Question

3. In the multiple regression model shown in the previous question, which one of the following statements is incorrect: (b) The sum of squared residuals is the square of the length of the vector ü (c) The residual vector is orthogonal to each of the columns of X (d) The square of the length of y is equal to the square of the length of y plus the square of the length of û by the Pythagoras theorem In all three of the following questions, the correct answer (e). However, the goal is not to spot the correct answer, but to understand why other answers are wrong. Write a sentence or two on each suggested answer that explains why that answer is correct or incorrect. 1. We have estimated the model wage Bo+Bi experience +u using OLS based on a sample of 4 observations. We know that the matrix of explanatory variables is 1 2 and we are told that the OLS residuals are: -1 We can immediately say that the residual vector: (a) is reported correctly because it sums to zero (b) is reported incorrectly because it should be a 2 x 1 vector (c) is reported correctly but shows a poor fit because X'û 0 (d) is reported correctly because it is linearly independent of columns of X (e) is reported incorrectly because it is not orthogonal to the second column of X 2·The multiple regression model in matrix form is where dimensions are specified below each vector and matrix. We denote the estimated model by in which β is the OLS estimate ofß and values given by This implies that: is the vector of OLS residuals. The vector of predicted Xß is the linear combination of columns of X that is closest to y. the OLS, У (b) Х'y 0 (c) X'u 0

Answer 1

1)
option E) is correct
sum x*u = 0
but here 1+1 -3-4 = -5 is not zero

2)

option E) is correct

3)

option B) is correct