A COMPANION TO Theoretical Econometrics

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


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


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*n-1 + 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* =



n odd


Подпись: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 f-ratio for a* in

(1 + L2)Vf = a* Vt-2 + 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 yt-2 Uf (2N )-1 X X S*,H (S*, j + S*,H)

T a 2 =------------- — = s=j. (31.30)

2 T 2 N

T-2 X yh (2N)-2XX (S*,j-1)2

f=1 s=1 j=1



If, for further expositional clarity, we assume that N is even, then using (31.28), we have

N N/2

XS*j-1(S*j + S*j-1) = X [S*,2/-2(S*2I-1 + S*2;-2) + S*,2i-1(S*2 + S*2-1)]

i= 1 i=1


= X [Ss,2 i-2( Ss,2i-1 + Ss,2i-2) - Ss,2i-1(Ss,2i - Ss,2i-1)]



X Ss, j -1(Ss, j - Ss, 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:


2 a*




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. 553-4). 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 half-cycle every § = 2 periods, and hence a full cycle every four periods. The appropriate alternative hypothesis is, therefore, two-sided. For this test regression,

T a* =






T -11 yt-1 ut


T-21 y2-1



Again referring to (31.27) and (31.28), we can see that


(2N)-11 [-S2,j-1(S1rj - V1) + S1J(S2J - S2,j-1)]






(2N)-21 (S2 j-1 + S2,j)



Thus, (31.35) converges to,



W1(r)dW2(r) -




[Ws (r)]2dr

Подпись: 0s =1

and consequently,


W1(r)dW2(r) -







Подпись:[Ws(r)]2 dr


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* yt-1 + a*yt-2 + £t (31.38)

because the regressors yt-1 and yt-2 can be shown to be asymptotically orthogonal (see, for instance, Ahtola and Tiao (1987) or Chan and Wei (1988) for more details).

Добавить комментарий

A COMPANION TO Theoretical Econometrics

Normality tests

Let us now consider the fundamental problem of testing disturbance normality in the context of the linear regression model: Y = Xp + u, (23.12) where Y = (y1, ..., …

Univariate Forecasts

Univariate forecasts are made solely using past observations on the series being forecast. Even if economic theory suggests additional variables that should be useful in forecasting a particular variable, univariate …

Further Research on Cointegration

Although the discussion in the previous sections has been confined to the pos­sibility of cointegration arising from linear combinations of I(1) variables, the literature is currently proceeding in several interesting …

Как с нами связаться:

тел./факс +38 05235  77193 Бухгалтерия
+38 050 512 11 94 — гл. инженер-менеджер (продажи всего оборудования)

+38 050 457 13 30 — Рашид - продажи новинок
e-mail: msd@msd.com.ua
Схема проезда к производственному офису:
Схема проезда к МСД

Партнеры МСД

Контакты для заказов шлакоблочного оборудования:

+38 096 992 9559 Инна (вайбер, вацап, телеграм)
Эл. почта: inna@msd.com.ua