Asymptotic properties
Consider the DGP of the seasonal random walk with initial values y§+s =… = y0 = 0. Using the notation of (31.6), the following § independent partial sum processes (PSPs) can be obtained:
n
Ssn = ^esj s = 1,…, §, n = 1,…, N (31.12)
j=i
where n represents the number of years of observations to time t. From the functional central limit theorem (FCLT) and the continuous mapping theorem (CMT) the appropriately scaled PSPs in (31.12) converge as N ^ ™ to
Ssn ^ oWs(r), (31.13)
where ^ indicates convergence in distribution, while Ws(r), s = 1,…, § are independent standard Brownian motions. Furthermore, the following Lemma collecting the relevant convergence results for seasonal unit root processes of periodicity § can be stated:
Lemma 1. Assuming that the DGP is the seasonal random walk in (31.1) with initial values equal to zero, e t ~ iid(0, о2) and T = §N, then from the CMT, as T ^ ro,
(a) 
T~1/2ytk ^ 
§1/2oLkWs 

T 
§ 
1 

(b) 
t 3/2 X yk 
^ §3/2 oX 
Wsdr 
t=1 
s=1 
0 

T 
§ 
1 

(c) 
T ~2 X ytiyt 
k ^ §2 °2 X 
Ws(Lk~Ws) dr k > 
t=1 
s=1 
0 

T 
§ 
1 

(d) 
T 1X yk C 
^ §102 X 
(LkWs) dWs 
t=1 
s=1 
0 
where k = 1,…, §, Ws(r) (s = 1 + (t – 1) mod §) are independent standard Brownian motions, L is the lag operator which shifts the Brownian motions between seasons (LkWs = Wk with Wk = W§+sk for s – k < 0) and Ws = Ws(r) for simplicity of notation.
It is important to note the circular property regarding the rotation of the Wk, so that after § lags of yt the same sum of § integrals emerges. The Lemma is established in Osborn and Rodrigues (1998).
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