Bayesian Stochastic Volatility (SV) Model with non-gaussian Errors

Transcription

1 rrors Bayesian Stochastic Volatility (SV) Model with non-gaussian Errors Seokwoo Lee 1, Hedibert F. Lopes 2 1 Department of Statistics 2 Graduate School of Business University of Chicago May 19, 2008

2 rrors Outline of Topics 1 Preliminaries Markov Chain Mote Carlo: Gibbs Sampling Bayesian Regression 2 SV model with Gaussian Errors Forward Filtering Backward Sampling with Kalman Filter Kim, Shephard, and Chib: FIX-SVM [5] 3 SVM with Non-Gaussian Errors: FLEX-SVM 4 Real Stock Data: S&P500 and 30 DOW JONES 5 Simulation Methodology

3 rrors Motivation rdata GE.txt Time Varying Volatility Volatility Clustering

4 rrors Standard Stochastic Volatility Model [3] Assumption ɛ t, h t are stochastically independent Log volatility h t is stationary AR(1) process Assume h 1 N(a 1, R 1 ) Equivalently, y t = e ht ɛ t ɛ t i.i.d. N(0, 1) (1) h t = ω + φ h t 1 + σ w w t w t N(0, 1) log y 2 t = h t + log ɛ 2 t log ɛ 2 t log χ 2 1 (2) h t = ω + φ h t 1 + w t w t N(0, σ 2 w )

5 rrors Preliminaries Markov Chain Mote Carlo: Gibbs Sampling MCMC In our standard SV model (Eq. 1), likelihood function is f (y θ) = f (y h, θ)f (h θ)dh where y = (y 1,, y n ) and h = (h 1,, h n ) The key issue is that this likelihood function is intractable. Instead, we are focusing on p(θ, h y) Markov Chain Mote Carlo procedures provides a way to sampling this density without directly computing the above complex likelihood function Posterior moments and marginal density can be estimated by averaging the relevant function over the sampled (simulated) variates

6 rrors Preliminaries Markov Chain Mote Carlo: Gibbs Sampling Gibbs Sampling Gibbs sampling generates a successive samples from the full conditional distributions. Algorithms proceeds by sampling each block from the full conditional distributionswhere the most recent values of the conditioning blocks are used in the simulation Gibbs for SV model: π(θ, h y) 1 Initialize h and θ 2 Sample h y, θ hard 3 Sample(Update) θ h, y easy 4 goto 2

7 rrors Preliminaries Bayesian Regression Sampling (ω, φ, σ 2 w h, y): Bayesian Regression h = (h 1,, h T 1 ) and h = (h 2,, h T ) X = (1 n 1, h) β = (ω, φ) System Equation: AR(1) h = β X + σ w w t where w t N(0, 1)

8 rrors Preliminaries Bayesian Regression Sampling (ω, φ, σ 2 w h, y): Bayesian Regression Priors: a 1, R 1, β 0, A, ν 0 and s0 2 are known hyperparameters. h 1 N(a 1, R1 1 ) β σw 2 N(β 0, σw 2 A 1 ) σ 2 w IG( ν 0 2, ν 0s ) Full Conditionals: σw 2 h 1:n, y 1:n IG( ν 1 2, ν 1s1 2 2 ) where ν 1 = ν 0 + (n 1)/2 ν 1 s1 2 = ( β β 0 ) A( β β 0 ) β = (X X + A) 1 (X h + Aβ 0 ) where X = (1 n 1, h 1:n 1 ) β σ 2 w, h 1:n, y 1:n N( β, σ 2 w (X X + A) 1 )

9 rrors SV model with Gaussian Errors Forward Filtering Backward Sampling with Kalman Filter Forwards Filtering Backwards Sampling with DLM Idea Jointly sample (h ω, φ, σ 2 w, y) using FFBS with Kalman Filter Kalman Filter (Dynamic Linear Model) Forward Filtering Backward Sampling (Simulation Smoothing)

