Automated Likelihood Based Inference for Stochastic Volatility Models using AD Model Builder. Oxford, November 24th 2008 Hans J.
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1 Automated Likelihood Based Inference for Stochastic Volatility Models using AD Model Builder Oxford, November 24th 2008 Hans J. Skaug (with Jun Yu and David Fournier)
2 Outline AD Model Builder ADMB Foundation Stochastic volatility models Implementation in AD Model Builder
3 AD Model Builder (ADMB) 4 th generation optimization/programming environment based on C++ and AD Developed by D. Fournier since around 1990 and sold by Otter Research Ltd. (Canada). Features: Template C++.exe Reverse mode AD and operator overloading Statistical functionality SD s by delta-method, profile likelihood, MCMC Mainly used within the fisheries stock assessment community
4 Global user base
5 The ADMB Foundation Non-profit organization Mission: to make ADMB free and open source software Promote ADMB use across scientific disciplines Extend ADMB capabilities
6 Stochastic volatility (SV) models Daily Pound/Dollar exchange rates (log-differences) Observation equation: X t = log(r t /r t-1 ), r t = Dollar-Pound exchange rate For time t = 1, K, T X = σ e ε, ε ~ N(0,1) h t /2 t x t t h = φh + ση, η ~ N(0,1) t t 1 t Latent process: y t Model parameters θ = ( σ, φ, σ ) x t 2 var( X t ht) = σ x exp( ht) Instance of nonlinear state-space model
7 t Daily Pound/Dollar exchange rates (log-differences) X
8 Likelihood estimation of θ Likelihood function High dimensional integral (dim=1000) Laplace approximation Based on a quadratic appoximation of log[p(x,,θ)]
9 J. Dongarra and F. Sullivan published a list of "The Top Ten Algorithms of the Century." (Computing in Science and Engineering, 2000) MCMC: Numerical integration method when applied to continious systems MCMC has revolutionized and dominated statistical computing since 1990
10 The need to integrate in Statistics Marginal density obtain from joint density = f ( x) f( x, h) du Statistical estimation/inference can only be based on observed quantities: x Latent random variables convenient in formulation of realistic models: h Revitalized interest in Laplace approximation (Rue et al., JRSS-B, to appear)
11 Other models with latent random variables Mixed models: Linear and GLMM Nonlinear regression with random effects State space models: Kalman filters Frailty models in survival analysis Models with spatial structure Semiparametric models (GAMS) Mark-recapture models with heterogeneity
12 Computational challenge Maximize wrt θ Gradient wrt θ needed - Nested optimization: Inner: h Outer: θ -Derivative of det(ω) 3rd order derivatives by AD Classical numerical optimization - 1st order AD useful Hessian by AD: Lifting the burden from the statistician s shoulders
13 ADMB-RE Extention of ADMB to allow applicatoin of Laplace approximation to arbitrary variables in program RE = Random Effects = latent variable Maximizes likelihood wrt θ Generic tool for latent variable models Approach explained in Skaug and Fournier (2006, Computational Statistics & Data Analysis) Skaug (AD 2004, Doing integration by differentiation )
14 Model parameters Target for Laplace approx.
15 Extention of the Basic SV model Skaug and Yu (2008, submitted) Automated Likelihood Based Inference for Stochastic Volatility Models With ADMB-RE it is easy to modify the model
16 SV-t model Model parameters θ = ( σ, φ, συ, ) x
17 Leverage effect Correlated
18 Dollar-Pound data; other data sets show larger likelihood differences Seconds Comparison with WinBUGS (generic MCMC tool) Computation time: 1.7 hours (Meyer and Yu, 2000)
19 Multivariate SV models 3 dimentional model T = 945 Computaton time: 711 seconds
20 Conclusion ADMB simplifies the implementation of SV models, and state-space models more generally Many numerical technicalities are dealt with generically; hidden from the user
21 Some references AD and statistics Skaug, H.J. (2002). Automatic differentiation to facilitate maximum likelihood estimation in nonlinear random effects models. Journal of Computational and Graphical Statistics. 11 p Skaug, H.J., Fournier D.A. (2006) Automatic Approximation of the Marginal Likelihood in non-gaussian Hierarchical Models. Computational Statistics and Data Analysis. 51:
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