Time series-calculate white term residual values from observed X (t) values

[vikky]

If X(t)=e(t) +0.6e(t-1) and we define a backward shift operator such that
B X(t)=X(t-1) and
B^2 X(t)=X(t-2)
then after this how do we get get
X(t)=(1+0.6B)e(t)

I maybe missing something very obvious.
Want to calculate the white term residual values from observed X(t )values

 

[Viki2010]

If X(t)=e(t) +0.6e(t-1) and we define a backward shift operator such that
B X(t)=X(t-1) and
B^2 X(t)=X(t-2)
then after this how do we get get
X(t)=(1+0.6B)e(t)

I maybe missing something very obvious.
Want to calculate the white term residual values from observed X(t )values

You get this from the first equation

X(t) = e(t) + 0.6 * B * e(t) = e(t) * (1 + B * 0.6)

 

[vikky]

How do you go from 0.6e(t-1) in the original equation to to 0.6B)e(t) by introducing a backward shift operator?

 

[Viki2010]

How do you go from 0.6e(t-1) in the original equation to to 0.6B)e(t) by introducing a backward shift operator?

0.6 * e(t-1) = 0.6 * B * e(t)

 

[John Lee]

If X(t)=e(t) +0.6e(t-1) and we define a backward shift operator such that
B X(t)=X(t-1) and
B^2 X(t)=X(t-2)
then after this how do we get get
X(t)=(1+0.6B)e(t)

I maybe missing something very obvious.
Want to calculate the white term residual values from observed X(t )values

The backwards shift operator, B, shifts anything back one period in time and so it can be applied to the et's as well as the Xt's.

hence Be(t) = e(t-1).

(Just in case it is this issue that's causing the problem!)

 

[vikky]

thanks a ton john!that was my problem...I could not figure out why we were using it for e(t) as well...cheers