (2010). Wavelet Multiresolution Analysis of Financial Time Series. Dissertation, Universitas Wasaensis, No. 223, Finland: Vaasan yliopisto. Sheng, Y. (2000). Wavelet Transform. In A. D. Poularikas (Ed.), The Transforms and Applications Handbook (2 nd ed.). Boca Raton, Fla.: CRC Press LLC Schwendener, A. (2010). The estimation of financial markets by means of a regime-switching model. Dissertation, University of St. Gallen. Bamberg: Difo-Druck GmbH Tan, C. (2009). Financial Time Series Forecasting Using Improved Wavelet Neural Network. Master thesis, Aarhus University, Denmark. Torrence, C. & Compo, G. P. (1998). A Practical Guide to Wavelet Analysis. Bulletin of American Meteorological Society, 79, Valens, C. (1999). A Really Friendly Guide to Wavelets. Working paper. Retrieved April 24, 2015, from Yao, S.J., Song, Y.H., Zhang, L.Z. & Cheng, X.Y. (2000). Wavelet transform and neural networks for short-term electrical load forecasting. Energy Conversion & Management, 41, Zhan, B.-L. & Dong, Z.-Y. (2001). An adaptive neural-wavelet model for short term load forecasting. Electric Power Systems Research, 59,