A COMPANION TO Theoretical Econometrics
Extensions to the HEGY approach
Ghysels, Lee, and Noh (1994), or GLN, consider further the asymptotic distribution of the HEGY test statistics for quarterly data and present some extensions. In particular, they propose the joint test statistics F(l 1 П 12 П 13 П l4) and F(l 2 П 13 П l4), the former being an overall test of the null hypothesis yt ~ SI(1) and the latter a joint test of the seasonal unit roots implied by the summation operator 1 + L + L2 + L3. Due to the two-sided nature of all F-tests, the alternative hypothesis in each case is that one or more of the unit root restrictions is not valid. Thus, in particular, these tests should not be interpreted as testing seasonal integration against stationarity for the process. From the asymptotic independence of t*., i =
1,. .., 4, it follows that F(l 1 П 12 П 13 П l4) has the same asymptotic distribution as 1 X4= 1(tft. )2, where the individual asymptotic distributions are given by (31.52), (31.56), (31.59) and (31.60). Hence, F(l 1 П 12 П 13 П l4) is asymptotically distributed as the simple average of the squares of each of two Dickey-Fuller distributions, a DHF distribution with S = 2 and (31.60). It is straightforward to see that a similar
expression results for F(l 2 П 13 П l4), which is a simple average of the squares of a Dickey-Fuller distribution, a DHF distribution with S = 2 and (31.60).
GLN also observe that the usual test procedure of Dickey and Fuller (DF) (1979) can validly be applied in the presence of seasonal unit roots. However, this validity only applies if the regression contains sufficient augmentation. The essential reason derives from (31.39), so that the SI(1) process ASyy = et can be written as
Ayt = а 1У-1 + фАум + ... + Фз-Ау t-s+i + e t (31.61)
with a 1 = 0 and ф1 = ... = ф§-1 = -1. With (31.61) applied as a unit root test regression, t&i asymptotically follows the usual DF distribution, as given in (31.52). See Ghysels, Lee, and Siklos (1993), Ghysels et al. (1994) and Rodrigues (2000a) for a more detailed discussion.
Beaulieu and Miron (1993) and Franses (1991) develop the HEGY approach for the case of monthly data.3 This requires the construction of at least seven transformed variables, analogous to yW/t, y{T)t, and yP)/t used in (31.41) to (31.43), and the estimation of twelve coefficients ni (i = 1,..., 12). Beaulieu and Miron present the asymptotic distributions, noting that the t-type statistics corresponding to the two real roots of +1 and -1 each have the usual Dickey-Fuller form, while the remaining coefficients correspond to pairs of complex roots. In the Beaulieu and Miron parameterization, each of the five pairs of complex roots leads to a tft. with a DHF distribution (again with S = 2) and a tfti with the distribution (31.60).
Although both Beaulieu and Miron (1993) and Franses (1991) discuss the use of joint F-type statistics for the two coefficients corresponding to a pair of complex roots, neither considers the use of the F-tests as in Ghysels et al. (1994) to test the overall A12 filter or the eleven seasonal unit roots. Taylor (1998) supplies critical values for these overall joint tests in the monthly case.
Kunst (1997) takes an apparently different approach to testing seasonal integration from that of HEGY, but his approach is easily seen to be related to that of DHF. Kunst is primarily concerned with the distribution of a joint test statistic. Although apparently overlooked by Kunst, it is easy to see that his joint test is identical to the joint test of all coefficients which arises in the HEGY framework. Ghysels and Osborn (2000) and Osborn and Rodrigues (1998) discuss in detail the equivalence between the tests proposed by Kunst and prior existing tests. Also, comparison of the percentiles tabulated by Ghysels et al. (1994) and Kunst with S = 4 are effectively identical.4 Naturally, these results carry over to the monthly case, with the HEGY F-statistic of Taylor (1998) being equivalent to that of Kunst with S = 12. Kunst does, however, provide critical values for other cases, including S = 7, which is relevant for testing the null hypothesis that a daily series is seasonally integrated at a period of one week.
