A COMPANION TO Theoretical Econometrics
Deterministic seasonality
A common practice is to attempt the removal of seasonal patterns via seasonal dummy variables (see, for example, Barsky and Miron, 1989; Beaulieu and Miron, 1991; Osborn, 1990). The interpretation of the seasonal dummy approach is that seasonality is essentially deterministic so that the series is stationary around seasonally varying means. The simplest deterministic seasonal model is
s
yt = 1 bstms + ef (31.14)
s=1
where 5s t is the seasonal dummy variable which takes the value 1 when t falls in season s and et ~ iid(0, о2). Typically, yt is a first difference series in order to account for the zero frequency unit root commonly found in economic time series. When a model like (31.14) is used, the coefficient of determination (R1) is often computed as a measure of the strength of the seasonal pattern. However, as Abeysinghe (1991, 1994) and Franses, Hylleberg, and Lee (1995) indicate, the presence of seasonal unit roots in the DGP will have important consequences for R1.
To illustrate this issue, take the seasonal random walk of (31.1) as the DGP and assume that (31.14) is used to model the seasonal pattern. As is well known, the OLS estimates of ms, s = 1,..., S are simply the mean values of yt in each season. Thus, using the notation of (31.4),
1 T 1 N
ns = N 1&«yt = N1 ysn (31.15)
N t=1 N t=1
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where (as before) T and N are the total number of observations and the total number of complete years of observations available, respectively, and it is again assumed for simplicity that T = SN. As noted by Franses et al. (1995), the estimated seasonal intercepts diverge under the seasonal random walk DGP. In particular, the appropriately scaled ns converges to a normal random variable
where the latter follows from Banerjee et al. (1993, pp. 43-5) who show that P0W(r)dr = N(0, 1/3). For this DGP, the R1 from (31.14) has the non-degenerate asymptotic distribution,1
Consequently, high values for this statistic are to be anticipated, as concluded by Franses et al. These are spurious in the sense that the DGP contains no deterministic seasonality since E( yt) = 0 when the starting values for (31.1) are zero. Hence a high value of R[18] when (31.14) is estimated does not constitute evidence in favor of deterministic seasonality.
