Time series

Hello, Dogod11!

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

Quote:

An example of time series would be:

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

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

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

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

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

Then we have:

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

And we see the pattern:

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

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

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

Why did they derive this formula?

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