Semi-Parametric Seasonal Unit Root Tests

Castro, Tomás del Barrio; Rodrigues, Paulo M. M.; Taylor, Robert

doi:10.1017/s0266466617000135

Cited by 9 publications

(10 citation statements)

References 31 publications

(62 reference statements)

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“…In Lemma 1 we now provide a multivariate invariance principle for X n in . Where u Sn + s is serially uncorrelated this contains, as a particular case, the result given in Lemma A.1 of del Barrio Castro et al () (see Remark ), and will form the basic building block for the asymptotic results that will be provided in this article.…”

Section: Seasonally Near‐integrated Processesmentioning

confidence: 94%

“…In particular, we could permit u Sn + s to follow a stationary and invertible ARMA process thereby allowing non‐zero stable roots to occur and these could be identified with any observable spectral frequency. Doing so would introduce additional nuisance parameters, deriving from the stationary autocorrelation, into the large sample results given in Lemma 1 (see del Barrio Castro et al , , for details on how our key results given below in and Lemma 1 would need to be modified when u Sn + s is serially correlated) but would not alter the conclusions drawn from them regarding the integrational properties of the aggregated data. Moreover, the implied ARMA dynamics in the time aggregated data arising from stationary and invertible ARMA shocks in the original data is already well documented in the literature; see, for example, Chapter 20 of Wei () for an excellent summary.…”

Section: Seasonally Near‐integrated Processesmentioning

confidence: 99%

“…Remark For the case where a (near‐) integrated component is present at all of the zero and seasonal frequencies, such that h k = 1, for k = 0,…,⌊ S /2⌋, it follows from Lemma A.1 of del Barrio Castro et al (), noting that the circulant matrices

C_{k} : = C i r c ([, \cos ([], 0),) \cos ([], ω_{k}), \cos ([], 2 ω_{k}), …, (], \cos ([], ((), S - 1) ω_{k}))

, k = 1,…, S ∗ , which appear in Lemma A.1 of del Barrio Castro et al () can be written as

C_{k} = \frac{1}{2} false(C_{k}^{-} + C_{k}^{+} false)

, that

N^{- 1 false/ 2} X_{⌊ r N ⌋} \Rightarrow \frac{σ}{S} ([], C_{0} {boldJ}_{c_{0}} ((), r) + C_{S false/ 2} {boldJ}_{c_{S false/ 2}} ((), r) + true \sum_{i = 1}^{S^{*}} ((), C_{k}^{-} + C_{k}^{+}) {boldJ}_{c_{k}} ((), r)), r \in false[0, 1 false] .

The result in is a special case of Lemma 1. Notice that in this example each of the ψ 0 , ψ S /2 ,

ψ_{k}^{-}

and

ψ_{k}^{+}

, k = 1,…, S ∗ , weights is equal to 1/ S .…”

Section: Seasonally Near‐integrated Processesmentioning

confidence: 99%

“…Definition 1.1 of Gregoir (, p. 48). These transformation vectors, also known as demodulation operators, were introduced by Granger and Hatanaka () and have also been employed by, among others, Gregoir () and del Barrio Castro et al (). As we will see in the next section, temporal aggregation can alter the form of these demodulation operators, and this can lead to the presence of near‐integrated behaviour at different spectral frequencies to those observed in the data before aggregation.…”

Section: Seasonally Near‐integrated Processesmentioning

confidence: 99%

“…In what follows we will adopt the familiar trigonometric deterministic seasonal form and follow del Barrio Castro et al () by focusing attention on the following three cases of practical relevance, although other possibilities, including seasonal mean shifts, could also be analysed using the same framework:…”

Section: The Impact Of Temporal Aggregationmentioning

confidence: 99%

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Temporal Aggregation of Seasonally Near‐Integrated Processes

