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

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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.