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Inference on common trends in functional time series

Morten Ørregaard Nielsen, Won-Ki Seo, Dakyung Seong

TL;DR

The paper develops ADI, variance-ratio–based inference for unit roots and cointegration in time series valued in a Hilbert space, allowing for unknown, possibly infinite, ambient dimensions. By projecting onto a finite slack extractor and analyzing generalized eigenvalues of projected variance operators, it provides limit theory, sequential testing procedures, and eigenvalue-ratio estimators to determine the dimension of the nonstationary subspace and test subspace hypotheses, without relying on parametric VAR models. The authors establish asymptotic results for VR and inverse VR statistics, accommodate deterministic components, and demonstrate the approach with Monte Carlo simulations and empirical analyses of corporate yield curves and labor-market indices. The work offers a versatile framework for nonstationary functional time series, high-dimensional cointegration, and nonstationary dynamic functional factor models, with practical guidance on implementation and interpretation in complex economic data.

Abstract

We study statistical inference on unit roots and cointegration for time series in a Hilbert space. We develop statistical inference on the number of common stochastic trends embedded in the time series, i.e., the dimension of the nonstationary subspace. We also consider tests of hypotheses on the nonstationary and stationary subspaces themselves. The Hilbert space can be of an arbitrarily large dimension, and our methods remain asymptotically valid even when the time series of interest takes values in a subspace of possibly unknown dimension. This has wide applicability in practice; for example, to cointegrated vector time series that are either high-dimensional or of finite dimension, to high-dimensional factor models that include a finite number of nonstationary factors, to cointegrated curve-valued (or function-valued) time series, and to nonstationary dynamic functional factor models. We include two empirical illustrations to the term structure of interest rates and labor market indices, respectively.

Inference on common trends in functional time series

TL;DR

The paper develops ADI, variance-ratio–based inference for unit roots and cointegration in time series valued in a Hilbert space, allowing for unknown, possibly infinite, ambient dimensions. By projecting onto a finite slack extractor and analyzing generalized eigenvalues of projected variance operators, it provides limit theory, sequential testing procedures, and eigenvalue-ratio estimators to determine the dimension of the nonstationary subspace and test subspace hypotheses, without relying on parametric VAR models. The authors establish asymptotic results for VR and inverse VR statistics, accommodate deterministic components, and demonstrate the approach with Monte Carlo simulations and empirical analyses of corporate yield curves and labor-market indices. The work offers a versatile framework for nonstationary functional time series, high-dimensional cointegration, and nonstationary dynamic functional factor models, with practical guidance on implementation and interpretation in complex economic data.

Abstract

We study statistical inference on unit roots and cointegration for time series in a Hilbert space. We develop statistical inference on the number of common stochastic trends embedded in the time series, i.e., the dimension of the nonstationary subspace. We also consider tests of hypotheses on the nonstationary and stationary subspaces themselves. The Hilbert space can be of an arbitrarily large dimension, and our methods remain asymptotically valid even when the time series of interest takes values in a subspace of possibly unknown dimension. This has wide applicability in practice; for example, to cointegrated vector time series that are either high-dimensional or of finite dimension, to high-dimensional factor models that include a finite number of nonstationary factors, to cointegrated curve-valued (or function-valued) time series, and to nonstationary dynamic functional factor models. We include two empirical illustrations to the term structure of interest rates and labor market indices, respectively.
Paper Structure (39 sections, 12 theorems, 88 equations, 4 figures, 7 tables)

This paper contains 39 sections, 12 theorems, 88 equations, 4 figures, 7 tables.

Table of Contents

  1. Introduction
  2. I(1) time series and stochastic trends in Hilbert space
  3. ADI variance ratio tests
  4. Tests based on VR($d_L$,1) with $d_L \geq 2$
  5. Tests based on VR($d_L$,0) with $d_L \geq 1$
  6. Inverse VR tests
  7. Estimation of slack extractor
  8. Inclusion of deterministic components
  9. Determination of dimension of nonstationary subspace
  10. Estimation via sequential testing
  11. Eigenvalue ratio estimator
  12. Inference on subspaces
  13. Monte Carlo study
  14. Empirical applications
  15. Corporate bond yield curves
  16. ...and 24 more sections

Key Result

Theorem 3.1

Suppose that Assumptions assum1, assumvr1, and assumkernel hold and define $\widetilde{\mu}_j = n_T \mu_j$, where $\{\mu_j\}_{j=1}^{\mathrm{K}}$ are the eigenvalues from eqgev with $d_L \geq 2$, $d_R = 1$, and where $h_m,a_m,c_m$ are given in Assumption assumkernel(ii) and eqcm. Then

Figures (4)

  • Figure 1: Eigenvalue requirements for the proposed tests, $\mathrm{K}=\mathrm{K}(\mathbbm{d}_0)=\mathbbm{d}_0+2$
  • Figure 2: HQM yield curves
  • Figure 3: Time series of $\langle X_t,\widehat{\nu}_j \rangle$
  • Figure 4: Time series of $\langle \Delta X_t,\widehat{\nu}_j \rangle$

Theorems & Definitions (33)

  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 3.1
  • Remark 3.2
  • Remark 3.3
  • Remark 3.4
  • Remark 3.5
  • Theorem 3.1
  • Corollary 3.1
  • ...and 23 more