1. ## Time series

I have a book series of time and says in part:

An example of time series would be:

$\displaystyle Y_t = 0.8Y_{t -1} + \epsilon_t,$ $\displaystyle t = 1, 2 ...$

Then they say, this series can also be expressed as:

$\displaystyle Y_t = \displaystyle\sum_{i=1}^t{0.8^{t - i}} \epsilon_i$

How do they do and why they develop the series?

Thank you very much.

Greetings.

Dogod

2. Hello, Dogod11!

I'm not familiar with Time Series.
I've had to come up with my own theories.

An example of time series would be:

$\displaystyle Y_t \:=\: 0.8Y_{t -1} + \epsilon_t, \;\;t = 1, 2 \hdots$

Then they say, this series can also be expressed as:

. . $\displaystyle Y_t \:=\: \displaystyle\sum_{i=1}^t{0.8^{t - i}} \epsilon_i$

How do they do it and why they develop the series?

It only makes sense if $\displaystyle Y_0 = 0$ . . . the initial quantity is zero.

Then we have:

. . $\displaystyle \begin{array}{cccccccccc} Y_0 &=& 0 \\ Y_1 &=& 0.8(0) + \epsilon_1 &=& \epsilon_1\\ Y_2 &=& 0.8(\epsilon_1) + \epsilon_2 &=& 0.8\epsilon_1 + \epsilon_2 \\ Y_3 &=& 0.8(0.8\epsilon_1 + \epsilon_2) + \epsilon_3 &=& 0.8^2\epsilon_1 + 0.8\epsilon_2 + \epsilon_3 \\ Y_4 &=& 0.8(0.8^2\epsilon_1 + 0.8\epsilon_2 + \epsilon_3) + \epsilon_4 &=& 0.8^3\epsilon_1 + 0.8^2\epsilon_2 + 0.8\epsilon_3 + \epsilon_4\end{array}$

And we see the pattern:

. . $\displaystyle Y_t \;=\;0.8^{t-1}\epsilon_1 + 0.8^{t-2}\epsilon_2 + 0.8^{t-3}\epsilon_3 + \hdots + 0.8^2\epsilon_{t-2} + 0.8\epsilon_{t-1} + \epsilon_t$

which can be written: .$\displaystyle \displaystyle \sum^t_{i=1} 0.8^{t-i}\epsilon_i$

~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

Why did they derive this formula?

So we can find, say, the 20th term
. . without cranking out the first 19 terms.

3. Thank you very much