# Richard Davis: The Use of Shape Constraints for Modeling Time Series of Counts

**Time: **
Mon 2019-10-14 15.15 - 16.15

**Location: **
F11, KTH

**Participating: **
Richard Davis, Columbia University

### Abstract

For many formulations of models for time series of counts, the specification of a family of probability mass functions relating the observation \(Y_t\) at time t to a state variable \(X_t\) must be explicitly specified. Typical choices are the Poisson and negative binomial distributions. One of the principal goals of this research is to relax this parametric framework and assume that the requisite pmf is a one-parameter exponential family in which the reference distribution is unknown but log-concave. This class of distributions includes many of the commonly used pmfs. The serial dependence in the model is governed by specifying the evolution of the conditional mean process. The particular link function used in the exponential family model depends on the specification of the reference distribution. Using this semi-parametric model formulation, we are able to extend the class of observation-driven models studied in Davis and Liu (2016). In particular, we show there exists a stationary and ergodic solution to the state-space model. In this new semi-parametric framework, we compute and maximize the likelihood function over both the parameters associated with the mean function and the reference measure subject to a concavity constraint. On top of this we can “smooth” the pmf using the Skellam distribution in order to obtain an estimated distribution defined on all the non-negative integers. In general, the smooth version has better performance than existing methods. The estimator of the mean function and the conditional distribution are shown to be consistent and perform well compared to a full parametric model specification. Further limit theory in other situations will be described. The finite sample behavior of the estimators are studied via simulation and empirical examples are provided to illustrate the methodology. This is joint work with Jing Zhang and Thibault Vatter.