In this paper, we present a new integer-valued time series model based on Poisson-weighted exponential innovations and a negative binomial thinning operator. One of its notable characteristics is its ability to handle over-dispersed integer-valued time series. Two different approaches are employed for estimating the related parameters. Furthermore, simulation tests are carried out to demonstrate the accuracy of these approaches. Finally, we compare our model with fair competitors using some count time series data and show that it is statistically superior.