Abstract
We introduce a new class of integer-valued self-exciting threshold models, which is based on the binomial autoregressive model of order one as introduced by McKenzie (Water Resour Bull 21:645–650, 1985. doi:10.1111/j.1752-1688.1985.tb05379.x). Basic probabilistic and statistical properties of this class of models are discussed. Moreover, parameter estimation and forecasting are addressed. Finally, the performance of these models is illustrated through a simulation study and an empirical application to a set of measle cases in Germany.
Notes
The density-dependent models by Weiß and Pollett (2014) might also be understood as special SET models with \(N+1\) regimes.
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Acknowledgments
The authors thank the referees for carefully reading the article and for their comments, which greatly improved the article. This work was supported by Portuguese funds through the CIDMA—Center for Research and Development in Mathematics and Applications, and the Portuguese Foundation for Science and Technology (FCT-Fundação para a Ciência e a Tecnologia), within project UID/MAT/04106/2013.
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Appendices
Appendix 1: Proofs
1.1 Unconditional mean and variance
The unconditional mean (6) is a direct consequence of (4):
For the unconditional variance (7), consider first
as well as (note that \(E[I_{t-1}(1-I_{t-1})\cdot Y]=0\))
Insertion of (30) and (31) into \(\sigma _X^2 = E\left[ V[X_{t} | X_{t-1}] \right] + V\left[ E[X_{t} | X_{t-1} ] \right] \) and reordering gives
This completes the proof of the variance formula (7).
To get the properties of the LSET-BAR(1) model, we have to insert \(r := r_{1} =r_{2}\) into the Eqs. (6) and (7). We start with the mean (10):
The derivation of the variance (11) is more tedious:
This completes the proof.
1.2 Binomial index of dispersion
First, we consider the denominator of the \(\hbox {BID}\) (15) for the case of the LSET model (\(r_{1}=r_{2}=r\)). Using (10), we obtain
Then we look for matching terms in the numerator. First note that
Using this together with (11) and (32), we find
Bringing the results together leads to (16) for the \(\hbox {BID}\). Note that only the last term, \(\frac{2r}{1+r}\ldots \), might become negative.
If \(r=0\), (16) reduces to
1.3 Autocovariance function
By the law of total covariance, we obtain
which proves (18).
Appendix 2: Tables
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Möller, T.A., Silva, M.E., Weiß, C.H. et al. Self-exciting threshold binomial autoregressive processes. AStA Adv Stat Anal 100, 369–400 (2016). https://doi.org/10.1007/s10182-015-0264-6
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DOI: https://doi.org/10.1007/s10182-015-0264-6