Table of Links Abstract and 1. Introduction Abstract and 1. Introduction 2. Preliminaries 2. Preliminaries 3. The GAR(1) Model 3.1. Model and Stationary Solution 3.1. Model and Stationary Solution 3.2. Identifability 3.2. Identifability 4. Estimation of model parameters and 4.1. Fréchet mean 4. Estimation of model parameters and 4.1. Fréchet mean 4.2. Concentration parameter 4.2. Concentration parameter 5. Testing for the absence of serial dependence 5. Testing for the absence of serial dependence 6. Numerical experiments 6. Numerical experiments 6.1. R with multiplicative noise 6.1. R with multiplicative noise 6.2. Univariate distributions with a density 6.2. Univariate distributions with a density 6.3. SPD Matrices 6.3. SPD Matrices 7. Application 7. Application 8. Acknowledgement 8. Acknowledgement Appendix A. General results in Hadamard spaces Appendix A. General results in Hadamard spaces Appendix B. Proofs Appendix B. Proofs Reference Reference 7. Application Analyzing consumer inflation expectations brings insights into how everyday perceptions shape broader economic trends (Dietrich et al., 2022; Meeks and Monti, 2023). The Survey of Consumer Expectations (SCE) is a monthly survey maintained by the Federal Reserve Bank of New York collecting information on households’ expectations on a broad variety of economic topics between June 2013 and November 2022, see Armantier et al. (2017). We focus our attention on the inflation expectation question, in which each consumer is asked to provide a distribution representing their belief for the 12-months ahead inflation. The survey respondents are presented with pre-defined bins over which they can distribute percentage points, defining a histogram of their beliefs. Each month, an average of approximately 1300 response histograms are available, which we aggregate by first taking the individual’s median belief and approximating the median belief density via kernel density estimator with a Gaussian kernel and using Scott’s rule (Scott, 1992) for the choice of the bandwidth, resulting in a time-series of T = 114 elements in D([−12, 12]) displayed in the left panel of Figure 6. 8. Acknowledgement This work has received funding from the European Union’s Horizon 2020 research and innovation program under the Marie Sk lodowska-Curie grant agreement No 956107, ”Economic Policy in Complex Environments (EPOC)”. Appendix A. General results in Hadamard spaces We start by stating results available in Hadamard spaces that will be used in the rest of the Appendix. Appendix B. Proofs Consistency of the mean estimator Consistency of the mean estimator Plugin this bound in the infinite sum, this gives Using this bound in the sum gives Appendix B.1. 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(1) Matthieu Bult´e, Department of Mathematical Sciences, University of Copenhagen, and Faculty of Business Administration and Economics, Bielefeld University;
(2) Helle Sørensen, Department of Mathematical Sciences, University of Copenhagen. Authors: Authors: (1) Matthieu Bult´e, Department of Mathematical Sciences, University of Copenhagen, and Faculty of Business Administration and Economics, Bielefeld University; (2) Helle Sørensen, Department of Mathematical Sciences, University of Copenhagen. This paper is available on arxiv under CC BY 4.0 DEED license. This paper is available on arxiv under CC BY 4.0 DEED license. available on arxiv available on arxiv

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