General Studies · Linear Regression

Suppose we fit the linear model Y = Xβ + ε using least squares, where Y = (Y1, Y2, ..., Yn)T and ε = (ε1, ε2, ..., εn)T. Here X is an n × p non-stochastic matrix with full column rank and εi values are i.i.d. with mean 0 and variance 1. For i ∈ {1, 2, ..., n}, let Ŷi be the fitted value of Yi and let ε̂i be the corresponding residual. For r, s ∈ {1, 2, ..., n} where r ≠ s, which of the following must be true?