Modeling multivariable time series using regular and singular autoregressions

Abstract

The primary aim of this thesis is to study the modeling of high-dimensional time series with periodic missing observations. This study is very important in different branches of science and technology such as: econometric modeling, signal processing and systems and control. For instance, in the field of econometric modeling, it is crucial to provide proper models for national economies to help policy makers with decision making and policy adjustments. These models are built upon available high-dimensional data sets, which are not usually collected at the same rate. For example, some data such as, the employment rate are available on a monthly basis while some others like the gross domestic product (GDP) are collected quarterly. Motivated by applications in econometric modeling, we mainly consider systems, which have two sets of measurement streams, one stream being available at all times and the other one is observed every N-th time. There are two major issues involved with modeling of high-dimensional time series with periodic missing observations, namely, the curse of dimensionality and missing observations. Generalized dynamic factor models (GDFMs), which have been recently introduced in the field of econometric modeling, are exploited to handle the curse of dimensionality phenomenon. Furthermore, the blocking technique from systems and control is used to tackle issues associated with the missing observations. In this thesis, we consider a class of GDFMs and assume that there exists an underlying linear time-invariant system operating at the highest sample rate and our task is to identify this model from the available mixed frequency measurements. To this end, we first provide a very detailed study about zeros of linear systems with alternate missing measurements. Zeros of this class of linear systems are examined when the parameter matrices of a minimal state space representation of a transfer function matrix corresponding to the underlying high frequency system assume generic values. Under this setting, we then illustrate situations under which linear systems with missing observations are completely zero-free. It is worthwhile noting that the obtained condition is very common in an econometric modeling context. Then we apply this result and assume that the underlying high frequency system has an autoregressive (AR) structure. Next, we study identifiability of AR systems from those population second order moments, which can be observed in principle. We propose the method of modified extended Yule-Walker equations to show that the set of identifiable AR systems is an open and dense subset of the associated parameter space i.e. AR systems are generically identifiable. Show more