A theorem of Kalman and minimal state-space realization of Vector Autoregressive Models

TL;DR

This paper introduces a minimal AR-state-space realization for vector autoregressive models, utilizing Kalman's theorem, and explores estimation methods, invariance properties, and applications in time series analysis.

Contribution

It develops a new minimal AR-state-space realization framework for VAR models based on Kalman's theorem, including estimation techniques and invariant properties.

Findings

Derived a minimal AR-state-space form for VAR models.

Established invariance properties of the likelihood function.

Provided estimation examples and discussed configuration selection.

Abstract

We introduce a concept of $autoregressive$ (AR)state-space realization that could be applied to all transfer functions $\boldsymbol{T}(L)$ with $\boldsymbol{T}(0)$ invertible. We show that a theorem of Kalman implies each Vector Autoregressive model (with exogenous variables) has a minimal $AR$-state-space realization of form $\boldsymbol{y}_t = \sum_{i=1}^p\boldsymbol{H}\boldsymbol{F}^{i-1}\boldsymbol{G}\boldsymbol{x}_{t-i}+\boldsymbol{\epsilon}_t$ where $\boldsymbol{F}$ is a nilpotent Jordan matrix and $\boldsymbol{H}, \boldsymbol{G}$ satisfy certain rank conditions. The case $VARX(1)$ corresponds to reduced-rank regression. Similar to that case, for a fixed Jordan form $\boldsymbol{F}$, $\boldsymbol{H}$ could be estimated by least square as a function of $\boldsymbol{G}$. The likelihood function is a determinant ratio generalizing the Rayleigh quotient. It is unchanged if $\boldsymbol{G}$ is replaced by $\boldsymbol{S}\boldsymbol{G}$ for an invertible matrix $\boldsymbol{S}$ commuting with $\boldsymbol{F}$. Using this invariant property, the search space for maximum likelihood estimate could be constrained to equivalent classes of matrices satisfying a number of orthogonal relations, extending the results in reduced-rank analysis. Our results could be considered a multi-lag canonical-correlation-analysis. The method considered here provides a solution in the general case to the polynomial product regression model of Velu et. al. We provide estimation examples. We also explore how the estimates vary with different Jordan matrix configurations and discuss methods to select a configuration. Our approach could provide an important dimensional reduction technique with potential applications in time series analysis and linear system identification. In the appendix, we link the reduced configuration space of $\boldsymbol{G}$ with a geometric object called a vector bundle.