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.
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 ?