Manfred Deistler

Consistency and relative efficiency of subspace methods

We give a consistency proof for two subspace methods. We then show the asymptotic equivalence of a special subspace method and the initial estimate proposed by Hannan and Rissanen. Finally, a simulation study comparing two subspace methods and the maximum-likelihood method is performed.

Statistical analysis of novel subspace identification methods

Abstract In this paper four subspace algorithms which are based on an initial estimate of the state are considered. Three novel algorithms are introduced and compared with an algorithm which is essentially equal to the N4SID algorithm by Van Overschee and De Moor. For the algorithms considered a consistency result is proved. In a simulation study the relative (statistical) efficiency of these algorithms in relation to the maximum likelihood algorithm is investigated.

Consistency and asymptotic normality of some subspace algorithms for systems without observed inputs

Asymptotic normality for a class of subspace algorithms, which estimate the state in a first step, is derived. Expressions for the asymptotic variance are given. Linear systems with unobserved white noise inputs are considered. A class of subspace estimates for the system matrices obtained by estimating the state in the first step is analyzed. The main result presented here states asymptotic normality of subspace estimates. In addition, a consistency result for the system matrix estimates is given. An algorithm to compute the asymptotic variances of the estimates is presented. In a final section the implications of the result are discussed.

Estimation of vector ARMAX models

The asymptotic properties of maximum likelihood estimates of a vector ARMAX system are considered under general conditions, relating to the nature of the exogenous variables and the innovation sequence and to the form of the parameterization of the rational transfer functions, from exogenous variables and innovations to the output vector. The exogenous variables are assumed to be such that the sample serial covariances converge to limits. The innovations are assumed to be martingale differences and to be nondeterministic in a fairly weak sense. Stronger conditions ensure that the asymptotic distribution of the estimates has the same covariance matrix as for Gaussian innovations but these stronger conditions are somewhat implausible. With each ARMAX structure may be associated an integer (the McMillan degree) and all structures for a given value of this integer may be topologised as an analytic manifold. Other parameterizations and topologisations of spaces of structures as analytic manifolds may also be considered and the presentation is sufficiently general to cover a wide range of these. Greater generality is also achieved by allowing for general forms of constraints.

Linear dynamic errors-in-variables models: Some structure theory

Abstract This paper gives a survey of some recent results on problems of identifiability (or, more general, of the relation between the observations and certain system characteristics) for linear dynamic errors-in-variables models. For a large part of the paper the noise components are assumed to be mutually uncorrelated. After the general problem statement, a rather complete analysis of the single-input–single-output case is given. Also the case of three variables and the case where the number of inputs is equal to the number of outputs are discussed in detail. Finally, the use of higher-order cummulant spectra for identifiability is investigated.

A Structure Theory for Linear Dynamic Errors-in-Variables Models

We deal with problems connected with the identification of linear dynamic systems in situations when inputs and outputs may be contaminated by noise. The case of uncorrelated noise components and the bounded noise case is considered. If also the inputs may be contaminated by noise, a number of additional complications in identification arise, in particular the underlying system is not uniquely determined from the population second moments of the observations. A description of classes of observationally equivalent systems is given, continuity properties of mappings relating classes of observationally equivalent systems to the spectral densities of the observations are derived and the classes of spectral densities corresponding to a given maximum number of outputs are studied.

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.

Linear errors-in-variables models

In this paper we are concerned with the statistical analysis of systems, where both, inputs and outputs, are contaminated by errors. Models of this kind are called error-in-variables (EV) models. Let x t * . and y t * denote the “true” inputs and outputs respectively and let xt and yt denote the observed inputs and outputs, then the situation can be illustrated as follows: Thereby ut and vt are the errors of the inputs and the outputs respectively.

Identifiability and Consistent Estimability in Econometric Models

In this paper the concepts of identifiability and of exact and consistent estimability for econometric models are introduced in a rather general framework, and the relations between these concepts are investigated. As a main result we obtain conditions under which identifiability and consistency are equivalent almost everywhere.

Generalized linear dynamic factor models - a structure theory

In this paper we present a structure theory for generalized linear dynamic factor models (GDFM¿s). Emphasis is laid on the so-called zeroless case. GDFM¿s provide a way of overcoming the ¿curse of dimensionality¿ plaguing multivariate time series modelling, provided that the single time series are similar. They are used in modelling and forecasting for financial and macroeconomic time series.