# ratio, MSE of basic demand to MSE of aggregated demand for MA(1) process

I have a basic demand series that follow MA(1) process,I've applied non-overlapping aggregation approach and then SES on both basic and aggregated series to obtain forecasts, 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 at time t, ft is forecast of basic series, FT is aggregated forecast and m is aggregation level and alpha is smoothing constant. this ratio shows 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 understand why aggregation does not work for high negative Theta,I was wondering if you guide me. bests Roji

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Simple exponential smoothing, although a widely-used forecasting technique, is not always optimal. It can be shown that it is optimal for an IMA(1,1) process:

$y_t - y_{t-1} = e_t - \theta e_{t-1}$

which is obviously not the same as the processes, either aggregated or otherwise, that you are dealing with. Consequently, you can't expect too much!

More generally, as $\theta$ gets more negative, the correlation between two consecutive $y_t$ gets larger (more positive). Aggregating highly positively correlated series doesn't do as much for you as aggregating less positively correlated series, all other things being equal, which they never are. As a thought experiment, consider a case where the autocorrelation = 1, say, $y_{t+1} = y_t = 0$; clearly no level of aggregation will do you any good when forecasting. Combining this with the suboptimal SES forecasting approach is resulting in the interesting table you've included in the question.

However, other forecasting approaches may give you substantially different results. If you choose one that is optimal for an MA(1) process, and why not do so, then there won't be any point in aggregating, as you cannot gain information about the structure of the errors by aggregation.

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many many thanks for your answer you said Aggregating highly positively correlated series doesn't do as much for you as aggregating less positively correlated series, but why? why aggregation works like this? could you please explain what do you mean by saying all other things? –  Roji Jan 27 '12 at 9:34