  We study the stability of fixed points in the two-loop renormalization group
for the random field O($N$) spin model in $4+\epsilon$ dimensions. We solve the
fixed-point equation in the 1/N expansion and $\epsilon$ expansion. In the
large-N limit, we study the stability of all fixed points. We solve the
eigenvalue equation for the infinitesimal deviation from the fixed points under
physical conditions on the random anisotropy function. We find that the fixed
point corresponding to dimensional reduction is singly unstable and others are
unstable or unphysical. Therefore, one has no choice other than dimensional
reduction in the large-N limit. The two-loop $\beta$ function enables us to
find a compact area in the $(d, N)$ plane where the dimensional reduction
breaks down. We calculate higher-order corrections in the 1/N and $\epsilon$
expansions to the fixed point. Solving the corrected eigenvalue equation
nonperturbatively, we find that this fixed point is singly unstable also for
sufficiently large $N$ and the critical exponents show a dimensional reduction.
