VARIMAX

262. VARIMAX

Vector ARIMA with exogenous regressors

VARIMAX (𝑝, 𝑑, 𝑞)

𝚽 (𝐵) {(1 - 𝐵)}^{𝑑} 𝒚_{𝑡} = 𝒄 + 𝑩 𝒙_{𝑡} + 𝚯 (𝐵) 𝜺_{𝑡}

$𝒚_{𝑡} \in ℝ^{𝐾}$ = endogenous vector series
$𝒙_{𝑡} \in ℝ^{𝑘}$ = exogenous regressor vector
$𝑩 \in ℝ^{𝐾 \times 𝑘}$ = matrix of regression coefficients
Each of the $𝐾$ series can depend on the same set of $𝑘$ exogenous variables.

Parameters: $𝚽_{1}, \dots, 𝚽_{𝑝}$ , $𝚯_{1}, \dots, 𝚯_{𝑞}$ , $𝑩$ ( $𝐾 \times 𝑘$ ), $𝒄$ , $𝚺$
Orders: $𝑝$ , $𝑑$ , $𝑞$ , $𝐾$ , $𝑘$ (regressors)

Example:

VARIMAX (1, 0, 1), 𝐾 = 2, 𝑘 = 1

Given

Orders: $𝑝 = 1$ , $𝑑 = 0$ , $𝑞 = 1$ , $𝐾 = 2$ , $𝑘 = 1$ (one exog)
Coefficient matrices:
$𝚽_{1} = [\begin{matrix} 0.4 & 0.2 \\ 0.2 & 0.4 \end{matrix}] 𝚯_{1} = [\begin{matrix} 0.4 & 0.2 \\ 0.2 & 0.4 \end{matrix}] 𝑩 = [\begin{matrix} 0.5 \\ 0.5 \end{matrix}]$
Intercept: $𝒄 = [\begin{matrix} 0 \\ 0 \end{matrix}]$
Initial conditions: $𝒚_{0} = [\begin{matrix} 0 \\ 0 \end{matrix}]$ , $𝜺_{0} = [\begin{matrix} 0 \\ 0 \end{matrix}]$
Endogenous $𝒚_{𝑡} = [\begin{matrix} 𝑦_{1, 𝑡} \\ 𝑦_{2, 𝑡} \end{matrix}]$ :

$𝑡$	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16
${y_(1\,)}_{𝑡}$	12	10	8	11	14	12	9	13	16	14	11	15	18	16	13	17

$𝑡$	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16
${y_(2\,)}_{𝑡}$	8	9	7	6	10	11	9	8	12	13	11	10	14	15	13	12

Exogenous $𝑧_{𝑡}$ :

$𝑡$	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16
$z_{𝑡}$	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16

Step 1 — formula

Substitute orders into the VARIMAX recursion (with $𝑑 = 0$ , no differencing):

(𝑰 - 𝚽_{1} 𝐵) 𝒚_{𝑡} = 𝒄 + 𝑩 𝑧_{𝑡} + (𝑰 + 𝚯_{1} 𝐵) 𝜺_{𝑡}

Forecast (set $𝜺_{𝑡} = 𝟎$ ):

{\hat{𝒚}}_{𝑡} = 𝒄 + 𝚽_{1} 𝒚_{𝑡 - 1} + 𝑩 𝑧_{𝑡} + 𝚯_{1} 𝜺_{𝑡 - 1}

Three contributions: AR product, exogenous regression $𝑩 𝑧_{𝑡}$ , MA product.

Componentwise:

{\hat{𝑦}}_{1, 𝑡} = 𝑐_{1} + 0.4 𝑦_{1, 𝑡 - 1} + 0.2 𝑦_{2, 𝑡 - 1} + 0.5 𝑧_{𝑡} + 0.4 𝜀_{1, 𝑡 - 1} + 0.2 𝜀_{2, 𝑡 - 1}

{\hat{𝑦}}_{2, 𝑡} = 𝑐_{2} + 0.2 𝑦_{1, 𝑡 - 1} + 0.4 𝑦_{2, 𝑡 - 1} + 0.5 𝑧_{𝑡} + 0.2 𝜀_{1, 𝑡 - 1} + 0.4 𝜀_{2, 𝑡 - 1}

Innovation:

𝜺_{𝑡} = 𝒚_{𝑡} - {\hat{𝒚}}_{𝑡}

Step 2 — apply at $𝑡 = 2$

Compute the three contributions separately, then sum.

