ComputationalToolsforComparing AsymmetricGARCHModelsviaBayes Factors. RicardoS.Ehlers

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1 ComputationalToolsforComparing AsymmetricGARCHModelsviaBayes Factors RicardoS.Ehlers Laboratório de Estatística e Geoinformação- UFPR ehlers ehlers@leg.ufpr.br II Workshop on Statistical Methods in Finance May 15, 2008

2 GARCH Model The GARCH(p,q) model, y t = ǫ t ht, ǫ t D(0, 1) h t p q = ω + α i yt i 2 + β i h t i. i=1 i=1 h t :conditionalvarianceof y t given {y t 1, y t 2,... } ǫ t :i.i.d.errors D(0, 1): denotes a distribution with mean zero and variance 1 ω > 0 α i 0, i = 1,...,p β i 0, i = 1,...,q p i=1 α i + q i=1 β i < 1 Ricardo Ehlers GARCH Models via Bayes Factors 2

3 The conditional likelihood function of the model, l(θ) = n t=s+1 h 1/2 t p ǫ (y t / h t ), s = max(p, q), θ = (θ 0, θ 1,..., θ p+q )= (ω, α 1,..., α p, β 1,..., β q ). Model comparison, error dist. parameters stardard normal θ symmetric standardized t θ = (θ, ν) standardizedged θ = (θ, ν) skewnormal θ = (θ, γ) asymmetric skewt θ = (θ, ν, γ) (Fernandez&Steel,JASA1998) skewged θ = (θ, ν, γ) Ricardo Ehlers GARCH Models via Bayes Factors 3

4 Figure 1: Density functions of the standard normal, standardized Student-t with 5 degrees offreedom,laplaceandgedwith ν = 1.5 f(x) N(0,1) Student t(5) Laplace GED(1.5) x Ricardo Ehlers GARCH Models via Bayes Factors 4

5 Figure 2: Density functions of the skew normal, skew Student-t and skew GED. f(x) Skew Normal(1.5) Skew Student (5,0.6) Skew GED(1.5,2) x Ricardo Ehlers GARCH Models via Bayes Factors 5

6 Bayesian Inference Bayesian inference is based on Bayes theorem: π(θ) L(y θ) p(θ) y: observed data and θ: model parameters. In words posterior distribution likelihood prior distribution. We want to make inferences about a function g(θ) computing its posterior mean E π [g(θ)] = g(θ)π(θ)dθ typically analytically intractable. Ricardo Ehlers GARCH Models via Bayes Factors 6

7 Priors for GARCH Model GED: ν U(0, 2) Student-t: ν Exp(0.1) Skewnormal,skew tandskewged: γ N(0, )truncatedto γ > 0 E(γ) 1and P(0 < γ < 1) θ j U(0, 1), j = 1,...,p + q,restrictedtothestationaryregion θ 0 U(0, ȳ 2 )where ȳ 2 = (1/n) y 2 t. Ricardo Ehlers GARCH Models via Bayes Factors 7

8 Markov chain Monte Carlo Generate θ 1,...,θ m π(θ)(targetdistribution) definingtransitiondensities P(θ t, θ t+1 ). Giventherealizations {θ (t), t = 0, 1,... }then,undercertain conditions, θ (t) t π(θ) and 1 n n t=1 g(θ (t) i ) n E π (g(θ i )) a.s. Ricardo Ehlers GARCH Models via Bayes Factors 8

9 The Metropolis-Hastings Algorithm Startingat θ 0 ateachiteration t = 1, 2,... 1.Sampleacandidatevalue φ q( θ t ). 2.Set θ t+1 = φwithprobability α(φ, θ) = min { 1, π(φ) π(θ) q(θ t } φ) q(φ θ t ) otherwiseset θ t+1 = θ t. qisarbitrarybutinpractice... q(φ θ t ) = q(φ), q(φ θ t ) = q( φ θ t ), q(φ θ t )symmetric. Ricardo Ehlers GARCH Models via Bayes Factors 9

