Inference on the Dimension of the Nonstationary Subspace in Functional Time Series

Abstract

We propose a statistical procedure to determine the dimension of the nonstationary subspace of cointegrated functional time series taking values in the Hilbert space of square-integrable functions defined on a compact interval. The procedure is based on sequential application of a proposed test for the dimension of the nonstationary subspace. To avoid estimation of the long-run covariance operator, our test is based on a variance ratio-type statistic. We derive the asymptotic null distribution and prove consistency of the test. Monte Carlo simulations show good performance of our test and provide evidence that it outperforms the existing testing procedure. We apply our methodology to three empirical examples: age-specific U.S. employment rates, Australian temperature curves, and Ontario electricity demand.

Functional Principal Component Analysis of Cointegrated Functional Time Series

“…However, we already have a few available methods to determine ϕ from functional observations, e.g. Chang, Kim and Park (2016), Nielsen, Seo and Seong (2019), and Li, Robinson and Shang (2020a). In addition to those, it will be shown in Section 4 that the asymptotic results to be given in this section lead to a novel KPSS-type testing procedure to determine ϕ.…”

“…Assumption M-(i) is employed for mathematical proofs, which may not be restrictive in practice. Assumption M-(ii) implies that dim(H N ) < ∞, which is commonly assumed in the recent literature on cointegrated FTS for statistical analysis based on eigendecomposition of the sample covariance operator, see e.g., Chang, Kim and Park (2016) and Nielsen, Seo and Seong (2019). If ϕ = 0, {X t } t≥1 is stationary and H N = {0}; this is an uninteresting case for our study of cointegrated FTS except when we examine the null hypothesis of ϕ = 0 in Section 4.…”

“…This means that our test can be used as an alternative to the existing tests of stationarity of FTS such as Horváth, Kokoszka and Rice (2014), Kokoszka and Young (2016), and Aue et al (2020). Our selection of hypotheses in (4.1) is the opposite of that used for the existing tests proposed by Chang, Kim and Park (2016) and Nielsen, Seo and Seong (2019) in the sense that the alternative hypothesis in those tests is set to…”

“…REMARK 4.2. In Chang, Kim and Park (2016) and Nielsen, Seo and Seong (2019), ϕ is determined by testing the following hypotheses: for a pre-specified positive integer ϕ max and ϕ 0 = ϕ max , ϕ max − 1, . .…”

“…Based on the results, we propose a modified FPCA methodology that guarantees (i) a more asymptotically efficient estimator for P N and (ii) a more convenient asymptotic limit which leads to novel statistical tests for examining hypotheses on cointegration. To elaborate more on the latter point, we first develop a KPSStype test for the dimension of the unit root component, ϕ, which is distinctive from the existing tests developed for the same purpose by Chang, Kim and Park (2016) and Nielsen, Seo and Seong (2019). Then the proposed test is applied to examine various hypotheses about cointegration.…”