Temporal Aggregation of Seasonally Near‐Integrated Processes

Abstract

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

On cointegration for processes integrated at different frequencies

“…As noted by Gregoir (1999a, 1999a,2006) and del Barrio Castro et al . (2018,2019), (5) is equivalent to where Note that, here it is clearly shown that is the starting value of and that is the starting value of . The demodulator operator of (5) therefore shifts the zero frequency peak of the complex‐valued process to frequency leading to a complex‐valued .…”

“…where e −i𝜔t is the demodulator operator and x − 0 is assumed to be O p (1) (with x − 0 been part of the starting value of x − t that is, e −i𝜔t x − 0 ). As noted by Gregoir (1999a, 2006) and del Barrio Castro et al (2018, 2019, ( 5) is equivalent to…”

Spurious Relationships for Nearly Non-Stationary Series

“…We note that in this paper, we consider Situations 3 and 4 because when two autoregressive processes in which one is associated to the zero frequency; that is, the AR(1) with a positive coefficient in our paper, and the other is associated to the Nyquist frequency (π); that is, the AR(1) with a negative coefficient in our paper that has power at frequency π and completes a cycle every 2 observations, are independent or even asymptotically orthogonal. Readers may refer to Johansen and Schaumburg (1999), Ghysels and Osborn (2001), and del Barrio Castro et al (2018, 2019 for more information. Readers may also refer to seasonal unit root tests, see, for example, del Barrio Castro et al (2012) and Smith et al (2009), and cointegration for processes integrated at different frequencies, see, for example, del Barrio Castro et al (2020) with properties that are related to the series we are using in our paper.…”

Asymptotic Properties of Estimators for Seasonally Cointegrated State Space Models Obtained Using the CVA Subspace Method

“…Finally note that temporal aggregation from hourly data to daily data implies that the frequency for hourly data aliases to the frequency in the daily data. Therefore the higher evidence of a unit root at found in daily data might be a consequence of the unit root at frequency found for hourly data, compare [ 37 ].…”