Table of Links
3. The GAR(1) Model
3.1. Model and Stationary Solution
4. Estimation of model parameters and 4.1. Fréchet mean
5. Testing for the absence of serial dependence
6.1. R with multiplicative noise
6.2. Univariate distributions with a density
Appendix A. General results in Hadamard spaces
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
Plugin this bound in the infinite sum, this gives
Using this bound in the sum gives
Appendix B.1. Uniform convergence of LT
Again using that Ω is bounded, the average is also bounded and we obtain the desired result.
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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