VAR

258. VAR

Vector autoregression

VAR (𝑝)

𝚽 (𝐵) 𝒙_{𝑡} = 𝒄 + 𝜺_{𝑡}

𝚽 (𝐵) = 𝑰 - 𝚽_{1} 𝐵 - \dots - 𝚽_{𝑝} 𝐵^{𝑝}

i.e.,

𝒙_{𝑡} = 𝒄 + 𝚽_{1} 𝒙_{𝑡 - 1} + \dots + 𝚽_{𝑝} 𝒙_{𝑡 - 𝑝} + 𝜺_{𝑡}

Each component depends on $𝑝$ lags of all $𝐾$ series.

Parameters: $𝚽_{1}, \dots, 𝚽_{𝑝}$ ( $𝐾^{2}$ each), $𝒄$ ( $𝐾$ ), $𝚺$ ( $𝐾 (𝐾 + 1) / 2$ )
Orders: $𝑝$ , $𝐾$ (series)

Example:

VAR (1), 𝐾 = 2

Given

Order: $𝑝 = 1$ , $𝐾 = 2$
Coefficient matrix:
$𝚽_{1} = [\begin{matrix} 0.4 & 0.2 \\ 0.2 & 0.4 \end{matrix}]$
Intercept: $𝒄 = [\begin{matrix} 0 \\ 0 \end{matrix}]$
Two series stacked into a vector $𝒙_{𝑡} = [\begin{matrix} 𝑥_{1, 𝑡} \\ 𝑥_{2, 𝑡} \end{matrix}]$ :

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

Step 1 — formula

Substitute $𝑝 = 1$ into the VAR $(𝑝)$ recursion:

𝒙_{𝑡} = 𝒄 + 𝚽_{1} 𝒙_{𝑡 - 1} + 𝜺_{𝑡}

Forecast (set $𝜺_{𝑡} = 𝟎$ ):

{\hat{𝒙}}_{𝑡} = 𝒄 + 𝚽_{1} 𝒙_{𝑡 - 1}

Componentwise (write out the matrix-vector product):

[\begin{matrix} {\hat{𝑥}}_{1, 𝑡} \\ {\hat{𝑥}}_{2, 𝑡} \end{matrix}] = [\begin{matrix} 𝑐_{1} \\ 𝑐_{2} \end{matrix}] + [\begin{matrix} 0.4 & 0.2 \\ 0.2 & 0.4 \end{matrix}] [\begin{matrix} 𝑥_{1, 𝑡 - 1} \\ 𝑥_{2, 𝑡 - 1} \end{matrix}]

{\hat{𝑥}}_{1, 𝑡} = 𝑐_{1} + 0.4 𝑥_{1, 𝑡 - 1} + 0.2 𝑥_{2, 𝑡 - 1}

{\hat{𝑥}}_{2, 𝑡} = 𝑐_{2} + 0.2 𝑥_{1, 𝑡 - 1} + 0.4 𝑥_{2, 𝑡 - 1}

Innovation:

𝜺_{𝑡} = 𝒙_{𝑡} - {\hat{𝒙}}_{𝑡}

Step 2 — apply at $𝑡 = 2$

Plug in $𝒄 = 𝟎$ , $𝚽_{1}$ , $𝒙_{1} = [\begin{matrix} 12 \\ 8 \end{matrix}]$ :

{\hat{𝑥}}_{1, 2} = 0.4 \cdot 12 + 0.2 \cdot 8 = 4.8 + 1.6 = 6.4

{\hat{𝑥}}_{2, 2} = 0.2 \cdot 12 + 0.4 \cdot 8 = 2.4 + 3.2 = 5.6

𝜺_{2} = [\begin{matrix} 10 \\ 9 \end{matrix}] - [\begin{matrix} 6.4 \\ 5.6 \end{matrix}] = [\begin{matrix} 3.6 \\ 3.4 \end{matrix}]

