Statistical inference for nonlinear dynamic model by Markov chain Monte Carlo method

在宏观经济数据的计量经济学分析中，由于微观经济数据不容易获得，个别特征的动态结构或未观察到的变量被忽略，这些随机效应也被认为在聚合微观经济数据后被抵消。然而，有人指出，忽略这些随机效应或未观察到的变量(潜变量)会导致模型参数估计的偏差。最近，微观经济数据开始被披露，例如描述个体特征动态结构的面板数据。使用这些微观经济数据，我们能够对个人经济行为的真实结构进行建模。自1999年以来，各种计量经济学模型被提出用于处理潜在变量的动态建模。当潜在变量较多时，传统的最大似然估计需要重复计算高度多维的数值积分。我们需要使用…更多的超级计算机来进行这样的计算，否则我们不得不以牺牲计算精度为代价来近似可能性。甚至在某些情况下，数值最大化步骤不能收敛到似然函数的最大值。虽然基于对潜在变量的稳健性的方法已经提出了诸如GEE或GMM等方法来估计这些模型，但是这些估计方法被认为是低效的。为了获得模型参数的边际后验分布，众所周知，MCMC估计方法可以通过仿真方法提供精确的多维积分。MCMC方法是计算机密集型的，但这些计算可以由PC完成(请注意，我们不需要超级计算机)。当模型中存在潜在变量或引入辅助变量(对于数据增广方法)时，MCMC样本在某些模型中甚至可能加速收敛到目标分布(后验分布)。我们考虑了各种非线性动态模型：经济周期依赖的持续期模型、具有杠杆效应的随机波动模型、马尔可夫切换和重尾误差(基于混合正态分布)以及外汇市场的随机波动模型。我们首先为这些模型提出了简单的采样方法(如一次采样一个参数的单移动采样器)，并详细阐述了多移动采样器(对一组参数进行采样)以提高收敛到目标后验分布的速度。较少

In the econometric analysis of macroeconomic data, dynamic structures of individual characteristics or unobserved variables have been ignored since microeconomic data were not easily available and these random effects are also considered to be cancelled out after aggregating microeconomic data. However, it has been pointed out that ignoring these random effects or unobserved variables (latent variables) would lead to the bias in the estimation of model parameters. Recently, microeconomic data have started to become disclosed such as panel data which describes dynamic structure of individual characteristics. Using these microeconomic data, we are able to model true structure of individual economic behavior. Various econometric models are proposed to deal with dynamic modeling of latent variables since 1999's.When there are many latent variables, the conventional maximum likelihood estimation requires the repeated evaluations of highly multidimensional numerical integration. We need to u … More se supercomputers to conduct such computations or we have to approximate the likelihood at the expense of computational accuracies. There are even some cases in which the numerical maximization step fails to converge to the maximum of the likelihood functions. Although alternative approaches such as GEE or GMM methods have been proposed to estimate these models based on methods which are robust to the existence of latent variables, those estimation methods are known to be inefficient.In this project, we take Bayesian approach and proposed efficient Markov chain Monte Carlo (MCMC) estimation method for various statistical and econometric nonlinear dynamic models. To obtain marginal posterior distribution of model parameters, the MCMC estimation method is known to provide accurate multidimensional integration using simulation method. The MCMC method is computer intensive, but these computations can be done by PC's (note that we do not need supercomputers). When there exist latent variables in the models or we introduce auxiliary variables (for the data augmentation method), the convergence of MCMC samples to the target distribution (posterior distribution) may even be accelerated in some models.We considered various nonlinear dynamic models : duration models for business cycle dependence, stochastic volatility model with leverage effects, Markov switching and heavy-tailed errors (based on mixture of normal distributions), and stochastic volatility model for foreign exchange markets. We first proposed simple sampling methods (such as single-move sampler which samples one parameter at a time) for these models and elaborate the multi-move samplers (which samples a block of parameters) to improve the speed of the convergence to the target posterior distributions. Less