Chan, Joshua and Strachan, Rodney (2012): Estimation in NonLinear NonGaussian State Space Models with PrecisionBased Methods.

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Abstract
In recent years state space models, particularly the linear Gaussian version, have become the standard framework for analyzing macroeconomic and financial data. However, many theoretically motivated models imply nonlinear or nonGaussian specifications or both. Existing methods for estimating such models are computationally intensive, and often cannot be applied to models with more than a few states. Building upon recent developments in precisionbased algorithms, we propose a general approach to estimating highdimensional nonlinear nonGaussian state space models. The baseline algorithm approximates the conditional distribution of the states by a multivariate Gaussian or t density, which is then used for posterior simulation. We further develop this baseline algorithm to construct more sophisticated samplers with attractive properties: one based on the acceptreject MetropolisHastings (ARMH) algorithm, and another adaptive collapsed sampler inspired by the crossentropy method. To illustrate the proposed approach, we investigate the effect of the zero lower bound of interest rate on monetary transmission mechanism.
Item Type:  MPRA Paper 

Original Title:  Estimation in NonLinear NonGaussian State Space Models with PrecisionBased Methods 
Language:  English 
Keywords:  integrated likelihood; acceptreject MetropolisHastings; crossentropy; liquidity trap; zero lower bound 
Subjects:  C  Mathematical and Quantitative Methods > C3  Multiple or Simultaneous Equation Models ; Multiple Variables > C32  TimeSeries Models ; Dynamic Quantile Regressions ; Dynamic Treatment Effect Models ; Diffusion Processes ; State Space Models E  Macroeconomics and Monetary Economics > E5  Monetary Policy, Central Banking, and the Supply of Money and Credit > E52  Monetary Policy C  Mathematical and Quantitative Methods > C1  Econometric and Statistical Methods and Methodology: General > C15  Statistical Simulation Methods: General C  Mathematical and Quantitative Methods > C1  Econometric and Statistical Methods and Methodology: General > C11  Bayesian Analysis: General 
Item ID:  39360 
Depositing User:  Joshua Chan 
Date Deposited:  10. Jun 2012 12:49 
Last Modified:  14. Mar 2015 18:31 
References:  Andrieu, C. , K. K. Berthelsen, A. Doucet, and G. O. Roberts. The expected auxiliary variable method for Monte Carlo simulation. Technical report, 2007. Andrieu, C., A. Doucet and R. Holenstein. Particle Markov chain Monte Carlo methods. Journal of the Royal Statistical Society Series B, 72:269�342, 2010. Carter, C. K., and R. Kohn. On Gibbs sampling for state space models. Biometrika, 81:541�553, 1994. Chan, J. C. C., and I. Jeliazkov. MCMC estimation of restricted covariance matrix. Journal of Computational and Graphical Statistics, 18:457480, 2009a. Chan, J. C. C., and I. Jeliazkov. Efficient simulation and integrated likelihood estimation in state space models. International Journal of Mathematical Modeling and Numerical Optimisation, 1:101120, 2009b. Chib, S., and E. Greenberg. Understanding the MetropolisHastings algorithm. The American Statistician, 49(4):327�335, 1995. Chib, S., and I. Jeliazkov. Acceptreject MetropolisHastings sampling and marginal likelihood estimation. Statistica Neerlandica, 59:3044, 2005. Chib, S., F. Nardari, and N. Shephard. Markov chain Monte Carlo methods for stochastic volatility models. Journal of Econometrics, 108(2):281316, 2002. Cogley, T. and T. J. Sargent. Evolving postWorld War II inflation dynamics, NBER Macroeconomic Annual, 16, 331373, 2001. Cogley, T., and T. J. Sargent. Drifts and volatilities: monetary policies and outcomes in the post WWII US. Review of Economic Dynamics, 8(2):262302, 2005. de Jong, P., and N. Shephard. The simulation smoother for time series models. Biometrika, 82:339350, 1995. Doucet, A., and A. M. Johansen. A tutorial on particle �ltering and smoothing: Fifteen years later. In D. Crisan and B. Rozovskii, editors, The Oxford Handbook of Nonlinear Filtering. Oxford University Press, Oxford, 2011. Doucet, A., N. De Freitas, and N.J. Gordon, editors. Sequential Monte Carlo Methods in Practice. Springer, New York, 2001. Durbin, J., and S. J. Koopman. Monte Carlo maximum likelihood estimation for nonGaussian state space