Liu Xinyi Thesis A New Markov Switching Volatility Model

3,405 views

Published on

Ph.D. Thesis, A New Markov Switching Volatility Model

0 Comments
0 Likes
Statistics
Notes
  • Be the first to comment

  • Be the first to like this

No Downloads
Views
Total views
3,405
On SlideShare
0
From Embeds
0
Number of Embeds
30
Actions
Shares
0
Downloads
0
Comments
0
Likes
0
Embeds 0
No embeds

No notes for slide

Liu Xinyi Thesis A New Markov Switching Volatility Model

  1. 1. A New Hidden Markov-Switching Volatility Model Oct/2008 in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Economics Nanyang Technological University Author Liu Xinyi Division of Economics, the School of Humanity and Social Science Thesis Supervisor Wang Peiming
  2. 2. Abstract: The thesis proposes and applies a two-state hidden Markov-switching model for financial time series featured with periodic structure breaks in volatility. The expected return, volatility and state transition probability are determined by three link functions respectively, whose coefficients are further governed by the hidden state. The proposed model particularly emphasizes on the parallel structure of the two states. The parallel structure separates the INTER-state and INTRA-state dynamics, enhances greater transparency, balances the memory of both recent and distant history, provides more consistent economic implication, and greatly simplifies and stabilizes the EM algorithm. We further discuss its estimation, inference, standard errors of the parameter estimate, forecasting, model selection and implementation, especially our innovations in those issues. The Monte Carlo experiments suggest that the proposed estimation method is accurate and reliable, the choice of the initial state probability has little effect on proposed model, and the information matrix calculated numerically is stable and reliable. We then apply the proposed model to forecast the conditional distribution of the weekly return of the S&P 500 index. It is found that the volatility at the tranquil state is best described as a constant while the volatility at the turbulent state is more closely correlated with the lagged intraweek ranges. Eventually, we give the economic implication on the clustered volatility and its interplay with the expected return. KEYWORDS: HIDDEN MARKOV-SWITCHING MODEL, PARALLEL STRUCTURE, VOLATILITY, LINK FUNCTION, EM ALGORITHM, FORWARD-BACKWARD ALGORITHM, RESCALING, INFORMATION MATRIX OF THE MISSING DATA, SUPPLEMENTED EM ALGORITHM, RANGE. 2
  3. 3. LIST OF TABLES ............................................................................................................ 7 0 18 7 LIST OF FIGURES .......................................................................................................... 8 1 19 7 LIST OF ABBREVIATIONS ......................................................................................... 10 2 10 8 LIST OF SYMBOLS...................................................................................................... 12 3 11 8 4 1. INTRODUCTION.................................................................................................................. 15 12 8 1.1. Volatility Observations ................................................................................................................... 15 5 13 8 1.2. Motivations and Oversights ........................................................................................................... 18 6 14 8 1.3. Thesis Contributions ...................................................................................................................... 21 7 15 8 1.3.1. Addressing the observations .................................................................................................................... 21 8 16 8 1.3.2. Addressing the Oversights ....................................................................................................................... 22 9 17 8 1.3. Thesis Outline ................................................................................................................................ 24 1 0 18 8 1 1 2. THE PROPOSED HIDDEN MARKOV-SWITCHING VOLATILITY MODEL .................. 27 19 8 2.1. Model Overview............................................................................................................................. 27 1 2 10 9 2.3. Special cases and relationships with existing models .................................................................... 31 1 3 11 9 2.4. The link functions .......................................................................................................................... 33 1 4 12 9 2.5. The likelihood function .................................................................................................................. 34 1 5 13 9 2.6. Initial state probabilities ................................................................................................................. 34 1 6 14 9 1 7 3. LITERATURE REVIEW AND MODEL COMPARISONS................................................... 36 15 9 3.1. Hidden Markov-switching model: A brief history.......................................................................... 37 1 8 16 9 3.2. Seminal work on HMM in Econometrics....................................................................................... 37 1 9 17 9 3.3. Prominent volatility models ........................................................................................................... 39 2 0 18 9 3.3.1. The i.i.d. mixture distribution .................................................................................................................. 39 2 1 19 9 3.3.2. Classic [G]ARCH models ....................................................................................................................... 41 2 2 20 0 3.3.3. [G]ARCH with mixture normal innovation ............................................................................................. 44 2 3 21 0 3.3.4. Regime-switching volatility models ........................................................................................................ 45 2 4 22 0 3.4. Two classes of popular Markov-switching [G]ARCH approaches ................................................ 49 2 5 23 0 3.4.1. The Hamilton/Gray/Klaassen approach ................................................................................................... 49 2 6 24 0 3.4.1.2. Non-parallel structure and potential limitations ............................................................................... 55 2 7 25 0 3
