1. A linear regression model is given below:
Yi = β1 + β2Xi + ui 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(ui)).
2. The linear regression model is given below:
yi = β0 + β1xi + ui 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-1i-1∑nxi2/i-1∑n(xi - x-)2).
3. The Linear regression model is given below:
Yi = β1 + β2Xi + ui i = 1, 2, ... n
The following information is derived from a sample of observations:
∑Yi = 1110, ∑Xi = 1680, ∑Yi2 = 133,300, ∑Xi2 = 315,400 and ∑XiYi - 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).