Collaborative Research: Perturbation Methods for Markov-Switching Models

摘要提案题目：合作研究：马尔可夫切换模型的摄动方法。提案编号：SES - 1223198本提案旨在开发新的宏观经济学动态模型的计算和估计工具，并将这些工具应用于相关政策问题。动态模型已成为现代宏观经济学的标准工具。因为它们的建立是为了分析经济如何随着时间的推移而演变，它们被用来研究增长和商业周期，设计货币和财政政策，或调查金融和劳动力市场的总体方面，以及许多其他任务。然而，应用宏观经济研究者的工具箱中仍然缺少许多工具。例如，经济学家对政策制度变化的影响没有很好的理解。也就是说，经济政策系统地执行的方式的变化与一个政权内部的变化相反，或者对未来政策的信念如何影响家庭和企业的当前行为。本建议的目标是为这些任务提供一些必要的工具，并说明如何利用这些工具来解决公共政策设计和评估中的重要问题。因此，这项新研究的大部分可能对宏观经济学家，更一般地说，对其他领域也采用动态模型的研究人员具有正外部性。特别是，该建议侧重于如何使用摄动方法从第一原理开始求解马尔可夫切换理性期望（MSRE）模型，即从描述经济主体行为的非线性最优性条件集开始，而不是像以前的文献那样从线性化条件集开始。通常用于自然科学和经济学的摄动方法，为难以分析的模型建立了近似解。MSRE模型允许不同的可能的政策制度（例如，一个鹰派的中央银行家和一个鸽派的中央银行家）随着时间的推移而演变和转换（鹰派的中央银行家之后，有一定的可能性，鸽派的中央银行家等等）。提出了用摄动求解策略函数一阶泰勒展开式的代数方程组。接下来，它展示了基于奇异值分解（SVD）算法并用于常量参数情况的传统方法如何不适用于MSRE模型。相反，该提案使用Gröbner基方法。然后，研究了如何检验确定性，并以具有名义刚性的商业周期模型为例进行了求解。最后，该建议指出，微扰方法也允许我们找到MSRE模型解的高阶近似。美国和国外的决策机构可以应用本建议中提出的方法。例如，联邦储备委员会和几个地区联邦储备银行、国际货币基金组织、欧洲中央银行、英格兰银行以及奥地利、加拿大、德国、意大利、日本、西班牙和瑞典（仅举几例）的中央银行正在积极制定和估计动态宏观经济模型，用于政策分析和预测，这些模型可以从提案中提出的扩展类型中受益。此外，经济学专业正在积累证据，证明这类模型具有良好的预测性能，即使与专业经济学家的判断性预测相比也是如此。本提案概述的更新和更好的工具是为了帮助联邦储备委员会和其他决策机构开发更灵活的模型而设计的，这些模型将有助于在美国实施有效的公共政策。最后，新的计算技术的发展在其他经济学领域（如国际经济学、工业组织或劳动经济学）具有潜在的应用。在其他社会科学领域，研究人员希望使用灵活而强大的工具来解决和估计动态模型。

AbstractProposal Title: Collaborative Research: Perturbation Methods for Markov-Switching ModelsProposal Number: SES - 1223198 This proposal aims to develop new tools for the computation and estimation of dynamic models in macroeconomics and for the application of those tools to relevant policy questions. Dynamic models have become a standard instrument in modern macroeconomics. Because they are built to analyze how the economy evolves over time, they are used to study growth and business cycles, to design monetary and fiscal policy, or to investigate the aggregate aspects of financial and labor markets, among many other tasks. However, many instruments are still missing in the toolbox of the applied macroeconomic researcher. For instance, economists do not have a good understanding of the effects of changes in policy regimes ?that is, variations in the way in which economic policy is systematically conducted in opposition to changes within one regime- or how the beliefs about future policies affect current behavior by households and firms.The goal of this proposal is to provide some of the required tools for these tasks and show how they can be used to address important questions in the design and evaluation of public policy. Consequently, much of this new research may have positive externalities for other economists within macro and, more generally, for researchers in other fields where dynamic models are also employed. In particular, this proposal focuses on how to use perturbation methods to solve Markov switching rational expectations (MSRE) models starting from first principles, that is, from the set of non-linearized optimality conditions that describe the behavior of the economic agents, rather than from the set of linearized ones, as the literature has previously done. Perturbation methods, commonly used in natural sciences and economics, built approximated solution to models that are analytically intractable. MSRE models allow for different possible policy regimes (for instance, a hawkish central banker and a dovish central banker) that evolve and switch over time (hawkish central bankers are followed, with some probability, by dovish central bankers and so on).The proposal derives the set of algebraic equations to be solved to find the first-order Taylor expansion to the policy functions using a perturbation. Next, it shows how the traditional approach, based on singular value decomposition (SVD) algorithms and used in the constant parameter case, does not work in the case of MSRE models. Instead, the proposal uses a Gröbner basis method. Then, it studies how to check for determinacy and, as an example, it solves a business cycle model with nominal rigidities. Finally, the proposal points out how the perturbation approach also allows us to find higher-order approximations to the solution of MSRE models. Policy-making institutions in the U.S. and abroad can apply the methods presented in this proposal. For instance, the Federal Reserve Board and several regional Federal Reserve Banks, the International Monetary Fund, the European Central Bank, the Bank of England, and the central banks of Austria, Canada, Germany, Italy, Japan, Spain, and Sweden (just to name a few) are actively formulating and estimating dynamic macroeconomic models for policy analysis and forecasting that can benefit from the type of extensions presented in the proposal. Moreover, the economics profession is accumulating evidence of the good forecasting performance of this class of models, even when compared with judgmental predictions from staff economists. The newer and better tools that this proposal outlines are designed explicitly for the purpose of helping the Federal Reserve Board and other policy-making institutions to develop more flexible models that will contribute to the implementation of an effective public policy in the U.S. Finally, the development of new computation techniques has potential applications in other fields of economics (such as international economics, industrial organization, or labor economics), and other social sciences where researchers want to solve and estimate dynamic models using flexible, yet powerful tools.