代写ECON0060 Problem Set 1代做Statistics统计

日期：2024-03-21 12:13

ECON0060

Problem Set 1

Question 1

Consider a system of seemingly unrelated regressions

Here, yig and uig are scalars and xig is a Kg vector for g = 1, . . . , G. As in the lecture we introduce the G-vectors yi = (yi1, . . . , yiG)' and ui = (ui1, . . . , uiG)' , and the ( Kg)-vector β and G × ( Kg) matrix Xi as follows

Let . The OLS and GLS estimator read

Show that  if one of the following two conditions is satisfied:

(a) if Σ is a diagonal matrix, i.e. Σ = diag(, . . . , ), with = E(|Xi) > 0.

(b) if the regressors in all G equations are identical, i.e. xig = xi for all g = 1, . . . , G.

Question 2

Consider the two seemingly unrelated regression equations

Here, the first equation only contains a constant regressor, while the second contains a constant and an additional regressor xi . Assume that

across i = 1, . . . , n. Let −1 < ρu < 1 and −1 < ρx < 1. Also define

The OLS and GLS estimator for β are given by

(a) Assume ρx = 0. Are and consistent estimators for β? Explain.

(b) Assume ρx = 0. Calculate and compare the asymptotic variances of and .

(you can use the results from the lecture notes).

(c) Assume ρx ≠ 0. Are and consistent estimators for β? Explain. Are the components and consistent for β1 and β21?

Question 3

Consider the system of two simultaneous equations

where yi1 and yi2 are endogenous variables and xi is a single exogenous regressors. There are four scalar structural parameters α1, α2, β1 and β2. In matrix notation the structural equations can be re-written as

where yi = (yi1, yi2)', ui = (ui1, ui2)', Γ is a 2 × 2 matrix and ∆ is a 1 × 2 vector. The reduced form. model can be written as

where Π = ∆Γ−1 = (π1, π2) is the 1×2 vector of reduced form. parameters and .

(a) Express Γ and ∆ in terms of α1, α2, β1 and β2.

(b) Calculate det(Γ). Under what condition on α1 and α2 is Γ invertible? (Hint: Γ is invertible if det(Γ) ≠ 0). In the following we assume that Γ is invertible.

(c) Express Π in terms of α1, α2, β1 and β2.

(d) Assume that β1 = 0 and π2 ≠ 0. Show that α1 is identified and find an expression for α1 in terms of π1 and π2.

(e) Assume that α1 = −α2, β1 = β2, and π1 + π2 ≠ 0. Show that all structural parameters are identified and find an expression for α1, α2, β1 and β2 in terms of π1 and π2.

Question 4

Consider the following panel data model with two time periods and one regressor

where i = 1, . . . , n indexes individuals, and t = 1, 2 indexes time periods. We assume that xi1, xi2, αi , εi1 and εi2 are all mutually independent, are all independently distributed across i, and that . and εit ∼ N (0, σε 2 ). We assume that .

For each i we define the two-vectors yi = (yi1, yi2)', xi = (xi1, xi2)' and ui = (ui1, ui2)', and the 2×2 matrix . Consider the following two estimators for the scalar parameter β

(a) Show that is consistent as n → ∞.

(b) Find an expression for Σ in terms of , and . Calculate the inverse of Σ.

(c) Find expressions for the asymptotic variance of the two estimators in terms of only , and (you can use the general result from the lecture). Which estimator has the smaller asymptotic variance?