The moduli space of curves has a compactification whose boundary can be understood as the product of moduli spaces of curves of lower genus. Therefore (perhaps naively) one might hope that there exists a compactification of $A_g$ whose boundary can be understood as in terms of the moduli of abelian varities of lower dimension. Is there any such compactification?

The moduli space of curves has a compactification whose boundary can be understood as the product of moduli spaces of curves of lower genus. Therefore (perhaps naively) one might hope that there exists a compactification of $A_g$ whose boundary can be understood as in terms of the moduli of abelian varieties of lower dimension. Is there any such compactification?