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In the book «Introduction to Time Series and Forecasting», by Brockwell and Davis, the authors state in section 1.5.2.1 (3rd edition) something which I summarise below.

Sample: $\{x_1,..., x_n\}$.

We first estimate the trend by applying a Moving Average Filter (MAF) chosen specially to eliminate seasonality and to dampen the noise. If the period $d$ is even, i.e. $d=2q$, then $$\hat m_t=\frac{0.5x_{t-q}+x_{t-q+1}+...+x_{t+q-1}+0.5x_{t+q}}{2q}$$ If period is odd, then use the simple MAF: $$\hat m_t=\frac{\sum_{j=-q}^{q}x_{t+q}}{2q+1}$$

Why does the first estimation eliminate seasonality? Is it because it considers the terms $x_{t-q},x_{t+q}$?

Afterwards, the authors also state something as:

After deseasonalizing the data, we reestimate the trend from the deseasonalized data. This reestimation is done in order to have a parametric form for the trend that can be extrapolated for the purposes of prediction and simulation.

In what precise way is the trend reestimation helpful for the purposes of prediction and simulation? and why not just use the 1st trend estimation?

Any help would be appreciated.

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1.5.2.1 is an attempt to obtain an equally weighted (except for the two endpoints) central average over 2 seasons (not 3 or 4 or any other ) thus obtaining a centralized replacement value that is supposedly unaffected by seasonal activity premising no anomalies. This anachronistic approach was first suggested in the 1940's by the innovative Julius Shiskin to purge the data of seasonality in order to more clearly see the trend GIVEN that the data could be characterized in simple T+S+I or T*S+I terms and that there were no level shifts. It was programmed because it was simple and thus became gospel because it had been programmed and was simple to understand thus it became the answer to all analyses/characterizations of time series data. It is the basis for X-11 .

G.E.P. Box and G. Jenkins essentially generalized this approach (i.e. extracting the signal hiding somewhere in the data ) and I once heard him say that he would have preferred ARIMA modelling (i.e. the Box-Jenkins approach ) to be called X-12 as it was an evolution/enhancement of X-11.

You can get the airline series ( a good example) from here https://www.mathworks.com/help/stats/examples/time-series-regression-of-airline-passenger-data.html?requestedDomain=www.mathworks.com and an updated discussion here http://autobox.com/cms/index.php/blog slide 44 and here .. my 2009 detailed discussion http://www.autobox.com/pdfs/vegas_ibf_09a.pdf slide 13

Answered by another old man !

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  • $\begingroup$ Dear IrishStat, thanks for your answer. However, could you expand a bit more on the second part of my question? $\endgroup$ – An old man in the sea. Nov 25 '17 at 8:27
  • $\begingroup$ I am unaware of any textbook example " of a small example where this is made clear" .Is this what you mean by the second part of your question. If not please restate your "second part" $\endgroup$ – IrishStat Nov 25 '17 at 11:15
  • $\begingroup$ I've just edited my question, restating my second part. I hope it's clearer now. Any help would be appreciated. $\endgroup$ – An old man in the sea. Nov 25 '17 at 14:51
  • $\begingroup$ "In what precise way is the trend reestimation helpful for the purposes of prediction and simulation? and why not just use the 1st trend estimation?" The first estimate assumed no seasonal factors by having adjusted/cleansed for seasonal factors we now can get a clearer (more correct) estimate of the trend. Now after getting this better estimate of the trend we detrend AND then again get better estimates of the seasonal factors. We do this until we stop and then we don't do it anymore because we stopped. $\endgroup$ – IrishStat Nov 25 '17 at 15:47

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