Count time series data is pervasive in many facets of scientific and social research. In this context, we developed an innovative method for modeling count data using the integer autoregressive (INAR) process; prespecifying, in the same family, the marginal distributions and the innovations. Our novelty approach is more natural and intuitive as on counting phenomena is easier to identify the distribution of the marginals and the innovations than that of the count series of the thinning operator. In our model, the count series is an innate consequence of the marginals and innovations, so the thinning operator arises naturally. Advantages are the analytic-mathematical simplifications, unrestricted parameter space, and interesting stochastic properties such as the time reversibility. In this work, we introduce and explore a special first-order INAR process with both marginal and innovation geometric distributed. The inference is via CLS, Yule-Walker and maximum likelihood; estimators are consistent and asymptotically normal. The performance of the estimators is checked using Monte Carlo methods. Applications to real datasets are provided and comparison with competing models presented.