A COMPANION TO Theoretical Econometrics

Testing the Seasonal Unit Root Null Hypothesis

In this section we discuss the test procedures proposed by Dickey et al. (1984) and Hylleberg, Engle, Granger, and Yoo (HEGY) (1990) to test the null hypo­thesis of seasonal integration. It should be noted that while there are a large number of seasonal unit root tests available (see, for example, Rodrigues (1998) for an extensive survey), casual observation of the literature shows that the HEGY test is the most frequently used procedure in empirical work. For simplicity of presentation, throughout this section we assume that augmentation of the test regression to account for autocorrelation is unnecessary and that presample starting values for the DGP are equal to zero.

3.1 The Dickey-Hasza-Fuller test

The first test of the null hypothesis yt ~ SI(1) was proposed by Dickey, Hasza, and Fuller (DHF) (1984), as a direct generalization of the test proposed by Dickey and Fuller (1979) for a nonseasonal AR(1) process. Assuming that the process is known to be a SAR(1), then the DHF test can be parameterized as

Asyt = a syt_s + e t. (31.18)

The null hypothesis of seasonal integration corresponds to as = 0, while the alternative of a stationary stochastic seasonal process implies as < 0. The appro­priately scaled least squares bias obtained from the estimation of as under the null hypothesis is

2

t - S

t=1

and the associated t-statistic is

1 T

T X yt-SC

T t=1

where o is the usual degrees of freedom corrected estimator of o. Similarly to the usual Dickey-Fuller approach, the test is typically implemented using

The asymptotic distribution of the DHF statistic given by (31.22) is non-standard, but is of similar type to the Dickey-Fuller t-distribution. Indeed, it is precisely the Dickey-Fuller t-distribution in the special case S = 1, when the test regression

(31.18) is the usual Dickey-Fuller test regression for a conventional random walk. It can also be seen from (31.22) that the distribution for the DHF t-statistic depends on S, that is on the frequency with which observations are made within each year. On the basis of Monte Carlo simulations, DHF tabulated critical values of TaS and t7s for various T and S. Note that the limit distributions presented as functions of Brownian motions can also be found in Chan (1989), Boswijk and Franses (1996) and more recently in Osborn and Rodrigues (1998). To explore the dependence on S a little further, note first that

1

Ws(r)dWs(r) = l{[Ws(1)]2 - 1}, (31.23)

J 0

where [Ws(1)]2 is x2(1) (see, for example, Banerjee et al., 1993, p. 91). The numer­ator of (31.22) involves the sum of S such terms which are mutually independent and hence

which is half the difference between a x2(S) statistic and its mean of S. It is well known that the Dickey-Fuller t-statistic is not symmetric about zero. Indeed,

Fuller (1996, p. 549) comments that asymptotically the probability of (in our nota­tion) 7 < 0 is 0.68 for the nonseasonal random walk because Pr[x2(1) < 1] = 0.68. In terms of (31.22), the denominator is always positive and hence Pr[%2(S) < S] dictates the probability that t7s is negative. With a seasonal random walk and quarterly data, Pr[ %2(4) < 4] = 0.59, while in the monthly case Pr[%2(12) < 12] = 0.55. Therefore, the preponderance of negative test statistics is expected to de­crease as S increases. As seen from the percentiles tabulated by DHF, the disper­sion of t7s is effectively invariant to S, so that the principal effect of an increasing frequency of observation is a reduction in the asymmetry of this test statistic around zero.