Essays on cointegration analysis in the state space framework

Abstract

Cointegration analysis is by now a standard tool in multivariate time series analysis with application ranging from economics to climate science. It was formalized by Soren Johansen and Katarina Juselius and their co-authors for VAR processes. This dissertation, consisting of three chapters corresponding to three articles written in collaboration with my co-authors Professor Dietmar Bauer, Patrick de Matos Ribeiro and Professor Martin Wagner, extends the cointegration theory to VARMA processes using a representation by state space systems.

Chapter 1 focuses on theoretical results regarding the sets of transfer functions corresponding to VARMA systems with similar cointegrating properties, summarized in the so-called state space unit root structure. We develop and discuss different parameterizations for vector autoregressive moving average processes with arbitrary unit roots and (co)integration orders and discuss their topological properties. The general results are exemplified in detail for the empirically most relevant cases, the (multiple frequency or seasonal) I(1) and the I(2) case.

In Chapter 2 we show that the Johansen framework for testing hypotheses on the cointegrating ranks and spaces for MFI(1) processes can be extended to the class of VARMA processes and introduce a state space error correction representation. The estimated cointegrating vectors are asymptotically mixed Gaussian and pseudo likelihood ratio tests for the cointegrating ranks have the same distributions under the null hypothesis in the VARMA case as in the VAR case. In a simulation study our tests outperform the tests by Johansen and Schaumburg in small samples.

In Chapter 3 we develop estimation and inference techniques for I(2) cointegrated VARMA processes cast in state space format. We show consistency and derive the asymptotic distributions of estimators maximizing the Gaussian pseudo likelihood function. Furthermore, we discuss hypothesis tests for the state space unit root structure, leading to the well-known limiting distributions for VAR I(2) processes. Again, a small simulation study shows favorable results for small samples, with our test leading to better performance in determining these integer parameters.