  We measure the radial profiles of the stellar velocity dispersions,
$\sigma(R)$, for 90 early-type galaxies (ETGs) in the MASSIVE survey, a
volume-limited integral-field spectroscopic (IFS) galaxy survey targeting all
northern-sky ETGs with absolute $K$-band magnitude $M_K < -25.3$ mag, or
stellar mass $M_* > 4 \times 10^{11} M_\odot$, within 108 Mpc. Our wide-field
107" $\times$ 107" IFS data cover radii as large as 40 kpc, for which we
quantify separately the inner (2 kpc) and outer (20 kpc) logarithmic slopes
$\gamma_{\rm inner}$ and $\gamma_{\rm outer}$ of $\sigma(R)$. While
$\gamma_{\rm inner}$ is mostly negative, of the 56 galaxies with sufficient
radial coverage to determine $\gamma_{\rm outer}$ we find 36% to have rising
outer dispersion profiles, 30% to be flat within the uncertainties, and 34% to
be falling. The fraction of galaxies with rising outer profiles increases with
$M_*$ and in denser galaxy environment, with 10 of the 11 most massive galaxies
in our sample having flat or rising dispersion profiles. The strongest
environmental correlations are with local density and halo mass, but a weaker
correlation with large-scale density also exists. The average $\gamma_{\rm
outer}$ is similar for brightest group galaxies, satellites, and isolated
galaxies in our sample. We find a clear positive correlation between the
gradients of the outer dispersion profile and the gradients of the velocity
kurtosis $h_4$. Altogether, our kinematic results suggest that the increasing
fraction of rising dispersion profiles in the most massive ETGs are caused (at
least in part) by variations in the total mass profiles rather than in the
velocity anisotropy alone.
