General matrix notation of a VAR(p)

This page just shows the details for different matrix notations of a VAR(p) process with k variables.

Var(p)

y_t = c + A₁y_t − 1 + A₂y_t − 2 + ⋯ + A_py_t − p + e_t, Where y is a k x 1 vector of variables of length T and A is a k x p matrix.

Large matrix notation

$$\begin{bmatrix}y_{1,t} \\ y_{2,t}\\ \vdots \\ y_{k,t}\end{bmatrix}=\begin{bmatrix}c_{1} \\ c_{2}\\ \vdots \\ c_{k}\end{bmatrix}+ \begin{bmatrix} a_{1,1}^1&a_{1,2}^1 & \cdots & a_{1,k}^1\\ a_{1,1}^1&a_{1,2}^1 & \cdots & a_{1,k}^1\\ \vdots& \vdots& \ddots& \vdots\\ a_{1,1}^1&a_{1,2}^1 & \cdots & a_{1,k}^1 \end{bmatrix} \begin{bmatrix}y_{1,t-1} \\ y_{2,t-1}\\ \vdots \\ y_{k,t-1}\end{bmatrix} + \cdots + \begin{bmatrix} a_{1,1}^p&a_{1,2}^p & \cdots & a_{1,k}^p\\ a_{1,1}^p&a_{1,2}^p & \cdots & a_{1,k}^p\\ \vdots& \vdots& \ddots& \vdots\\ a_{1,1}^p&a_{1,2}^p & \cdots & a_{1,k}^p \end{bmatrix} \begin{bmatrix}y_{1,t-p} \\ y_{2,t-p}\\ \vdots \\ y_{k,t-p}\end{bmatrix} + \begin{bmatrix}e_{1,t} \\ e_{2,t}\\ \vdots \\ e_{k,t}\end{bmatrix}$$

Equation by equation notation

Rewriting the y variables one to one gives:

y_1, t = c₁ + a_1, 1¹y_{1, t − 1} + a_1, 2¹y_{2, t − 1} + ⋯ + a_1, k¹y_{k, t − 1} + ⋯ + a_p, 1¹y_{1, t − p} + a_p, 2¹y_{2, t − p} + ⋯ + a_p, k¹y_{k, t − p} + e_1, t  y_2, t = c₂ + a_1, 1²y_{1, t − 1} + a_1, 2²y_{2, t − 1} + ⋯ + a_1, k²y_{k, t − 1} + ⋯ + a_p, 1²y_{1, t − p} + a_p, 2²y_{2, t − p} + ⋯ + a_p, k²y_{k, t − p} + e_1, t   ⋮  =   ⋮ y_k, t = c_k + a_1, 1^ky_{1, t − 1} + a_1, 2^ky_{2, t − 1} + ⋯ + a_1, k^ky_{k, t − 1} + ⋯ + a_p, 1^ky_{1, t − p} + a_p, 2^ky_{2, t − p} + ⋯ + a_p, k^ky_{k, t − p} + e_1, t 

Concise matrix notation

One can rewrite a VAR(p) with k variables in a general way

Y = BZ + U 

Where:

$$Y= \begin{bmatrix}y_{1,1} &y_{1,2} & \cdots & y_{1,T} \\ y_{2,1} &y_{2,2} & \cdots & y_{2,T}\\ \vdots& \vdots &\vdots &\vdots \\ y_{k,1} &y_{k,2} & \cdots & y_{k,T}\end{bmatrix}$$

$$=\begin{bmatrix} c_{1} & A_{1,1}^{p=1}&A_{1,2}^{p=1}&\cdots &A_{1,k}^{p=1}&\cdots & A_{1,1}^{p=p}&A_{1,2}^{p=p}&\cdots &A_{1,k}^{p=p}\\ c_{2} & A_{2,1}^{p=1}&A_{2,2}^{p=1}&\cdots &A_{2,k}^{p=1}&\cdots & A_{2,1}^{p=p}&A_{2,2}^{p=p}&\cdots &A_{2,k}^{p=p}\\ \vdots& \vdots &\vdots &\ddots &\vdots& \ddots& \vdots &\vdots &\ddots &\vdots\\ c_{k} & A_{k,1}^{p=1}&A_{k,2}^{p=1}&\cdots &A_{k,k}^{p=1} &\cdots & A_{k,1}^{p=p}&A_{k,2}^{p=p}&\cdots &A_{k,k}^{p=p} \end{bmatrix} \begin{bmatrix} 1&1&\cdots&1\\ y_{1,1}&y_{1,2}&\cdots&y_{1,T-1}\\ y_{2,1}&y_{2,2}&\cdots&y_{2,T-1}\\ \vdots & \vdots & \vdots&y\vdots\\ y_{k,1}&y_{k,2}&\cdots&y_{k,T-1}\\ \vdots & \vdots & \vdots&\vdots\\ \vdots & \vdots & \vdots&\vdots\\ y_{1,1-p}&y_{1,2-p}&\cdots&y_{1,T-p}\\ y_{2,1-p}&y_{2,2-p}&\cdots&y_{1,T-p}\\ \vdots & \vdots & \vdots&\vdots\\ y_{k,1-p}&y_{k,2-p}&\cdots& y_{k,T-p} \end{bmatrix}$$

$$+ U= \begin{bmatrix} u_{1,1}&u_{1,2}&\cdots&u_{1,T}\\ u_{2,1}&u_{2,2}&\cdots&u_{1,T}\\ \vdots&\vdots&\ddots&\vdots\\ u_{k,1}&u_{k,2}&\cdots&u_{k,T} \end{bmatrix}$$

References

• Helmut Lütkepohl. New Introduction to Multiple Time Series Analysis. Springer. 2005.