Bayesian Estimation of the Markov-Switching … · Bayesian Estimation of the Markov-Switching...

Motivation Our contribution Outline MS-GARCH(1,1) Bayesian estimation Application Conclusion

Bayesian Estimation of theMarkov-Switching GARCH(1,1) Modelwith Student-t Innovations

David Ardia

Department of Quantitative EconomicsUniversity of Fribourg, [email protected]

R/Rmetrics User and Developer Workshop, Meielisalp, July 2007

David Ardia University of Fribourg


MOTIVATIONWhy using MS-GARCH models?

• High persistence with GARCH models (structural breaks);• Markov-switching ARCH [Hamilton and Susmel, 1994];• Markov-switching GARCH [Gray, 1996; Klaassen, 2002;

Haas et al., 2004];• Very flexible models. Better volatility forecasts.



MOTIVATIONSWhy using the Bayesian approach?

• ML (EM) estimation is difficult (local max);• MCMC methods can explore the full posterior;• Model discrimination is possible (w.r.t. the number of

states);• Probabilistic statements.



MOTIVATIONSWhy using R?

• Quick and easy coding;• C or Fortran implementation to speed up calculations;• Use of the coda (or boa) library to check the MCMC output;• Nice plots (legend, symbols, etc...);• ...



OUR CONTRIBUTION

• MCMC scheme for MS-GARCH(1, 1) model of Haas et al.[2004] with Student-t innovations:

• Model parameters are updated by block;• The state variables are updated in a multi-move manner;• The degrees of freedom parameter is generated via an

efficient rejection technique.

• Application to real data set. In-sample and out-sampleperformance analysis.



OUTLINE

1 MS-GARCH(1, 1) model

2 Bayesian estimation

3 Application

4 Conclusion



MS-GARCH(1, 1) MODELConditional variance process

Haas et al. [2004] hypothesize K separate GARCH(1, 1)processes for the conditional variance:

hkt.= αk

0 + αk1y2

t−1 + βkhkt−1 for k = 1, . . . ,K.

This formulation has practical and conceptual advantages:• Allows to generate the states in a multi-move manner;• Interpretation of the variance dynamics in each regime;• Theoretical results on single-regime GARCH(1, 1)

available.



MS-GARCH(1, 1) MODELModel specification

The MS-GARCH(1, 1) model with Student-t innovations may bewritten as follows:

yt = εt(%hstt )1/2 for t = 1, . . . ,T

εtiid∼ S(0, 1, ν)

%.= ν − 2ν

where the latent process {st} with state space {1, . . . ,K} isassumed to be a stationary, irreducible Markov process withtransition matrix P.



MS-GARCH(1, 1) MODELModel specification (cont.)

Equivalent specification (via data augmentation) to perform theBayesian estimation in a convenient manner:

yt = εt(ωt%hstt )1/2 for t = 1, . . . ,T

εtiid∼ N (0, 1)

ωtiid∼ IG

(ν

2,ν

2

).



BAYESIAN ESTIMATIONSimulating from the joint posterior

Our MCMC sampler can be decomposed as follows:

(s1 · · · sT) using FFBSP using Gibbs

(α10 α

20 · · ·αK

0 α11 α

21 · · ·αK

1 ) using M-H(β1 · · ·βK) using M-H(ω1 · · ·ωT) using Gibbsν using efficient rejection



BAYESIAN ESTIMATIONLabel switching

• Likelihood function and the joint prior are invariant torelabeling the states;

• The joint posterior distribution will also be invariant;• Multimodality (K! modes);• Need an identification constraint. We use the permutation

sampler of Frühwirth-Schnatter [2001].



APPLICATIONData set

• Demeaned daily log-returns of the SMI;• Total of 3’800 observations;• The first 2’500 log-returns are used for the estimation;• The remaining data are used in a forecasting performance

analysis.



APPLICATIONEstimation

• Single-regime and two-state Markov-switching models;• Asymmetric GJR(1, 1) specification of Glosten et al. [1993]:

hkt.= αk

0 +(αk

1I{yt−1>0} + αk2I{yt−1<0}

)y2

t−1 + βkhkt−1;

• Joint posterior sample of size 10’000.



APPLICATIONPosterior results for the single-regime model

• High persistence of the conditional variance process;• Presence of the leverage effect: P(α2 > α1 | y) = 0.999;• Conditional leptokurtosis;• Unconditional variance exists. Posterior mean 1.179

[1.173,1.189]. Empirical variance 1.136.