Castro

Rodrigues

Taylor

2019

Journal Time Series Analysis

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We investigate the implications that temporally aggregating, either by average sampling or systematic (skip) sampling, a seasonal process has on the integration properties of the resulting series at both the zero and seasonal frequencies. Our results extend the existing literature in three ways. First, they demonstrate the implications of temporal aggregation for a general seasonally integrated process with S seasons. Second, rather than only considering the aggregation of seasonal processes with exact unit roots at some or all of the zero and seasonal frequencies, we consider the case where these roots are local-to-unity such that the original series is near-integrated at some or all of the zero and seasonal frequencies. These results show, among other things, that systematic sampling, although not average sampling, can impact on the non-seasonal unit root properties of the data; for example, even where an exact zero frequency unit root holds in the original data it need not necessarily hold in the systematically sampled data. Moreover, the systematically sampled data could be near-integrated at the zero frequency even where the original data is not. Third, the implications of aggregation on the deterministic kernel of the series are explored. 7 In other words, Q is a constant that determines the outcome of the filters at a specific frequency, for instance, for the zero, Nyquist and harmonic frequencies the filters are (1 + L + · · · + L Q−1 ), sin(Q ∕2)∕ sin( ∕2)e −i(Q+1) ∕2 , and sin(Q k ∕2)∕ sin( k ∕2)e i(Q+1) k ∕2 and sin(Q k ∕2)∕ sin( k ∕2)e −i(Q+1) k ∕2 respectively. wileyonlinelibrary.com/journal/jtsa

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Section: Seasonally Near‐integrated Processesmentioning

confidence: 94%

Section: Seasonally Near‐integrated Processesmentioning

confidence: 99%

C_{k} : = C i r c ([, \cos ([], 0),) \cos ([], ω_{k}), \cos ([], 2 ω_{k}), …, (], \cos ([], ((), S - 1) ω_{k}))

, k = 1,…, S ∗ , which appear in Lemma A.1 of del Barrio Castro et al () can be written as

C_{k} = \frac{1}{2} false(C_{k}^{-} + C_{k}^{+} false)

, that

N^{- 1 false/ 2} X_{⌊ r N ⌋} \Rightarrow \frac{σ}{S} ([], C_{0} {boldJ}_{c_{0}} ((), r) + C_{S false/ 2} {boldJ}_{c_{S false/ 2}} ((), r) + true \sum_{i = 1}^{S^{*}} ((), C_{k}^{-} + C_{k}^{+}) {boldJ}_{c_{k}} ((), r)), r \in false[0, 1 false] .

The result in is a special case of Lemma 1. Notice that in this example each of the ψ 0 , ψ S /2 ,

ψ_{k}^{-}

and

ψ_{k}^{+}

, k = 1,…, S ∗ , weights is equal to 1/ S .…”

Section: Seasonally Near‐integrated Processesmentioning

confidence: 99%

Section: Seasonally Near‐integrated Processesmentioning

confidence: 99%

Section: The Impact Of Temporal Aggregationmentioning

confidence: 99%

See 3 more Smart Citations

Temporal Aggregation of Seasonally Near‐Integrated Processes

Castro

Rodrigues

Taylor

2019

Journal Time Series Analysis

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Non-parametric seasonal unit root tests under periodic non-stationary volatility

Göğebakan

Eroğlu

2022

Comput Stat

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This paper presents a new non-parametric seasonal unit root testing framework that is robust to periodic non-stationary volatility in innovation variance by making an extension to the fractional seasonal variance ratio unit root tests of Eroğlu et al. (Econ Lett 167:75–80, 2018). The setup allows for both periodic heteroskedasticity structure of Burridge and Taylar (J Econ 104(1):91–117, 2001) and non-stationary volatility structure of Cavaliere and Taylor (Econ Theory 24(1):43-71, 2008). We show that the limiting null distributions of the variance ratio tests depend on nuisance parameters derived from the underlying volatility process. Monte Carlo simulations show that the standard variance ratio tests can be substantially oversized in the presence of such effects. Consequently, we propose wild bootstrap implementations of the variance ratio tests. Wild bootstrap resampling schemes are shown to deliver asymptotically pivotal inference. The simulation evidence depicts that the proposed bootstrap tests perform well in practice and essentially correct the size problems observed in the standard fractional seasonal variance ratio tests, even under extreme patterns of heteroskedasticity. Supplementary Information The online version contains supplementary material available at 10.1007/s00180-022-01211-w.

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