In the linear regression quantile model, the conditional quantile of the response,
Y, given
x is
QY|x(τ)≡x′β(τ). Though
QY|x(τ) must be monotonically increasing, the Koenker–Bassett regression quantile estimator,
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, is not monotonic outside a vanishingly small neighborhood of
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. Given a grid of mesh
δn, let
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be the linear interpolation of the values of
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along the grid. We show here that for a range of rates,
δn,
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will be strictly monotonic (with probability tending to one) and will be asymptotically equivalent to
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in the sense that
n1/2 times the difference tends to zero at a rate depending on
δn.