Near Seasonal Integration
As noted in Section 3.1 for the DHF test, Pr[t7s < 0] = Pr[x2(S) < S] seems to be converging to 1/2 as S increases. However, for the periodicities typically considered this probability always exceeds 1/2. This phenomenon indicates that a standard normal distribution may not be a satisfactory approximation when the characteristic root is close to 1 and the sample size is moderate, as Chan and Wei (1987) point out. It is also a well established fact that the power of unit root tests is quite poor when the parameter of interest is in the neighborhood of unity (see, for example, Evans and Savin (1981, 1984) and Perron (1989)). This suggests a distributional gap between the standard distribution typically assumed under stationarity and the function of Brownian motions obtained when the DGP is a random walk. To close this gap, a new class of models have been proposed, which allow the characteristic root of a process to be in the neighborhood of unity. This type of process is often called near integrated. Important work concerning near integration in a conventional AR(1) process includes Bobkoski (1983), Cavanagh (1986), Phillips (1987, 1988), Chan and Wei (1987), Chan (1988, 1989), and Nabeya and Perron (1994). In the exposition of the preceding sections, it has been assumed that the DGP is a special case of
Vt = Фе Vt-s + e t (31.62)
with фs = 1 and y-S+1 = … = y0 = 0. In this section we generalize the results by considering a class of processes characterized by an autoregressive parameter фs close to 1.
Analogously to the conventional near integrated AR(1), a noncentrality parameter c can be considered such that
ф§ = ec/N – 1 + N = 1 + IIе. (31.63)
This characterizes a near seasonally integrated process, which can be locally stationary (c < 0), locally explosive (c > 0), or a conventional seasonal random walk (c = 0). This type of near seasonally integrated process has been considered by Chan (1988, 1989), Perron (1992), Rodrigues (2000b), and Tanaka (1996). Similarly to the seasonal random walk, when the DGP is given by (31.62) and (31.63), and assuming that the observations are available for exactly N (N = T/S) complete years, then
Ssn = X eN Є.,и = X e( n4) c i s = 1, . . . , S. (31.64)
This indicates that each season represents a near integrated process with a common noncentrality parameter c across seasons. One of the main features of a process like (31.62) with ф§ = ec /N, is that
N/2ysn = NWSsn ^ °2lsc(r), s = 1, . . . , S, (31.65)
where Ssn is the PSP corresponding to season s and Jsc(r) is a Ornstein-Uhlenbeck process and not a Brownian motion as in the seasonal random walk case. Note that, as indicated by, for example, Phillips (1987) or Perron (1992), this diffusion process is generated by the stochastic differential equation
dJsc(r) = cJsc(r)dr + dWs(r),
and Jsc(0) = 0.
Applying results given by Phillips (1987), and following analogous steps to those underlying Section 3.1, yields:
T,* . . s=u
where Jsc(r) and Ws(r), s = 1,…, S, are independent Ornstein-Uhlenbeck processes and standard Brownian motions, respectively. Similarly, the f-statistic converges to
(Ф* – )
A more detailed analysis appears in Chan (1988, 1989), Perron (1992), Rodrigues (2000b) and Tanaka (1996). The result in (31.69) is the asymptotic power function for the DHF f-test. It is straightforward to observe that the distributions in (31.21) and (31.22) are particular cases of (31.68) and (31.69) respectively with c = 0.
The examination of the HEGY procedure in a near seasonally integrated framework is slightly more involved. As indicated by Rodrigues (2000b), ( 1 – (1 + n )L4) can be approximated by,
The results provided by Jeganathan (1991), together with the orthogonality of the regressors in the HEGY test regression, yield the distributions of the HEGY statistics in the context of a near seasonally integrated process. Rodrigues (2000b) establishes the limit results for the HEGY test regression. One important result also put forward by Rodrigues (2000b) is that the distributions are still valid when we allow different noncentrality parameters for each factor in (31.70).
We have considered only the simple seasonal random walk case, which was used to present the general properties of seasonally integrated processes. It should be noted, however, that the effect of nonzero initial values and drifts on the distributions of the seasonal unit root test statistics can easily be handled substituting the standard Brownian motions by demeaned or detrended independent Brownian motions.
Among other issues not considered are the implications of autocorrelation and mean shifts for unit root tests. The first is discussed in detail in Ghysels ef al. (1994), Hylleberg (1995), and Rodrigues and Osborn (1999). It is known that strong MA components can distort the power of these procedures. To a certain extent, however, these distortions can be corrected by augmenting the test regression with lags of the dependent variable.
The negative impact of mean shifts on the unit root test procedures, was noted by Ghysels (1991). Recently, Smith and Otero (1997) and Franses and Vogelsang
(1998) have shown, using artificial data, that the HEGY test is strongly affected
by seasonal mean shifts. This led Franses and Vogelsang to adapt the HEGY
test so as to allow for deterministic mean shifts (Smith and Otero also present
relevant critical values for the HEGY procedure in this context).
* We would like to thank three referees for their valuable comments.
1 The proof of the result appears in the Appendix to the companion working paper Ghysels, Osborn, and Rodrigues (1999).
2 Notice that the unit roots of a monthly seasonal random walk are:
1 -1, ±Ь л (1 ± V3 i), л (1 ± V3 і), л ( V3 ± i), і ( л/3 ± i).
The first is, once again, the conventional nonseasonal, or zero frequency, unit root. The remaining 11 seasonal unit roots arise from the seasonal summation operator 1 + L + L2 + … + L11 and result in nonstationary cycles with a maximum duration of one year. As can be observed, this monthly case implies five pairs of complex roots on the unit circle.
3 The reparameterization of the regressors proposed for monthly data by Beaulieu and Miron (1993) is typically preferred because, in contrast to that of Franses (1991), the constructed variables are asymptotically orthogonal.
4 Once again, due to the definition of his F-type statistic, the Kunst (1997) percentiles have to be divided by 4 to be comparable with those of Ghysels et al. (1994). In the monthly case, the Kunst values have to be divided by 12 for comparison with Taylor (1998). Since these percentiles are obtained from Monte Carlo simulations, they will not be identical across different studies.