How to calculate the average age of observations in forecasting models of various types?

Question

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).

Answer 1

The average age of observations in a model is calculating by attributing age 0 for the current observation at time $t$, age 1 for the first past observation $t-1$, age 2 to the second past observation or the observation $z_{t-2}$, and so on.

In the end, if the model goes back to observation $z_{t-m}$ and it has one observation for all the intermediate periods between $z_t$ and $z_{t-m}$, the average age $AA$ will be:

$\large AA = \large \frac{\frac{(r_0 + r_m).m}{2} }{m+1}$

Knowing that, by the one of the criteria above, $AA(r_0)$ = 0, one can write:

$\large AA = \large \frac{\frac{(r_m).m}{2} }{m+1}$

where $r_m$ is the age of the last observation, and $m$ is the index of the last observation. So, if the last observation is $z_6$, $r_m$ = 6, and $m$ = 6.