10 Random Walk Metropolis for the GARCH 1.SetinitialvaluesfortheGARCHparameters θ (0) andtransformto φ (0) = log(θ (0) /(1 θ (0) )). 2.Atiteration j,generateavector φ N(φ (j 1), τσ). 3.If p+q j=1 eφ j /(1 + e φ j ) < 1set φ (j) = φ withprobability { α(φ, φ ) = min 1, l(φ )p(φ } ), l(φ)p(φ) otherwiserejectthemoveandset φ (j) = φ (j 1). 4. Repeat until convergence. Ricardo Ehlers GARCH Models via Bayes Factors 10

11 Bayesian Approach to Model Uncertanty Supposewehave Kdifferentmodels M 1,...,M K.Foreachmodel M i : p(y θ i, M i ):likelihoodgiventheobservations p(θ i M i ):apriordistributionfor θ i p(m i ):apriorprobabilityfor M i. Ricardo Ehlers GARCH Models via Bayes Factors 11

12 Applications of Bayes theorem. Within-modelposterior p(θ i y, M i ) = p(y θ i, M i )p(θ i M i ) p(y M i ) Marginallikelihood, p(y M i ) = p(y θ i, M i )p(θ i M i )dθ i Joint posterior distribution, π(m i, θ i ) = p(y θ i, M i ) p(θ i M i ) p(m i ) K p(y θ i, M i ) p(θ i M i ) p(m i )dθ i j=1 Posteriormodelprobabilities, p(m i y) = p(y M i)p(m i ) K p(y M j ) p(m j ) j=1 Ricardo Ehlers GARCH Models via Bayes Factors 12

13 Pairwise comparison, p(m i y) p(m j y) }{{} Posterior Odds = p(y M i) p(y M j ) }{{} Bayes Factor p(m i) p(m j ) }{{} Prior Odds Posterior model probabilities, p(m i y) = [ K j=1 ] 1 p(m j ) B ji p(m i ) where B ji = p(y M j) p(y M i ). Ricardo Ehlers GARCH Models via Bayes Factors 13

14 Approximating the marginal likelihood Chib and Jeliazkov(JASA, 2001) p(y M i ) = p(y θ i, M i )p(θ i M i ) π(θ i y, M i ) α(θ, φ)q(φ θ)π(θ y) = α(θ, θ)q(θ θ )π(θ y) π(θ y) = α(θ, θ )q(θ θ)π(θ y)dθ α(θ, θ)q(θ θ )dθ N 1 N g=1 α(θ(g), θ )q(θ θ (g) ) J 1 J j=1 α(θ, θ (j) ) Ricardo Ehlers GARCH Models via Bayes Factors 14

15 Preliminary results on simulated data Figure3:1000valuesofasimulatedARCH(2)serieswith terrors. z Time Ricardo Ehlers GARCH Models via Bayes Factors 15

16 normal t GED skewnormal skewt skewged c a[1] a[2] nu gamma Post. prob Ricardo Ehlers GARCH Models via Bayes Factors 16

17 Figure 4: Trace plot, estimated density plot and autocorrelations for MCMC output(normal errors) Iterations N = 5000 Bandwidth = Autocorrelation Iterations N = 5000 Bandwidth = Autocorrelation Iterations N = 5000 Bandwidth = Autocorrelation Lag Lag Lag Ricardo Ehlers GARCH Models via Bayes Factors 17

18 A(Small) Toolbox for MCMC BUGS(Bayesian inference Using Gibbs Sampling): WinBUGS, GeoBUGS, OpenBUGS. ( JAGS(Just Another Gibbs Sampler). (www-fis.iarc.fr/ martyn/software/jags). R resources: boa, coda, mcmc, MCMCpack, bayessurv, bayesm, bayesmix, dlm, etc. Updated information: MCMC Preprint Service ( mcmc). Ricardo Ehlers GARCH Models via Bayes Factors 18

Riemann Manifold Methods in Bayesian Statistics

Riemann Manifold Methods in Bayesian Statistics Ricardo Ehlers ehlers@icmc.usp.br Applied Maths and Stats University of São Paulo, Brazil Working Group in Statistical Learning University College Dublin September 2015 Bayesian inference is based on Bayes