I am trying to write the following as a sum $$\phi^{5}*Y_{t-10} + \phi^{4}*e_{t-8} + \phi^3*e_{t-6} + \phi^2*e_{t-4} +e_t$$

I let the 10 in the index of Y be $k$. In following the pattern of this sum I am trying to rewrite the sum in general terms of t and k. So far I have the following $$\phi^{k/2}*Y_{t-k} + \sum_{j = ?}^{k-2} \phi^{j/2}*e_{t-j} + e_t$$

I know that the subscript is skipping by two's and I left the last $e_t$ out of the summation because it jumps from $e_t$ to $e_{t-4}$. As you can see I am having a difficult time getting the $j$ in $e_{t-j}$ to increase by 2 to $k-2$.

Note: The beginning sum is a recursion process of a AR(2) time series model but since I need help on only rewriting the sum I decided to put this question on Math Stack.

Any help is greatly appreciated.


1 Answer 1


Term with $\phi$ seems to be missing so we can write it as $$ \phi^n*Y_{t-2n} + \sum_{k=0}^{n-1} \phi^k e_{t-2k} - \phi*e_{t-2} = \phi^n*Y_{t-2n} + \sum_{k=2}^{n-1} \phi^k e_{t-2k} + e_t.$$


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