Robust tests for deterministic seasonality and seasonal mean shifts

We develop tests for the presence of deterministic seasonal behaviour and seasonal mean shifts in a seasonally observed univariate time series. These tests are designed to be asymptotically robust to the order of integration of the series at both the zero and seasonal frequencies. Motivated by the approach of Hylleberg, Engle, Granger and Yoo, we base our approach on linear filters of the data that remove any potential unit roots at the frequencies not associated with the deterministic component(s) under test. Test statistics are constructed using the filtered data such that they have well defined limiting null distributions regardless of whether the data are either integrated or stationary at the frequency associated with the deterministic component(s) under test. In the same manner as Vogelsang, Bunzel and Vogelsang and Sayginsoy and Vogelsang, we scale these statistics by a function of an auxiliary seasonal unit root statistic. This allows us to construct tests that are asymptotically robust to the order of integration of the data at both the zero and seasonal frequencies. Monte Carlo evidence suggests that our proposed tests have good finite sample size and power properties. An empirical application to UK gross domestic product indicates the presence of seasonal mean shifts in the data.