APPLICATIONPosterior results for the Markov-switching model

• Presence of leverage effect in both states;• Conditional leptokurtosis but posterior mean and median

slightly larger than for the single-regime model;• Infrequent mixing between states;• Posterior mean of the unconditional variance is 0.56

[0.557,0.563] in state 1 and 2.00 [1.992,2.012] in state 2;• Posterior mean of the unconditional variance is 1.134

[1.128,1.139]. Empirical variance 1.136.



APPLICATIONMisspecification tests

• Probability integral transforms [see Diebold et al., 1998];• Test of autocorrelation and autocorrelation of squares;• Joint test for zero mean, unit variance, zero skewness, and

the absence of excess kurtosis;• No evidence of misspecification at the 5% significance

level for both models.



APPLICATIONDeviance information criterion

• Alternative to AIC and BIC, as well as LR which are notconsistent in a Markov-switching context;

• The DIC consists of two terms: a component thatmeasures the goodness-of-fit and a penalty term forincreasing model complexity (effective number ofparameters);

• Smallest DIC is preferred.

Model DIC D pD

GJR 6770.4 6765.6 4.76[6769.9,6770.8] [6765.3,6765.8] [4.49,4.93]

MS-GJR 6713.3 6704.4 8.84[6712.6,6713.8] [6793.9,6794.9] [8.49,9.04]

[•]: 95% confidence interval obtained by bootstrap.



APPLICATIONModel likelihood

• Estimate the model likelihood for the two models;• Bridge sampling of Meng and Wong [1996].

Model ln p(y)

GJR -3408.04 (2.644)MS-GJR -3389.66 (3.191)

ln p(y): bridge sampling; (•) numericalstandard error (×100).



APPLICATIONForecasting performance analysis

• We forecast the one-day ahead VaR (backtest);• Quantile of interest that corresponds to the probability

associated to a certain extreme loss;• Compute the predictive VaR by simulation;• Test the joint hypothesis of independence and

unconditional coverage of the VaR [Christoffersen, 1998].



APPLICATIONForecasting performance analysis (cont.)

• We consider the GJR and MS-GJR models;• Also a “rolling” GJR model:

• 750 log-returns used to estimate the model;• Next 50 log-returns used as a forecasting window;• The methodology fulfills the recommendations of the BIS in

the use of internal models.

• Test the models over the 1’300 out-of-sample observations.



APPLICATIONForecasting performance analysis (cont.)

• MS-GJR and rolling GJR outperform the static GJR model;• MS-GJR and rolling GJR perform equally well;• However, the MS-GJR model has two advantages:

• Can anticipate structural breaks in the conditional varianceprocess through the filtering probabilities;

• MS-GJR needs only to be estimated once.

• Rolling GJR is merely and ad-hoc approach.



CONCLUSION

• MS-GARCH more flexible than GARCH;• Bayesian estimation has many advantages;• We provide a new block updating scheme for performing

the Bayesian estimation for the MS-GARCH model of Haaset al. [2004] with Student-t innovations;

• Better in-sample and out-sample performance than singleregime GARCH.


References References

REFERENCES

Christoffersen PF (1998). “Evaluating Interval Forecasts.” International Economic Review, 39(4), 841–862.Symposium on Forecasting and Empirical Methods in Macroeconomics and Finance.

Diebold FX, Gunther TA, Tsay AS (1998). “Evaluating Density Forecasts with Applications to Financial RiskManagement.” International Economic Review, 39(4), 863–883.

Dueker MJ (1997). “Markov Switching in GARCH Processes and Mean-Reverting Stock-Market Volatility.” Journal ofBusiness and Economic Statistics, 15(1), 26–34.

Frühwirth-Schnatter S (2001). “Markov Chain Monte Carlo Estimation of Classical and Dynamic Switching andMixture Models.” Journal of the American Statistical Association, 96(453), 194–209.

Glosten LR, Jaganathan R, Runkle DE (1993). “On the Relation between the Expected Value and the Volatility ofthe Nominal Excess Return on Stocks.” Journal of Finance, 48(5), 1779–1801.

Gray SF (1996). “Modeling the Conditional Distribution of Interest Rates as a Regime-Switching Process.” Journalof Financial Economics, 42(1), 27–62.

Haas M, Mittnik S, Paolella MS (2004). “A New Approach to Markov-Switching GARCH Models.” Journal ofFinancial Econometrics, 2(4), 493–530.

Hamilton JD, Susmel R (1994). “Autoregressive Conditional Heteroskedasticity and Changes in Regime.” Journal ofEconometrics, 64(1–2), 307–333.

Klaassen F (2002). “Improving GARCH Volatility Forecasts with Regime-Switching GARCH.” Empirical Economics,27(2), 363–394.

Meng XL, Wong WH (1996). “Simulating Ratios of Normalizing Constants via a Simple Identity: a TheoreticalExploration.” Statistica Sinica, 6, 831–860.


References References


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