  This paper investigates extremal quantiles under two-way cluster dependence.
We demonstrate that the limiting distribution of the unconditional intermediate
order quantiles in the tails converges to a Gaussian distribution. This is
remarkable as two-way cluster dependence entails potential non-Gaussianity in
general, but extremal quantiles do not suffer from this issue. Building upon
this result, we extend our analysis to extremal quantile regressions of
intermediate order.
