Abstract
For functional time series with physical dependence, we construct confidence bands for its mean function. The physical dependence is a general dependence framework, and it slightly relaxes the conditions of m-approximable dependence. We estimate functional time series mean functions via a spline smoothing technique. Confidence bands have been constructed based on a long-run variance and a strong approximation theorem, which is satisfied with mild regularity conditions. Simulation experiments provide strong evidence that corroborates the asymptotic theories. Additionally, an application to S&P500 index data demonstrates a non-constant volatility mean function at a certain significance level.
Highlights
MSC 2010 subject classifications: Primary 62G08, 62G15
A central issue in functional time series analysis is taking into account the temporal dependence of the observations, i.e. the dependence between curves of {Yi(t), i ≤ m} and {Yj(t), j ≥ m + l} for any l ∈ Z
Functional time series retains the merit of functional observations, while it is more flexible than purely i.i.d curves by allowing a dependence structure
Summary
In this paper we use the following notation. Let F denote L2([0, 1]); the Hilbert space on a compact domain of square integrable functions with the norm ||x||22 =. In the literature in functional time series, m-approximable dependence has been extensively used, see [1] and [13] among the others. In their framework, weak dependence is quantified by a summability condition which intuitively states that the function g decays so fast that the impact of shocks far back in the past is so small that they can be replaced by their independent copies, with only a small change in the distribution of the process. For stationary functional time series, Lp-m-approximable dependence ensures the physical dependence.
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