  We describe a fast computation method for leave-one-out cross-validation
(LOOCV) for $k$-nearest neighbours ($k$-NN) regression. We show that, under a
tie-breaking condition for nearest neighbours, the LOOCV estimate of the mean
square error for $k$-NN regression is identical to the mean square error of
$(k+1)$-NN regression evaluated on the training data, multiplied by the scaling
factor $(k+1)^2/k^2$. Therefore, to compute the LOOCV score, one only needs to
fit $(k+1)$-NN regression only once, and does not need to repeat
training-validation of $k$-NN regression for the number of training data.
Numerical experiments confirm the validity of the fast computation method.
