2. We describe an alternative way of studying how a dependent random variable y depends on independent random variables $1, ..., I'm. We have / observations. The value of y in the jth observation is denoted by y, and the value of a; in the jth observation is denoted by aj. Therefore, the observations can be summarized by the following table where each column corresponds to a column and each row corresponds to an observation: C12 C21 122 $2m 1/2 CNI IN2 INm UN As in the case of least squares, assume that the columns of the above matrix are linearly independent. Consider the following optimization problem: BERm min > ly; -23Bi - X;232 -... -IjmPm. 1=1 (a) [5 marks] First, we try to get an idea about what the minimization is doing. For this purpose, consider the simplest case where m = 1 and 2,1 = 1 for every j. What is the interpretation of 81 in this case? (Try generating some concrete y1, ..., y, plot the objective function and see how it behaves and where the minimum is.) Remember that we are seeking only for a global minimum. (b) [7 marks] Next, we want to show that the problem always has a solution. Here is a proof suggested by Daniel. Proof. For j = 1, ..., N, let uj = (Xj1, T;2, ..., Ijm). Then the objective function can be written as