A skew INAR(1) process on <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$${\mathbb {Z}}$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi mathvariant="double-struck">Z</mi> </math> </EquationSource> </InlineEquation>

<Para ID="Par1">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 <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$${\mathbb {Z}}$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi mathvariant="double-struck">Z</mi> </math> </EquationSource> </InlineEquation> (that is, including both negative and positive integers). Our aim in this paper is to introduce a first-order, integer-valued autoregressive process on <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$${\mathbb {Z}}$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi mathvariant="double-struck">Z</mi> </math> </EquationSource> </InlineEquation> with skew discrete Laplace marginals (Kozubowski and Inusah, Ann Inst Stat Math 58:555–571, <CitationRef CitationID="CR13">2006</CitationRef>). 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, <CitationRef CitationID="CR5">2010</CitationRef>). 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. Copyright Springer-Verlag Berlin Heidelberg 2015