  We study the Land\'e $g$-factor of conduction electrons in the $L$-valley of
bulk GaAs and AlAs by using a three-band $\mathbf{k}\cdot\mathbf{p}$ model
together with the tight-binding model. We find that the $L$-valley $g$-factor
is highly anisotropic, and can be characterized by two components, $g_{\perp}$
and $g_{\|}$. $g_{\perp}$ is close to the free electron Land\'e factor but
$g_{\|}$ is strongly affected by the remote bands. The contribution from remote
bands on $g_{\|}$ depends on how the remote bands are treated. However, when
the magnetic field is in the Voigt configuration, which is widely used in the
experiments, different models give almost identical $g$-factor.