AR part: $𝚽_{1} 𝒚_{1} = 𝚽_{1} [\begin{matrix} 12 \\ 8 \end{matrix}] = [\begin{matrix} 0.4 (12) + 0.2 (8) \\ 0.2 (12) + 0.4 (8) \end{matrix}] = [\begin{matrix} 6.4 \\ 5.6 \end{matrix}]$

Exog part: $𝑩 𝑧_{2} = [\begin{matrix} 0.5 \\ 0.5 \end{matrix}] \cdot 2 = [\begin{matrix} 1 \\ 1 \end{matrix}]$

MA part: $𝚯_{1} 𝜺_{1}$ . With $𝜺_{0} = 𝟎$ , the $𝑡 = 1$ step gives ${\hat{𝒚}}_{1} = 𝑩 𝑧_{1} = [\begin{matrix} 0.5 \\ 0.5 \end{matrix}]$ and $𝜺_{1} = [\begin{matrix} 12 \\ 8 \end{matrix}] - [\begin{matrix} 0.5 \\ 0.5 \end{matrix}] = [\begin{matrix} 11.5 \\ 7.5 \end{matrix}]$ .

𝚯_{1} 𝜺_{1} = [\begin{matrix} 0.4 (11.5) + 0.2 (7.5) \\ 0.2 (11.5) + 0.4 (7.5) \end{matrix}] = [\begin{matrix} 6.1 \\ 5.3 \end{matrix}]

Sum: ${\hat{𝒚}}_{2} = [\begin{matrix} 6.4 \\ 5.6 \end{matrix}] + [\begin{matrix} 1 \\ 1 \end{matrix}] + [\begin{matrix} 6.1 \\ 5.3 \end{matrix}] = [\begin{matrix} 13.5 \\ 11.9 \end{matrix}]$

$𝜺_{2} = [\begin{matrix} 10 \\ 9 \end{matrix}] - [\begin{matrix} 13.5 \\ 11.9 \end{matrix}] = [\begin{matrix} - 3.5 \\ - 2.9 \end{matrix}]$

Step 3 — iterate

Three contributions per row: AR $𝚽_{1} 𝒚_{𝑡 - 1}$ , exog $𝑩 𝑧_{𝑡}$ , MA $𝚯_{1} 𝜺_{𝑡 - 1}$ . Values rounded to 4 decimal places.