10 rrors SV model with Gaussian Errors Forward Filtering Backward Sampling with Kalman Filter Dynamic Linear Model (DLM) Model: y t = F t β t + ɛ t, where ɛ t iid N(0, Vt ) β t = G t β t 1 + w t, where w t iid N(0, Wt ) y t : sequence of observations F t : vector of explanatory variables β t : d-dimensional state vector G t : d d evolution matrix β 1 N(a 1, R 1 ) v t w t

11 rrors SV model with Gaussian Errors Forward Filtering Backward Sampling with Kalman Filter Sequential Inference β t β t 1 N(G t β t 1, W t ) Posterior at t 1 β t 1 y 1:t 1 N(m t 1, C t 1 ) Prior at t β t y 1:t 1 N(a t, R t ) a t = G t m t 1 R t = G t C t 1 G t +W t Predictive at t y t y 1:t 1 N(f t, Q t ) f t = F t a t Q t = F t R t F t + V t

12 rrors SV model with Gaussian Errors Forward Filtering Backward Sampling with Kalman Filter Sequential Inference: Filtering Since, p(β t y 1:t ) = p(β t y t, y 1:t 1 ) p(y t β t )p(β t y 1:t 1 ) Posterior at t β t y 1:t N(m t, C t ) where m t = a t + A t e t C t = R t + A t A tq t A t = R t F t Qt 1 e t = y t f t By induction, these distributions are valid for all times

13 rrors SV model with Gaussian Errors Forward Filtering Backward Sampling with Kalman Filter Forward Filtering Backward Sampling p(β y) = p(β n,, β 1 y 1:n ) = p(β n y 1:n )p(β n 1 β n, y 1:n )p(β n 2 β n 1, β n, y 1:n ) p(β 2 β 3,, β n, y 1:n ) = p(β n y 1:n )p(β n 1 β n, y 1:n )p(β n 2 β n 1, y 1:n ) p(β 2 β 3, y 1:n ) Also, p(β n 1 β n, y 1:n ) = p(β n 1 β n, y 1:n 1, y n ) = p(β n 1 β n, y 1:n 1 ) In general, p(β t β t+1, y 1:n ) = p(β t β t+1, y 1:t ) for 1 < t < n

14 rrors SV model with Gaussian Errors Forward Filtering Backward Sampling with Kalman Filter FFBS (β t, β t+1 ) given y 1:t is bivariate normal under Gasussian assumption Conditional mean and covariance matrix of (β t, β t+1 ) given y 1:t are readily available from Kalman Filtering. [ ] ( [ ] [ ] ) βt mt Ct G N, t C t β t+1 a y t+1 G t C t R t+1 1:t Consequently, where p(β t β t+1, y 1:t ) N(m t, C t ) m t = m t + G t C t R 1 t+1 [β t+1 a t+1 ] C t = C t G t C t R 1 t+1 C t G t

15 rrors SV model with Gaussian Errors Kim, Shephard, and Chib: FIX-SVM [5] SV model as DLM Recalling log y 2 t := y t = h t + v t v t := log ɛ 2 t log χ 2 1 h t = ω + φ h t 1 + w t w t N(0, σ 2 w ) where E(log ɛ 2 t ) = 1.27 and Var(log ɛ 2 t ) = 4.9 Indeed, very close to DLM except for log ɛ 2 t is non-gaussian

16 rrors SV model with Gaussian Errors Kim, Shephard, and Chib: FIX-SVM [5] Comparison between Approximations density log chi^2_1 mixture of normals N( 1.27, 4.9) x Figure: log χ 2 1 vs N( 1.27, 4.9) and mixture of 7 normals It is obvious single Normal approximation is not good enough

17 rrors SV model with Gaussian Errors Kim, Shephard, and Chib: FIX-SVM [5] Approximation by Normal mixture 7 log χ 2 1 π k N(µ k, τk 2 ) k=1 µ k τk π k Table: 7 mixture normal component Figure 1 suggests mixture approximation is more appropriate.

