  The presence of a dipolar statistical anisotropy in the spectrum of cosmic
microwave background (CMB) fluctuations was reported by the Wilkinson Microwave
Anisotropy Probe (WMAP), and has recently been confirmed in the Planck 2013
analysis of the temperature anisotropies. At the same time, the Planck 2013
results report a stringent bound on the amplitude of the local-type
non-Gaussianity. We show that the non-linear effect of the dipolar anisotropy
generates not only a quadrupole moment in the CMB but also a local-type
non-Gaussianity. Consequently, it is not easy to build models having a large
dipolar modulation and at the same time a sufficiently small quadrupole and
level of local bispectral anisotropy to agree with the present data. In
particular, most models proposed so far are almost excluded, or are at best
marginally consistent with observational data. We present a simple alternative
scenario that may explain the dipolar statistical anisotropy while satisfying
the observational bounds on both the quadrupole moment and local-type
non-Gaussianity.
