Pseudo maximum likelihood estimation of cointegrated multiple frequency I(1) VARMA processes using the state space framework
Creators: name: Ribeiro, Patrick De Matos
Publisher: name: Technische Universität Dortmund
Year: 2020
Subjects: subject: Kointegration
subject: Cointegration
subject: MFI(1) processes
subject: State space systems
subject: 310
subject: Kointegration
subject: ARMA-Modell
subject: Vektor-autoregressives Modell
subject: Zustandsraum

description: Since the seminal contribution of Clive W.J. Granger that introduced the concept of cointegration, the modeling of multivariate (economic) time series with models and methods that allow for unit roots and cointegration has become standard econometric practice with applications ranging from macroeconomics through finance to climate science. With some early exceptions most authors focus on the VAR framework, most notably Johaansen who developed vector error correction models for the empirically most relevant cases, the I(1) and the I(2) case. Limiting cointegration analysis to VAR processes may be too restrictive. For several reasons discussed in this theses it may be advantageous to use the more general VARMA framework. However, cointegration analysis in theVARMA framework is complicated, in particular in the case of higher integration orders or multiple unit roots. One possibility to overcome the difficulties for the cointegration analysis of VARMA processes is the usage of the state space framework. This dissertation provides important tools for cointegration analysis in the state space framework, namely a continuous parameterization and a pseudo maximum likelihood estimator for the multiple frequency I(1) case. Chapter 1 discusses the parameterization of state space processes of arbitrary integration orders. Since the state space representation of a stochastic process is not unique, a canonical form is necessary which selects one unique state space representation. Since this canonical form places restrictions on the system matrices, not all entries of the matrices are free parameters. Some entries are restricted to be zero or depend on other entries. The parametrization is based on the canonical form of Bauer and Wagner (2012), which is particuarly well suited for cointegration analysis. Since there is no continuous parameterization for all state space systems of a given system order, we partition the set of all systems into subsets on which a continuous parameterization is possible. For this we use a multi-index which is chosen in such a way that properties like the unit roots, integration orders and dimensions of the cointegrating spaces remain constant in each subset. In addition to deriving a continuous parametrization, which is almost everywhere continuously invertible, we find a generic subset which is open and dense in the set of all integrated processes with a state space representation of a given system order. Additionally, we discuss the topological structure of the subsets, defining a partial ordering of the multi-indices. Finally, we discuss the implementation of hypotheses on the cointegrating ranks and spaces in the parametrization for the empirically most relevant cases, the multiple frequency I(1) and the I(2) case. We show that all hypotheses commonly tested for VAR processes in these cases can be implemented in the state space framework. This potentially allows for the derivation of pseudo likelihood ratio tests for these hypotheses. Chapter 2 examines pseudo maximum likelihood estimation for multiple frequency I(1) processes. We derive the likelihood function for MFI(1) processes and show that the pseudo maximum likelihood estimator is consistent under relatively mild conditions. Additionally, we show that setting the starting values of the state process to zero does not affect the asymptotic properties of the pseudo maximum likelihood estimator. For the case of a correctly chosen multi-index we additionally derive the asymptotic distribution of the pseudo maximum likelihood estimator, providing the ground work for pseudo likelihood ratio tests. \\ Finally, Chapter 3 consists of an useful tutorial for the analysis of economic time series using the state space framework. Using the analysis of King, Plosser, Stock and Watson (1991) as an illustrative example, we demonstrate that all economically relevant questions examined by these authors can also be analyzed using the state space framework. The analysis of King, Plosser, Stock and Watson (1991) is based on quarterly US economic data from 1949 to 1988. We compare the methods developed for the state space framework, namely the pseudo maximum likelihood estimator from Chapter 2 and the tests based upon it to the methods used by King, Plosser, Stock and Watson (1991), i.e., the DOLS estimator of and the tests for the cointegrating rank of Stock and Watson (1988) and to the vector error correction model for I(1) processes by Johansen (1995). The results obtained with the three different approaches differ, which indicates that the results of empirical applications to time series of dimension six or more of sample sizes below two or three hundred should be interpreted with care. Additionally, we test the robustness of the vector error correction model and the state space framework by repeating the analysis on an extended data set with quarterly US economic data from 1949 to 2018 and on the subset with data from 1989 to 2018. The results of both approaches differ for the three data sets. This may be a hint that there are structural breaks in the economic time series.