Estimation and inference theory for (co)integrated processes in the state space representation

虽然（协）集成过程的估计和规范理论在向量自回归（VAR）设置中得到了很好的研究，但对于向量自回归移动平均（VARMA）过程框架中的相应理论以及一阶（l(1)），二阶（l(2)）以及季节集成（MFI(1)）过程的经验相关集成过程的等效状态空间表示知之甚少。VARMA过程，特别是其状态空间表示，由于其与动态随机一般均衡（DSGE）模型的解有很强的联系，最近在（经验）宏观经济学中获得了很多关注。对于具有（co）积分变量的DSGE模型的计量分析，需要对具有约束的状态空间模型进行估计和推理理论。通常使用的无限制VAR近似不包含模型所隐含的限制。此外，它们不适用于非可逆系统的分析，这可能会给结构冲击的识别带来问题。此外，对于具有大量内生变量的模型，不受限制的VAR系统需要大量参数，而使用更灵活的状态空间系统可以大大减少这些参数。此外，对l(1)、l(2)以及MFI(1)系统的处理消除了对数据去趋势化和去季节化的需要，因此可以在建模过程中纳入有关变量长期行为的有价值的信息。因此，该项目的主要目标是发展估计和推理理论，其中(i)允许合并对变量动态特性的限制（例如由积分特性和（多项式）协整关系的存在引起），（ii）允许对不可逆转系统的分析，以及（iii）最佳地利用状态空间系统的灵活性。这一目标将通过以下方式实现：(i)基于申请人最近开发的单位根过程的规范形式开发参数化，允许纳入经济理论引起的限制，（ii）为给定整数参数（例如状态空间的维度）导出拟极大似然估计的渐近结果，并开发潜在经济理论所隐含的限制的测试。（iii）寻找用于初始化拟似然函数最大值的一致估计，（iv）定义并彻底评估用于指定整数参数的算法。另一个主要成就是在工具箱中实现方法（在MATLAB和R中）。这些目标将通过结合Dietmar Bauer的状态空间建模能力与Martin Wagner的估计理论和协整分析的经济应用的深刻知识来实现。

While estimation and specification theory for (co-)integrated processes is well-studied in the vector-autoregressive (VAR) setting, still little is known about corresponding theory in the framework of vector-autoregressive-moving-average (VARMA) processes and the equivalent state space representation for the empirically relevant integrated processes of order one (l(1)), order two (l(2)), as well as seasonally integrated (MFI(1)) processes. VARMA processes and in particular their state space representation have recently gained a lot of attention in (empirical) macroeconomics due to their strong connection to the solutions of dynamic stochastic general equilibrium (DSGE) models.For the econometric analysis of DSGE models with (co)integrated variables, theory for estimation and inference is needed for state space models with restrictions. Commonly used unrestricted VAR approximations do not encompass the restrictions implied by the models. Furthermore they are not suitable for the analysis of non-invertible systems, which may be problematic for the identification of structural shocks. Moreover, unrestricted VAR systems for models with a large number of endogenous variables entail the need for a large number of parameters which can be substantially reduced with the use of the more flexible class of state space systems. Additionally, the treatment of l(1), l(2), as well as MFI(1) systems eliminates the need for de-trending and de-seasonalizing the data, therefore allowing to incorporate valuable information on the long-run behavior of the variables in the modelling process.Consequently, the main goal of the project is to develop estimation and inference theory which (i) allows to incorporate the restrictions on the dynamic properties of the variables (induced for instance by integration properties and the presence of (polynomial) co-integrating relations), (ii) admits the analysis of non-invertible systems and (iii) optimally exploits the flexibility of state space systems.This goal will be accomplished by (i) developing a parametrization based on the canonical form for unit root processes recently developed by the applicants allowing to incorporate the restrictions induced by economic theory, (ii) deriving asymptotic results for quasi-maximum likelihood estimators for given integer parameters (such as the dimension of the state space) and developing tests for restrictions implied by underlying economic theory, (iii) finding consistent estimators for initializing the maximization of the quasi likelihood function, (iv) defining and thoroughly evaluating algorithms for the specification of integer parameters. Another main achievement is the implementation of the methods in toolboxes (in MATLAB and R). These goals will be achieved by combining the state-space modeling competences of Dietmar Bauer with the profound knowledge on estimation theory and economic application of cointegration analyis of Martin Wagner.