Estimation and Elimination of both Trend and Seasonality with a Moving Average Filter 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.
 A: 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 !
