3-oxoacyl-(acyl-carrier-protein) reductase

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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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