18 rrors SV model with Gaussian Errors Kim, Shephard, and Chib: FIX-SVM [5] SV model as DLM with Fixed Mixture: FIX-SVM Let z 1,, z n be latent indicators corresponding to the observation innovations such that z t {1,, 7}. (v t z t = i) N(µ i, τi 2 ) P(v t = i) = π i Then, conditional on {z t } n t=1, the models becomes DLM: log y 2 t = h t + v t v t N(µ zt, τ 2 z t ) h t = ω + φ h t 1 + w t w t N(0, σ 2 w ) h 1 N(a 1, R 1 ) where µ zt and τ 2 z t are presented in Table 1 (h ω, φ, σ 2 w, y) can be jointly sampling by using FFBS as discussed.

19 rrors SV model with Gaussian Errors Kim, Shephard, and Chib: FIX-SVM [5] Gibbs sampling for FIX-SVM 1 Initialize h 1 2 Jointly sample h 2:n by FFBS with given h 1, θ and σ 2 w 3 Sample (z i v i ) {1,, K} with p(z i = j v i ) = π j p N (v i µ j,τj 2 ) 7 l=1 π l p N (v i µ l,τl 2 ) 4 Update σ 2 w θ, h, y, z and 5 Update θ σ 2 w, h, y, z by Bayesian Regression as discussed 6 go to step 2

20 rrors SV model with Gaussian Errors Kim, Shephard, and Chib: FIX-SVM [5] Simulation Study: Heavy tailed Innovations ɛ t t 2 yt Time Figure: Simulated 4000 observations with Heavy tailed innovation Error n obs = 4000 ω = , φ = , σ 2 w =

21 phi Histogram of phi phi Index Series phi Lag sig Histogram of sig sig Index Series sig Lag om Histogram of om om Index Series om Lag rrors SV model with Gaussian Errors Kim, Shephard, and Chib: FIX-SVM [5] FIX-SVM Parameter Estimation: ɛ t t 2 INCORRECT! ACF Frequency ACF Frequency Frequency ACF FIX-SVM mean std 5% 95% True φ σ w ω

22 rrors SV model with Gaussian Errors Kim, Shephard, and Chib: FIX-SVM [5] FIX-SVM E(h t ) vs true h t : ɛ t t 2 INCORRECT! FLEX SVM estimated ht(black solid) versus true ht(red dotted) Time Figure: FIX-SVM E(h t ) and true h t with t 2 innovations

23 rrors SVM with Non-Gaussian Errors: FLEX-SVM SVM with Non-Gaussian Errors: FLEX-SVM Observation Wrong assumption of distribution of innovation errors appears to cause incorrect inferences about h, ω, φ, σ w Idea Eliminate the assumption that ɛ t N(0, 1) Estimate the density of ɛ t only depending on data Inference ω, φ, σ w + Density estimation

24 rrors SVM with Non-Gaussian Errors: FLEX-SVM FLEX-SVM Mechanism: Learning the unknown density of ɛ t by the mixture π i K i=1 N(µ i, τ 2 i ) {µ i, τ i, π i } K i=1 are dynamically estimated through MCMC (Gibbs) based on the observations Regarding K is fixed e.g., K = 10 or K = 14

25 rrors SVM with Non-Gaussian Errors: FLEX-SVM Density Estimation by Mixture Model Given observation {v 1,, v n } probability density of v can be as; p(v i γ) = K i=1 π i p N (v i µ i, τ 2 i ) Latent group classifiers, z i, is introduced per observation v i : i is classified in group j when z i = j Using latent indicators such that (v i z i = j) N(µ j, τj 2), (v, z γ) has the following joint density; [ K ] n p(v, z γ) = p(v z, γ)p(z γ) = p N (v i µ i, τi 2 ) p(z i γ) j=1 i I j i=1 where I j = {t z t = j}.

