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Math Help - Time series

  1. #1
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    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
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  2. #2
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    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:

    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}<br />
Y_0 &=& 0 \\<br />
Y_1 &=& 0.8(0) + \epsilon_1 &=& \epsilon_1\\<br />
Y_2 &=& 0.8(\epsilon_1) + \epsilon_2 &=& 0.8\epsilon_1 + \epsilon_2 \\<br />
Y_3 &=& 0.8(0.8\epsilon_1 + \epsilon_2) + \epsilon_3 &=& 0.8^2\epsilon_1 + 0.8\epsilon_2 + \epsilon_3 \\<br />
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.

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  3. #3
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    Thank you very much
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