Traditional crash count models, such as the Poisson and negative binomial models, do not account for the temporal correlation of crash data. In reality, crashes that occur in the same time frame are likely to share unobserved effects that may have been excluded from the model. If the temporal correlation of crash data is ignored, the estimated parameters can be biased and less precise. Therefore, there is a need to extend the standard crash count data models by incorporating temporal dependence. Whereas the literature for modeling time series count data is well developed, its applications for traffic crash data are limited. A particularly flexible model for the time series of counts is the negative binomial integer-valued generalized autoregressive conditional heteroscedastic (NBINGARCH) model, which properly accounts for the overdispersion, nonnegativity, and integer-valued features of count data. In this paper, the NBINGARCH model is extended to incorporate covariates so that the relationship between a time series of counts and correlated external factors may be properly modeled. The improved performance of the NBINGARCH model is demonstrated through a simulation study and an application to monthly driving under the influence (DUI) fatal crashes in Texas between 2003 and 2009. In addition, the relationship between monthly vehicle miles traveled (VMT) and gasoline prices in Texas is also examined. Ultimately, gasoline prices had no significant effect on DUI fatal crashes in Texas during that time period, and VMT had a positive effect.