A moment-based notion of time dependence for functional time series

Tempered functional time series

“…In an apparently independent work, Salish and Gleim (2019) defined long memory FTS in terms of the asymptotic behavior of the 281 autocovariances of the scores 𝜉 t,l in the expansion X t (u) = 𝜇(u) + ∑ ∞ l=1 𝜉 t,l 𝜈 l (u), where the 𝜈 l form an orthonormal basis. These two approaches use, in fact, similar general principles because the power law decay of the kernels b j (⋅, ⋅) is quantified in Li et al (2020) in terms of the decay of suitable projections, and chief examples in Salish and Gleim (2019) are functional moving averages of the form (1.1). Another related work is Van Delft and Eichler (2020) where an optimal Cramér representation of a large class of processes, including long memory processes defined in the functional spectral domain, is derived.…”

“…The LUR models have proven useful for analyzing economic data exhibiting bubbles, see for example, Phillips andYu(2011)andPhillips et al (2015a,b).When 𝜆 is sample size dependent, the kernels b j (⋅, ⋅) = b Nj (⋅, ⋅) must depend on the sample size N to permit local unit root behavior extending (1.3) and (4.2). In contrast to the work of Li et al (2020) and Salish and Gleim (2019), we thus work with a triangular array of FTS. Our assumptions are formulated in the general framework proposed by Salish and Gleim (2019).…”

“…In contrast to the work of Li et al (2020) and Salish and Gleim (2019), we thus work with a triangular array of FTS. Our assumptions are formulated in the general framework proposed by Salish and Gleim (2019). We do not need to work with non-normalized covariance kernels as Li et al (2020) because for each N, the time series we study do not have long memory in their sense.…”

“…An important difference relative to the univariate model (3.6) is that the observations we consider are functions, not scalars, and hence the short memory part of the model must be defined in terms of a sequence of integral operators rather than scalars. Our work is related to Salish and Gleim (2019) with a difference that their model has one low frequency parameter, d ∈ (0, 1∕2), and our model has two low frequency parameters, d ∈ (0, ∞) and 𝜆 ∈ (0, ∞). Moreover, in our setting 𝜆 = 𝜆 N is sample size-dependent.…”

“…Within the framework of local to unit root fractional processes, it is convenient to adopt assumptions similar in spirit to those used by Salish and Gleim (2019). We now proceed to formulate our assumptions.…”