  We prove the convergence of moments of the number of directions of affine
lattice vectors that fall into a small disc, under natural Diophantine
conditions on the shift. Furthermore, we show that the pair correlation
function is Poissonian for any irrational shift in dimension 3 and higher,
including well-approximable vectors. Convergence in distribution was already
proved in the work of Str\"ombergsson and the second author, and the principal
step in the extension to convergence of moments is an escape of mass estimate
for averages over embedded $\operatorname{SL}(d,\mathbb{R})$-horospheres in the
space of affine lattices.
