A COMPANION TO Theoretical Econometrics

The Hylleberg-Engle-Granger-Yoo test

It is well known that the seasonal difference operator As = 1 - Ls can always be factorized as

1 - Ls = (1 - L)(1 + L + L2 + ... + Ls-1). (31.39)

Hence, (31.39) indicates that an SI(1) process always contains a conventional unit root and a set of § - 1 seasonal unit roots. The approach suggested by Hylleberg et al. (1990), commonly known as HEGY, examines the validity of A§ through exploiting (31.39) by testing the unit root of 1 and the § - 1 separate nonstationary roots on the unit circle implied by 1 + L + ... + L§-1. To see the implications of this factorization, consider the case of quarterly data (§ = 4) where

1 - L4 = (1 - L)(1 + L + L2 + L3)

= (1 - L)(1 + L)(1 + L2). (31.40)

Thus, A4 = 1 - L4 has four roots on the unit circle,2 namely 1 and -1 which occur at the 0 and n frequencies respectively, and the complex pair ± i at the frequencies П and - y. Hence, in addition to the conventional unit root, the quarterly case implies three seasonal unit roots, which are -1 and the complex pair ±i.

Corresponding to each of the three factors of (31.40), using a Lagrange appro­ximation, HEGY suggest the following linear transformations:

y(1),t = (1 + L)(1 + Ll)Vt = yt + yt-1 + yt-2 + yt-3 (31.41)

y(2),t = - (1 - L)(1 + L2)yt = - yt + yt-1 - yt-2 + yt-3 (31.42)

y(3),t = -(1 - L)(1 + L)yt = - yt + yt-2. (31.43)

By construction, each of the variables in (31.41) to (31.43) accepts all the factors of A4 except one. That is, assumes the factors (1 + L) and (1 + L2), y{T)t assumes (1 - L) and (1 + L2), while yP)/t assumes (1 - L) and (1 + L). The test regression for quarterly data suggested by HEGY has the form:

A4yt = п 1 У(1),м + П2y(2),t-1 + П3y(3),t-2 + П4y(3),t_1 + £t, t = 1, 2,..., T (31.44)

where y(1)t, y(2)t, and yP),t are defined in (31.41), (31.42), and (31.43), respectively. Note that these regressors are asymptotically orthogonal by construction. The two lags of y(3),t arise because the pair of complex roots ±i imply two restrictions on a second order polynomial 1 + <^1L + ф2Ь1, namely ф1 = 0 and ф2 = 1 (see Sec­tion 3.2). The overall null hypothesis yt ~ SI(1) implies п 1 = п2 = п3 = п4 = 0 and hence A4yt = Et as for the DHF test. The HEGY regression (31.44) and the associ­ated asymptotic distributions can be motivated by considering the three factors of A4 = (1 _ L)(1 + L)(1 + L2) one by one. Through the variable yW/t, we may consider the DGP

y(1),t = y(1),t_1 + Ef (31.45)

Therefore, when yt is generated from a seasonal random walk, ym, t has the proper­ties of a conventional random walk process and hence, with initial values equal to zero,

t-1

y(1),t = X£t-j.
j=0

Since A1 y(1),t = A 4yt, the Dickey-Fuller test regression for the

A4 yt = П 1y(1),t_1 + E t,

where we test п 1 = 0 against п 1 < 0. Considering

then from Lemma 1 and (31.13) it can be observed that under the seasonal random walk null hypothesis

and

where Щ7)(г) = X4=^(г). Substituting (31.49) and (31.50) into (31.48) gives:

1

[W(4(r )]2 dr

0

The associated {-statistic, which is commonly used to test for the zero frequency unit root, can be expressed as

W*(r)dW*(r)

[W *(r )]2 dr

0

where W *(r) = W(1)(r)/2. Division by 2 is undertaken here so that W *(r) is stan­dard Brownian motion, whereas W^(r) is not. Therefore, (31.52) is the conven­tional Dickey-Fuller {-distribution, tabulated by Fuller (1996).

Similarly, based on (31.42), the seasonal random walk (31.1) implies

- (1 + %(2),t = E{. (31.53)

Notice the "bounce back" effect in (31.53) which implies a half cycle for уй/{ every period and hence a full cycle every two periods. Also note that (31.53) effectively has the same form as (31.26). Testing the root of -1 implied by (31.53) leads to a test of ф2 = 1 against ф2 < 1 in

- (1 + Ф2%(2),{ = Є{.

Equivalently, defining п 2 = ф2 - 1 and again using (31.42) yields:

A4 У{ = П2У(2),{-1 + Є{, (31.54)

with null and alternative hypotheses п 2 = 0 and п 2 < 0, respectively. Under the null hypothesis, and using analogous reasoning to Section 3.2 combined with Lemma 1, we obtain

і

W*2(r)dW * (r)

[W *(r )]2 dr

0

where the Brownian motion W(2)(r) = W1(r) - W2(r) + W3(r) - W4(r) is standardized as W*(r) = W(2)(r)/2. Like (31.52), (31.56) is the conventional Dickey-Fuller dis­tribution tabulated by Fuller (1996). It is important to note that, as indicated by Chan and Wei (1988) and Fuller (1996), the distributions of the least squares bias and corresponding t-statistic when the DGP is an AR(1) with a -1 root are the "mirror image" of those obtained when testing the conventional random walk. However, in (31.55) and (31.56), this mirror image is incorporated through the design of the HEGY test regression in that the linear transformation of yP),t is defined with a minus sign as -(1 - L)(1 + L2).

Finally, from (31.43) it follows that yt ~ SI(1) as in (31.1) with S = 4 implies

- (1 + L2)ym = є f. (31.57)

This process implies a "bounce back" after two periods and a full cycle after four. This process has the complex root form identical to (31.25). Hence, the results presented for that process carry over directly for this case. Noting again that -(1 + L2)y(3), t = A4y t, we can test ф3 = 1 and ф4 = 0 in

- (1 + ф4L + фэ^Ур), t = є t

through the regression

A4Vt = П 3У(3),(-2 + П 4У (3), t-1 + Є (31.58)

with п 3 = ф3 - 1 and п4 = - ф4. Testing against stationarity implies null and alter­native hypotheses of п 3 = 0 and п 3 < 0. However, while п4 = 0 is also indicated under the null hypothesis, the alternative here is п4 Ф 0. The reasoning for this two-sided alternative is precisely that for the test regression (31.34) and (31.58) has the same form as (31.38). Therefore, using similar arguments to Section 3.2, and noting that the "mirror image" property discussed there is incorporated through the minus sign in the definition of y(3),t, it can be seen that

[W*(r)]2 dr +

where W*(r) = [W1(r) - W3(r)]/V2 and W*(r) = [W2(r) - W4(r)] V2 are indepen­dent standard Brownian motions. Note that the least squares bias T* 3 and Tl4 can also be obtained from (31.32) and (31.36).

HEGY suggest that n 3 and n4 might be jointly tested, since they are both asso­ciated with the pair of nonstationary complex roots ± i. Such joint testing might be accomplished by computing F(*3 П l4) as for a standard F-test, although the distribution will not, of course, be the standard F-distribution. Engle, Granger, Hylleberg, and Lee (1993) show that the limiting distribution of F(*3 П l4) is identical to that of -1[t|3 + t|J, where the two individual components are given in (31.59) and (31.60). More details can be found in Smith and Taylor (1998) or Osborn and Rodrigues (1998).

Due to the asymptotic orthogonality of the regressors in (31.47), (31.54) and (31.58), these can be combined into the single test regression (31.44) without any effect on the asymptotic properties of the coefficient estimators.