Good to see the Bureau of Economic Analysis being so transparent in the link below. Residual seasonality is the ultimate seasonal adjustment bogey man. So it is important that any seasonally adjusted estimates, especially something as high profile as GDP, are assessed for residual seasonality. Basically, if a seasonally adjusted series is still seasonal, then the job hasn’t been done properly and there is some systematic calendar related variation still hanging around.
This typically occurs when aggregate estimates are derived from more detailed seasonally adjusted estimates, and then small amounts of “seasonality” can add up. It also occurs when the seasonal adjustment approach has not been applied in an optimal way.
The full BEA briefing note from May 2015 notes that:
“Each spring, BEA conducts an extensive review–receiving updated seasonally adjusted data from the agencies that supply us with data used in our calculation of GDP. Most of the data the feeds into GDP is seasonally adjusted by the source agency, not BEA. At the same time, BEA examines its own seasonal factors for those series that BEA seasonally adjusts itself.”
Full article here: http://blog.bea.gov/2015/05/22/residual-seasonality-gdp/
So, given that a lot of the estimates are supplied to BEA, it has to hope that these inputs are top notch or there could be issues. At least with some form of a residual seasonality test, they would be able to pick these issues up.
One way to check for residual seasonality is to seasonally adjust the seasonally adjusted output. The premise being that this will identify any additional seasonality as it is treating the seasonally adjusted estimates as the non-seasonal series. This approach does have its own problems depending on which seasonal adjustment approach is used, as a double application of seasonal adjustment methods can not be as powerful in its detection of the seasonal cycles that it has already removed. The HEGY approach for testing for unit roots is a good one to use for checking for residual seasonality (link: https://ideas.repec.org/a/eee/econom/v44y1990i1-2p215-238.html).