Statistica Sinica 26 (2016), 1673-1707

THEORY AND INFERENCE FOR A CLASS OF

NONLINEAR MODELS WITH APPLICATION

TO TIME SERIES OF COUNTS

Richard A. Davis and Heng Liu

Columbia University and Google Inc.

Abstract: This paper studies theory and inference related to a class of time series models that incorporates nonlinear dynamics. It is assumed that the observations follow a one-parameter exponential family of distributions given an accompanying process that evolves as a function of lagged observations. We employ an iterated random function approach and a special coupling technique to show that, under suitable conditions on the parameter space, the conditional mean process is a geometric moment contracting Markov chain and that the observation process is absolutely regular with geometrically decaying coefficients. Asymptotic theory of the maximum likelihood estimates of the parameters is established under some mild assumptions. These models are applied to two examples; the first is the number of transactions per minute of Ericsson stock and the second is related to return times of extreme events of Goldman Sachs Group stock.

Key words and phrases: Absolute regularity, ergodicity, geometric moment contraction, iterated random functions, one-parameter exponential family, time series of counts.