Problem 8

Suppose we have a sample of n observations in the form of

(x1,1, x1,2, y1),(x2,1, x2,2, y2), · · · ,(xn,1, xn,2, yn),

where xi,1 and xi,2 are the input features and yi is the response variable for sample i. We want to find the least square estimators for β1 and β2 in the following model:

y = β1x1 + β2x2.

(a) Write Sum of Square of Error (SS) formulation for this problem.

(b) Consider the following system of two equations, where β1 and β2 are the unknowns:

What is the solution of the above system (for β1 and β2) in terms of a, b, c, d, e, and f.

(c) Find the lease square estimators of β1 and β2. (Hint: you may need to use the results of parts (a) and (b) ).

(d) Use the formulas from part (c) to compute βˆ 1 and βˆ 2 for the following data set.