How to check if a series has been seasonally adjusted correctly? I am a bit puzzled here and would like to understand how to check if a time series has been seasonally adjusted correctly using X-13 Arima. 
After seasonally adjusting time series using X13-ARIMA procedure from US census bureau, 
why does the auto.arima model still show a seasonal component of (0, 0, 12) ? 
Graph of the series or the acf/pacf donot show any seasonal component as such.
Questions :
Does this imply that the series has not been seasonally adjusted properly ? 
Are there visual cues or  tests to check if the series has been adjusted properly, specially in the case of stock economic series ?
How can we (or should we) remove the remaining seasonal component ? 
(x_t - x_(t-12)) filter seems to be the applicable but I am hesitant in applying this filter again after seasonal adjustment, as forecast::nsdiffs() doesnot imply any stochastic seasonality or any other reason. 
Should the (0, 0, 12) component be of concern if we are working with seasonally adjusted series for further analysis ?
 A: As a complement to the plot of the sample autocorrelations that you already made, you can plot the periodogram of the seasonally adjusted series to 
check if there are peaks at the seasonal frequencies.
You should also look at the following tables in the output file returned by X-13ARIMA-SEATS:


*

*F 3. Monitoring and Quality Assessment Statistics: Do these measures suggest an acceptable performance of the model? Some of these indicators will give 
you a measure about the quality of the estimated seasonal component.

*D 8.A  F-tests for seasonality: The program performs some tests for the presence of seasonality. You can run the program for the seasonally adjusted series that you have obtained and check if seasonality is significant according to these tests.
If no seasonality is present in your seasonally adjusted series, then you can 
set the arguments max.P, max.D and max.Q to zero when using auto.arima.
Edit:
If I remember correctly, the details about these test statistics and the diagnostic tests are given in this book.
In this website of the US Census Bureau you will find helpful documentation. See in particular 
FAQs number 10 and 11, which are related to your question and sketch some of the tests and measures reported in the tables mentioned above.
A: I have found that the seasonally adjusted series often has seasonal structure as the method used is very simplistic. A review of the seasonally adjusted series should exhibit no seasonal structure i.e. the acf should be free of structure and no seasonal spikes should be evident. Note that the acf can be downwards biased by anomalous data thus no structure may be a false negative. Put your series through an aggressive test such as conducted by AUTOBOX  or any good automatic ARIMA package and examine the suggested model for both stochastic and deterministic seasonal structure.
