Codependent VAR models and the pseudo-structural form

Abstract

This paper investigates whether codependence restrictions can be uniquely imposed on VAR models via the so-called pseudo-structural form used in the literature. Codependence of order q is given if a linear combination of autocorrelated variables eliminates the serial correlation after q lags. Importantly, maximum likelihood estimation and likelihood ratio testing are only possible if the codependence restrictions can be uniquely imposed. Applying the pseudo-structural form, our study reveals that this is not generally the case, but that unique imposition is guaranteed in several important special cases.

Appendix: numerical examples

Consider a three-dimensional version of the VAR in (1) of order three with

$$\begin{aligned} A_1&= \begin{pmatrix} 0.00&\quad 0.50&\quad 0.00\\ 0.00&\quad 0.40&\quad 0.00\\ a_{31,1}&\quad 0.00&\quad a_{33,1} \end{pmatrix}, \quad A_2= \begin{pmatrix} 0.00&\quad 0.36&\quad 0.00\\ 0.00&\quad 0.00&\quad 0.00\\ 0.00&\quad 0.00&\quad 0.00 \end{pmatrix}, \quad \mathrm{and}\\ A_3&= \begin{pmatrix} 0.00&\quad a_{12,3}&\quad 0.00\\ 0.00&\quad 0.00&\quad 0.00\\ 0.00&\quad 0.00&\quad 0.00 \end{pmatrix}. \end{aligned}$$

We assume $a_{31,1} \ne 0$ and $a_{33,1}\ne 0$. If $a_{12,3}=-0.16$, then $\delta _0 = (1 -\!1 \,0)^\prime$ is a codependence vector which generates codependence of order $q=2$ with respect to $y_t$. We have $\gamma _0 = (1 -\!1\;0 \;0 \cdots 0)^\prime$, $\gamma _1=\gamma _0^\prime \varvec{A} = (0 \; 0.1 \;0 \;0 \; 0.36 \; 0 \; 0 -\!0.16 \; 0)^\prime$, and $\gamma _2=\gamma _1^\prime \varvec{A}=(0\;0.4\;0\;0 -\!0.16\;0\;0\;0\;0)^\prime$ with $\gamma _2^\prime \varvec{A}=\varvec{0}$. Here, $\varvec{A}$ is the corresponding companion matrix of the three-dimensional VAR($3$). The vectors consisting of the first three elements of $\gamma _1$ and $\gamma _2$ represent the vectors labeled by $\delta _1$ and $\delta _2$ in Sect. 2.2. Obviously, $\delta _1$ and $\delta _2$ are linearly dependent meaning that the codependence vector $\delta _0$ imposes additional constraints on the first two MA parameter matrices. Hence, the current setup represents an example for $q = 2 < 3 = n$ with linear dependence of codependence and companion restrictions.

Let us extend the previous VAR to the lag order of four with

$$\begin{aligned} A_4 = \begin{pmatrix} 0.00&\quad -0.12&\quad 0.00\\ 0.00&\quad 0.00&\quad 0.00\\ 0.00&\quad 0.00&\quad 0.00 \end{pmatrix} \end{aligned}$$

and set $a_{12,3} = 0.14$. Then, the codependence vector $\delta _0 = (1 -\!1 \;0)^\prime$ is associated with a codependence order $q=3$. We obtain $\gamma _0 = (1 -\!1 \;0 \;0 \cdots 0)^\prime$, $\gamma _1=\gamma _0^\prime \varvec{A} = (0 \; 0.1 \;0 \;0 \; 0.36 \; 0 \; 0 \; 0.14 \; 0 \; 0 -\!0.12 \; 0)^\prime$, and $\gamma _2 = \gamma _1^\prime \varvec{A}= (0 \; 0.4 \;0 \;0 \; 0.14 \; 0 \; 0$ $-\!0.12 \; 0 \cdots \; 0)^\prime$, $\gamma _3 = \gamma _2^\prime \varvec{A}= (0 \; 0.3 \;0 \;0 -\!0.12 \; 0 \cdots \; 0)^\prime$ with $\gamma _3^\prime \varvec{A} = \varvec{0}$. Now, $\varvec{A}$ is the companion matrix related to the three-dimensional VAR(4). Obviously, the vectors $\delta _1$, $\delta _2$, and $\delta _3$ linearly depend on each other such that we have an example for the case of $q \ge n$ with linear dependence of codependence and companion restrictions.

We make two final remarks. First, the considered VAR processes are stable if $|a_{33,1}|<1$. Second, the values of $a_{31,1}$ and $a_{33,1}$ do not affect the properties of codependence and linear dependence with respect to the considered VAR models.