# 程序代写代做代考 Alastair Hall ECON61001: Autumn 2020 Econometric Methods

Alastair Hall ECON61001: Autumn 2020 Econometric Methods

Problem Set for Tutorial 3

The first question considers the construction of one-sided tests.

1. Consider the regression model

and let βˆT denote the OLS estimator of β0. Assume Assumptions CA1-CA6 (see Lecture 1)

y = Xβ0 + u

hold. Suppose it is desired to test H0 : β0,i ≤ β∗,i versus H1 : β0,i > β∗,i.

(a) Suggest an appropriate decision rule for performing this hypothesis test at the 100α% significance level.

(b) Consider the following regression model from Lectures 1 & 2:

yt = controls + βbelt,0 ∗ beltt + βmph,0 ∗ mpht + u,

with the set of controls being those used in (2.18) in the Lecture Notes. Use the coefficient estimator in (2.18) in Example 2.2 and the standard error reported in Example 2.3 of the Lecture Notes to test: H0 : βmph,0 ≤ 0 versus H1 : βmph,0 > 0 using a 5% significance level. Interpret the outcome of the test in terms of the impact of this law.

The next question relates to testing hypotheses involving linear restrictions on β0. 2. Consider the linear regression model

y = Xβ0 + u

with k = 5. In lectures, we discussed procedures for testing the null hypothesis Rβ0 = r.

Show that the following null hypotheses fit within this framework. (a)H0:β0,2 −β0,4 =0.

(b) H0 : β0,1 + β0,2 = β0,3 and1 + β0,4/3 = β0,5. Verify that both cases satisfy the restriction rank(R) = nr.

In the lecture, we discussed testing H0 : β0,i = β∗,i using a t-statistic. In this question, we con- sider testing this hypothesis using a F-statistic and show that the results of the two approaches are identical.

3. Consider the linear regression model

y = Xβ0 + u 1

(a) Verify that the restriction β0,i = β∗,i can be written equivalently as Rβ0 = r where R is the1×kvectorequaltothetheith rowofIk andr=β∗,i.

(b) Define

t-stat = βˆT,i − β∗,i σˆT √mi,i

where mi,i is the (i, i)th element of (X′X)−1, and recall this is the t-statistic used to test

H0 : β0,i = β∗,i. Show that

where F is the F- statistic for testing Rβ0 = r using the definitions of R, r in part (a).

4. Consider the linear regression model

y = Xβ0 + u

where y is the T × 1 vector containing the observable dependent variable, X is a T × k matrix of observable explanatory variables and u is the T × 1 vector of unobservable errors, and let Assumptions CA1-CA6 from the lectures hold. Let βˆR,T denote the RLS estimator based on the linear restrictions Rβ = r where where R is a nr × k matrix of pre-specified constants with rank equal to nr and r is a nr × 1 vector of pre-specified constants that is,

βˆR,T = βˆT − (X′X)−1R′{R(X′X)−1R′}−1(RβˆT − r),

where βˆT is the OLS estimator of β0.

(a) Assuming that Rβ0 = r, show that E[βˆR,T ] = β0 and V ar[βˆR,T ] = σ02D where

D = (X′X)−1 − (X′X)−1R′{R(X′X)−1R′}−1R(X′X)−1. (b) Show that if Rβ0 ̸= r then E[βˆR,T ] ̸= β0.

Hint: you can take advantage of the properties of the OLS estimator in answering these questions.

(t-stat)2 = F

In this question, you verify the properties of the RLS estimator discussed in lectures.

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