2. THE SEASONAL UNIT ROOT FRAMEWORK

2.1. The Seasonal Model

Following Hylleberg et al. (1990), Beaulieu and Miron (1993), Smith and Taylor (1998, 1999a), and Nabeya (2001a), inter alia, consider the scalar process {x_Sn+s}, observed with seasonal periodicity S, written as the sum of a purely deterministic component, μ_Sn+s, and a purely stochastic process, v_Sn+s, namely,

where

in (2.2) is an Sth-order autoregressive (AR(S)) polynomial in the conventional lag (backshift) operator, L^kx_Sn+s ≡ x_Sn+s−k, k = 0,1,…. The driving shocks {u_Sn+s} of (2.2) are assumed to follow an AR(p), 0 ≤ p < ∞ process, namely, φ(L)u_Sn+s = ε_Sn+s, the roots of

all lying outside the unit circle, |z| = 1, with ε_Sn+s ∼ i.i.d.(0,σ²), with finite fourth moments. Exact assumptions on the initial conditions v_S+s, s = 1 − S,…,0, are delayed until later. In what follows we use the notation T ≡ SN to denote the total sample size.

The specification (2.1)–(2.3) allows for the presence of deterministic mean effects in {x_Sn+s} through μ_Sn+s. For the purposes of this paper, we follow Smith and Taylor (1999a) and consider the following six cases of interest.

Case 1. No intercept, no trend: γ_s* = 0, β_s* = 0, s = 1 − S,…,0.

Case 2. Constant intercept, no trend: γ_s* = γ, β_s* = 0, s = 1 − S,…,0.

Case 3. Seasonal intercepts, no trend: β_s* = 0, s = 1 − S,…,0.

Case 4. Constant intercept, constant trend: γ_s* = γ, β_s* = β, s = 1 − S,…,0.

Case 5. Seasonal intercepts, constant trend: β_s* = β, s = 1 − S,…,0.

Case 6. Seasonal intercepts, seasonal trends: as in (2.3) with γ_s* and β_s* unrestricted.

2.2. The Seasonal Unit Root Hypotheses

In this paper we are concerned with the behavior of tests for seasonal unit roots in the AR(S) polynomial, α(L), against near seasonally integrated alternatives; that is, the null hypothesis of interest is

whereas, following Tanaka (1996, pp. 355–356), Rodrigues (2001), and Nabeya (2000, 2001b), inter alia, the near seasonally integrated alternative we consider is of the form

where c is a fixed nonpositive constant. Notice that

reduces to

for c = 0.

Under

of (2.4) the data generating process (DGP) (2.1)–(2.3) of {x_Sn+s} is that of a seasonally integrated process (with or without drifts according to the form of μ_Sn+s), admitting unit roots at both the zero frequency, ω₀ ≡ 0, and at each of the seasonal spectral frequencies, ω_k ≡ 2πk/S, k = 1,…,[S/2] , [·] denoting the integer part of its argument. Under

of (2.5) the process {x_Sn+s} is locally stable at each of the zero and seasonal frequencies if c < 0. Although we constrain c to be nonpositive, all of the analysis that follows also holds for positive c, in which case the process is locally explosive.

Denoting

, we may factorize the polynomial α(L) under

of (2.5) as

where the lag polynomial

corresponds to the zero frequency ω₀ ≡ 0 and the lag polynomial ω_k^c(L) corresponds to the harmonic seasonal frequencies (ω_k,2π − ω_k) and is defined by

k = 1,…,S*, where S* ≡ (S/2) − 1 (if S is even) or [S/2] (if S is odd), together with

corresponding to the Nyquist frequency ω_S/2 ≡ π, when S is even. Moreover, as demonstrated in Rodrigues (2001) for the case of S = 4, taking a Taylor series expansion about one on each of the factors of (2.6) allows α(L) of (2.2) to be written as

Consequently,

of (2.5) is correspondingly partitioned as

, where the hypothesis

corresponds to a local to unit root at the zero frequency ω₀ = 0, whereas

yields a local to unit root at the Nyquist frequency ω_S/2 = π, where S is even. A pair of complex conjugate local to unit roots at the harmonic seasonal frequencies (ω_k,2π − ω_k) is obtained under

, k = 1,…,S* (see also Gregoir, 2001). Notice therefore that the particular local alternative,

of (2.5), that we have considered imposes a common noncentrality parameter, c, on each of the zero and seasonal frequencies. However, as we see subsequently, this involves no loss of generality.

