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{{Howto|date=August 2013}}
 
This page shows the details for different matrix notations of a [[vector autoregression]] process with ''k'' variables.
 
==Var(''p'')==
{{Main|Vector autoregression}}
 
:<math>y_t =c + A_1 y_{t-1} + A_2 y_{t-2} + \cdots + A_p y_{t-p} + e_t, \, </math>
Where  each <math>y_{i}</math> is a ''k''&nbsp;×&nbsp;1 vector and each <math> A_i </math> is a ''k''&nbsp;×&nbsp;''k'' matrix.
 
==Large matrix notation==
 
:<math>\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_{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}
+ \cdots +
\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}
 
+ \begin{bmatrix}e_{1,t} \\ e_{2,t}\\ \vdots \\ e_{k,t}\end{bmatrix}</math>
 
==Equation by equation notation==
 
Rewriting the ''y'' variables one to one gives:
 
<math>y_{1,t} = c_{1} + a_{1,1}^1y_{1,t-1} + a_{1,2}^1y_{2,t-1} +\cdots + a_{1,k}^1y_{k,t-1}+\cdots+a_{1,1}^py_{1,t-p}+a_{1,2}^py_{2,t-p}+ \cdots +a_{1,k}^py_{k,t-p} + e_{1,t}\,</math>
 
<math>y_{2,t} = c_{2} + a_{2,1}^1y_{1,t-1} + a_{2,2}^1y_{2,t-1} +\cdots + a_{2,k}^1y_{k,t-1}+\cdots+a_{2,1}^py_{1,t-p}+a_{2,2}^py_{2,t-p}+ \cdots +a_{2,k}^py_{k,t-p} + e_{2,t}\,</math>
 
<math>\qquad\vdots</math>
 
<math>y_{k,t} = c_{k} + a_{k,1}^1y_{1,t-1} + a_{k,2}^1y_{2,t-1} +\cdots + a_{k,k}^1y_{k,t-1}+\cdots+a_{k,1}^py_{1,t-p}+a_{k,2}^py_{2,t-p}+ \cdots +a_{k,k}^py_{k,t-p} + e_{k,t}\,</math>
 
==Concise matrix notation==
 
One can rewrite a VAR(''p'') with ''k'' variables in a general way which includes ''T+1'' observations <math>y_0</math> through <math>y_T</math>
 
:<math> Y=BZ +U \, </math>
 
Where:
:<math> Y=
\begin{bmatrix}y_{p} & y_{p+1} & \cdots & y_{T}\end{bmatrix} =
\begin{bmatrix}y_{1,p} & y_{1,p+1} & \cdots & y_{1,T} \\ y_{2,p} &y_{2,p+1} & \cdots & y_{2,T}\\
\vdots& \vdots &\vdots &\vdots \\  y_{k,p} &y_{k,p+1} & \cdots & y_{k,T}\end{bmatrix} </math>
 
:<math> 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}
</math>
 
:<math>
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}
</math>
 
and
 
:<math>U=  
\begin{bmatrix}
e_{p} & e_{p+1} & \cdots & e_{T}
\end{bmatrix}=
\begin{bmatrix}
e_{1,p} & e_{1,p+1} & \cdots & e_{1,T} \\
e_{2,p} & e_{2,p+1} & \cdots & e_{2,T} \\
\vdots & \vdots & \ddots & \vdots \\
e_{k,p} & e_{k,p+1} & \cdots & e_{k,T}
\end{bmatrix}.
</math>
 
One can then solve for the coefficient matrix ''B'' (e.g. using an  [[ordinary least squares]] estimation of <math> Y \approx BZ</math>)
 
==References==
{{Reflist}}
*{{Cite book |first=Helmut |last=Lütkepohl |title=New Introduction to Multiple Time Series Analysis |publisher=Springer |location=Berlin |year=2005 |isbn=3540401725 }}
 
[[Category:Econometrics]]
[[Category:Multivariate time series analysis]]

Latest revision as of 22:44, 24 August 2013

Template:Howto

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

Var(p)

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yt=c+A1yt1+A2yt2++Apytp+et,

Where each yi is a k × 1 vector and each Ai is a k × k matrix.

Large matrix notation

[y1,ty2,tyk,t]=[c1c2ck]+[a1,11a1,21a1,k1a2,11a2,21a2,k1ak,11ak,21ak,k1][y1,t1y2,t1yk,t1]++[a1,1pa1,2pa1,kpa2,1pa2,2pa2,kpak,1pak,2pak,kp][y1,tpy2,tpyk,tp]+[e1,te2,tek,t]

Equation by equation notation

Rewriting the y variables one to one gives:

y1,t=c1+a1,11y1,t1+a1,21y2,t1++a1,k1yk,t1++a1,1py1,tp+a1,2py2,tp++a1,kpyk,tp+e1,t

y2,t=c2+a2,11y1,t1+a2,21y2,t1++a2,k1yk,t1++a2,1py1,tp+a2,2py2,tp++a2,kpyk,tp+e2,t

yk,t=ck+ak,11y1,t1+ak,21y2,t1++ak,k1yk,t1++ak,1py1,tp+ak,2py2,tp++ak,kpyk,tp+ek,t

Concise matrix notation

One can rewrite a VAR(p) with k variables in a general way which includes T+1 observations y0 through yT

Y=BZ+U

Where:

Y=[ypyp+1yT]=[y1,py1,p+1y1,Ty2,py2,p+1y2,Tyk,pyk,p+1yk,T]
B=[cA1A2Ap]=[c1a1,11a1,21a1,k1a1,1pa1,2pa1,kpc2a2,11a2,21a2,k1a2,1pa2,2pa2,kpckak,11ak,21ak,k1ak,1pak,2pak,kp]
Z=[111yp1ypyT1yp2yp1yT2y0y1yTp]=[111y1,p1y1,py1,T1y2,p1y2,py2,T1yk,p1yk,pyk,T1y1,p2y1,p1y1,T2y2,p2y2,p1y2,T2yk,p2yk,p1yk,T2y1,0y1,1y1,Tpy2,0y2,1y2,Tpyk,0yk,1yk,Tp]

and

U=[epep+1eT]=[e1,pe1,p+1e1,Te2,pe2,p+1e2,Tek,pek,p+1ek,T].

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

References

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