26 rrors SVM with Non-Gaussian Errors: FLEX-SVM Priors for Mixture Models Define Priors: I j := {t z t = j} where j {1,, K} n j := I j n j v j = i I j v i n j s 2 j = i I j (v i v j ) 2 µ j N(µ 0j, s 2 0j ) τ 2 j N(n 0j /2, n 0j τ 2 0j /2) π i Drichlet(α 0 ) where α 0 = (α 01,, α 0K ) where µ 0j, s 0j, τ 0j, n 0j, α 0 are the known hyperparameters.

27 rrors SVM with Non-Gaussian Errors: FLEX-SVM Full Conditional for Mixture Models Full conditionals: [µ i τi 2, z, v] N(µ, τ 2 ) where τ 2 = n i τ 2 i + s 2 0i and µ = τ 2 (τ 2 i n i v i + s 2 0i µ 0i ) ) [τi 2 µ i, z, v] IG ((n 0i + n i )/2, (n 0i τ0i 2 + n isi 2)/2 [π γ, z, v] Drichlet(α 0 + n) where n = (n 1,, n K ) [z i γ, v] {1,, K} with p(z i = j γ, v i ) = π j p N (v i µ j,τ 2 j ) K l=1 π l p N (v i µ l,τ 2 l )

28 rrors SVM with Non-Gaussian Errors: FLEX-SVM Gibbs Sampling for FLEX-SVM Core procedures 1 FFBS h 2 Update ω, φ, σ by Bayesian Regression 3 Estimte density by updating mixture components {π i, µ i, τ i } K i=1

29 rrors SVM with Non-Gaussian Errors: FLEX-SVM Gibbs sampling for FLEX-SVM 1 Initialize h 1:n, θ = (ω, φ) and σ 2 w. 2 Initialize {µ i, τ i } K i=1 3 Make initial assignment of z 1:n 4 FFBS h 1:n y 1:n, z 1:n, {µ i, τ i, π i } K i=1, θ 5 Sample(update) θ and σ 2 w using Bayesian Regression 6 Compute n = {n 1,, n K } 7 Sample {µ i } K i=1 {τ i 2, π i} K i=1, v 1:n, z 1:n by the full conditionals 8 Sample {τi 2}K i=1 {π i, µ i } K i=1, v 1:n, z 1:n by the full conditionals 9 Sample {π i } K i=1 v 1:n, z 1:n, {µ i, τi 2}K i=1 by the full conditionals 10 Sample z 1:n v 1:n, {π i, µ i, τi 2}K i=1 by the full conditionals 11 go to 4

30 rrors SVM with Non-Gaussian Errors: FLEX-SVM FLEX-SVM Innovation Density Estimation: ɛ t t Emperical True Prior dist eps Figure: FLEX-SVM density log ɛ 2 t estimation

31 phi Histogram of phi phi Index Series phi Lag sig Histogram of sig sig Index Series sig Lag om Histogram of om om Index Series om Lag rrors SVM with Non-Gaussian Errors: FLEX-SVM FLEX-SVM Parameter Estimation: ɛ t t 2 CORRECT! ACF ACF ACF Frequency Frequency Frequency Figure: FLEX-SVM parameter estimation with t 2 innovations(φ, σ w, ω)

32 rrors SVM with Non-Gaussian Errors: FLEX-SVM FLEX-SVM E(h t ) vs true h t : ɛ t t 2 CORRECT! FLEX SVM estimated ht(black solid) versus true ht(red dotted) with normal Time Figure: FLEX-SVM E(h t ) and true h t with t 2 innovations

33 rrors Real Stock Data: S&P500 and 30 DOW JONES S&P500 return series rdata sp500r.txt

34 rrors Real Stock Data: S&P500 and 30 DOW JONES Equivalent Model Instead of the model introduced from the beginning, the finance researcher often used the following parsimonious model. (I ) y t = e ht/2 ɛ t h t+1 = ω + φ h t + w t (II ) y t = β e ht/2 ɛ t h t+1 = φ h t + w t where ( φ ) β = exp 2(1 ω) In the later section, we fit the models with the parsimonious model (II) instead of model (I).

