A skew INAR(1) process on Z\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {Z}}$$\end{document}

Integer-valued time series models have been a recurrent theme considered in many papers in the last three decades, but only a few of them have dealt with models on Z\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {Z}}$$\end{document} (that is, including both negative and positive integers). Our aim in this paper is to introduce a first-order, integer-valued autoregressive process on Z\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {Z}}$$\end{document} with skew discrete Laplace marginals (Kozubowski and Inusah, Ann Inst Stat Math 58:555–571, 2006). For this, we define a new operator that acts on two independent latent processes, similarly as made by Freeland (Adv Stat Anal 94:217–229, 2010). We derive some joint and conditional basic properties of the proposed process such as characteristic function, moments, higher-order moments and jumps. Estimators for the parameters of our model are proposed and their asymptotic normality is established. We run a Monte Carlo simulation to evaluate the finite-sample performance of these estimators. In order to illustrate the potential for practice of our process we apply it to a real data set about stock market.