Inference in Unstable Time Series Models

计量时间序列模型参数的不稳定性在理论上是合理的，在经验上是广泛存在的现象。过去的研究主要集中在测试模型的稳定性上。这个建议关注的是自然的下一步：如何在不稳定的时间序列模型中进行推理？第一个项目涉及一般似然模型中参数时变路径的有效推断，包括参数路径的有效估计的特殊情况。路径的知识在许多方面都是最重要的：(i)它有助于理解不稳定性的来源，（ii）路径的终点与有效的预测密切相关，（iii）经济理论有时暗示着特定形式的不稳定性，因此对路径的推断成为对经济理论的检验。主要结果是，通过处理分数向量序列，在忽略不稳定性的通常的极大似然估计中进行评估，作为高斯局部水平模型，可以获得对路径的渐近有效推断。这特别意味着一个有效的路径估计器可以通过对分数序列应用一个简单的卡尔曼平滑来获得。第二个项目解决了当其他参数时变时，如何在广义矩量方法（GMM）框架中对稳定参数子集进行有效推断的问题。当优化代理适应策略变化时，部分稳定模型自然产生，这在其简化形式的行为方程（如欧拉方程）中引起时间变化。该研究的许多可能的应用之一是如何对稳定的结构参数进行推理，描述技术和偏好。主要的——也许令人惊讶的——结果是，忽略时间变化的标准GMM推理在参数的稳定子集上仍然是渐近有效的。第三个不相关的项目开发了“健壮的”长期方差估计器。这些项目解决了应用时间序列计量经济学家的一阶问题。这些方法是创新的，并且采用了仔细的论证来解决技术难题。从计量经济学理论的角度来看，这些结果可以说代表了对不稳定时间序列模型理解的实质性进展。有些结果是通用的，可以在其他上下文中使用。更广泛的影响：虽然基于技术上复杂的论点，但这项研究的主要结果非常直接适用。考虑到实践中参数不稳定性的普遍存在，本研究应因此对实证研究产生重大影响。这包括对经济理论、预测和政策分析的测试。将通过在大学和会议上的演讲以及免费提供的计算机代码，促进该项目的想法和结果的传播。最后，通过与研究生的合作和研究援助，该提案对学生培训有直接影响。

Instabilities in the parameters of econometric time series models are a theoretically plausible and empirically widespread phenomenon. Past research has largely focused on testing the stability of models. This proposal is concerned with the natural next step: How does one do inference in unstable time series models? The first project concerns efficient inference on the time varying path of the parameters in general likelihood models, including the special case of efficient estimation of the parameter path. Knowledge of the path is of primary interest for many purposes: (i) it helps understand the source of the instability, (ii) the endpoint of the path is intimately linked to efficient forecasting and (iii) economic theory sometimes implies specific forms of instability, such that inference on the path becomes a test of economic theory. The main result is that asymptotically efficient inference on the path may be obtained by treating the sequence of score vectors, evaluated at the usual maximum likelihood estimator that ignores the instability, as a Gaussian local level model. This in particular implies that an efficient path estimator can be obtained by applying a simple Kalman smoother to the sequence of scores. The second project addresses the question of how to conduct valid inference on a subset of stable parameters in a Generalized Methods of Moments (GMM) framework when other parameters are time varying. Partially stable models arise naturally when optimizing agents adapt to policy changes, which induces time variation in their reduced form behavioral equations, such as Euler equations. One of many possible applications of this research is how to conduct inference on the stable structural parameters, describing technology and preferences. The main-and maybe surprising-result is that standard GMM inference, ignoring the time variation, remains asymptotically valid on the stable subset of parameters. A third and unrelated project develops 'robust' long-run variance estimators. The projects address first order problems for applied time series econometricians. The approaches are innovative, and careful arguments are employed to solve the technical difficulties. From an econometric theory point of view, the results arguably represent substantial advances in the understanding of unstable time series models. Some of the results are generic and may be used in other contexts. Broader Impacts: While based on technically sophisticated arguments, the main results of this research are very straightfoward to apply. Given the prevalence of parameter instability in practice, this research should therefore have a major impact on empirical research. This includes tests of economic theory, forecasting and policy analysis. Dissemination of the project's ideas and results will be facilitated through presentations at universities and conferences, as well as freely available computer code. Finally, through collaborations with graduate students and research assistance, the proposal has a direct impact on student training.