Efficient Analysis of Non-Linear and Non-Gaussian State-Space Representations

研究人员建议开发数值方法，用于在涉及非线性和/或非高斯状态空间模型的应用中实现似然评估和过滤。这些模型的特点是它们对可观测变量（通常是受测量误差影响的观测变量）和不可观测状态变量的相互作用的表示。这些模型在广泛的应用中充当主力，从简化形式的表示，例如，资产收益行为，动态随机一般均衡（DSGE）模型的代表服从基于可能性的估计和推理。过滤的目的是推断状态变量的时间t行为，给定通过时间t的可观察变量的测量，条件是指定的模型，但无条件是过去的状态行为。似然性评估涉及关于通过时间t测量的可观察变量的概率评估，同样以模型为条件，但以状态的当前和过去行为为条件。这两个目标都需要计算必要的积分来消除推理对状态实现的依赖性。当所讨论的模型是线性的并且随机新息是高斯的时，实现似然估计和滤波所需的积分可以通过卡尔曼滤波器解析地计算。然而，理论和经验的考虑往往需要扩展超出线性/高斯框架。例如，与DSGE模型的线性近似相关联的近似误差可以在相应的似然表示中赋予显著的误差。高度精确的非线性模型近似几乎可以消除相关的似然近似误差，但是采用这种模型近似当然需要偏离线性。无论是从线性或正常渲染所需的积分分析棘手的;他们的计算，因此需要使用数值方法。该项目的目标是开发数值方法，实现似然评估和过滤，是简单的实施，并提供准确和数值有效的近似目标积分。这一目标将通过实施有效的重要性抽样（EIS）技术来实现，该技术通过设计产生目标被积函数的最佳全局近似值。在状态空间表示的背景下，由于目标被积体的解析表示的不可用，EIS的实现变得复杂。 因此，该项目的目标？EIS滤波器的开发更广泛的影响：迄今为止获得的结果表明，EIS过滤器可以提供显着的改善，在精度和数值效率超过现有的技术。 结果已经获得了从应用程序到两个模型的研究，以前在过滤文献中的EIS过滤器减少了几个数量级的相对于粒子滤波器的目标被积体的数值标准误差，同时减少计算时间。此外，与EIS滤波器相关的似然近似是模型参数的连续函数，大大简化了模型估计的问题。这些结果表明，通过开发强大的算法实现EIS过滤器在一般的状态空间环境（包括那些涉及高维状态空间和奇异的过渡密度），该项目可以大大拓宽的线性/正常的枷锁可以放松的问题的范围。

The investigators propose to develop numerical methods for achieving likelihood evaluation and filtering in applications involving non-linear and/or non-Gaussian state space models. The hallmark of these models is their representation of the interaction of observable variables (typically observed subject to measurement error) and unobservable state variables. The models serve as workhorses in a broad spectrum of applications ranging from reduced-form representations of, e.g., asset-return behavior, to representations of dynamic stochastic general equilibrium (DSGE) models amenable to likelihood-based estimation and inference. The objective of filtering is to infer time-t behavior of state variables given measurement of the observable variables through time t, conditional on the specified model, but unconditional on the past behavior of the state. Likelihood evaluation involves probability assessments regarding the observable variables measured through time t, again conditional on the model but unconditional on current and past behavior of the state. Both objectives require the calculation of integrals necessary for eliminating the dependence of inferences on realizations of the state.When the model in question is linear and stochastic innovations are Gaussian, the integrals required for achieving likelihood evaluation and filtering can be calculated analytically via the Kalman filter. However, theoretical and empirical considerations often necessitate extensions beyond linear/Gaussian frameworks. For example, approximation errors associated with linear approximations of DSGE models can impart significant errors in corresponding likelihood representations. Highly accurate non-linear model approximations can all but eliminate associated likelihood approximation errors, but the adoption of such model approximations of course entails a departure from linearity. Departures from either linearity or normality render required integrals as analytically intractable; their calculation thus entails the use of numerical methods.The objective of the project is to develop numerical methods for achieving likelihood evaluation and filtering that are straightforward to implement, and that deliver accurate and numerically efficient approximations of targeted integrals. The objective will be pursued through the implementation of efficient importance sampling (EIS) techniques, which by design yield optimal global approximations of targeted integrands. In the context of state-space representations, implementation of EIS is complicated by the unavailability of analytical representations of targeted integrands. Thus the objective of the project ? development of the EIS filter ? amounts to the development of techniques for overcoming this complication in a broad range of settings.Broader Impacts: Results obtained to date indicate that the EIS filter can deliver dramatic improvements in accuracy and numerical efficiency over existing techniques. Results have been obtained from applications to two models studied previously in the filtering literature for which the EIS filter reduces numerical standard errors of targeted integrands by several orders of magnitude relative to the particle filter, while entailing reduced computational times. Moreover, likelihood approximations associated with the EIS filter are continuous functions of model parameters, easing greatly the problem of model estimation. These results suggest that by developing robust algorithms for implementing the EIS filter in general state-space environments (including those involving high dimensional state spaces and singular transition densities), this project can broaden substantially the range of problems for which the shackles of linearity/normality can be loosened.