When we're dealing with seasonal data - e.g., quarterly data - we need to distinguish between "deterministic seasonality" and "stochastic seasonality". The first type of seasonality is what we try to remove when we "seasonally adjust" the series. It's also what we're trying to account for when we include seasonal dummy variables in a regression model.
On the other hand, "stochastic seasonality" refers to unit roots at the seasonal frequencies. This is a whole different issue, and it's been well researched in the time-series econometrics literature.
This distinction is similar to that between a "deterministic trend" and a "stochastic trend" in annual data. The former can be removed by "de-tending" the series, but the latter refers to a unit root (at the zero frequency).
The most widely used procedure for testing for seasonal unit roots is that proposed by Hylleberg et al. (HEGY) (1990), and extended by Ghysels et al. (1994).
In my graduate-level time-series course we always look at stochastic seasonality. Recently, Nicolas Ronderos has written a new "Add-in" for EViews to make it easy to implement the HEGY testing procedure (see here). This will certainly save some coding for EViews users.
Of course, stochastic seasonality can also arise in the case of monthly data - this is really messy - see Beaulieu and Miron (1993). In the case of half-yearly data, the necessary theoretical framework and critical values are developed and illustrated by Feltham and Giles (2003)
And if you have unit roots at the seasonal frequencies in two or more time-series, you might also have seasonal cointegration. The seminal contribution relating to this is by Engle et al. (1993), and an short empirical application is provided by Reinhardt and Giles (2001)
I plan to illustrate the application of seasonal unit root and cointegration tests in a future blog post.
(Also, note the comment from Jack Lucchetti, below, that draws attention to a HEGY addon for Gretl, written by Ignacio Diaz Emparanza.)
Beaulieu, J. J., and J. A. Miron, 1993. Seasonal unit roots in aggregate U.S. data. Journal of Econometrics, 55, 305-328.
Engle, R. F., C. W. J. Granger, S. Hyleberg, H. S. Lee, 1993. Seasonal cointegration: The Japanese consumption function. Journal of Econometrics, 55, 275-298.
Feltham, S. G. and D. E. A. Giles, 2003. Testing for unit roots in semi-annual data. in D.E.A. Giles
(ed.), Computer-Aided Econometrics. Marcel Dekker, New York, 175-208. (Pre-print here.)
Ghysels, E., H. S. Lee, and J. Noh, 1994. Testing for unit roots in seasonal time series: Some theoretical extensions and a Monte Carlo investigation. Journal of Econometrics, 62, 415-442.
Hylleberg, S., R. F. Engle, C. W. J. Granger, and B. S. Yoo, 1990. Seasonal integration and cointegration. Journal of Econometrics, 44, 215-238.
Reinhardt, F. S. and D. E. A. Giles, 2001. Are cigarette bans really good economic policy?. Applied Economics, 33, 1365-1368. (Pre-print here.)