35 rrors Real Stock Data: S&P500 and 30 DOW JONES SP500: FIX-SVM Parameter Estimation Histogram of phi Series phi Density phi ACF phi Time Lag Histogram of sig.w Series sig.w Density sig.w ACF sig.w Time FIX mean std 5% 95% φ σ w Lag

36 rrors Real Stock Data: S&P500 and 30 DOW JONES SP500: FLEX-SVM innovation density estimation log(mm) flex log X^2 mm x x

37 rrors Real Stock Data: S&P500 and 30 DOW JONES SP500: FLEX-SVM/FIX-SVM Tail Behavior k FIX-SVM: P(v > k) # Obs FLEX-SVM: P(v > k) # Obs E E Table: Comparison of Tail behavior with different threshholds

38 rrors Real Stock Data: S&P500 and 30 DOW JONES SP500: FLEX-SVM Paramter Estimation Histogram of phi Series phi Density phi ACF phi Time Lag Histogram of sig.w Series sig.w Density sig.w ACF sig.w Time FLEX mean std 5% 95% φ σ w Lag

39 rrors Real Stock Data: S&P500 and 30 DOW JONES SP: FLEX-SVM/FIX-SVM Parameter Est. Comparison density.default(x = phi1) density.default(x = sig.w1) Density Density N = 1781 Bandwidth = N = 1781 Bandwidth = FIX FLEX mean std 5% 95% mean std 5% 95% φ σ w

40 rrors Real Stock Data: S&P500 and 30 DOW JONES SP500: FLEX-SVM/FIX-SVM Model Comparison(h t ) eht E(exp(ht/2)) with flex E(exp(ht/2)) with fixed E(ht) with flex E(ht) with fixed Index

41 flex log X^ flex log X^2 x x flex log X^ flex log X^2 x x x x flex log X^ flex log X^2 x x x x flex log X^ flex log X^2 x x x x rrors Real Stock Data: S&P500 and 30 DOW JONES DOW JONES mm mm log(mm) mm mm mm log(mm) mm (a) AXP (b) AXP (c) BA (d) BA mm mm log(mm) mm mm mm log(mm) mm (e) GM (f) GM (g) IBM (h) IBM

42 flex log X^ x flex log X^ flex log X^2 x x x flex log X^ flex log X^2 x x x x flex log X^ x x x rrors Real Stock Data: S&P500 and 30 DOW JONES DOW JONES mm mm log(mm) mm mm mm log(mm) mm (i) JNJ (j) JNJ (k) MO (l) MO mm mm log(mm) mm (m) MRK (n) MRK

43 rrors Simulation Methodology High Performance MCMC Engine Language & Libraries C/C++ (FFBS, core Gibbs sampler) Rmath, GSL, ATLAS(BLAS+LAPACK) R plot, GNU plot Achieve average 4000 CPM (cycles per minute) for FIX-SVM, 2000CPM FLEX-SVM Future Work: Distributed Computing Construct Sim-Grid Parallel simulation with batch job(such as PBS queue or Condor)

44 rrors Simulation Methodology Performance Comparison # Obs R C/C sec 8.71 sec sec sec Table: FFBS performance benchmark C/C++ against R: 3000 sweep Data Set iter FIX-SVM FLEX-SVM GE (2100) s (5.559 m) s (12.28 m) S&P(6107) s (12.20 m) 1667 s (27.78 m) Table: Full SVM estimation performance benchmark with S&P500(6107 obs) and GE(2500 obs)

45 rrors Conclusion and Future Work Conclusion and Future Work Conclusion Novel FLEX-SVM is presented to correctly estimate the density of innovations by dynamically learn the parameters of a mixture components The data analysis suggested FLEX-SVM apperas better approach than FIX-SVM in the presence of non-gaussian, particulary heavy tail innovations The desirable precision of estimation achieved by sampling sufficiently large variates from MCMC by the virtue of customized high-performance MCMC engine

