Generalized Autoregressive Multivariate Models: From Binary to Poisson

TL;DR

This paper introduces a comprehensive framework for binary autoregressive time series, demonstrating convergence to Poisson models under rare-events scaling, with theoretical guarantees and empirical validation on financial data.

Contribution

It develops a novel binary autoregressive model with GARCH-like dynamics, proves stationarity and convergence to Poisson processes, and provides maximum likelihood estimation methods.

Findings

Binary autoregressive models can approximate Poisson processes under rare-events scaling.

Existence and uniqueness of stationary solutions are established.

Empirical analysis on S&P 100 data supports theoretical results.

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.