  We propose a goodness-of-fit test for a class of count time series models
with covariates which includes the Poisson autoregressive model with covariates
(PARX) as a special case. The test criteria are derived from a specific
characterization for the conditional probability generating function and the
test statistic is formulated as a $L_2$ weighting norm of the corresponding
sample counterpart. The asymptotic properties of the proposed test statistic
are provided under the null hypothesis as well as under specific alternatives.
A bootstrap version of the test is explored in a Monte--Carlo study and
illustrated on a real data set on road safety.
