Using the HEGY Procedure When Not All Roots Are Present

Empirical studies have shown little evidence to support the presence of all unit roots present in the filter in quarterly seasonal time series. This paper analyses the performance of the Hylleberg, Engle, Granger and Yoo (1990) (HEGY) procedure when the roots under the null are not all present. We exploit the Vector of Quarters representation and cointegration relationship between the quarters when factors 4 (1 L),(1+ L), (1+ L2 ), (1 L2 ) and (1+ L + L2 + L3 ) are a source of nonstationarity in a process in order to obtain the distribution of tests of the HEGY procedure when the underlying processes have a root at the zero, Nyquist frequency, two complex conjugates of frequency / 2 and two combinations of the previous cases. We show both theoretically and through a Monte-Carlo analysis that the t-ratios and and the F-type tests used in the HEGY procedure have the same distribution as under the null of a seasonal random walk when the root(s) is/are present, although this is not the case for the t-ratio tests associated with unit roots at frequency 1 t 2 t / 2 .