Assignment-

Q1. Consider a simple linear regression model without an intercept:

Y = βX + U, where Y, X, and U are n-vectors, and β is the unknown scalar regression coefficient. Assume that E (U|X) = 0 and E (UU'|X) = σ²I_n.

(a) Show that the OLS estimator of β is

β^ˆ = X'Y/X'X.

(b) Define U^ˆ = Y - βX^ˆ. For each of the following statements, explain if it is true or false:

(i) E(U_iX_i) = 0 for all i = 1, . . . , n.

(ii) EU_i = 0 for all i = 1, . . . , n.

(iii) _i=1∑ⁿU^ˆ_iX_i = 0.

(iv) _i=1∑ⁿU^ˆ_i = 0.

(v) _i=1∑ⁿU_iX_i = 0.

(vi) _i=1∑ⁿ U_i = 0.

(c) Find Var (β^ˆ|X).

(d) Consider the following estimator of β:

β^˜ = Y^¯/X^¯,

where

Y^¯ = 1/n _i=1∑ⁿ Y_i and X^¯ = 1/n _i=1∑ⁿX_i,

(assume that with probability one, X^¯ ≠ 0). Is β^˜ unbiased?

(e) Find Var (β^˜|X).

(f) Without relying on the Gauss-Markov Theorem, show that β^˜ is less efficient than β^ˆ. Hint: Using the Cauchy-Schwartz inequality, show that

(_i=1∑ⁿX_i)² ≤ n _i=1∑ⁿX_i².

Q2. Suppose that Assumptions A1, A2, and A4 of the Classical Linear Regression model hold, i.e.

Y = Xβ + U, β ∈ R^k

E(U|X) = 0,

rank(X) = k,

however,

E(UU'|X) = Ω,

where Ω is an n × n, positive definite and symmetric matrix, but different from σ²In.

(a) Derive the conditional variance (given X) of the OLS estimator β^ˆ = (X'X)^-1X'Y.

(b) Derive the conditional variance (given X) of the GLS estimator β^˜ = (X'Ω^-1X)^-1X'Ω^-1Y.

(c) Without relying on the Gauss-Markov Theorem, show that

Var(β^ˆ|X) - Var(β^˜|X) ≥ 0

(in the positive semi-definite sense). Hint: Show

(Var(β^˜|X))^-1 - (Var(β^ˆ| X))^-1 ≥ 0

by showing that the expression on the left-hand side depends on a symmetric and idempotent matrix of the form I_n - H(H'H)^-1H' for some n × k matrix H of rank k.

Q3. Consider the GLS estimator β^˜ defined in the previous question.

(a) Show that β^˜ satisfies U^˜'Ω^-1X = 0, where U^˜ = Y - Xβ^˜.

(b) Using the result in (a), show that the generalized squared distance function Q(b) = (Y - Xb)'Ω^-1(Y - Xb) can be written as

Q(b) = U^˜'Ω^-1U^˜ + (β^˜ - b)'X'Ω^-1X(β^˜ - b).

(c) Using the result in (b), show that β^˜ minimizes Q(b).

Need answers to question 2 and question 3. Lecture notes can be provided.