Consider the model For i = 1,...,n and j = 1,...,ri, where we have r i replicates at each vector (1, X i1 ,...,X ip ) of independent variables. Assume that ϵ ij are uncorrelated random variables with...

Consider the model

For i = 1,...,n and j = 1,...,ri, where we have r_i
replicates at each vector (1, X_i1,...,X_ip) of independent variables. Assume that ϵ_ij
are uncorrelated random variables with mean zero and variance σ². Consider also the model

(a) Show that the weighted least squares estimator of (β₀
β₁
··· β_p) in Model

(b) is also the ordinary least squares estimator of (β₀
β₁
··· β_p) in Model (a).

(b) Show also that they coincide with the ordinary least squares estimates in the rescaled model: