# 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 ?

• You may be interested in this answer, which is related to a similar question. – javlacalle Oct 2 '14 at 17:23

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.

• Thank you for the reply. There are no peaks in the periodogram and that is why was concerned as to why auto.arima is still showing that seasonal component. Is there a bias towards detecting seasonality ? Would you please provide a little more detail in reading the diagnostics table and what exactly to look for ? The X-13 output is giving the diagnostics in a tabular form. Any way to get them in a log file like X-12 ? – Anusha Sep 29 '14 at 7:27
• I used the option for stock series when creating the spec file for stock series. Are there any other details to specify in that case ? – Anusha Sep 29 '14 at 7:28
• Like I said, you can run the program on the seasonally adjusted data and look at output line "D 8.A F-tests for seasonality", at the end it is explicitly stated whether seasonality is present or not and the significance level. If it is concluded that there is no seasonality I wouldn't concern about the seasonal MA paremeter and would set max.P, max.D and max.Q to zero. – javlacalle Sep 29 '14 at 18:57
• They seem to have changed the format of the files returned. There is a matrix of output stats returned but not a detailed file like in X-12. Could you tell which option of output files to select to get the file with the above tests ? Many thanks. – Anusha Sep 29 '14 at 20:16
• I get a detailed file with extension "out" (not "log"). – javlacalle Sep 29 '14 at 20:33

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.