Vector Series (VAR Models)

Vector Auto-regression of order 1 VAR(1)

\[X_t = AX_{t-1}+a_t\]

where \(A_{k\times k}\) is the coefficient matrix and \(a_t\) is a K-dimensional white noise process with time-invariant positive definite covariance matrix \(E(aa')=\Sigma\)

We can repeatedly substitute the VAR(1) above to

\[y_t = \sum A^i u_{t-i}\]

For the above VMA(\(\infty\)) process to be stationary. \(A^i\) must converge to zero, i.e. the all K eigenvalues of \(A\) must be less than 1.

Then, the VAR(p) model is

\[y_t = A_0 + \sum_1^p A_i y_{t-i} +a_t\]

OR the compact form

\[A(B)y_t = a_t\]

Then, to be stationary, all the roots of \(\det(I_k - A_1B-...-A_pB^p)=0\) are greater than 1 in absolute value.

Companion form

We can check the stability of a VAR(p) model using its companion form.

\[\xi_t = A\xi_{t-1}+v_t\]

where \(\xi_t = \begin{bmatrix} y_t \\ ... \\ y_{t-p+1} \end{bmatrix} , A = \begin{bmatrix} A_1 & A_2 & ... & A_{p-1} & A_p \\ I & 0 & ... & 0 & 0 \\ 0 & I & ... & 0 & 0 \\ ... & ... & ... & ... & ... \\ 0 & 0 & ... & I & 0 \end{bmatrix}, v_t = \begin{bmatrix} a_t \\ 0 \\...\\0\end{bmatrix}\)

If the moduli of the eigenvalues of \(A\) are less than 1, then the process is stable

Example

\[ \begin{bmatrix}y_1 \\y_2\end{bmatrix}_t = \begin{bmatrix}5 \\ 10\end{bmatrix} + \begin{bmatrix}0.5 &-0.2 \\-0.2 & -0.5\end{bmatrix} \begin{bmatrix}y_1 \\y_2\end{bmatrix}_{t-1} + \begin{bmatrix}0.3 &-0.7 \\-0.1 & 0.3\end{bmatrix} \begin{bmatrix}y_1 \\y_2\end{bmatrix}_{t-2} + \begin{bmatrix}a_1 \\a_2\end{bmatrix}_t \]

Then, its companion matrix \(A\) will be

\[\begin{bmatrix} 0.5 & 0.2 & -0.3 & -0.7 \\ -0.2 & -0.5 & -0.1 & 0.3 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \end{bmatrix}\]

Order selection

sequential likelihood ratio tests of VAR(p) vs. VAR(p-1)
BigVAR: dimension reduction methods for multi time series