Collaborative Research: Identification, Estimation, and Inference of DSGE Models

本项目发展了动态随机一般均衡（DSGE）模型的识别理论，并研究了其对估计和推理的影响。DSGE模型现在已经达到了可以分析重要宏观经济问题的复杂程度。虽然这些模型中的参数过去需要校准，但在过去二十年中，数值上的进步使得估计具有多达100个参数的模型成为可能。然而，研究人员意识到，由于识别问题，并非所有感兴趣的参数都可以一致地估计：不同的结构参数可能导致无法区分的结果。尽管认识到这一识别问题，但尚未制定一种程序，以系统的方式告诉我们有多少参数是可识别的，如果是，则是哪些参数。该项目的第一个目标是研究DSGE模型可识别的条件。分析是非标准的，因为识别的经典条件依赖于在DSGE模型中通常不成立的假设。我们的建议是使用DSGE模型的紧密结构，以建立新的等级和顺序条件进行识别。这些条件尚未在文献中提出。重点是协方差平稳过程，第一步是明确两个动力系统可以具有观测等效谱密度的意义。根据控制理论的结果，证明等效DSGE模型的马尔可夫（脉冲响应）参数和误差方差必须通过相似变换相关联。然后使用这些限制来建立条件，在这些条件下，DSGE模型可以从可观测的光谱中识别出来。我们将证明，即使DSGE模型的简化形式参数不是可识别的，它也是可识别的。将制定正式的鉴定条件，以明确地处理测量误差的存在。本课题的第二个目标是研究DSGE模型参数的估计。一个众所周知的事实是，不能始终如一地估计不可识别的参数。这对频率分析和贝叶斯分析都有重要意义，因为当使用平坦先验时，局部不识别会导致后验的奇怪行为。尽管这个问题很重要，但关于非识别DSGE模型的全信息估计的文献相对较少。DSGE约简形式参数无法从频谱中识别的结果对仅使用自协方差子集的有限信息估计具有重要意义。该项目将提供由动态方程和恒等式组成的“可识别简化形式”的完整表征。对于完全信息估计和有限信息估计的挑战是降阶的误差方差。该项目将为奇异系统开发新的估计方法，而不会丢弃信息或增加随机误差。该项目的第三个目标是关注DSGE模型中的推理。部分确定的模型为测试带来了挑战性的问题。该项目侧重于两个问题。首先，如何检验奇异系统（如DSGE模型）的统计假设，当感兴趣的参数是点识别的，而干扰参数是点识别的。其次，当感兴趣的参数本身只有集合识别时，如何在动态和可能的奇异模型中进行推理。这些问题具有挑战性，但在DSGE框架之外也具有相关性。智力优势：目前，没有正式的识别结果，DSGE模型允许比内生变量更少的冲击。这个项目提供了易于评估的等级和顺序条件，以便从业者可以检查识别。未识别约简形式参数时结构参数的估计与推理，以及奇异系统的集识别都是新的研究课题。研究结果将对计量经济学理论作出新的贡献。更广泛的影响：DSGE模型越来越多地用于政策分析，因此这项工作的影响超出了对线性动力系统的更好的方法理解。该工作也与需求系统等其他奇异模型的估计和推理有关。部分识别的动态模型的推理具有普遍的意义。计算机代码将提供给科学界用于非商业、研究和教育目的。

This project develops the theory for identification of dynamic stochastic general equilibrium (DSGE) models and studies its implications for estimation and inference. DSGE models have now reached the level of sophistication to permit analysis of important macroeconomic issues. Whereas the parameters in these models used to be calibrated, numerical advances in the last two decades have made it possible to estimate models with as many as a hundred parameters. Researchers are, however, aware that not all the parameters of interest can be consistently estimated because of identification problems: that different structural parameters can lead to indistinguishable outcomes. In spite of the recognition of this identification issue, a procedure has yet to be developed that tells us in a systematic manner how many parameters are identifiable, and if so which ones.The first goal of this project is to study conditions under which a DSGE model is identifiable. The analysis is nonstandard as classical conditions for identification rely on assumptions that do not generally hold in DSGE models. Our proposal is to use the tight structure of DSGE models in order to establish new rank and order conditions for identification. Such conditions have not yet been proposed in the literature. T he focus is on covariance stationary process and the first step is to make precise the sense in which two dynamical systems can have observationally equivalent spectral densities. Adapting results from control theory, it is shown that the Markov (impulse response) parameters and the error variance of equivalent DSGE models must be related through a similarity transformation. These restrictions are then used to establish conditions under which DSGE models are identifiable from the spectrum of the observables. We will show that a DSGE model is identifiable even when its reduced form parameters are not. Formal identification conditions will be developed to explicitly handle the presence of measurement errors. The second goal of this project is to study the estimation of DSGE model parameters. It is a well known fact that parameters that are not identifiable cannot be consistently estimated. This has important implications for both frequentist as well as Bayesian analysis as local non-identification leads to strange behavior of posteriors when flat priors are used. In spite of the importance of this problem, the literature on full information estimation of non-identified DSGE models is relatively small. The result that DSGE reduced form parameters are not identifiable from the spectrum has important implications for limited information estimation which uses only a subset of the autocovariances. The project will provide a complete characterization of an 'identifiable reduced form' consisting of dynamic equations and identities. The challenge for both full and limited information estimation is an error variance of reduced rank. The project will develop new estimation methods for singular systems without throwing away information or adding stochastic errors.The third goal of the project is to focus on inference in DSGE models. Partially identified models pose challenging problems for testing. The project focuses on two issues. First, how to test statistical hypothesis in singular systems such as DSGE models when the parameters of interest are point identified but the nuisance parameters are not. Second, how to conduct inference in dynamic and possibly singular models when the parameters of interest are themselves only set identified. These problems are challenging, but are also relevant outside of the DSGE framework.Intellectual Merit: Currently, there exists no formal identification results for DSGE models that allow for fewer shocks than endogenous variables. This project provides easy to evaluate rank and order conditions for identification that practitioners can check. Estimation and inference of structural parameters when the reduced form parameters are not identified, and set identification in singular systems are both new research topics. The results will be a new contribution to econometric theory.Broader Impacts: DSGE models are increasingly used in policy analysis, so the impact of this work goes beyond a better methodological understanding of linear dynamical systems. The work is also relevant to estimation and inference of other singular models such as demand systems. Inference of partially identified dynamic models is of general interest. The computer code will be made available to the scientific community for non-commercial, research, and educational purposes.