[Paper Review] Cointegrated Density-Valued Linear Processes

Won‐Ki Seo|arXiv (Cornell University)|Oct 21, 2017

Monetary Policy and Economic Impact20 references3 citations

TL;DR

This paper develops a cointegrated linear process framework for density-valued time series using Bayes Hilbert spaces, resolving non-negativity and linearity issues in traditional $L^2$-based approaches. It establishes an I(1) representation theory, adapts functional unit root testing for attractor space estimation, and demonstrates with empirical applications to earnings and wage densities that cointegration exists in the attractor space of dimension one.

ABSTRACT

In data rich environments we may sometimes deal with time series that are probability density-function valued, such as observations of cross-sectional income distributions over time. To apply the methods of functional time series analysis to such observations, we should first embed them in a linear space in which the essential properties of densities are preserved under addition and scalar multiplication. Bayes Hilbert spaces provide one way to achieve this embedding. In this paper we investigate the use of Bayes Hilbert spaces to model cointegrated density-valued linear processes. We develop an I(1) representation theory for cointegrated linear processes in a Bayes Hilbert space, and adapt existing statistical procedures for estimating the corresponding attractor space to a Bayes Hilbert space setting. We revisit empirical applications involving earnings and wage densities to illustrate the utility of our approach.

Motivation & Objective

• To resolve the incompatibility between nonnegativity of probability densities and linear operations in $L^2(K)$-based functional time series models.
• To develop a cointegration theory for density-valued time series in Bayes Hilbert spaces, preserving essential density properties under addition and scalar multiplication.
• To extend the Granger-Johansen representation theorem to infinite-dimensional Bayes Hilbert spaces for AR(p) processes.
• To adapt functional unit root testing and attractor space estimation procedures to the Bayes Hilbert space setting.
• To empirically demonstrate the method using cross-sectional earnings and wage density time series.

Proposed method

• Embeds probability density-valued time series into a Bayes Hilbert space, where densities are represented via the centred log-ratio (clr) transformation to ensure linearity and closure under operations.
• Applies the Granger-Johansen representation theorem in a Bayes Hilbert space to characterize I(1) processes and derive necessary and sufficient conditions for I(1) solutions in AR(p) models.
• Uses functional principal component analysis (FPCA) on clr-transformed densities to estimate the attractor space and identify the cointegrating relationship.
• Employs a sequential functional unit root test based on Chang et al. (2016) to determine the dimension of the attractor space, testing $R=1,2,\dots,R_{\max}$.
• Estimates the attractor space as the span of the leading eigenfunction from FPCA of clr-transformed densities, then back-transforms via $\operatorname{clr}^{-1}$ to interpret in the original density space.
• Perturbs the stationary mean in the direction of the leading cointegrating vector to visualize the long-run equilibrium relationship in the density space.

Experimental results

Research questions

• RQ1 Can cointegration be meaningfully defined for density-valued time series in a functional space that preserves the nonnegativity and normalization of densities?
• RQ2 Does the standard $L^2(K)$ framework allow for nontrivial cointegrated density processes, or is the nonnegativity constraint incompatible with integrated behavior?
• RQ3 Can the Granger-Johansen representation theorem be extended to infinite-dimensional Bayes Hilbert spaces for AR(p) processes with density-valued innovations?
• RQ4 What statistical procedures can reliably estimate the dimension and structure of the attractor space for cointegrated density-valued processes?
• RQ5 How can the long-run equilibrium relationship in density-valued time series be visualized and interpreted empirically?

Key findings

• The attractor space for the cross-sectional earnings density time series is one-dimensional, as $R=1$ is not rejected at the 5% significance level, while $R=2$ is rejected at the 1% level.
• The attractor space for the cross-sectional wage density time series is also one-dimensional, with $R=1$ failing to be rejected at the 5% level and $R=2$ rejected at the 1% level.
• The leading eigenfunction from FPCA of clr-transformed densities captures the dominant mode of variation in the cointegrating relationship, and its $\operatorname{clr}^{-1}$-image defines the primary direction of long-run equilibrium.
• Functional unit root test statistics for $R=1$ are $\tau_1^T = 0.03638$ for earnings and $0.05174$ for wages, both below critical values, supporting a one-dimensional attractor space.
• Perturbation of the stationary mean in the direction of the leading cointegrating vector reveals the long-run equilibrium path, with clear shifts in the shape of the density distributions.
• The use of Bayes Hilbert spaces resolves the fundamental incompatibility between linear operations and the nonnegativity of densities, enabling valid cointegration analysis in functional time series of densities.