Alexander Filler

Generalized Linear Dynamic Factor Models: An Approach via Singular Autoregressions

We consider generalized linear dynamic factor models. These models have been developed recently and they are used for high dimensional time series in order to overcome the “curse of dimensionality”. We present a structure theory with emphasis on the zeroless case, which is generic in the setting considered. Accordingly the latent variables are modeled as a possibly singular autoregressive process and (generalized) Yule Walker equations are used for parameter estimation.

Properties of blocked linear systems

This paper presents a systematic study on the properties of blocked linear systems that have resulted from blocking discrete-time linear time invariant systems. The main idea is to explore the relationship between the blocked and the unblocked systems. Existing results are reviewed and a number of important new results are derived. Focus is given particularly on the zero properties of the blocked system as no such study has been found in the literature.

On the zeros of blocked linear systems with single and mixed frequency data

This paper studies properties of blocked systems resulting from blocking discrete linear systems with mixed frequency data. The focus is on the zeros of the blocked systems. We first establish results on the simpler single frequency case, where the unblocked linear systems have all data at the same frequency. In particular, an explicit relation between the system matrix of the unblocked linear systems and that of their corresponding blocked systems is derived. Based on this relation, it is shown that the blocked systems are zero free if and only if the related unblocked systems are zero free. Furthermore, it is illustrated that square systems have zeros generically, i.e. for generic parameter matrices, and the corresponding kernel is of dimension one. With the help of the results obtained for the single frequency case, we then identify a situation in which the blocked systems can be zero free.

Solutions of Yule‐Walker Equations for Singular AR Processes

A study is presented on solutions of the Yule-Walker equations for singular AR processes that are stationary outputs of a given AR system. If the Yule-Walker equations admit more than one solution and the order of the AR system is no less than two, the solution set includes solutions which define unstable AR systems. The solution set also includes one solution, the minimal norm solution, which defines an AR system whose characteristic polynomial has either only stable zeros (implying that only one stationary output exists for this system and it is linearly regular) or has stable zeros as well as zeros of unit modulus, (implying that stationary solutions of this system are a sum of a linearly regular process and a linearly singular process). The numbers of stable and unit circle zeros of the characteristic polynomial of the defined AR system can be characterized in terms of the ranks of certain matrices, and the characteristic polynomial of the AR system defined by the minimal norm solution has the least number of unit circle zeros and the most number of stable zeros over all possible solutions.

Autoregressive models of singular spectral matrices

This paper deals with autoregressive (AR) models of singular spectra, whose corresponding transfer function matrices can be expressed in a stable AR matrix fraction description D−1(q)B with B a tall constant matrix of full column rank and with the determinantal zeros of D(q) all stable, i.e. in |q|>1,q∈C. To obtain a parsimonious AR model, a canonical form is derived and a number of advantageous properties are demonstrated. First, the maximum lag of the canonical AR model is shown to be minimal in the equivalence class of AR models of the same transfer function matrix. Second, the canonical form model is shown to display a nesting property under natural conditions. Finally, an upper bound is provided for the total number of real parameters in the obtained canonical AR model, which demonstrates that the total number of real parameters grows linearly with the number of rows in W(q).

AR models of singular spectral matrices

This paper deals with autoregressive models of singular spectra. The starting point is the assumption that there is available a transfer function matrix W(q) expressible in the form D−1(q)B for some tall constant matrix B of full column rank and with the determinantal zeros of D(q) all stable. It is shown that, even if this matrix fraction representation of W(q) is not coprime, W(q) has a coprime matrix fraction description of the form D̃−1(q)[Im 0]T . It is also shown how to characterize the equivalence class of all autoregressive matrix fraction descriptions of W(q), and how canonical representatives can be obtained. A canonical representative can be obtained with a minimal set of row degrees for the submatrix of D̃(q) obtained by deleting the first m rows. The paper also considers singular autoregressive descriptions of nested sequences of Wp(q), p = p0, p0+1, …, where p denotes the number of rows, and shows that these canonical descriptions are nested, and contain a number of parameters growing linearly with p.

Properties of Blocked Linear Systems

Abstract This paper presents a systematic study on the properties of blocked linear systems that are resulted from blocking discrete-time linear time invariant systems. The main idea is to explore the relationship between the blocked and the unblocked systems. Existing results are reviewed and a number of important new results are derived. Focus is given particularly on the zero properties of the blocked system as no such a study has been found in the literature.

Singular autoregressions for Generalized Dynamic Factor Models

We consider Generalized Linear Dynamic Factor Models in a stationary context, where the latent variables and thus the static and dynamic factors are the sum of a linearly regular and a linearly singular stationary process and the noise process is linearly regular. The linearly singular component may be useful for modeling e.g. business cycles or seasonal fluctuations in the observed variables. We present a structure theory for this case. The emphasis is laid on the autoregressive case. In general the stationary solutions of the autoregressive models considered here consist of a linearly regular and a linearly singular part. The linearly singular part corresponds to the homogeneous solution of a system having stable roots as well as roots of modulus one. We discuss the solutions of the Yule Walker equations for this case.