Models for autoregressive processes of bounded counts: How different are they?
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Models for autoregressive processes of bounded counts: How different are they? Hee-Young Kim1 · Christian H. Weiß 2
· Tobias A. Möller2
Received: 25 September 2019 / Accepted: 11 March 2020 © The Author(s) 2020
Abstract We focus on purely autoregressive (AR)-type models defined on the bounded range {0, 1, . . . , n} with a fixed upper limit n ∈ N. These include the binomial AR model, binomial AR conditional heteroscedasticity (ARCH) model, binomial-variation AR model with their linear conditional mean, nonlinear max-binomial AR model, and binomial logit-ARCH model. We consider the key problem of identifying which of these AR-type models is the true data-generating process. Despite the volume of the literature on model selection, little is known about this procedure in the context of nonnested and nonlinear time series models for counts. We consider the most popular approaches used for model identification, Akaike’s information criterion and the Bayesian information criterion, and compare them using extensive Monte Carlo simulations. Furthermore, we investigate the properties of the fitted models (both the correct and wrong models) obtained using maximum likelihood estimation. A real-data example demonstrates our findings. Keywords Binomial autoregressive models · Count time series · Model adequacy · Model selection · Parameter estimation
1 Introduction Count time series occur in various fields, including investigations of natural phenomena (e. g., rare disease occurrences, animal sightings, and severe weather events) and in economic contexts (e. g., monitoring the number of transactions). Significant progress
Electronic supplementary material The online version of this article (https://doi.org/10.1007/s00180020-00980-6) contains supplementary material, which is available to authorized users.
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Christian H. Weiß [email protected]
1
Division of Economics and Statistics, National Statistics, Korea University, Sejong, South Korea
2
Department of Mathematics and Statistics, Helmut Schmidt University, PO box 700822, 22008 Hamburg, Germany
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H.-Y. Kim et al.
has been made in modeling count time series over the last 20 years; see Weiß (2018) for a recent survey. The most popular stationary count time series models are arguably the integer-valued autoregressive moving-average (INARMA) models, dating back to McKenzie (1985) and Al-Osh and Alzaid (1987), which use a probabilistic operator called binomial thinning. The simplest INARMA model is the integer-valued autoregressive model of order one, abbreviated as INAR(1). The model is defined as X t = α ◦ X t−1 +εt , with t ∈ N = {1, 2, . . .}, where {εt } is an innovation process composed of independent and identically distributed (i. i. d.) non-negative integer-valued random variables (i. e., having the range N0 = {0, 1, . . .}) with finite mean and variance. The symbol “◦” denotes X the binomial thinning operator (Steutel and van Harn Yi , where {Y j } are i. i. d. Bernoulli random variables 1979), defined by α ◦ X = i=1 with P(Y j = 1) = α = 1 − P(Y j = 0).
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