Alternative representation of time series

Question

In a paper I am reading, it refers to the following time series model:

$$ Y_t=\rho Y_{t-1}+e_t $$ Where $ \lvert\rho\rvert < 1$

It goes on to say that this process can be represented in the following way:

$$ Y_t=\sum^{\infty}_{h=0}\rho ^h e_{t-h}=\sum^{\infty}_{\tau=-\infty} e_{t} I(\tau \le t)\rho^{t-\tau} $$

Where $I()$ is an indicator function.

I understand that the first expression can be found by telescoping the initial expression, but how do they reach the final expression, and if possible, what is the intuition of representing it this way?

Thanks

Answer 1

Using the substitution $\tau{}={}t-k$ implies:

i) if $\,\,\,0\le k <\infty\,,\,\,$ then $\,\,-\infty<\tau\le t$;

ii) $Y_t=\displaystyle \sum^{\infty}_{h=0}\rho ^h e_{t-h}{}={}\displaystyle \sum^{t}_{\tau=-\infty} e_{\tau} \rho^{t-\tau}{}={}\displaystyle \sum^{\infty}_{\tau=-\infty} e_{\tau} I(\tau \le t)\rho^{t-\tau}\,\,.$