  4. 4. 3.4.2. The Haas, Mittnik and Paolella approach ................................................................................................ 60 2 8 26 0 3.4.2.1 Model Overview ............................................................................................................................... 60 2 9 27 0 3.4.2.2. Improvements/Limitations ............................................................................................................... 62 3 0 28 0 3.5. Strengths of the proposed parallel hidden Markov-switching volatility model.............................. 65 3 1 29 0 3.6. HMM empirical studies.................................................................................................................. 69 3 2 20 1 3.6.1. Business Cycles....................................................................................................................................... 70 3 3 21 1 3.6.2. Interest Rates and Yield Curves............................................................................................................... 70 3 4 22 1 3.6.3. Foreign Exchange Rates .......................................................................................................................... 71 3 5 23 1 3.6.4. Stock Index.............................................................................................................................................. 72 3 6 24 1 3.6.5. Default probabilities and credit spread .................................................................................................... 74 3 7 25 1 3.6.6. Commodity Prices ................................................................................................................................... 75 3 8 26 1 4. MODEL ESTIMATION AND FORECAST........................................................................... 77 3 9 27 1 4.1. The EM algorithm .......................................................................................................................... 78 4 0 28 1 4.1.1. Overview ................................................................................................................................................. 78 4 1 29 1 4.1.2. The E-Step ............................................................................................................................................... 80 4 2 20 2 4.1.3. The parallel M-Step ................................................................................................................................. 84 4 3 21 2 4.1.4. Algorithm Summary ................................................................................................................................ 87 4 4 22 2 4.2. The quasi-Newton algorithm.......................................................................................................... 89 4 5 23 2 4.3. A hybrid approach .......................................................................................................................... 91 4 6 24 2 4.4. Standard errors of the estimated parameters .................................................................................. 93 4 7 25 2 4.4.1. Information matrix by numerical difference ............................................................................................ 93 4 8 26 2 4.4.2. The Supplemented EM algorithm............................................................................................................ 95 4 9 27 2 4.4.3. Parametric bootstrap ................................................................................................................................ 99 5 0 28 2 4.5. Monte Carlo experiments ............................................................................................................. 100 5 1 29 2 4.5.1. Objectives.............................................................................................................................................. 100 5 2 20 3 4.5.2. The choice of covariate and “true” coefficients ..................................................................................... 100 5 3 21 3 4.5.3. Experiment 1: Flat initial state probabilities .......................................................................................... 105 5 4 22 3 4.5.4. Experiment 2: Stochastic initial state probabilities ................................................................................ 107 5 5 23 3 4.5.5. Results and Observations....................................................................................................................... 108 5 6 24 3 4.6. Forecasting with HMS-V ............................................................................................................. 114 5 7 25 3 5. APPLICATION & EMPIRICAL STUDY ............................................................................ 116 5 8 26 3 5.1. Choice of covariate ...................................................................................................................... 117 5 9 27 3 4
  5. 5. 5.1.1. Range vs. observable range .................................................................................................................... 117 6 0 28 3 5.1.2. Forecasting range.................................................................................................................................... 119 6 1 29 3 5.2. Data .............................................................................................................................................. 119 6 2 20 4 5.3. Model selection criteria................................................................................................................ 120 6 3 21 4 5.3.1. Likelihood ratio test............................................................................................................................... 121 6 4 22 4 5.3.2. AIC and BIC.......................................................................................................................................... 121 6 5 23 4 5.3.3. MSE, MAD and integrated volatility..................................................................................................... 122 6 6 24 4 5.3.5. The c.d.f. of out-of-sample forecast....................................................................................................... 126 6 7 25 4 5.4. HMS-V without covariate: Bull vs. bear, a qualitative assessment .............................................. 127 6 8 26 4 5.5. HMS-V with covariates: Quantitative fine-tuning ....................................................................... 