  Let $\{X_k,k\in{\mathbb{Z}}\}$ be an autoregressive process of order $q$.
Various estimators for the order $q$ and the parameters ${\bolds
\Theta}_q=(\theta_1,...,\theta_q)^T$ are known; the order is usually determined
with Akaike's criterion or related modifications, whereas Yule-Walker, Burger
or maximum likelihood estimators are used for the parameters
${\bolds\Theta}_q$. In this paper, we establish simultaneous confidence bands
for the Yule--Walker estimators $\hat{\theta}_i$; more precisely, it is shown
that the limiting distribution of ${\max_{1\leq i\leq
d_n}}|\hat{\theta}_i-\theta_i|$ is the Gumbel-type distribution $e^{-e^{-z}}$,
where $q\in\{0,...,d_n\}$ and $d_n=\mathcal {O}(n^{\delta})$, $\delta >0$. This
allows to modify some of the currently used criteria (AIC, BIC, HQC, SIC), but
also yields a new class of consistent estimators for the order $q$. These
estimators seem to have some potential, since they outperform most of the
previously mentioned criteria in a small simulation study. In particular, if
some of the parameters $\{\theta_i\}_{1\leq i\leq d_n}$ are zero or close to
zero, a significant improvement can be observed. As a byproduct, it is shown
that BIC, HQC and SIC are consistent for $q\in\{0,...,d_n\}$ where
$d_n=\mathcal {O}(n^{\delta})$.
