# Time series

• Sep 17th 2010, 06:30 PM
Dogod11
Time series
I have a book series of time and says in part:

An example of time series would be:

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

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 and why they develop the series?

Thank you very much.

Greetings.

Dogod
• Sep 17th 2010, 07:02 PM
Soroban
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.

• Sep 18th 2010, 06:08 AM
Dogod11
Thank you very much