131 6 9 27 4 5.5.1. Link functions and covariates................................................................................................................ 131 7 0 28 4 5.5.2. Time varying sensitivity to recent shocks.............................................................................................. 133 7 1 29 4 5.6. Model Comparisons (empirical study) ......................................................................................... 146 7 2 20 5 5.6.1. Comparison with single regime models................................................................................................. 146 7 3 21 5 5.6.1.1. Comparison using statistical loss functions.................................................................................... 146 7 4 22 5 5.6.1.2. Comparison by closer examinations of two episodes..................................................................... 148 7 5 23 5 5.6.2. Comparison with MS-GARCH.............................................................................................................. 159 7 6 24 5 5.6.3. Summary ............................................................................................................................................... 164 7 7 25 5 6. ECONOMIC IMPLICATION OF THE EMPIRICAL STUDIES.......................................................... 172 7 8 26 5 6.1. Asymmetric business cycle, velocity of time and alternative time scales .................................... 172 7 9 27 5 6.2. Recursive loop of expectation and realization, self-reinforcing or self-correcting ...................... 176 8 0 28 5 6.3. Asymmetric surprises, sensitivities, correlations and risk premia................................................ 178 8 1 29 5 6.3.1. Fewer surprises at the tranquil state....................................................................................................... 179 8 2 20 6 6.3.2. More surprises at the turbulent state ...................................................................................................... 182 8 3 21 6 6.3.3. Greater sensitivity at the turbulent state................................................................................................. 184 8 4 22 6 6.3.4. Increased equity return correlation at the turbulent state ....................................................................... 188 8 5 23 6 6.3.5. Greater risk premia requested by the investor at the turbulent market................................................... 189 8 6 24 6 7. THESIS SUMMARY ................................................................................................................ 190 8 7 25 6 ADDITIONAL TABLES ......................................................................................................... 192 8 8 26 6 ADDITIONAL FIGURES........................................................................................................ 194 8 9 27 6 APPENDIX I............................................................................................................................ 197 9 0 28 6 App1.1. The property of the mixture regression ................................................................................. 197 9 1 29 6 5
  6. 6. App1.2. A sufficient identifiability condition...................................................................................... 198 9 2 20 7 9 3APPENDIX II .......................................................................................................................... 204 21 7 9 4APPENDIX III ......................................................................................................................... 208 22 7 9 5APPENDIX IV......................................................................................................................... 210 23 7 9 6APPENDIX V .......................................................................................................................... 212 24 7 9 7APPENDIX VI......................................................................................................................... 215 25 7 9 8APPENDIX VII........................................................................................................................ 216 26 7 9 9APPENDIX VIII ...................................................................................................................... 218 27 7 APPENDIX IX ............................................................................................................. 221 10 0 28 7 11 0 APPENDIX X .......................................................................................................................... 22429 7 12 0 APPENDIX XI......................................................................................................................... 231 20 8 REFERENCES............................................................................................................. 235 13 0 21 8 6
  7. 7. LIST OF TABLES 0 14 0 Table 1 ................................................................................................................................................... 54 22 8 Comparison between Gray’s (1996) and Klaassen’s (2002) model 15 0 Table 2 ................................................................................................................................................. 110 23 8 The result of the Monte Carlo study 1 with fixed flat initial state probabilities 16 0 Table 3 ................................................................................................................................................. 11124 8 The result of the Monte Carlo study 1 with fixed flat initial state probabilities, additional check 17 0 Table 4 ................................................................................................................................................. 112 25 8 Monte Carlo study 2 with random initial state probabilities 18 0 Table 5 ................................................................................................................................................. 113 26 8 Monte Carlo study 2 with random initial state probabilities, additional check 19 0 Table 6 ................................................................................................................................................. 120 27 8 Descriptive statistics of the weekly percentage returns and the percentage observed intraweek ranges 10 1 Table 7 ................................................................................................................................................. 