Session 5B: A worked example EGARCH model

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1 Session 5B: A worked example EGARCH model John Geweke Bayesian Econometrics and its Applications August 7, worked example EGARCH model August 7, / 6

2 EGARCH Exponential generalized autoregressive conditional heteroskedasticity (EGARCH) model The ARCH class of models has been (and continues to be) important in pricing nancial assets and derivatives. EGARCH is one of the most successful members of the class. Model here re ects recent improvements (Durham and Geweke in review). Multiple volatility factors Non-normal return distributions Likelihood and posterior are very badly behaved Many local modes (up to 576 here) due to label-switching; widely regarded as impossible. Example demonstrates robustness due to data tempering in the algorithm, huge speedup factors. worked example EGARCH model August 7, / 6

3 Model and data Sequence of observed asset returns fy t g Evolution of volatility factors v kt = α k v k,t + β k jε t j (/π) / + γ k ε t (k =,..., K ) Then with y t = µ Y + σ Y exp! K v kt / ε t k= p(ε t ) = I p i φ(x i ; µ i, σ i ); E (ε t ) =, var (ε t ) =. i= Data: S&P 5 index closing value p t, //99-3/3/; y t = log (p t /p t ). worked example EGARCH model August 7, 3 / 6

4 SIMD-compatible code Task is to evaluate p (y t j y :t, θ). Using the arithmetic in the previous two slides, transform θ to µ Y, σ Y ; (α k, β k, γ k ) (k =,..., K ) ; p i, µ i, σ i (i =,..., I ). Compute v kt (k =,..., K ) using v kt = α k v k,t + β k jε t j (/π) / + γ k ε t (k =,..., K ) terms ε t and v k,t from evaluation of p (y t j y :t, θ). Compute h t = σ Y exp K k v kt / and ε t = (y t µ Y ) /h t. Initial condition: v k = (k =,..., K ) =) ε = (y µ Y ) /σ Y Evaluate " I p (y t j y :t, θ) = (π) / ht exp p i (ε t i= µ i ) /σ i #. worked example EGARCH model August 7, 4 / 6

5 Data S&P 5 index closing value p t on trading days from January, 99 (t = ) through March 3, (t = T = 5). The returns are y t = log (p t /p t ) (t =,..., T ). worked example EGARCH model August 7, 5 / 6

6 Performance Model Time (secs.) Cycles L log ML NSE EGARCH(, ) , EGARCH(, ) ,7.55. EGARCH(, ) , EGARCH(, ) , EGARCH(, 3) , EGARCH(3, ) , EGARCH(3, 3) , EGARCH(3, 4) , EGARCH(4, 3) , EGARCH(4, 4) , J = 64; N = 496; J N = 6, 44; R = 55 worked example EGARCH model August 7, 6 / 6

7 Sensitivity to number of Metropolis steps R in phase M Metropolis Compute Log Numerical steps time Cycles Marginal Standard R (seconds) L Likelihood Error (NSE ) , , , , , , , , J = 6, N = 496, D =.5, D =. worked example EGARCH model August 7, 7 / 6

8 Interaction between choices of D and R ESS Metropolis Compute Log Numerical rule steps time Cycles Marginal Standard D R (seconds) L Likelihood Error (NSE ) , , , , , , , , , , , , J = 6, N = 496 worked example EGARCH model August 7, 8 / 6

9 Moment of interest: g (θ) = P (Y t+ <.3 j y :t, θ) t = March 3, 9 t = March 3, Posterior Posterior mean NSE RNE mean NSE RNE R = R = R = R = R = R = R = R = egarch_3, J = 64; N = 496; J N = 6, 44 worked example EGARCH model August 7, 9 / 6

10 Multimodal posterior distributions There are J! permutations of the factors and K! permutations of the normal components of ε t. 6 = in the preferred egarch 3 model 4 4 in the egarch_44 model for which the algorithm performed very smoothly. The prior distribution is also symmetric with respect to the factors and normal components of ε t. This leads to re ections or mirror images in the posterior distribution. These permutations present a severe challenge for Markov chain Monte Carlo (MCMC) and are a standard test for such algorithms. (Generic MCMC simply cannot handle the multimodality here.) worked example EGARCH model August 7, / 6

11 Performance of the algorithm Focus on the 3! permutations of parameters (p s, µ s, σ s ) of the normal mixture distribution Consider a parameter vector θ with 3 distinct values of the triplets (p s, µ s, σ s ) There are six distinct ways these could be assigned to the three components of the normal mixture. These permutations de ne six points θ u (u =,..., 6). Thus the posterior distribution has six mirror images in a high-dimensional space. These mirror images will also be evident in any marginal distribution, including two-dimensional marginal distributions. J worked example EGARCH model August 7, / 6

12 log( sigma ) log( sigma ) log( sigma ) log( sigma ) log( sigma ) log( sigma ) log( sigma ) log( sigma ) log( sigma ) t= t=6 t= log( sigma ) log( sigma ) log( sigma ) t=8 t=4 t=3 3 3 log( sigma ) log( sigma ) log( sigma ) t=36 t=4 t=48 log( sigma ) log( sigma ) log( sigma ) worked example EGARCH model August 7, / 6

13 p p p p p p p p p t= t=6 t= p.5 p.5 p t=8 t=4 t= p.5 p.5 p t=36 t=4 t= p.5 p.5 p worked example EGARCH model August 7, 3 / 6

14 Moment of interest: g (θ) = P (Y t+ <.3 j y :t, θ) t = March 3, 9 t = March 3, Posterior Posterior mean NSE RNE mean NSE RNE R = R = R = R = R = R = R = R = egarch_3, J = 6, N = 496, R = 55, D =.5, D =. worked example EGARCH model August 7, 4 / 6

15 Speedup Factors egarch_3 model SMC with J = 6, N = 4, 96, J N = 6, 44, R = 55 MCMC random walk Metropolis, iterated until SMC numerical standard error is matched Marginal Posterior moments Likelihood Type Type SMC particles 6,44 6,44 6,44 SMC time,796,796,796 MCMC iterations ! MCMC time u , 67! Speedup factor u 4, 5.68! J worked example EGARCH model August 7, 5 / 6

16 Conclusion SMC is very closely related to simulated annealing methods for function optimization Essentially the same algorithm can be used to: Conduct Bayesian inference Conduct classical inference (extremum estimators) Solve economic optimization problems Especially attractive for irregular functions, multiple local modes Solve for equilibria Can be used to determine existence and uniqueness In the next - years cheap parallel computing and SIMD algorithms will revolutionize the computational infrastructure of quantitative economics. worked example EGARCH model August 7, 6 / 6

Bayesian Modeling of Conditional Distributions

Bayesian Modeling of Conditional Distributions John Geweke University of Iowa Indiana University Department of Economics February 27, 2007 Outline Motivation Model description Methods of inference Earnings