A General Representation Theorem for Integrated Vector Autoregressive Processes

We study the algebraic structure of an I(d) vector autoregressive process, where d is restricted to be an integer. This is useful to characterize its polynomial cointegrating relations and its moving average representation, that is to prove a version of the Granger representation theorem valid for I(d) vector autoregressive processes.vector autoregressive processes; unit roots; Granger representation theorem; cointegration

Cointegration in functional autoregressive processes

This paper defines the class of $\mathcal{H}$-valued autoregressive (AR) processes with a unit root of finite type, where $\mathcal{H}$ is an infinite dimensional separable Hilbert space, and derives a generalization of the Granger-Johansen Representation Theorem valid for any integration order $d=1,2,\dots$. An existence theorem shows that the solution of an AR with a unit root of finite type is necessarily integrated of some finite integer $d$ and displays a common trends representation with a finite number of common stochastic trends of the type of (cumulated) bilateral random walks and an infinite dimensional cointegrating space. A characterization theorem clarifies the connections between the structure of the AR operators and $(i)$ the order of integration, $(ii)$ the structure of the attractor space and the cointegrating space, $(iii)$ the expression of the cointegrating relations, and $(iv)$ the Triangular representation of the process. Except for the fact that the number of cointegrating relations that are integrated of order 0 is infinite, the representation of $\mathcal{H}$-valued ARs with a unit root of finite type coincides with that of usual finite dimensional VARs, which corresponds to the special case $\mathcal{H}=\mathbb{R}^p$

Recent Developments in Cointegration

It is well known that inference on the cointegrating relations in a vector autoregression (CVAR) is difficult in the presence of a near unit root. The test for a given cointegration vector can have rejection probabilities under the null, which vary from the nominal size to more than 90%. This paper formulates a CVAR model allowing for multiple near unit roots and analyses the asymptotic properties of the Gaussian maximum likelihood estimator. Then two critical value adjustments suggested by McCloskey (2017) for the test on the cointegrating relations are implemented for the model with a single near unit root, and it is found by simulation that they eliminate the serious size distortions, with a reasonable power for moderate values of the near unit root parameter. The findings are illustrated with an analysis of a number of different bivariate DGPs

Inverting a matrix function around a singularity via local rank factorization

This paper proposes a recursive procedure, called the extended local rank factorization (elrf), that characterizes the order of the pole and the coefficients of the Laurent series representation of the inverse of a regular analytic matrix function around a given point. The elrf consists in performing a finite sequence of rank factorizations of matrices of nonincreasing dimension, at most equal to the dimension of the original matrix function. Each step of the sequence is associated with a reduced rank condition, while the termination of the elrf corresponds to a full rank condition; this last step reveals the order of the pole. The Laurent coefficients B n are calculated recursively as B_n = C n + sum_{k=1}^n D_k B_{n−k} , where C_n , D_k have simple closed form expressions in terms of the quantities generated by the elrf. It is also shown that the elrf characterizes the structure of Jordan pairs, Jordan chains, and the local Smith form. The procedure is easily cast in an algorithmic form, and a MATLAB implementation script is provided. It is further found that the elrf coincides with the complete reduction process (crp) in Avrachenkov, Haviv, and Howlett [SIAM J. Matrix Anal. Appl., 22 (2001), pp. 1175–1189]. Using this connection, the results on the elrf provide both an explicit recursive formula for B n implied by the crp, and the link between the crp and the structure of the local Smith form

Minimality of state space solutions of DSGE models and existence conditions for their VAR representation

Standard solution methods of DSGE models do not necessarily deliver minimal state space forms. When the ABCD form is non-minimal, the conditions in the literature are not necessary for the existence of a VAR representation of the observables. In this paper we present necessary and sufficient conditions that are valid in general, and hence can be applied to minimal and non-minimal ABCD forms. If the state space form is minimal, our conditions coincide with those in the literature. These results also clarify that it is possible to have unstable eigenvalues together with a (possibly finite) VAR representation of the observables.JRC.DDG.01-Econometrics and applied statistic