An Autoregressive Model for Time Series of Random Objects

Random variables in metric spaces indexed by time and observed at equally spaced time points are receiving increased attention due to their broad applicability. However, 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 and establish its asymptotic normality, thereby enabling hypothesis testing for the absence of serial dependence. Finally, we introduce a bootstrap 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.