46 rrors Conclusion and Future Work Future Work Regard the number of mixture components, K, as parameter. (transdimensional jump between different K) Incorporating with Particle Filters (non-linear structure, non-gaussian error) Construct Sim-Grid parallel simulation network to deal with a variety of data index simultaneously

47 rrors Reference Slides: Extra Meyer R. Berg A. and Yu J. Dic as a model comparison criterion for stochastic volatility model. Gammerman D. and Lopes H. F. Markov Chain Monte Carlo: Stochastic Simulation for Bayesian Inference. Chapman & Hall/CRC, Taylor S. J. Modelling Financial Time Series. John & Willey, Polson G Jacquier, E and E Rossi. Bayesian analysis of stochastic volatility model. The Review of Economic Studies, 65(3): , Jul Shephard N. Kim S. and Chib S. Stochastic volatility: Likelihood inference and comparison with arch models. The Review of Economic Studies, 65(3): , Jul West M. and Harrison J. Bayesian Forecasting and Dynamic Models. springer, 1998.

48 rrors Reference Slides: Extra Asset Pricing: Heston model An asset price that is a geometric Brownian motion: ds S = µdt + σdb s where µ and σ are unknown constants and B s is a Brownian motion under the risk-neutral measure. ( d log S = µ 1 2 σ2) dt + σdb s σ is not a constant but instead evolves as σ(t) = v(t) [ ] dv(t) = κ θ v(t) + γ v(t)db v κ, θ, γ > 0 (3) where B v is a Brownian motion under the risk-neutral measure having a constant correlation ρ with B s

49 rrors Reference Slides: Extra Asset Pricing: Heston Model We could discretize (3) as: log S(t i+1 ) = log S(t i ) + (µ 1 ) 2 σ2 ti t + v(t i ) B s [ ] v(t i+1 ) = v(t i ) + κ θ v(t i ) t + γ v(t i ) B v Simple way to approximate(simulate) the changes B s and B v in two correlated Brownian motion is to generate two independent standard normal z 1 and z 2 and take B s = t z and B s = t z where z = z 1 and z = ρz ρ 2 z 2

50 rrors Reference Slides: Extra Sampling (h ω, φ, σ 2 w, y): Individual Sampler Since, Thus, y t h t N(0, e ht ) and h t h t 1 N(ω + φh t 1, σ 2 w ) p(h t h t, y 1:n ) = p(y t h t )p(h t h t ) = p(y t h t )p(h t h t 1 )p(h t+1 h t ) = N(y t 0, e ht ) N(h t ω + φh t 1, σ 2 w ) N(h t+1 ω + φh t, σw 2 ) ( σ 2 ) = N(y t 0, e ht ) N h t ω t, (1 + φ 2 ) where ω t = ω + φ[(h t 1 ω)(h t+1 ω)] 1 + φ 2

51 rrors Reference Slides: Extra Since, p(y t h t ) exp{ h t 2 y 2 t 2 e ht } log p(y t h t ) 1 2 h t y 2 t 2 e ht Let 1 2 h t y t 2 2 e ht := log f (h t y t ), and observe that e ht is convex function, so can be bounded by a linear function of h t [5] log f (h t y t ) 1 2 h t y 2 t 2 {e ωt (1+ω t ) h t e ωt } := log g (h t y t ) Thus, p(h t h t )p(y t h t ) p(h t h t )f (y t h t ) σ 2 N(h t ω t, 1 + φ 2 )g (h t y t ) }{{} σ 2 N(h t ω t, 1 + φ 2 )

52 rrors Reference Slides: Extra Sampling (h ω, φ, σ 2 w, y): Individual Sampler Jacquire, Polson and Rossi [4] observed that where For t = 1,, n p(h t h t, y 1:n ) N(h t ω t, ω t = ω t + σ φ 2 ) σ 2 2(1 + φ 2 ) [y 2 t e ωt 1] 1 Sample the candidate h t N( ω t, σ 2 1+φ 2 ) 2 Accept h t with probability f ( h t y t, θ) { y 2 g ( h t y t, θ) = exp t [ e h t e ωt (1 + ω t ) + 2 h ]} ωt t e 3 If rejected, then return to step 1 and make a new proposal.

