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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 $n \leq4g+5$. In this paper we consider the complementary case. We calculate thecanonical divisor of $\overline{\mathcal{H}}_{g,n}$ and show that it iseffective for $n=4g+6$ and big for $n\leq 4g+7$. This leads us to conjecturethat $\overline{\mathcal{H}}_{g,n}$ has non-negative Kodaira dimension for $n =4g+6$ and is of general type for $n \geq 4g+7$.

On the Kodaira dimension of the moduli space of hyperelliptic curves with marked points