128 28 8 The regression result of the HMS-V model without covariate 11 1 Table 8 ................................................................................................................................................. 135 29 8 The regression result of the HMS-V model with covariate (Regression 02 and 03) 12 1 Table 9 ................................................................................................................................................. 136 20 9 The regression result of the HMS-V model with covariate (Regression 04 and 05) 13 1 Table 10 ............................................................................................................................................... 138 21 9 Sample ACFs and XCF, from May/13/ 1992 to Dec/06/1995 14 1 Table 11 ............................................................................................................................................... 139 22 9 Sample ACFs and XCF, from Jan/07/2004 to Jun/27/2007 15 1 Table 12 ............................................................................................................................................... 140 23 9 Sample ACFs and XCF, from January 08, 1997 to June 02, 2003 16 1 Table 13 ............................................................................................................................................... 147 24 9 Comparison with single regime GARCH models 17 1 Table 14 ............................................................................................................................................... 153 25 9 Comparison of HMS-V and GARCH-T, a closer look at Oct/07/1987 to Mar/02/1988 18 1 Table 15 ............................................................................................................................................... 157 26 9 Comparison of HMS-V and GARCH-T, a closer look at Jul/18/2007 to Nov/21/2007 19 1 Table 16 ............................................................................................................................................... 162 27 9 Comparison of HMS-V and MS-GARCH, a closer look at Oct/07/1987 to Mar/02/1988 10 2 Table 17 ............................................................................................................................................... 163 28 9 Comparison of HMS-V and MS-GARCH, a closer look at Jul/18/2007 to Nov/21/20 11 2 Table 18 ............................................................................................................................................... 193 29 9 The SEM algorithm and the comparison with other models 12 2 Table 19 ............................................................................................................................................... 225 30 0 Descriptive statistics of the percentage observed intraweek ranges 13 2 Table 20 ............................................................................................................................................... 233 31 0 Adding lagged terms of squared return hardly improves the goodness-of-fit of log range 7
  8. 8. LIST OF FIGURES 1 14 2 Figure 1 ................................................................................................................................................. 40 32 0 p.d.f. of MixN with different parameter settings 15 2 Figure 2 ................................................................................................................................................. 47 33 0 Simulated time series with structure breaks 16 2 Figure 3 ................................................................................................................................................. 48 34 0 Standard deviation estimated by GARCH (1,1) 17 2 Figure 4 ................................................................................................................................................. 55 35 0 INTER-state and INTRA-state dynamics of Hamilton/Gray/Klaassen (when the underlying hidden state is known) 18 2 Figure 5 ................................................................................................................................................. 57 36 0 Gray/Klaassen Model (when the underlying process is unknown and has to be estimated) 19 2 Figure 6 ................................................................................................................................................. 61 37 0 INTER-state and INTRA-state dynamics of Haas/Mittnik/Paolella (2004b), (when the underlying hidden state is known) 10 3 Figure 7 ................................................................................................................................................. 63 38 0 Haas, Mittnik, and Paolella’s (2004b) (when the underlying process is unknown and has to be estimated) 11 3 Figure 8 ................................................................................................................................................. 67 39 0 The proposed model (when the underlying process is unknown and has to be estimated) 12 3 Figure 9 ................................................................................................................................................. 92 30 1 Summarization of the hybrid EM and quasi-Newton approach 13 3 Figure 10 ............................................................................................................................................. 101 31 1 The VIX index by CBOE from Jan/02/1990 to Sep/15/2006 14 3 Figure 11 ............................................................................................................................................. 101 32 1 The Fed Target rate from Jan/02/1990 to Sep/15/2006 15 3 Figure 12 ............................................................................................................................................. 103 33 1 Simulated time series, the “true” standard deviations at each state 16 3 Figure 13 ............................................................................................................................................. 104 34 1 Simulated time series, the “true” state transition probabilities at each state 17 3 Figure 14 ............................................................................................................................................. 104 35 1 A representative simulated time series 18 3 Figure 15 ............................................................................................................................................. 