$𝑡$	$𝒚_{𝑡}$	$𝚽_{1} 𝒚_{𝑡 - 1}$	$𝑩 𝑧_{𝑡}$	$𝚯_{1} 𝜺_{𝑡 - 1}$	${\hat{𝒚}}_{𝑡} / 𝜺_{𝑡}$
1	$[\begin{matrix} 12 \\ 8 \end{matrix}]$	$[\begin{matrix} 0 \\ 0 \end{matrix}]$	$[\begin{matrix} 0.5 \\ 0.5 \end{matrix}]$	$[\begin{matrix} 0 \\ 0 \end{matrix}]$	$[\begin{matrix} 0.5 \\ 0.5 \end{matrix}] / [\begin{matrix} 11.5 \\ 7.5 \end{matrix}]$
2	$[\begin{matrix} 10 \\ 9 \end{matrix}]$	$[\begin{matrix} 6.4 \\ 5.6 \end{matrix}]$	$[\begin{matrix} 1 \\ 1 \end{matrix}]$	$[\begin{matrix} 6.1 \\ 5.3 \end{matrix}]$	$[\begin{matrix} 13.5 \\ 11.9 \end{matrix}] / [\begin{matrix} - 3.5 \\ - 2.9 \end{matrix}]$
3	$[\begin{matrix} 8 \\ 7 \end{matrix}]$	$[\begin{matrix} 5.8 \\ 5.6 \end{matrix}]$	$[\begin{matrix} 1.5 \\ 1.5 \end{matrix}]$	$[\begin{matrix} - 1.98 \\ - 1.86 \end{matrix}]$	$[\begin{matrix} 5.32 \\ 5.24 \end{matrix}] / [\begin{matrix} 2.68 \\ 1.76 \end{matrix}]$
4	$[\begin{matrix} 11 \\ 6 \end{matrix}]$	$[\begin{matrix} 4.6 \\ 4.4 \end{matrix}]$	$[\begin{matrix} 2 \\ 2 \end{matrix}]$	$[\begin{matrix} 1.424 \\ 1.24 \end{matrix}]$	$[\begin{matrix} 8.024 \\ 7.64 \end{matrix}] / [\begin{matrix} 2.976 \\ - 1.64 \end{matrix}]$
5	$[\begin{matrix} 14 \\ 10 \end{matrix}]$	$[\begin{matrix} 5.6 \\ 4.6 \end{matrix}]$	$[\begin{matrix} 2.5 \\ 2.5 \end{matrix}]$	$[\begin{matrix} 0.8624 \\ - 0.0608 \end{matrix}]$	$[\begin{matrix} 8.9624 \\ 7.0392 \end{matrix}] / [\begin{matrix} 5.0376 \\ 2.9608 \end{matrix}]$
6	$[\begin{matrix} 12 \\ 11 \end{matrix}]$	$[\begin{matrix} 7.6 \\ 6.8 \end{matrix}]$	$[\begin{matrix} 3 \\ 3 \end{matrix}]$	$[\begin{matrix} 2.6072 \\ 2.1918 \end{matrix}]$	$[\begin{matrix} 13.2072 \\ 11.9918 \end{matrix}] / [\begin{matrix} - 1.2072 \\ - 0.9918 \end{matrix}]$
7	$[\begin{matrix} 9 \\ 9 \end{matrix}]$	$[\begin{matrix} 7 \\ 6.8 \end{matrix}]$	$[\begin{matrix} 3.5 \\ 3.5 \end{matrix}]$	$[\begin{matrix} - 0.6812 \\ - 0.6381 \end{matrix}]$	$[\begin{matrix} 9.8188 \\ 9.6619 \end{matrix}] / [\begin{matrix} - 0.8188 \\ - 0.6619 \end{matrix}]$
8	$[\begin{matrix} 13 \\ 8 \end{matrix}]$	$[\begin{matrix} 5.4 \\ 5.4 \end{matrix}]$	$[\begin{matrix} 4 \\ 4 \end{matrix}]$	$[\begin{matrix} - 0.4599 \\ - 0.4286 \end{matrix}]$	$[\begin{matrix} 8.9401 \\ 8.9714 \end{matrix}] / [\begin{matrix} 4.0599 \\ - 0.9714 \end{matrix}]$
9	$[\begin{matrix} 16 \\ 12 \end{matrix}]$	$[\begin{matrix} 6.8 \\ 5.8 \end{matrix}]$	$[\begin{matrix} 4.5 \\ 4.5 \end{matrix}]$	$[\begin{matrix} 1.4297 \\ 0.4234 \end{matrix}]$	$[\begin{matrix} 12.7297 \\ 10.7234 \end{matrix}] / [\begin{matrix} 3.2703 \\ 1.2766 \end{matrix}]$
10	$[\begin{matrix} 14 \\ 13 \end{matrix}]$	$[\begin{matrix} 8.8 \\ 8 \end{matrix}]$	$[\begin{matrix} 5 \\ 5 \end{matrix}]$	$[\begin{matrix} 1.5634 \\ 1.1647 \end{matrix}]$	$[\begin{matrix} 15.3634 \\ 14.1647 \end{matrix}] / [\begin{matrix} - 1.3634 \\ - 1.1647 \end{matrix}]$
11	$[\begin{matrix} 11 \\ 11 \end{matrix}]$	$[\begin{matrix} 8.2 \\ 8 \end{matrix}]$	$[\begin{matrix} 5.5 \\ 5.5 \end{matrix}]$	$[\begin{matrix} - 0.7783 \\ - 0.7386 \end{matrix}]$	$[\begin{matrix} 12.9217 \\ 12.7614 \end{matrix}] / [\begin{matrix} - 1.9217 \\ - 1.7614 \end{matrix}]$
12	$[\begin{matrix} 15 \\ 10 \end{matrix}]$	$[\begin{matrix} 6.6 \\ 6.6 \end{matrix}]$	$[\begin{matrix} 6 \\ 6 \end{matrix}]$	$[\begin{matrix} - 1.1209 \\ - 1.0889 \end{matrix}]$	$[\begin{matrix} 11.4791 \\ 11.5111 \end{matrix}] / [\begin{matrix} 3.5209 \\ - 1.5111 \end{matrix}]$
13	$[\begin{matrix} 18 \\ 14 \end{matrix}]$	$[\begin{matrix} 8 \\ 7 \end{matrix}]$	$[\begin{matrix} 6.5 \\ 6.5 \end{matrix}]$	$[\begin{matrix} 1.0861 \\ 0.0598 \end{matrix}]$	$[\begin{matrix} 15.5861 \\ 13.5598 \end{matrix}] / [\begin{matrix} 2.4139 \\ 0.4402 \end{matrix}]$
14	$[\begin{matrix} 16 \\ 15 \end{matrix}]$	$[\begin{matrix} 10 \\ 9.2 \end{matrix}]$	$[\begin{matrix} 7 \\ 7 \end{matrix}]$	$[\begin{matrix} 1.0536 \\ 0.6589 \end{matrix}]$	$[\begin{matrix} 18.0536 \\ 16.8589 \end{matrix}] / [\begin{matrix} - 2.0536 \\ - 1.8589 \end{matrix}]$
15	$[\begin{matrix} 13 \\ 13 \end{matrix}]$	$[\begin{matrix} 9.4 \\ 9.2 \end{matrix}]$	$[\begin{matrix} 7.5 \\ 7.5 \end{matrix}]$	$[\begin{matrix} - 1.1932 \\ - 1.1543 \end{matrix}]$	$[\begin{matrix} 15.7068 \\ 15.5457 \end{matrix}] / [\begin{matrix} - 2.7068 \\ - 2.5457 \end{matrix}]$
16	$[\begin{matrix} 17 \\ 12 \end{matrix}]$	$[\begin{matrix} 7.8 \\ 7.8 \end{matrix}]$	$[\begin{matrix} 8 \\ 8 \end{matrix}]$	$[\begin{matrix} - 1.5919 \\ - 1.5597 \end{matrix}]$	$[\begin{matrix} 14.2081 \\ 14.2403 \end{matrix}] / [\begin{matrix} 2.7919 \\ - 2.2403 \end{matrix}]$