Suppose you have a time series $\ Z_t$ that is used as a forecasting model for $\tau$ steps ahead from origin $T$. $\ Z_t$ is defined as:
$\ Z_t = 0.05 Z_T + 0.10 Z_{T-1} + 0.15 Z_{T-2} + 0.20 Z_{T-3} + 0.15 Z_{T-4} + 0.10 Z_{T-5} + 0.05 Z_{T-6 }$
How to calculate the $\ average\ age$ of this series observations in this forecasting model?
Is it right to assume the average age does not depend on the weights in each $\ Z_{T-i}$? In this case, the age of observation $\ Z_T$ is zero, since it's the observation at the origin $\ T$. The age of observation $\ Z_{T-1}$ is 1 since it happened one time-step ago. And so, ages will be respectively, from $\ T$ to $\ T-6$: 0, 1, 2, 3, 4, 5, and 6. Is this right?
And once the$\ average\ age$ is found, how to calculate a smoothing coefficent $\alpha$ that will yield a exponential smoothing model with the $\ average\ age$ equal to model $\ Z_t$?
If $\ Z_t$ were a exponential smoothing model, I'd know the value of $\alpha $; hence, I'd be able to calculate the average age of observations using equation: ($ \ 1$ - $\alpha $)/$\alpha $. However, it seems to me that $\ Z_t$ is a simple $\ AR$ model (moving average).