  It is known that the moduli space $\overline{\mathcal{H}}_{g,n}$ of genus $g$
stable hyperelliptic curves with $n$ marked points is uniruled for $n \leq
4g+5$. In this paper we consider the complementary case. We calculate the
canonical divisor of $\overline{\mathcal{H}}_{g,n}$ and show that it is
effective for $n=4g+6$ and big for $n\leq 4g+7$. This leads us to conjecture
that $\overline{\mathcal{H}}_{g,n}$ has non-negative Kodaira dimension for $n =
4g+6$ and is of general type for $n \geq 4g+7$.
