Testing complex unit roots
Before proceeding to the examination of the procedure proposed by Hylleberg et al. (1990) it will be useful to consider some of the issues related to testing complex unit roots, because these are an intrinsic part of any SI(1) process.
The simplest process which contains a pair of complex unit roots is
Vt = – У t2 + ut, (31.25)
with ut ~ iid(0, о2). This process has S = 2 and, using the notation identifying the season s and year n, it can be equivalently written as
Vsn = – Vs, n1 + Usn s = 1, 2. (31.26)
Notice that the seasonal patterns reverse each year. Due to this alternating pattern, and assuming y0 = y1 = 0, it can be seen that
n1
Vt = S*n = X H)4n; = – S*n1 + uSn, (31.27)
1=0
where, in this case, n = [у1]. Note that S*n (s = 1, 2) are independent processes, one corresponding to each of the two seasons of the year. Nevertheless, the nature of the seasonality implied by (31.25) is not of the conventional type in that S* (for given s) tends to oscillate as j increases. Moreover, it can be observed from (31.27) that aggregation of the process over full cycles of two years annihilates the nonstationarity as S*n1 + S*n = usn. To relate these S*n to the S independent random walks of (31.6), let ej = (1)jusj which (providing the distribution of ut is symmetric) has identical properties. Then
S* = sn 


X (1)jusj = X£sj = Sjn,
.j=1 j=1
where Sjn is defined in (31.12). Analogously to the DHF test, the unit root process (31.25) may be tested through the fratio for a* in
(1 + L2)Vf = a* Vt2 + Uf. (31.29)
The null hypothesis is a* = 0 with the alternative of stationarity implying a* > 0. Then, assuming T = 2N, under the null hypothesis
T 2 N
T 1 X yt2 Uf (2N )1 X X S*,H (S*, j + S*,H)
T a 2 =———— — = s=j. (31.30)
2 T 2 N
T2 X yh (2N)2XX (S*,j1)2
f=1 s=1 j=1
and
(31.31)
If, for further expositional clarity, we assume that N is even, then using (31.28), we have
N N/2
XS*j1(S*j + S*j1) = X [S*,2/2(S*2I1 + S*2;2) + S*,2i1(S*2 + S*21)]
i= 1 i=1
N/2
= X [Ss,2 i2( Ss,2i1 + Ss,2i2) – Ss,2i1(Ss,2i – Ss,2i1)]
i=1
N
X Ss, j 1(Ss, j – Ss, j 1).
j=1
Thus, there is a "mirror image" relationship between the numerator of (31.30) and (31.31) compared with that of (31.19) and (31.20) with § = 2. The corresponding denominators are identical as (S*)2 = Sj Thus, by applying similar arguments as in the proof of Lemma 1:
T
2 a*
and
ta* a2 
which can be compared with (31.21) and (31.22) respectively. This mirror image property of these test statistics has also been shown by Chan and Wei (1988) and Fuller (1996, pp. 5534). One important practical consequence of (31.33) is that with a simple change of sign, the DHF tables with § = 2 apply to the case of testing a* = 0 in (31.29). Under the assumed DGP (31.25), we may also consider testing the null hypothesis a*1 = 0 against the alternative a* Ф 0 in 
(1 + L2)Vt = a* yt_1 + ut. 
The test here is not, strictly speaking, a unit root test, since the unit coefficient on L2 in (31.34) implies that the process contains two roots of modulus one, irrespective of the value of a*. Rather, the test of a* = 0 is a test of the null hypothesis that the process contains a halfcycle every § = 2 periods, and hence a full cycle every four periods. The appropriate alternative hypothesis is, therefore, twosided. For this test regression, 
T a* = 















W1(r)dW2(r) –
W2(r)dW1(r)
0
1
[Ws (r)]2dr
s =1

W1(r)dW2(r) –
W2(r)dW1(r)



1
[Ws(r)]2 dr
0
Indeed, the results for the distributions associated with the test statistics in (31.29) and (31.34) continue to apply for the test regression
(1 + L2)Vt = a* yt1 + a*yt2 + £t (31.38)
because the regressors yt1 and yt2 can be shown to be asymptotically orthogonal (see, for instance, Ahtola and Tiao (1987) or Chan and Wei (1988) for more details).
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