1. A linear regression model is given below:

Y_i = β₁ + β₂X_i + u_i                                i = 1, 2, ... N

Where the assumption of the classical linear regression model hold. Instead of the OLS estimator, the researcher decided to use the following estimator   

Β^{^}₂ = Y₃-Y₂/X₃-X₂

Where Y₂ and Y₃ are the second and third observation of the Y's and X₂ and X₃ are the second and third observation of X's.

a. Show that β^{^}₂ is a linear estimator.

b. Prove that β^{^}₂ is an unbiased estimator of the true regression slop parameter, β₂.

c. Derive the variance of β^{^}₂. (i.e., express the variance of β^{^}₂ in terms of X's and σ², where σ² = Var(u_i)).

2. The linear regression model is given below:

y_i = β₀ + β₁x_i + u_i                                i = 1, 2, .... n   

Let β^{^}₀ and β^{^}₁ be the OLS intercept and slope coefficient estimator, respectively, and let u^- be the sample average of the errors (not the residuals!).

a. Show that β^{^}₀ can be written as β^{^}₀ = β₀ + u^- - (β^{^}₁ - β₁)x^-.

b. Show that Var(β^{^}₀) = σ²/n + σ²(x^-)²/SSTx^-.

c. Show that Var(β^{^}₀) = (σ²n^-1_i-1∑ⁿx_i²/_i-1∑ⁿ(x_i - x^-)²).

3. The Linear regression model is given below:

Y_i = β₁ + β₂X_i + u_i                                i = 1, 2, ... n

The following information is derived from a sample of observations:

∑Y_i = 1110, ∑X_i = 1680, ∑Y_i² = 133,300, ∑X_i² = 315,400 and ∑X_iY_i - 204,200

a. Calculate β^{^}₁ and β^{^}₂.

b. Calculate SST, SSE, SSR and R².

c. Calculate the value for σ^{^2}, var(β^{^}₁) and var(β^{^}₂).