2.3. Regression-Based Seasonal Unit Root Tests

Following Hylleberg et al. (1990) and Smith and Taylor (1999a), inter alia, the regression-based approach to testing for seasonal unit roots in α(L) consists of two stages. First, one obtains the ordinary least squares (OLS) demeaned series

, where

is the fitted value from the OLS regression of x_Sn+s on the intercept and trend variables relevant to each of Cases 1–6, κ ∈ {1,…,6} indicating the case of interest. Notice that for Case 1,

will be zero, and hence x_Sn+s¹ = x_Sn+s, by definition. In what follows we assume that μ_Sn+s is not estimated under an overly restrictive case, such that the resulting unit root tests will be exact invariant to the parameters characterizing the mean function μ_Sn+s (see Burridge and Taylor, 2004).

Following Smith and Taylor (1999a, equation (4.17), p. 9), we then linearize α(L) in (2.2) around the seasonal unit roots exp(± i2πk/S), k = 0,…,[S/2] , to obtain the auxiliary regression equation

which may be estimated along Sn + s = p* + S + 1,…,T, p* ≥ p, omitting the term π_S/2 x_S/2,Sn+s−1^κ if S is odd, and where corresponding to the zero and seasonal frequencies ω_k = 2πk/S, k = 0,…,[S/2],

k = 1,…,S*, together with Δ_S x_Sn+s^κ ≡ x_Sn+s^κ − x_S(n−1)+s^κ. For the case of quarterly data, S = 4, the relevant transformations are x_0,Sn+s^κ ≡ (1 + L + L² + L³)x_Sn+s^κ, x_2,Sn+s^κ ≡ −(1 − L + L² − L³)x_Sn+s^κ, x_1,Sn+s^α,κ ≡ −L(1 − L²)x_Sn+s^κ, and x_1,Sn+s^β,κ ≡ −(1 − L²)x_Sn+s^κ.

It is the elements of π ≡ (π₀,π_S/2,π_1,α,….,π_S*,β)′, omitting π_S/2 where S is odd, from (2.11) that are of focal interest. From the characterization theorem of Smith and Taylor (1999a, p. 7), the following expressions for the elements of π, omitting that for π_S/2 where S is odd, obtain. This result is proved in the accompanying working paper, Rodrigues and Taylor (2003).

PROPOSITION 2.1. The elements of the parameter vector π from the test regression (2.11), when the DGP is (2.1)–(2.3) under

of (2.5), are given by

where δ_k* ≡ [1/sin(ω_k)][((1 + 2c/T)^1/2 − 1)² cos(ω_k)], k = 1,…,S*.

Remark 2.1. It follows immediately from (2.13) that Tπ₀ = φ(1)c + o(1) and Tπ_S/2 = φ(−1)c + o(1) (S even). Similarly, from (2.14) and (2.15), Tπ_k,α = 2c Re{φ(exp(−iω_k))} + o(1), and Tπ_k,β = 2c Im{φ[exp(−iω_k)]} + o(1), k = 1,…,S*, where we have used the result that lim_N→∞[T/sin(ω_k)]{[(1 + (2c/T) + O(T⁻²))^1/2 − 1]² cos(ω_k)} = 0, k = 1,…,S*.

Remark 2.2. Suppose that, instead of using a single noncentrality parameter c in (2.6), we used a set of frequency-specific noncentrality parameters (which may or may not be equal), that is,

, where the ω_j^c_j(L) are as defined in (2.7)–(2.9) but replacing c by c_j, j = 0,…,[S/2] . Then the results in Proposition 2.1 remain valid on replacing c in (2.13)–(2.15) by the appropriate noncentrality parameter c_j, j = 0,…,[S/2] . This result is proved in Rodrigues and Taylor (2003).

Noting that for c = 0 the expansion in (2.10) is exact,

¹

That is,

, where δ = 1(0) if S is even (odd).

it is clear from Proposition 2.1 that under

of (2.4), π = 0, regardless of the lag parameters characterizing φ(z). Under

of (2.5) with c < 0 and T finite, it can be shown from Proposition 2.1 that π₀, π_S/2 (S even) and π_k,α, k = 1,…,S*, are negative, regardless of φ(z), whereas the π_k,β, k = 1,…,S*, can either be zero or nonzero, depending on the form of φ(z). Consequently, to test

of (2.4) against the alternative of stationarity at one or more of the zero and seasonal frequencies, Hylleberg et al. (1990) and Smith and Taylor (1999a), inter alia, have suggested using standard regression t- and F-statistics from (2.11).

