I'm having some trouble with these two parts of this question:
The least squares regression of y (nx1) on the K coloumns of X (nxK) can be written as y = Xb + e, where b is the LS coefficient vector and e is the nx1 vector of LS residuals. The rank of X is 1 < K < n.
The first order conditions that determine b are: X'y = (X'X)b
1. Suppose y is quarterly beef consumption per capita. Data are available over the period 1988-2008 (21 years x 4 quarters = 84 observations.) Suppose K = 4 and the columns of X are Q1, Q2, Q3 and Q4, where Qj is a quarterly dummy variable: Qj = 1 in the jth quarter and zero otherwise. Interpret the four LS coefficients.
2. Suppose y is regressed on Q1...Q4 and X5 (K = 5), where X5 is real per capita income and your objective is to predict consumption in quarter 1 given a level of income equal to the first quarter's sample mean income. Demonstrate that the predicted level of consumption given by the LS equation is the first quarter's sample mean consumption.
For part one I'm trying to find what b= (X'X)^-1 X'y
I found that the X'X becomes a diagonal 4x4 matrix where the diagonal is Q1, Q2, Q3 and Q4. However I'm not sure how to move on from here.
Thanks for your help.