Step 3 — iterate

$𝑡$	$𝒙_{𝑡}$	${\hat{𝑥}}_{1, 𝑡} = 0.4 𝑥_{1, 𝑡 - 1} + 0.2 𝑥_{2, 𝑡 - 1}$	${\hat{𝑥}}_{2, 𝑡} = 0.2 𝑥_{1, 𝑡 - 1} + 0.4 𝑥_{2, 𝑡 - 1}$	$𝜺_{𝑡}$
2	$[\begin{matrix} 10 \\ 9 \end{matrix}]$	$0.4 (12) + 0.2 (8) = 6.4$	$0.2 (12) + 0.4 (8) = 5.6$	$[\begin{matrix} 3.6 \\ 3.4 \end{matrix}]$
3	$[\begin{matrix} 8 \\ 7 \end{matrix}]$	$0.4 (10) + 0.2 (9) = 5.8$	$0.2 (10) + 0.4 (9) = 5.6$	$[\begin{matrix} 2.2 \\ 1.4 \end{matrix}]$
4	$[\begin{matrix} 11 \\ 6 \end{matrix}]$	$0.4 (8) + 0.2 (7) = 4.6$	$0.2 (8) + 0.4 (7) = 4.4$	$[\begin{matrix} 6.4 \\ 1.6 \end{matrix}]$
5	$[\begin{matrix} 14 \\ 10 \end{matrix}]$	$0.4 (11) + 0.2 (6) = 5.6$	$0.2 (11) + 0.4 (6) = 4.6$	$[\begin{matrix} 8.4 \\ 5.4 \end{matrix}]$
6	$[\begin{matrix} 12 \\ 11 \end{matrix}]$	$0.4 (14) + 0.2 (10) = 7.6$	$0.2 (14) + 0.4 (10) = 6.8$	$[\begin{matrix} 4.4 \\ 4.2 \end{matrix}]$
7	$[\begin{matrix} 9 \\ 9 \end{matrix}]$	$0.4 (12) + 0.2 (11) = 7$	$0.2 (12) + 0.4 (11) = 6.8$	$[\begin{matrix} 2 \\ 2.2 \end{matrix}]$
8	$[\begin{matrix} 13 \\ 8 \end{matrix}]$	$0.4 (9) + 0.2 (9) = 5.4$	$0.2 (9) + 0.4 (9) = 5.4$	$[\begin{matrix} 7.6 \\ 2.6 \end{matrix}]$
9	$[\begin{matrix} 16 \\ 12 \end{matrix}]$	$0.4 (13) + 0.2 (8) = 6.8$	$0.2 (13) + 0.4 (8) = 5.8$	$[\begin{matrix} 9.2 \\ 6.2 \end{matrix}]$
10	$[\begin{matrix} 14 \\ 13 \end{matrix}]$	$0.4 (16) + 0.2 (12) = 8.8$	$0.2 (16) + 0.4 (12) = 8$	$[\begin{matrix} 5.2 \\ 5 \end{matrix}]$
11	$[\begin{matrix} 11 \\ 11 \end{matrix}]$	$0.4 (14) + 0.2 (13) = 8.2$	$0.2 (14) + 0.4 (13) = 8$	$[\begin{matrix} 2.8 \\ 3 \end{matrix}]$
12	$[\begin{matrix} 15 \\ 10 \end{matrix}]$	$0.4 (11) + 0.2 (11) = 6.6$	$0.2 (11) + 0.4 (11) = 6.6$	$[\begin{matrix} 8.4 \\ 3.4 \end{matrix}]$
13	$[\begin{matrix} 18 \\ 14 \end{matrix}]$	$0.4 (15) + 0.2 (10) = 8$	$0.2 (15) + 0.4 (10) = 7$	$[\begin{matrix} 10 \\ 7 \end{matrix}]$
14	$[\begin{matrix} 16 \\ 15 \end{matrix}]$	$0.4 (18) + 0.2 (14) = 10$	$0.2 (18) + 0.4 (14) = 9.2$	$[\begin{matrix} 6 \\ 5.8 \end{matrix}]$
15	$[\begin{matrix} 13 \\ 13 \end{matrix}]$	$0.4 (16) + 0.2 (15) = 9.4$	$0.2 (16) + 0.4 (15) = 9.2$	$[\begin{matrix} 3.6 \\ 3.8 \end{matrix}]$
16	$[\begin{matrix} 17 \\ 12 \end{matrix}]$	$0.4 (13) + 0.2 (13) = 7.8$	$0.2 (13) + 0.4 (13) = 7.8$	$[\begin{matrix} 9.2 \\ 4.2 \end{matrix}]$
17	—	$0.4 (17) + 0.2 (12) = 9.2$	$0.2 (17) + 0.4 (12) = 8.2$	—