Motivation Our contribution Outline MS-GARCH(1,1) Bayesian estimation Application Conclusion Bayesian Estimation of the Markov-Switching GARCH(1,1) Model with Student-t Innovations David Ardia Department of Quantitative Economics University of Fribourg, Switzerland [email protected]R/Rmetrics User and Developer Workshop, Meielisalp, July 2007 David Ardia University of Fribourg
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Bayesian Estimation of the Markov-Switching … · Bayesian Estimation of the Markov-Switching GARCH(1,1) ... Why using MS-GARCH models? ... Bayesian Estimation of the Markov-Switching
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• 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...);• ...
• 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.
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)
• 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
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
• 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
• 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);
• 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.
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