105 36 1 A few real world time series 19 3 Figure 16 ............................................................................................................................................. 130 37 1 Regression 01, in-sample (HMS-V without covariate) 10 4 Figure 17 ............................................................................................................................................. 131 38 1 Regression 01, out-of-sample (HMS-V without covariate) 11 4 Figure 18 ............................................................................................................................................. 138 39 1 Sample ACFs and XCFs, from May/13/1992 to Dec/06/1995 12 4 Figure 19 ............................................................................................................................................. 139 30 2 Sample ACFs and XCFs, from Jan/07/2004 to Jun/27/2007 13 4 Figure 20 ............................................................................................................................................. 141 31 2 Sample ACFs and XCFs, from Jan/08/1997 to Jun/02/2003 14 4 Figure 21 ............................................................................................................................................. 143 32 2 Regression 05, in-sample, volatility at each state 15 4 Figure 22 ............................................................................................................................................. 143 33 2 Regression 05, in-sample, transition probability 16 4 Figure 23 ............................................................................................................................................. 144 34 2 Regression 05, in-sample, state probability and volatility 17 4 Figure 24 ............................................................................................................................................. 144 35 2 Regression 05, out-of-sample, volatility at each state 18 4 Figure 25 ............................................................................................................................................. 145 36 2 Regression 05, out-of-sample, transition probability 19 4 Figure 26 ............................................................................................................................................. 145 37 2 Regression 05, out-of-sample, state probability and volatility 10 5 Figure 27 ............................................................................................................................................. 149 38 2 Volatility forecasting using different methods, from Jan/05/1983 to Dec/26/2001 11 5 Figure 28 ............................................................................................................................................. 150 39 2 Volatility forecasting using different methods, from Dec/26/2001 to Nov/21/2007 8
  9. 9. Figure 29 ............................................................................................................................................. 154 12 5 30 3 Volatility forecasting using different methods, from Oct/07/1987 to Mar/02/1988 Figure 30 ............................................................................................................................................. 154 13 5 31 3 Volatility forecasting and state probability, HMS-V model, from Oct/07/1987 to Mar/02/1988 Figure 31 ............................................................................................................................................. 155 14 5 32 3 imitated mean forecasting by HMS-V and GARCH-T model, from Oct/07/1987 to Mar/02/1988 Figure 32 ............................................................................................................................................. 158 15 5 33 3 Volatility forecasting using different methods, from Jul/25/2007 to Nov/21/2007 Figure 33 ............................................................................................................................................. 158 16 5 34 3 Volatility forecasting and state probability, HMS-V model, from Jul/25/2007 to Nov/21/2007 Figure 34 ............................................................................................................................................. 159 17 5 35 3 imitated mean forecasting by HMS-V and GARCH-T model, from Jul/25/2007 to Nov/21/2007 Figure 35 ............................................................................................................................................. 174 18 5 36 3 Asymmetric business cycle, velocity of time and alternative time scales Figure 36 ............................................................................................................................................. 178 19 5 37 3 A self-reinforcing or a self-defeating tranquil state Figure 37 ............................................................................................................................................. 186 10 6 38 3 The sharp decline in price and rising volatility are self-reinforcing Figure 38 ............................................................................................................................................. 187 11 6 39 3 De-leverage process for the short position is also self-reinforcing process of rising price and rising volatility Figure 39 ............................................................................................................................................. 194 12 6 30 4 Historical price of S&P500 stock index from Jan/05/1983 to Nov/21/2007 Figure 40 ............................................................................................................................................. 194 13 6 31 4 Historical weekly percentage return on S&P500 index from Jan/05/1983 to Nov/21/2007 Figure 41 ............................................................................................................................................. 195 14 6 32 4 Historical absolute weekly percentage return from Jan/05/1983 to Nov/21/2007 Figure 42 ............................................................................................................................................. 195 15 6 33 4 Fit the historical weekly percentage return on S&P500 index to a normal distribution Figure 43 ............................................................................................................................................. 