Let $X$ be a normal, projective complex variety with an anticanonical divisor $D$. Do the virtual Hodge numbers of the noncompact Calabi-Yau variety $X$ \ $D$ enjoy some sort of positivity property?

Being virtual, defined by inclusion-exclusion from complete varieties, they're not individually positive. What I'd most like is to hear "The Euler characteristic is nonnegative". But that's not true for $X$ a quintic 3-fold, $D$ empty.