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.

Contents

Var(p)

Main article: Vector autoregression
y_{t}=c %2B A_{1}y_{t-1} %2B A_{2}y_{t-2} %2B \cdots %2B A_{p}y_{t-p} %2B e_{t},

Where each y_{i} is a k x 1 vector and each A_{i} is a k x k 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}%2B \begin{bmatrix} a_{1,1}^1&a_{1,2}^1 & \cdots & a_{1,k}^1\\ a_{2,1}^1&a_{2,2}^1 & \cdots & a_{2,k}^1\\ \vdots& \vdots& \ddots& \vdots\\ a_{k,1}^1&a_{k,2}^1 & \cdots & a_{k,k}^1 \end{bmatrix} \begin{bmatrix}y_{1,t-1} \\ y_{2,t-1}\\ \vdots \\ y_{k,t-1}\end{bmatrix} %2B \cdots %2B \begin{bmatrix} a_{1,1}^p&a_{1,2}^p & \cdots & a_{1,k}^p\\ a_{2,1}^p&a_{2,2}^p & \cdots & a_{2,k}^p\\ \vdots& \vdots& \ddots& \vdots\\ a_{k,1}^p&a_{k,2}^p & \cdots & a_{k,k}^p \end{bmatrix} \begin{bmatrix}y_{1,t-p} \\ y_{2,t-p}\\ \vdots \\ y_{k,t-p}\end{bmatrix} %2B \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} = c_{1} %2B a_{1,1}^1y_{1,t-1} %2B a_{1,2}^1y_{2,t-1} %2B\cdots %2B a_{1,k}^1y_{k,t-1}%2B\cdots%2Ba_{1,1}^py_{1,t-p}%2Ba_{1,2}^py_{2,t-p}%2B \cdots %2Ba_{1,k}^py_{k,t-p} %2B e_{1,t}\,

y_{2,t} = c_{2} %2B a_{2,1}^1y_{1,t-1} %2B a_{2,2}^1y_{2,t-1} %2B\cdots %2B a_{2,k}^1y_{k,t-1}%2B\cdots%2Ba_{2,1}^py_{1,t-p}%2Ba_{2,2}^py_{2,t-p}%2B \cdots %2Ba_{2,k}^py_{k,t-p} %2B e_{2,t}\,

\qquad\vdots

y_{k,t} = c_{k} %2B a_{k,1}^1y_{1,t-1} %2B a_{k,2}^1y_{2,t-1} %2B\cdots %2B a_{k,k}^1y_{k,t-1}%2B\cdots%2Ba_{k,1}^py_{1,t-p}%2Ba_{k,2}^py_{2,t-p}%2B \cdots %2Ba_{k,k}^py_{k,t-p} %2B e_{k,t}\,

Concise matrix notation

One can rewrite a VAR(p) with k variables in a general way which includes T+1 observations y_{0} through y_{T}

Y=BZ %2BU \,

Where:

Y= \begin{bmatrix}y_{p} & y_{p%2B1} & \cdots & y_{T}\end{bmatrix} = \begin{bmatrix}y_{1,p} & y_{1,p%2B1} & \cdots & y_{1,T} \\ y_{2,p} &y_{2,p%2B1} & \cdots & y_{2,T}\\ \vdots& \vdots &\vdots &\vdots \\ y_{k,p} &y_{k,p%2B1} & \cdots & y_{k,T}\end{bmatrix}
B= \begin{bmatrix} c & A_{1} & A_{2} & \cdots & A_{p} \end{bmatrix} = \begin{bmatrix} c_{1} & a_{1,1}^1&a_{1,2}^1 & \cdots & a_{1,k}^1 &\cdots & a_{1,1}^p&a_{1,2}^p & \cdots & a_{1,k}^p\\ c_{2} & a_{2,1}^1&a_{2,2}^1 & \cdots & a_{2,k}^1 &\cdots & a_{2,1}^p&a_{2,2}^p & \cdots & a_{2,k}^p \\ \vdots & \vdots& \vdots& \ddots& \vdots & \cdots & \vdots& \vdots& \ddots& \vdots\\ c_{k} & a_{k,1}^1&a_{k,2}^1 & \cdots & a_{k,k}^1 &\cdots & a_{k,1}^p&a_{k,2}^p & \cdots & a_{k,k}^p \end{bmatrix}
Z= \begin{bmatrix} 1 & 1 & \cdots & 1 \\ y_{p-1} & y_{p} & \cdots & y_{T-1}\\ y_{p-2} & y_{p-1} & \cdots & y_{T-2}\\ \vdots & \vdots & \ddots & \vdots\\ y_{0} & y_{1} & \cdots & y_{T-p} \end{bmatrix} = \begin{bmatrix} 1 & 1 & \cdots & 1 \\ y_{1,p-1} & y_{1,p} & \cdots & y_{1,T-1} \\ y_{2,p-1} & y_{2,p} & \cdots & y_{2,T-1} \\ \vdots & \vdots & \ddots & \vdots\\ y_{k,p-1} & y_{k,p} & \cdots & y_{k,T-1} \\ y_{1,p-2} & y_{1,p-1} & \cdots & y_{1,T-2} \\ y_{2,p-2} & y_{2,p-1} & \cdots & y_{2,T-2} \\ \vdots & \vdots & \ddots & \vdots\\ y_{k,p-2} & y_{k,p-1} & \cdots & y_{k,T-2} \\ \vdots & \vdots & \ddots & \vdots\\ y_{1,0} & y_{1,1} & \cdots & y_{1,T-p} \\ y_{2,0} & y_{2,1} & \cdots & y_{2,T-p} \\ \vdots & \vdots & \ddots & \vdots\\ y_{k,0} & y_{k,1} & \cdots & y_{k,T-p} \end{bmatrix}

and

U= \begin{bmatrix} e_{p} & e_{p%2B1} & \cdots & e_{T} \end{bmatrix}= \begin{bmatrix} e_{1,p} & e_{1,p%2B1} & \cdots & e_{1,T} \\ e_{2,p} & e_{2,p%2B1} & \cdots & e_{2,T} \\ \vdots & \vdots & \ddots & \vdots \\ e_{k,p} & e_{k,p%2B1} & \cdots & e_{k,T} \end{bmatrix}.

One can then solve for the coefficient matrix B (e.g. using an ordinary least squares estimation of Y \approx BZ)

References