## 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 t-2 + 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, n-1 + Usn s = 1, 2. (31.26)

Notice that the seasonal patterns reverse each year. Due to this alternating pattern, and assuming y0 = y-1 = 0, it can be seen that

n-1

Vt = S*n = X H)4n-; = – S*n-1 + uSn, (31.27)

1=0

where, in this case, n = [у1]...

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