  We extend the well-known $\beta$-model for directed graphs to dynamic network
setting, where we observe snapshots of adjacency matrices at different time
points. We propose a kernel-smoothed likelihood approach for estimating $2n$
time-varying parameters in a network with $n$ nodes, from $N$ snapshots. We
establish consistency and asymptotic normality properties of our
kernel-smoothed estimators as either $n$ or $N$ diverges. Our results contrast
their counterparts in single-network analyses, where $n\to\infty$ is
invariantly required in asymptotic studies. We conduct comprehensive simulation
studies that confirm our theory's prediction and illustrate the performance of
our method from various angles. We apply our method to an email data set and
obtain meaningful results.
