Consider the following simple regression model for which

Question

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 h_ii = 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 y_i = β₀ - β₁x1_i + β₂x2_i + ∈, 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 s_e² 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))?