General matrix notation of a VAR(p)

This page shows the details for different matrix notations of a vector autoregression process with k variables.

Var(p)

Main article: Vector autoregression

$y_t=c+A_1{y}_{t-1}+A_2{y}_{t-2}+\cdots +A_p{y}_{t-p}+e_t,$

where each $y_i$ is a vector of length k and each $A_i$ is a k × k matrix.

What are the assumptions on the noise?

Large matrix notation

$\left[\begin{array}{c}{y}_{1,t}\\ y_{2,t}\\ ⋮\\ y_{k,t}\end{array}\right]=\left[\begin{array}{c}{c}_1\\ c_2\\ ⋮\\ c_k\end{array}\right]+\left[\begin{array}{cccc}{a}_{1,1}^1& a_{1,2}^1& \cdots & a_{1,k}^1\\ a_{2,1}^1& a_{2,2}^1& \cdots & a_{2,k}^1\\ ⋮& ⋮& \ddots & ⋮\\ a_{k,1}^1& a_{k,2}^1& \cdots & a_{k,k}^1\end{array}\right]\left[\begin{array}{c}{y}_{1,t-1}\\ y_{2,t-1}\\ ⋮\\ y_{k,t-1}\end{array}\right]+\cdots +\left[\begin{array}{cccc}{a}_{1,1}^p& a_{1,2}^p& \cdots & a_{1,k}^p\\ a_{2,1}^p& a_{2,2}^p& \cdots & a_{2,k}^p\\ ⋮& ⋮& \ddots & ⋮\\ a_{k,1}^p& a_{k,2}^p& \cdots & a_{k,k}^p\end{array}\right]\left[\begin{array}{c}{y}_{1,t-p}\\ y_{2,t-p}\\ ⋮\\ y_{k,t-p}\end{array}\right]+\left[\begin{array}{c}{e}_{1,t}\\ e_{2,t}\\ ⋮\\ e_{k,t}\end{array}\right]$

Equation by regression notation

Rewriting the y variables one to one gives:

$y_{1,t}=c_1+a_{1,1}^1{y}_{1,t-1}+a_{1,2}^1{y}_{2,t-1}+\cdots +a_{1,k}^1{y}_{k,t-1}+\cdots +a_{1,1}^p{y}_{1,t-p}+a_{1,2}^p{y}_{2,t-p}+\cdots +a_{1,k}^p{y}_{k,t-p}+e_{1,t}$

$y_{2,t}=c_2+a_{2,1}^1{y}_{1,t-1}+a_{2,2}^1{y}_{2,t-1}+\cdots +a_{2,k}^1{y}_{k,t-1}+\cdots +a_{2,1}^p{y}_{1,t-p}+a_{2,2}^p{y}_{2,t-p}+\cdots +a_{2,k}^p{y}_{k,t-p}+e_{2,t}$

$⋮$

$y_{k,t}=c_k+a_{k,1}^1{y}_{1,t-1}+a_{k,2}^1{y}_{2,t-1}+\cdots +a_{k,k}^1{y}_{k,t-1}+\cdots +a_{k,1}^p{y}_{1,t-p}+a_{k,2}^p{y}_{2,t-p}+\cdots +a_{k,k}^p{y}_{k,t-p}+e_{k,t}$

Concise matrix notation

One can rewrite a VAR(p) with k variables in a general way which includes T+1 observations $y_p$ through $y_T$

$Y=BZ+U$

where:

$Y=\left[\begin{array}{cccc}{y}_p& y_{p+1}& \cdots & y_T\end{array}\right]=\left[\begin{array}{cccc}{y}_{1,p}& y_{1,p+1}& \cdots & y_{1,T}\\ y_{2,p}& y_{2,p+1}& \cdots & y_{2,T}\\ ⋮& ⋮& ⋮& ⋮\\ y_{k,p}& y_{k,p+1}& \cdots & y_{k,T}\end{array}\right]$

$B=\left[\begin{array}{ccccc}c& A_1& A_2& \cdots & A_p\end{array}\right]=\left[\begin{array}{cccccccccc}{c}_1& a_{1,1}^1& a_{1,2}^1& \cdots & a_{1,k}^1& \cdots & a_{1,1}^p& a_{1,2}^p& \cdots & a_{1,k}^p\\ c_2& a_{2,1}^1& a_{2,2}^1& \cdots & a_{2,k}^1& \cdots & a_{2,1}^p& a_{2,2}^p& \cdots & a_{2,k}^p\\ ⋮& ⋮& ⋮& \ddots & ⋮& \cdots & ⋮& ⋮& \ddots & ⋮\\ c_k& a_{k,1}^1& a_{k,2}^1& \cdots & a_{k,k}^1& \cdots & a_{k,1}^p& a_{k,2}^p& \cdots & a_{k,k}^p\end{array}\right]$

$Z=\left[\begin{array}{cccc}1& 1& \cdots & 1\\ y_{p-1}& y_p& \cdots & y_{T-1}\\ y_{p-2}& y_{p-1}& \cdots & y_{T-2}\\ ⋮& ⋮& \ddots & ⋮\\ y_0& y_1& \cdots & y_{T-p}\end{array}\right]=\left[\begin{array}{cccc}1& 1& \cdots & 1\\ y_{1,p-1}& y_{1,p}& \cdots & y_{1,T-1}\\ y_{2,p-1}& y_{2,p}& \cdots & y_{2,T-1}\\ ⋮& ⋮& \ddots & ⋮\\ y_{k,p-1}& y_{k,p}& \cdots & y_{k,T-1}\\ y_{1,p-2}& y_{1,p-1}& \cdots & y_{1,T-2}\\ y_{2,p-2}& y_{2,p-1}& \cdots & y_{2,T-2}\\ ⋮& ⋮& \ddots & ⋮\\ y_{k,p-2}& y_{k,p-1}& \cdots & y_{k,T-2}\\ ⋮& ⋮& \ddots & ⋮\\ y_{1,0}& y_{1,1}& \cdots & y_{1,T-p}\\ y_{2,0}& y_{2,1}& \cdots & y_{2,T-p}\\ ⋮& ⋮& \ddots & ⋮\\ y_{k,0}& y_{k,1}& \cdots & y_{k,T-p}\end{array}\right]$

and

$U=\left[\begin{array}{cccc}{e}_p& e_{p+1}& \cdots & e_T\end{array}\right]=\left[\begin{array}{cccc}{e}_{1,p}& e_{1,p+1}& \cdots & e_{1,T}\\ e_{2,p}& e_{2,p+1}& \cdots & e_{2,T}\\ ⋮& ⋮& \ddots & ⋮\\ e_{k,p}& e_{k,p+1}& \cdots & e_{k,T}\end{array}\right].$

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

References

• Lütkepohl, Helmut (2005). New Introduction to Multiple Time Series Analysis. Berlin: Springer. ISBN 3540401725.