Explanatory variables are incorporated into the count data (integer-valued) autoregressive model of order one. The properties of the resulting model are studied for both univariate and panel data speciÞcations. Weighted and unweighted conditional least squares and conditional generalized method of moment estimators are introduced and evaluated by Monte Carlo experimentation. The results indicate that the least squares estimators perform well for realistically short time series. Tests against time dependent parameters are obtained and their properties are evaluated. Tests based on least squares perform well in small samples. An empirical illustration is included.