We study instrumental-variable designs where policy reforms strongly shift the distribution of an endogenous variable but only weakly move its mean. We formalize this by introducing distributional relevance: instruments may be “purely distributional" with
Var
⁡
(
𝔼
​
[
X
∣
Z
]
)

0
while
F
X
∣
Z
(
⋅
∣
Z
)
varies nontrivially. Within a triangular model, distributional relevance suffices for nonparametric identification of average structural effects via a control function constructed from
F
X
∣
Z
. We then propose Quantile Least Squares (Q–LS), which aggregates conditional quantiles of
X
given
Z
into an optimal mean-square predictor and uses this projection as an instrument in a linear IV estimator. We establish consistency, asymptotic normality, and the validity of standard 2SLS variance formulas, and we discuss regularization across quantiles. Monte Carlo designs show that Q–LS delivers well-centered estimates and near-correct size when mean-based 2SLS suffers from weak instruments. In Health and Retirement Study data, Q–LS exploits Medicare Part D–induced distributional shifts in out-of-pocket risk to sharpen estimates of its effects on depression.