An equaion like: X_t=rho*Xt-1 is called Difference Equation of order 1. It is straitforward to write the solution, which is: Xt=rho^t*X0.

When the order is higher, things will be complicated. For example: Xt=rho1*Xt-1+rho2*Xt-2....+rhop*Xt-p

The solution needs a littlle trick. We define C0=1, C1=rho1, C2=C1*rho1+rho2, C3=C2*rho1+C1*rho2+rho3,.....

Cn=Cn-1*rho1+Cn-2*rho2+...Cn-p*rhop.

Xt=rho1*Xt-1+rho2*Xt-2+...+rhop*Xt-p; times C0

Xt-1=rho1*Xt-2+rho2*Xt-3+...+rhop*Xt-1-p; times C1

Xt-2=rho1*Xt-3+rho2*Xt-4+...+rhop*Xt-2-p; times C2

......

X2=rho1*X1+rho2*X0+...+rhop*X2-p; times Ct-2

X1=rho1*X0+rho2*X-1+...+rhop*X1-p; times Ct-1

And then sum them up. The solution of Difference Equation of order p is:

Xt=summation(m=t to t+p-1){[summation(i=m-t+1 to p) Cm-i*rhoi]*Xt-m}

It will be much easier if the companian matrix idea is applied.

Define Yt=(Xt,Xt-1,...Xt-p)'

The difference equation of order p can be rewritten as Yt=A*Yt-1.

A=

| rho1, rho2,......,rhop |

|1,0,0....... 0|

|... |

|0,0,......... 1, 0|

So, Yt=A^tY0

The Difference Equaion has been widely used in Time Series Analysis!

[此贴子已经被作者于2006-9-12 7:09:20编辑过]