Part A- Consider the following simple regression model for which ∈i ∼ N(0, σ)
y1 = β0 + 0.5β1 + ∈1
y2 = β0 - β1 + ∈2
y3 = β0 + 0.5β1 + ∈3
a. Write the above model in matrix form.
b. Find the least squares estimates using vectors and matrices.
c. Find the variance-covariance matrix of β^.
d. Find the hat matrix. Verify that the sum of the diagonal elements of the hat matrix is equal to 2(i=1Σn hii = k + 1).
e. Generate your own data with n = 3 based on this model and verify that the estimates of β0 and β1 are those given by part (b).
Part B- Suppose that you need to fit the multiple regression model yi = β0 - β1x1i + β2x2i + ∈, where E(∈i) = 0, E(∈i∈i) = 0 for i ≠ j, and var(∈i) = σ2, to the following data:
|
Y
|
x1
|
x2
|
|
-43.6
|
27
|
34
|
|
3.3
|
33
|
30
|
|
-12.4
|
27
|
33
|
|
7.6
|
24
|
11
|
|
11.4
|
31
|
16
|
|
5.9
|
40
|
30
|
|
-4.5
|
15
|
17
|
|
22.7
|
26
|
12
|
|
-14.4
|
22
|
21
|
|
-28.3
|
23
|
27
|
It turns out that
a. Find the least squares estimator of β = (β0, β1, β2)'.
b. Find the variance-covariance matrix of the previous estimator.
c. Compute the estimate se2 of σ2.
d. Using your answers to parts (b) and (c) find the variances of β^0, β^1, and β^2.
e. Find the fitted value y^i and its variance.
f. What is the variance of the first residual (var(∈i))?