What is the best filter/way for deseasonalizing quarterly data? There are many deseasonalization techniques for deseasonalizing quarterly time series data:
1. Filter: Centered moving averages
2. Filter/way: automatic ARIMA selection using X-11-Auto , X-11 based deseasonalization
3. Filter/way: automatic ARIMA selection using TRAMO, SEATS based deseasonalization 
etc.
So, What is the best filter for deseasonalizing quarterly data?
Can one elaborate on this, please?
 A: Like I said in my comment above, this question is too broad to be addressed in a single answer, but a concise and hopefully informative summary can be made upon a few references that I selected.

Maravall (1995) [1] analyses and compares some of the statistical properties of 
three seasonal adjustment tools and methods: X-11, ARIMA-model based methods (in particular the Airline model, ARIMA(0,1,1)(0,1,1)) and the basic structural time series models (BSM) studied in Harvey (1989) [2]. Some of the main points of the paper are the following:


*

*Structural time series models can be expressed as a restricted ARIMA model. The paper shows that for the BSM these restrictions are satisfied by the implied structure of the X-11 filters and by the Airline model.

*The implied autocorrelation structure of the input time series is very similar in the three methods (numerical examples are given in the paper). However, the allocation of the overall dynamics across the components may differ (technical reasons are given in the paper).

*The components extracted by X-11 can be expected to be smoother than those based on the BSM. 

*The variance of the X-11 irregular component is around 10% larger than that in the BSM.

*The BSM guarantees that the models are uniquely identified (in fact, the model defines a predefined structure for each component).

*The decomposition into components (trend-cycle, seasonal and irregular) 
of the ARIMA model chosen for the input time series is not always feasible.
For example, the Airline model requires a positive or null seasonal moving-average coefficient for the model to admit a decomposition.


Despite the paper mentions different pros and cons of each method, it is 
a critic to structural time series models. According to the author, 
to allow the trend and seasonal components to include (separable) white noise is to introduce unnecessary ambiguity.

Section 3 in Durbin (2000) [3] (reproduced in Durbin and Koopman (2001, section 3.5) [4] emphasizes the advantages of structural time series models. 
ARIMA-model based methods rely on the assumption that the differenced series is stationary elimination of trend and seasonal by differencing. Although this approach may not be troublesome for forecasting, it may be a rigid framework for seasonal adjustment. When working with structural time series models a model for each component is explicitly specified and, hence, this approach is judged more transparent by the author.

As regards X-11 and similar non-parametric smoothing techniques, it is 
generally acknowledged the practical success of these methods. X-11 has been used for long time on a daily basis on a large number of time series. However, the theoretical appeal of these methods is sometimes deemed lower, perhaps because they are considered intuitively-based techniques (Durbin, 2000) [3].
A limitation of non-parametric methods is that they depend on fixed filters and does not allow the flexibility of model-based methods. Despite the empirical success of X-11, recent developments have moved onto the direction of ARIMA models. X-11 incorporates now ARIMA techniques in the program X-13ARIMA-SEATS.

The disparity of views and approaches mentioned in this answer may make you think that the choice of the seasonal adjustment program is even more critical than you thought. In practice, for many applications the results obtained with X-11, an ARIMA model or the BSM are very similar. I would recommend you choosing the method that you understand better or are more familiar with, this will allow you to make an informed choice of the parameters and options that are available in these programs.

References
[1] Maravall, A. (1995)
On Structural Time Series Models and the Characterization of Components.
Journal of Business & Economic Statistics, Vol. 3 No. 4.
Stable URL
Also available here.
[2] Harvey, A. C. (1989). 
Forecasting, Structural Time Series Models and the Kalman Filter. Cambridge University Press.
[3] Durbin, J.  (2000)
The Foreman Lecture: the State Space Approach to Time Series Analysis and its Potential for Official Statistics (with Discussion by Geoff Lee)
URL: http://onlinelibrary.wiley.com/doi/10.1111/1467-842X.00104/abstract.
[4] Durbin, J. and Koopman,  S. J.(2001) Time Series Analysis by State Space Methods. Oxford University Press. 
