Efficient tests of the seasonal unit root hypothesis

Abstract

In this paper we derive, under the assumption of Gaussian errors with known error covariance matrix, asymptotic local power bounds for unit root tests at the zero and seasonal frequencies for both known and unknown deterministic scenarios and for an arbitrary seasonal aspect. We demonstrate that the optimal test of a unit root at a given spectral frequency behaves asymptotically independently of whether unit roots exist at other frequencies or not. Optimal tests for unit roots at multiple frequencies are also developed. We also detail modified versions of the optimal tests which attain the asymptotic Gaussian power bounds under much weaker conditions. We further propose near-efficient regression-based seasonal unit root tests using local GLS de-trending which, in the case of single frequency unit root tests, are shown to have limiting null distributions and asymptotic local power functions of a known form. Monte Carlo simulations indicate that these tests perform well in finite samples. r

Semi-Parametric Seasonal Unit Root Tests

“…2 That is,δ 1 is the OLS estimator obtained from regressing y Sn+s onto z Sn+s,ξ . While, as in Rodrigues and Taylor (2007) Rodrigues and Taylor (2007, p.556 that an autoregressive approximation of order say p * is valid, the second step is to then expand the composite AR(p * +S) polynomial φ * (z) := α(z)φ(z) around the zero and seasonal frequency unit roots exp(±i2πk/S), k = 0, ..., S/2 , to obtain the augmented HEGY-type regression 3…”

“…Following Elliot, Rothenberg and Stock (1996) and Rodrigues and Taylor (2007), the initial conditions, x 1−S , ..., x 0 , are taken to be of o p (T 1/2 ). Relaxing this assumption will not alter the limiting null distributions of the test statistics we outline in this paper because tests based on data which have been de-trended under Cases 1, 2 or 3 will be exact similar with respect to the initial conditions; see Smith et al (2009).…”

“…, S * , yields a pair of complex conjugate unit roots at the harmonic seasonal frequencies (ω k , 2π − ω k ). In order to examine the asymptotic local power properties of the test procedures we discuss, we follow Rodrigues (2001) and Rodrigues and Taylor (2007) and focus attention on the alternative hypotheses of near integration at the zero, Nyquist and harmonic seasonal frequencies; that is,…”

“…In the first step one de-trends the data in order to yield tests which will be exact invariant (assuming µ Sn+s is not under-specified) to the elements of δ which characterise the deterministic component µ Sn+s in (2.1a). This can either be done using OLS de-trending, as in, for example, HEGY and Smith et al (2009), or by local GLS de-trending as in Rodrigues and Taylor (2007). We define the resulting de-trended data series as y ξ Sn+s := y Sn+s −δ τ z Sn+s,ξ where ξ = 1, 2 and 3 corresponds to the deterministic kernels defined in Cases 1, 2 and 3 above, and where τ = 1 indicates OLS de-trending and τ = 2 local GLS de-trending.…”

“…Corresponding results for the case of local GLS de-trending are given in Rodrigues and Taylor (2007).…”