  For a monomial ideal $I$, we consider the $i$th homological shift ideal of
$I$, denoted by $\text{HS}_i(I)$, that is, the ideal generated by the $i$th
multigraded shifts of $I$. Some algebraic properties of this ideal are studied.
It is shown that for any monomial ideal $I$ and any monomial prime ideal $P$,
$\text{HS}_i(I(P))\subseteq \text{HS}_i(I)(P)$ for all $i$, where $I(P)$ is the
monomial localization of $I$. In particular, we consider the homological shift
ideal of some families of monomial ideals with linear quotients. For any
$\textbf{c}$-bounded principal Borel ideal $I$ and for the edge ideal of
complement of any path graph, it is proved that $\text{HS}_i(I)$ has linear
quotients for all $i$. As an example of $\textbf{c}$-bounded principal Borel
ideals, Veronese type ideals are considered and it is shown that the
homological shift ideal of these ideals are polymatroidal. This implies that
for any polymatroidal ideal which satisfies the strong exchange property,
$\text{HS}_j(I)$ is again a polymatroidal ideal for all $j$.
  Moreover, for any edge ideal with linear resolution, the ideal
$\text{HS}_j(I)$ is characterized and it is shown that $\text{HS}_1(I)$ has
linear quotients.