Specifically, tests for the presence or otherwise of a unit root at the zero and Nyquist (S even) frequencies are conventional lower tailed regression t-tests, denoted t₀ and t_S/2, for the exclusion of x_0,Sn+s−1^κ and x_S/2,Sn+s−1^κ, respectively, from (2.11). Similarly, the hypothesis of a pair of complex unit roots at the kth harmonic seasonal frequency may be tested by the lower tailed t_k^α and two-tailed t_k^β regression t-tests from (2.11) for the exclusion of x_k,Sn+s−1^α,κ and x_k,Sn+s−1^β,κ, respectively, or by the regression F-test, denoted F_k, for the exclusion of both x_k,Sn+s−1^α,κ and x_k,Sn+s−1^β,κ from (2.11), k = 1,…,S*. Ghysels, Lee, and Noh (1994), Taylor (1998), and Smith and Taylor (1998, 1999a) also consider the joint frequency regression

-tests from (2.11), F_1…[S/2], for the exclusion of x_S/2,Sn+s−1^κ (S even) and {x_k,Sn+s−1^α,κ,x_k,Sn+s−1^β,κ}_k=1^S*, and F_0…[S/2], for the exclusion of x_0,Sn+s−1^κ, x_S/2,Sn+s−1^κ (S even) and {x_k,Sn+s−1^α,κ,x_k,Sn+s−1^β,κ}_k=1^S*. The former tests the null hypothesis of unit roots at all of the seasonal frequencies, whereas the latter tests the overall null,

.

Various percentiles from approximations to the finite sample null distributions of the preceding t- and F-statistics for certain choices of the seasonal aspect S, obtained by Monte Carlo simulation, assuming that {u_Sn+s} ∼ IN(0,1), appear in the literature; see, inter alia, Hylleberg et al. (1990, Tables 1a and 1b, pp. 226–277), Smith and Taylor (1998, Tables 1a and 1b, p. 276), Beaulieu and Miron (1993, Table A.1, pp. 325–326) and Ghysels et al. (1994, Tables C.1 and C.2, pp. 440–441). Asymptotic critical values are also provided in Beaulieu and Miron (1993, Table A.1, pp. 325–326) and Taylor (1998, Tables I and II, pp. 356–357).

3. ASYMPTOTIC REPRESENTATIONS

In this section we derive representations for the limiting distributions of the OLS seasonal unit root statistics computed from the test regression (2.11) for each of Cases 1–6 of (2.3), under the near seasonally integrated alternative,

of (2.5).

In what follows, what we assume about the initial conditions v_S+s, s = 1 − S,…,0, is important. One possibility, following Elliott, Rothenberg, and Stock (1996), is to assume that the starting values satisfy T^−1/2v_S+s →^p 0, s = 1 − S,…,0. One particular example of this is the so-called conditional case where v_S+s = u_S+s, s = 1 − S,…,0, so that the initial observation in each of the S seasons, x_S+s, has variance equal to that of the shocks. In contrast, Pantula, Gonzalez-Farias, and Fuller (1994) argue that “there are a modest number of situations” (p. 459) where one might reasonably assume that the conditional case applies. For macroeconomic data in particular, the conditional assumption seems untenable. Pantula et al. (1994) suggest instead the unconditional case, where the initial conditions obey the same data generating process as the rest of the process. Within our near seasonally integrated model we can contain both the conditional and unconditional cases within a more general framework by making the following assumption, which is the seasonal generalization of Assumption 3 of Canjels and Watson (1997, p. 185; see also Phillips and Lee, 1996).

Assumption 3.1. The initial conditions satisfy

, where λ ≥ 0 and m ∈ {0,1,…,m*}, m* finite. As noted in Canjels and Watson (1997), the unconditional case obtains taking the limit as λN → ∞. The conditional case obtains for λ = m = 0.

In the results given in Theorem 3.1, which follows, we will use the superscript ξ to indicate which of Cases 1–6 of μ_Sn+s of (2.3) hold. For the zero frequency ω₀ tests: Case 1: ξ = 0; Cases 2 and 3: ξ = 1; Cases 4–6: ξ = 2. For the seasonal frequency ω_k tests, k = 1,…,[S/2] : Cases 1, 2, and 4: ξ = 0; Cases 3 and 5: ξ = 1; Case 6: ξ = 2. For example, substituting ξ = 0 into the expression given in (3.1), which follows, for t₀ gives the limiting representation for the t₀ statistic under Case 1 (no deterministics) of μ_Sn+s, whereas ξ = 1 gives the limiting representation under Case 2 (nonseasonal intercept, no trend) and Case 3 (seasonal intercepts, no trend). Similarly, substituting ξ = 2 into (3.2) and (3.3) gives the limiting representations for the t_k^α and t_k^β statistics, respectively, under Case 6 (seasonal intercepts and seasonal trends) of μ_Sn+s.

We now state our main theorem. Some remarks follow.

THEOREM 3.1. Let the process {x_Sn+s} be generated by (2.1)–(2.3) and let the (unique) polynomial ψ(z) be defined such that ψ(z)φ(z) ≡ 1. Then, under

of (2.5) and Assumption 3.1, and denoting weak convergence of the associated probability measures by ⇒,

omitting

in (3.5) and (3.6) where S is odd. In (3.1), i = 0,S/2 if S is even and i = 0 if S is odd, whereas k = 1,…S* in (3.2)–(3.4). The nomenclature ξ is as defined following Assumption 3.1, a_k ≡ Im{ψ[exp(iω_k)]}, a_k* ≡ Im{φ[exp(−iω_k)]}, b_k ≡ Re{ψ[exp(iω_k)]}, b_k* ≡ Re{φ[exp(−iω_k)]}. Finally, the independent limiting processes J_i,c^ξ(r,λ), i = 0,S/2, and J_α,k,c^ξ(r,λ) and J_β,k,c^ξ(r,λ), k = 1,…,S*, are as defined in Definition A.1 of the Appendix.

Remark 3.1. Setting c = 0 and S = 4, the representations provided in Theorem 3.1 reduce to the limiting null representations provided in Burridge and Taylor (2001). Theorem 3.1 therefore generalizes the results of Burridge and Taylor to an arbitrary seasonal aspect, S, and to the near seasonally integrated case, c ≠ 0. For c = 0 and φ(z) = 1, the representations in Theorem 3.1 also reduce to those given in Smith and Taylor (1999a) and Nabeya (2001a). Moreover, notice that the representations in (3.1)–(3.6) for c = 0 do not depend on the initial conditions v_S+s, s = 1 − S,…,0, provided ξ > 0. Indeed, for each of Cases 3, 5, and 6 of μ_Sn+s of (2.3), the statistics also yield exact similar tests (for detailed discussion on this point, see Smith and Taylor 1998, 1999a).

Remark 3.2. The representations given in Theorem 3.1 delineate the asymptotic local power functions of the seasonal unit root tests from (2.11) in all cases indexed by a common noncentrality parameter c. Numerical tabulations of certain of these functions are given in Section 4, which allow us to investigate the relative power properties of the tests and also to quantify the degree of power loss incurred when (seasonal) intercepts and (seasonal) time trends are included in the test regression. Notice also, for example, from the results in Theorem 3.1 that the asymptotic local power functions of the seasonal frequency t_S/2 (S even), t_k^α, t_k^β and F_k, k = 1,…,S*, and F_1…[S/2] tests are not affected by including a nonseasonal intercept or time trend in (2.11) but are affected by the inclusion of seasonal intercepts and seasonal trends. Similarly, the asymptotic local power function of the t₀ test is not affected by the inclusion of seasonal intercepts (trends) vis-à-vis the case where a nonseasonal intercept (trend) is included in (2.11). The asymptotic local power function of the F_0…[S/2] test is, however, clearly affected by both seasonal and nonseasonal intercepts and time trends.

Remark 3.3. In practice, we would probably want to permit the noncentrality parameter to vary across the seasonal frequencies ω_k ≡ 2πk/S, k = 0,…,[S/2] , as in Remark 2.2. Just as argued in Rodrigues (2001, p. 80), the asymptotic orthogonality result stated in Remark A.1 of the Appendix ensures that the representations given in Theorem 3.1 remain appropriate in such cases on substituting c for c_k ≤ 0, k = 0,…,[S/2] , in (3.1) and (3.2), respectively, throughout and appropriately redefining Assumption 3.1.

