1. Consider n pairs of observations (Xi ,Yi ),i =1,...,n. If we want to fit the model, Y =2 +ß.X^2 + ?. find the least squares estimator for ß .

Question

This question could be asking us to do regression, but I do not really know. IT requires us to use the least square estimator.

Answer 1

John, I help you set this up. The model that is proposed is non linear (x²) so we want to transform it into a linear equation by the following steps (so you can use the linear least squares technique)

 

y-2=βx² we then take the natural log of both sides which gives

 

ln(y-2)=2βln(x) this equation is linear if we define v=ln(y-2) and w=ln(x) then the model becomes

 

v=2βw you can now use the standard procedure for finding the linear least squares estimate for the coefficient using the transformed variables v and w. Now if the least squares estimate is say α then α=2β so β=α/2.

Hope this helps.

Jim Stanton