An Autoregressive Model for Time Series of Random Objects

TL;DR

This paper develops an autoregressive model for time series data in metric spaces, specifically Hadamard spaces, enabling analysis of complex random objects over time with consistent estimation and hypothesis testing.

Contribution

It introduces a novel autoregressive model for random objects in Hadamard spaces, including estimation procedures and hypothesis testing methods with proven theoretical properties.

Findings

Consistent estimation of Fréchet mean and concentration parameter.

Asymptotic normality of the test statistic.

Simulation studies and real data application demonstrate effectiveness.

Abstract

Random variables in metric spaces indexed by time and observed at equally spaced time points are receiving increased attention due to their broad applicability. The absence of inherent structure in metric spaces has resulted in a literature that is predominantly non-parametric and model-free. To address this gap in models for time series of random objects, we introduce an adaptation of the classical linear autoregressive model tailored for data lying in a Hadamard space. The parameters of interest in this model are the Fr\'echet mean and a concentration parameter, both of which we prove can be consistently estimated from data. Additionally, we propose a test statistic for the hypothesis of absence of serial correlation and establish its asymptotic normality. Finally, we use a permutation-based procedure to obtain critical values for the test statistic under the null hypothesis. Theoretical results of our method, including the convergence of the estimators as well as the size and power of the test, are illustrated through simulations, and the utility of the model is demonstrated by an analysis of a time series of consumer inflation expectations.