53 rrors Reference Slides: Extra Quality of Approximation ht true h_t Filtered h_t with single normal Index Figure: Approximation by single Normal ht true h_t Filtered h_t with 7 normal mixtures Index Figure: Approximation by mixtures

54 rrors Reference Slides: Extra Density Estimation by Mixture Model Given observation {v 1,, v n } probability density of v can be as; p(v i γ) = K i=1 π i p N (v i µ i, τ 2 i ) where γ = {µ 1,..., µ K, σ1 2,, σ2 K, π 1,, π K } and p N (v i µ i, τi 2) is the normal PDF with mean µ and variance τi 2 Then, N [ K p(v γ) = π i p N (v i µ i, τi 2 ) ] j=1 i=1

55 rrors Reference Slides: Extra Simulation Study: (i) Normal Innovations ɛ t N(0, 1) yt y_t h_t Index Figure: Simulated Data 4000 observation with Normal innovation Error n obs = 4000 ω = , φ = , σ 2 w =

56 phi Histogram of phi phi Index Series phi Lag sig Histogram of sig sig Index Series sig Lag om Histogram of om om Index Series om Lag rrors Reference Slides: Extra FIX-SVM Parameter Estimation: ɛ t N(0, 1) ACF ACF ACF Frequency Frequency Frequency Figure: FIX-SVM parameter estimation with normal innovations(φ, σ w, ω)

57 rrors Reference Slides: Extra FIX-SVM E(h t ) vs true h t : ɛ t N(0, 1) FIX SVM estimated ht(black solid) versus true ht(red dotted) tht Index Figure: FIX-SVM E(h t ) and true h t with normal innovations

58 rrors Reference Slides: Extra FLEX-SVM Innovation Density Estimation: ɛ t N(0, 1) mm flex log X^2 mm x x Figure: FLEX-SVM density log ɛ 2 t estimation

59 phi Histogram of phi phi Index Series phi Lag sig Histogram of sig sig Index Series sig Lag om Histogram of om om Index Series om Lag rrors Reference Slides: Extra FLEX-SVM Parameter Estimation: ɛ t N(0, 1) ACF ACF ACF Frequency Frequency Frequency Figure: FLEX-SVM parameter estimation with normal innovations (φ, σ w, ω)

60 rrors Reference Slides: Extra FLEX-SVM E(h t ) vs true h t : ɛ t N(0, 1) FLEX SVM estimated ht(black solid) versus true ht(red dotted) Time Figure: FLEX-SVM E(h t ) and true h t with normal innovations

61 rrors Reference Slides: Extra DOWJONES(GE): 2516 obs rdata GE.txt

62 rrors Reference Slides: Extra FLEX-SVM Model innovation density estimation log(mm) flex log X^2 mm x x

63 rrors Reference Slides: Extra FLEX-SVM/FIX-SVM Model Comparison(parameter) density.default(x = phi1) density.default(x = sig.w1) Density Density N = 1001 Bandwidth = N = 1001 Bandwidth =

64 rrors Reference Slides: Extra FLEX-SVM/FIX-SVM Model Comparison(h t ) eht E(exp(ht/2)) with flex E(exp(ht/2)) with fixed E(ht) with flex E(ht) with fixed Index

65 rrors Reference Slides: Extra DIC comparison between FIX- and FLEX-SVM Dow Jone FIX FLEX Dow Jone FIX FLEX AA (ALCOA) JNJ AIG JPM AXP KO BA MCD C MMM CAT MO DD MRK DIS MSFT GE PFE GM PG HD T HON UTX HPQ VZ IBM WMT INTC XOM SP