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I have a time series that follow an MA(1) process , I do non-overlapping aggregation with aggregation level $m=2,3,\dots$, and then I calculate autocorrelation of basic and aggregated series, for negative value of moving average parameter, aggregated autocorrelation becomes close to zero, means that the aggregated series want to be like white noise, I was wondering if anybody can explain the reasons behind this behavior?

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up vote 4 down vote accepted

One way to analyze this is to start with the MA(1) expression $y_t = e_t - \theta e_{t-1}$, then write out the expressions for $(y_1 + y_2)$ and $(y_3 + y_4)$ in terms of the $e_t$. You get:

$y_1 + y_2 = e_2 + (1-\theta)e_1 - \theta e_0$

$y_3 + y_4 = e_4 + (1-\theta)e_3 - \theta e_2$

Note that the only common term between the two expressions is $e_2$; the first expression has $e_1$ and $e_0$, which don't appear in the second, and the second has $e_4$ and $e_3$, which don't appear in the first. If you find the correlation explicitly, you get:

$\rho_{12,34} = -\theta / (1 + \theta^2 + (1-\theta)^2)$

For just $y_1$ and $y_2$, we have $y_1 = e_1 - \theta e_0$ and $y_2 = e_2 - \theta e_1$. Here we have $e_1$ in both, but fewer other terms. The correlation here is

$\rho_{12} = -\theta / (1+\theta^2)$.

Comparing the denominators of the two expressions for the correlation, you can see the first has the extra (non-negative) term $(1-\theta)^2$ in it. Since the numerators are the same, for all $\theta \ne 0 \space \text{or}\space 1$, the correlation between $y_t+y_{t+1}$ and $y_{t+2}+y_{t+3}$ will be closer to 0 than the correlation between $y_t$ and $y_{t+1}$.

As you aggregate more and more, this effect will get larger; the only common term is $e_{t+\tau}$, where you are aggregating $y_t \dots y_{t+\tau}$ and $y_{t+\tau+1} \dots y_{t+2\tau}$, but the variability of the aggregates increases as the number of terms making up the aggregation increases, so the correlation decreases.

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may I ask one more question please? I have applied SES as a forecasting method on both basic and aggregated series, and then I disaggregated forecasts resulted from aggregated series and finally I calculate ratio of MSEbasic/MSEdagg, where MSEbasic=Var(dt-ft) and MSEdagg=Var(dt-(FT/m), dt is basic series, ft is forecast of basic series, FT is aggregated forecast and m is aggregation level. this ratio show that for highly negative value of Theta, basic approach works better and for the rest of Theta parameter aggregated approach works better, I'm trying to find why? –  Roji Jan 26 '12 at 9:40
Thank you so much for your help. –  Roji Jan 26 '12 at 11:13
(+1) Nicely presented. –  cardinal Jan 26 '12 at 13:59
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