models. Biometrika, 84:669�684, 1997. Durbin, J., and S. J. Koopman. A simple and efficient simulation smoother for state space time series analysis. Biometrika, 89:603615, 2002. FernandezVillaverde, J., and J. F. RubioRamirez. Estimating macroeconomic models: A likelihood approach. Review of Economic Studies, 74(4):10591087, 2007. Flury, T., and N. Shephard. Bayesian inference based only on simulated likelihood: particle filter analysis of dynamic economic models. Economics Series Working Papers 413, University of Oxford, Department of Economics, 2008. FrüwirthSchnatter, S. Data augmentation and dynamic linear models. Journal of Time Series Analysis, 15:183202, 1994. FrühwirthSchnatter, S., and R. Frühwirth. Auxiliary mixture sampling with applications to logistic models. Computational Statistics & Data Analysis, 51(7):35093528, 2007. FrühwirthSchnatter, S., and H. Wagner. Auxiliary mixture sampling for parameterdriven models of time series of counts with applications to state space modelling. Biometrika, 93:827841, 2006. Geweke, J. Bayesian inference in econometric models using Monte Carlo integration. Econometrica, 57(6):13171339, 1989. Iwata, S. and S. Wu. Estimating monetary policy effects when interest rates are close to zero. Journal of Monetary Economics, Elsevier, vol. 53(7):13951408, 2006. Jungbacker, B., and S. J. Koopman. Monte Carlo estimation for nonlinear nonGaussian state space models. Biometrika, 94:827839, 2008. Keith, J. M., D. P. Kroese, and G. Y. Sofronov. Adaptive independence samplers. Statistics and Computing, 18:409420, 2008. Kim, S., N. Shepherd, and S. Chib. Stochastic volatility: Likelihood inference and comparison with ARCH models. Review of Economic Studies, 65(3):361393, 1998. Koop, G. Bayesian Econometrics. Wiley & Sons, New York, 2003. Koop G., R. LéonGonzález and R. W. Strachan. On the Evolution of Monetary Policy. Journal of Economic Dynamics and Control 33, 9971017, 2009. Kroese, D. P., T. Taimre, and Z. I. Botev. Handbook of Monte Carlo Methods. John Wiley & Sons, New York, 2011. McCausland, W. J.. The HESSIAN method (Highly Efficient State Smoothing, In A Nutshell). University of Montreal Department of Economics Working Paper Series, 200803, 2008. McCausland, W. J., S. Millera, and D. Pelletier. Simulation smoothing for statespace models: A computational efficiency analysis. Computational Statistics and Data Analysis, 55:199�212, 2011. Pourahmadi, M. Joint meancovariance models with applications to longitudinal data: Unconstrained parameterisation. Biometrika, 86:677690, 1999. Pourahmadi, M. Maximum likelihood estimation of generalised linear models for multivariate normal covariance matrix. Biometrika, 87:425435, 2000. Primiceri, G. E. Time varying structural vector autoregressions and monetary policy. Review of Economic Studies, 72(3):821852, 2005. Reifschneider, D. and J. C. Williams. Three lessons for monetary policy in a low in�flation era. Journal of Money, Credit and Banking 32:936966, 2000. Roberts, G. O., and J. S. Rosenthal. General state space Markov chains and MCMC algorithms. Probability Surveys, 1:20�71, 2004. Rubinstein, R. Y., and D. P. Kroese. The CrossEntropy Method: A Unified Approach to Combinatorial Optimization MonteCarlo Simulation, and Machine Learning. SpringerVerlag, New York, 2004. RubioRamirez, J. F., and J. FernandezVillaverde. Estimating dynamic equilibrium economies: linear versus nonlinear likelihood. Journal of Applied Econometrics, 20(7):891910, 2005. Rue, H., S. Martino, and N. Chopin. Approximate Bayesian inference for latent Gaussian models by using integrated nested Laplace. Journal of the Royal Statistical Society Series B, 71:319392, 2009. Shephard, N., and M. K. Pitt. Likelihood analysis of nonGaussian measurement time series. Biometrika, 84:653667, 1997. Sims, C. and T. Zha, Were there regime switches in macroeconomic policy? American Economic Review, 96, 5481, 2006. Smith, M., and R. Kohn. Parsimonious covariance matrix estimation for longitudinal data. Journal of the American Statistical Association, 97:1141�1153, 2002. Strickland, C. M., C. S. Forbes, and G. M.Martin. Bayesian analysis of the stochastic conditional duration model. Computational Statistics and Data Analysis, 50:22472267, 2006. Tierney, L. Markov chains for exploring posterior distributions. The Annals of Statistics, 22(4):17011728, 1994. 
URI:  http://mpra.ub.unimuenchen.de/id/eprint/39360 