Generalized Autoregressive Multivariate Models: From Binary to Poisson
Anna Bykhovskaya, Nour Meddahi
TLDR
This paper introduces a GARCH-type framework for binary time series, demonstrating how their aggregates converge to Poisson autoregressions.
Key contributions
- Presents a GARCH-type framework for binary autoregressive time series, accommodating nonlinearities and dependencies.
- Establishes existence and uniqueness of stationary solutions via a coupling argument tailored for binary data.
- Shows that aggregates of binary processes converge to a Poisson autoregression under rare-events scaling.
- Provides a micro-foundation for the widely used Poisson count model, supported by empirical S&P 100 data.
Why it matters
This paper offers a robust framework for modeling binary time series with complex dependencies. Crucially, it bridges binary and Poisson models, providing a theoretical micro-foundation for Poisson autoregression. This enhances our understanding and modeling capabilities for count data.
Original Abstract
This paper presents a framework for binary autoregressive time series in which each observation is a Bernoulli variable whose success probability evolves with past outcomes and probabilities, in the spirit of GARCH-type dynamics, accommodating nonlinearities, network interactions, and cross-sectional dependence in the multivariate case. Existence and uniqueness of a stationary solution is established via a coupling argument tailored to the discontinuities inherent in binary data. A key theoretical result, further supported by our empirical illustration on S&P 100 data, shows that, under a rare-events scaling, aggregates of such binary processes converge to a Poisson autoregression, providing a micro-foundation for this widely used count model. Maximum likelihood estimation is proposed and illustrated empirically.
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