196 16 6 34 4 Split return data, for in-sample and out-of-sample inference Figure 44 ............................................................................................................................................. 223 17 6 35 4 The price of Citigroup Inc at the official trading hours from Dec/26/2007 to Dec/28/2007 Figure 45 ............................................................................................................................................. 223 18 6 36 4 The price of Citigroup Inc at the official trading hours from Dec/26/2007 to Dec/28/2007 Figure 46 ............................................................................................................................................. 226 19 6 37 4 The log observed intraweek range and some preliminary data analysis (Sample ACFs and PACFs) Figure 47 ............................................................................................................................................. 228 10 7 38 4 ARMA model and observed log range, in-sample Figure 48 ............................................................................................................................................. 228 11 7 39 4 ARMA model and observed log range, in-sample, diagnosis of residuals Figure 49 ............................................................................................................................................. 229 12 7 30 5 ARMA model and observed log range, out-of-sample Figure 50 ............................................................................................................................................. 230 13 7 31 5 ARMA model and observed log range, out-of-sample, diagnosis of residuals Figure 51 ............................................................................................................................................. 234 14 7 32 5 QQ plot of the residuals to study if F-test can be used 9
  10. 10. LIST OF ABBREVIATIONS 2 ACF Autoregressive function AIC Akaike information criterion BIC Bayesian information criterion CARR Conditional autoregressive range DGP Sata generating process EM The expectation and maximization algorithm FX Foreign exchange rate [G]ARCH [Generalized] autoregressive conditional heteroskedasity EGARCH Exponential GARCH IGARCH Integrated GARCH GARCH-MixN GARCH model with mixture normal innovations GARCH-T GARCH model whose innovation follows Student’s distribution HMS-V The proposed hidden Markov-switching volatility model J-B test Jarque-Bera test KL Kullback-Leibler information measure MAD Mean absolute deviation MS (HMM) Hidden Markov-switching model MLE Maximum likelihood estimator 10
  11. 11. MS-[G]ARCH Hidden Markov-switching [G]ARCH model MSE Mean square error MixN Mixture normal diffusion SEM The supplemented expectation and maximization algorithm XCF Cross-sectional correlation function 11
  12. 12. LIST OF SYMBOLS 3 Vectors and matrices will be denoted with boldfaces (e.g. C and β ); a parameter vector is always a column vector; and sets and spaces will be denoted with calligraphic letters (e.g. V ). I ( x) Indicator function, which equals to 1 if the statement x is true and 0 other wise T A vector of a time series indices, T = {t : t = 1, ,T} Δt The state at time t s i ,t An indicator function of the state variable Δ t , s i ,t := I ( Δ t = i ) , that is, the realization of a Bernoulli trial; s i ,t = 1 if Δ t = i , and s i ,t = 0 if Δ t ≠ i s i ,t The “true” rate of a successful Bernoulli trial s i ,t , s i ,t := E (s ) i ,t s i ,t The estimated rate of a successful Bernoulli trial s i ,t , s i ,t := E (s ) = E (s ) i ,t i ,t ℜm The m -dimensional Euclidean space. β Estimation of the parameter vector β β (l +1) The estimate of the parameter β at the ( l + 1) th iteration. β (l +1) − β (l ) The distance between the two matrices, β ( l +1) and β ( l ) , for the 12
  13. 13. sizes of two matrices are both i × j , i j β (l +1) − β (l ) = ∑∑ β k(l,h 1) − β k(l,h + ) k =1 h =1 θ (l +1) = (θ ( ) ) l A mapping from θ ( l ) to θ (l +1) , where is the rule L The lag operator XT A matrix with the index T , T = 1, ,T x' Transpose of the vector x x>0 All elements of vector x are positive elementwise 0a×b The zero matrix with size a × b 1a×b The unity matrix with size a × b diag ( c1 , , cm ) The diagonal matrix with diagonal elements ( c1 , , cm ) tr ( S ) The trace of the square matrix S I The identity matrix ψt All the available information up to, and including, time t Et ( ε | x ) The expectation of the variable ε at time t based on the information x Vart ( ε | x ) The variance of the variable ε at time t based on the information x φ (• μ ,σ 2 ) Probability density function of the normal distribution N μ , σ 2 ( ) Φ (• μ ,σ 2 ) Cumulative density function of the normal distribution N μ , σ 2 ( ) I com The information matrix of the complete data I obs The information matrix of the observable data 13
  14. 14. I miss The information matrix of the missing data I The Fisher information matrix f (• • , •) i A function f i with 2 inputs LF (θ ) A likelihood function whose parameter is θ LLF (θ ) A log likelihood function whose parameter is θ LLFobs (θ ) The log likelihood function of the observed data LLFcom (θ ) The log likelihood function of the complete data ∇LLF (θ ) The Gradient of the log likelihood function ∇ 2 LLF (θ ) The Hessian of the log likelihood function 14
  15. 15. 1. INTRODUCTION 4 Volatility, the most extensively used measure of uncertainty, is fundamental to much of modern finance theory. Quantifying and forecasting volatility is essential to financial asset pricing, portfolio optimization and risk management. Over the past two decades, a diverse range of theoretical and empirical research has been carried out on modeling the volatility of time series. Following the widespread popularity of a number of financial instruments, whose values are intrinsically linked to their corresponding volatilities, the development of volatility models and estimation frameworks has emerged as active research areas. The proposed model is a further extension of those presented in existing literature. It uniquely emphasizes the parallel structure of the Markov states. This parallel structure separates the INTER-state and INTRA-state dynamics, allows for greater transparency, balances the memory of both recent and distant history, provides more consistent economic implication as well as greatly simplifying and stabilizing the EM algorithm. 1.1. Volatility Observations 1 2 It is widely recognized that the volatilities of most financial asset returns are time-varying and typically display, at least, the following six stylized characteristics:- 15

×