Remark 3.4. From (3.1)–(3.6), it is seen that the limiting distributions of the t₀, t_S/2 (S even), F_k, k = 1,…,S*, F_1…[S/2], and F_0…[S/2] statistics do not depend on the serial correlation nuisance parameters {φ_i}_i=1^p under

of (2.5), whereas those of the harmonic frequency t-statistics t_k^α and t_k^β, k = 1,…S*, do, in general. We investigate this dependence for the case of an AR(1) error process in Section 4. An obvious exception occurs when φ(z) is expressible as φ(z^S), that is, φ(z) is a purely seasonal polynomial. In this case the limiting distributions of the t_k^α and t_k^β statistics in (3.2) and (3.3) are invariant to the parameters characterizing φ(z), k = 1,…S*. Other exceptions can occur at specific frequencies. For example, if S is an integer multiple of four, then the limiting distributions of the t_S/4^α and t_S/4^β statistics, corresponding to the harmonic frequency pair (π/2,3π/2), will be invariant to the parameters of φ(z), provided φ(z) is expressible as φ(z²). Notice that in the serially uncorrelated case, φ(z) = 1, a_k = 0, k = 1,…,S*, and b_k = 1, k = 0,…,[S/2] . For the special case of S = 4, c = 0, and λ = 0, these results reproduce those proved in Burridge and Taylor (2001).

Remark 3.5. It can be seen from (3.1) of Theorem 3.1 that for a given value of ξ, t₀ and t_S/2 have identical limiting representations, and hence asymptotic local power functions, under

of (2.5). These limiting distributions are also seen to be independent, by virtue of the independence of J_0,c^ξ(r,λ) and J_S/2,c^ξ(r,λ), ξ ∈ {0,1,2}. For c = 0 and ξ = 1,2, these coincide with the corresponding limiting null representations for the augmented Dickey–Fuller (ADF) t-statistic provided in Theorem 10.1.3, p. 561, and Theorem 10.1.6, pp. 567–568, of Fuller (1996), respectively. For ξ = 0 the representations obtained for λ = 0 coincide with the representation given in Corollary 10.1.1.5 of Fuller (1996, p. 554), noting that Fuller imposes zero starting values on his analysis. For c < 0 and ξ = 2 they replicate the representation given in Canjels and Watson (1997, Footnote 13, p. 192) for the asymptotic power function of the conventional ADF test. For λ = 0 the representation in (3.1) replicates those given for the ADF test in, inter alia, Chan and Wei (1987), Nabeya and Tanaka (1990), Perron (1989), Phillips (1987), and Elliott et al. (1996), with graphs of the associated asymptotic power function for ξ = 0,1,2 provided in Figures 1, 2, and 3, respectively, of Elliott et al. (1996, pp. 822–824).

Remark 3.6. Representations (3.2)–(3.5) of Theorem 3.1 demonstrate that, under

of (2.5), the t_k^α and t_l^α, k ≠ l, and t_k^β and t_l^β, k ≠ l, k,l = 1,…,S*, statistics are asymptotically mutually independent and are also asymptotically independent of t₀ and t_S/2; the F-statistics F_k, k = 1,…,S*, possess mutually independent and identical limiting representations and are asymptotically independent of the t₀ and t_S/2 statistics; and the F-statistic F_1…[S/2] is asymptotically independent of the zero frequency t₀ statistic.

Remark 3.7. The limiting representation in (3.2) for c = λ = 0 can be shown to coincide with that of the limiting null distribution of the t-statistic of Dickey, Hasza, and Fuller (1984) for the case of a biannual (S = 2) seasonal process. Moreover, for c ≤ 0, the stated representation in (3.2) for the t_k^α statistic when λ = 0, a_k = 0 and b_k = 1, k = 1,…,S*, reduces to that provided by Chan (1989, Theorem 1, p. 282) and Perron (1992, Theorem 2, p. 127) for ξ = 0 and Tanaka (1996, pp. 355–362) and Nabeya (2000, 2001b) for ξ = 0,1,2 for the limiting distribution of the Dickey et al. (1984) statistic for S = 2.