13 (1) Consider the simple regression model y = Bo + B₁x + u under the first four Gauss-Markov assumptions. For some function g(x), for example g(x) = x² or g(x) = log(1 + x²), define Z₁ = g(x). Define a slope estimator as mes show anoder B₁ = ( (z − 2)x))/(2(z. - 2)x). - Show that B₁ is linear and unbiased. Remember, because E(ulx) = 0, you can treat both Xi as nonrandom in your derivation. (ii) Add the homoskedasticity assumption, MLR.5. Show that and Zi Var(3,) = o²($(z, − 2)²)/(2(2, − 2)x;). (iii) Show directly that, under the Gauss-Markov assumptions, Var(3₁) ≤ Var(B₁), where B, is the OLS estimator. [Hint: The Cauchy-Schwartz inequality in Math Refresher B implies that (n-¹2 (2,₁ − 2)(x − 3))² = (n-¹ (z − 2)²)(n-¹ § (x, − 1)²);
13 (1) Consider the simple regression model y = Bo + B₁x + u under the first four Gauss-Markov assumptions. For some function g(x), for example g(x) = x² or g(x) = log(1 + x²), define Z₁ = g(x). Define a slope estimator as mes show anoder B₁ = ( (z − 2)x))/(2(z. - 2)x). - Show that B₁ is linear and unbiased. Remember, because E(ulx) = 0, you can treat both Xi as nonrandom in your derivation. (ii) Add the homoskedasticity assumption, MLR.5. Show that and Zi Var(3,) = o²($(z, − 2)²)/(2(2, − 2)x;). (iii) Show directly that, under the Gauss-Markov assumptions, Var(3₁) ≤ Var(B₁), where B, is the OLS estimator. [Hint: The Cauchy-Schwartz inequality in Math Refresher B implies that (n-¹2 (2,₁ − 2)(x − 3))² = (n-¹ (z − 2)²)(n-¹ § (x, − 1)²);
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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Problem 13
![algebra, take the expectation treating x3 and ₁1 as nonrandom.]
12 The following equation represents the effects of tax revenue mix on subsequent employment growth
for the population of counties in the United States:
growth
where growth is the percentage change in employment from 1980 to 1990, sharep is the share of prop-
adini serty taxes in total tax revenue, share, is the share of income tax revenues, and shares is the share of
sales tax revenues. All of these variables are measured in 1980. The omitted share, share, includes
fees and miscellaneous taxes. By definition, the four shares add up to one. Other factors would include
expenditures on education, infrastructure, and so on (all measured in 1980).
Why must we omit one of the tax share variables from the equation?
Give a careful interpretation of B₁.
(i)
(ii)
13 (1)
=
Bo + Bisharep + B₂share, + B3shares + other factors,
Consider the simple regression model y = Bo + B₁x + u under the first four Gauss-Markov
assumptions. For some function g(x), for example g(x) = x² or g(x) = log(1 + x²), define
Z₁ = g(x). Define a slope estimator as
15 anos
³₁ = ( (z − 2)y.)/(2(2₁ − 2)x.).
-
Show that B₁ is linear and unbiased. Remember, because E(ulx) = 0, you can treat both x, and zi
as nonrandom in your derivation.
(ii) Add the homoskedasticity assumption, MLR.5. Show that
(E80.
Var(3₁) = ²(2(z, - 2)²)/(2(2, - 2)x.)".
(iii) Show directly that, under the Gauss-Markov assumptions, Var(3₁) ≤ Var(B₁), where B₁ is the
OLS estimator. [Hint: The Cauchy-Schwartz inequality in Math Refresher B implies that
(n-¹(²₁-
(2,-2)(x − 3))² = (n-¹(2₁ - 2)²)(n-¹(x - 1)³):
notice that we can drop x from the sample covariance.]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fff6f9a4d-2084-42ea-abf6-3325fa2ca826%2Fc8609e9f-539e-48f9-bd45-7a7b9825834a%2F07nqn_processed.jpeg&w=3840&q=75)
Transcribed Image Text:algebra, take the expectation treating x3 and ₁1 as nonrandom.]
12 The following equation represents the effects of tax revenue mix on subsequent employment growth
for the population of counties in the United States:
growth
where growth is the percentage change in employment from 1980 to 1990, sharep is the share of prop-
adini serty taxes in total tax revenue, share, is the share of income tax revenues, and shares is the share of
sales tax revenues. All of these variables are measured in 1980. The omitted share, share, includes
fees and miscellaneous taxes. By definition, the four shares add up to one. Other factors would include
expenditures on education, infrastructure, and so on (all measured in 1980).
Why must we omit one of the tax share variables from the equation?
Give a careful interpretation of B₁.
(i)
(ii)
13 (1)
=
Bo + Bisharep + B₂share, + B3shares + other factors,
Consider the simple regression model y = Bo + B₁x + u under the first four Gauss-Markov
assumptions. For some function g(x), for example g(x) = x² or g(x) = log(1 + x²), define
Z₁ = g(x). Define a slope estimator as
15 anos
³₁ = ( (z − 2)y.)/(2(2₁ − 2)x.).
-
Show that B₁ is linear and unbiased. Remember, because E(ulx) = 0, you can treat both x, and zi
as nonrandom in your derivation.
(ii) Add the homoskedasticity assumption, MLR.5. Show that
(E80.
Var(3₁) = ²(2(z, - 2)²)/(2(2, - 2)x.)".
(iii) Show directly that, under the Gauss-Markov assumptions, Var(3₁) ≤ Var(B₁), where B₁ is the
OLS estimator. [Hint: The Cauchy-Schwartz inequality in Math Refresher B implies that
(n-¹(²₁-
(2,-2)(x − 3))² = (n-¹(2₁ - 2)²)(n-¹(x - 1)³):
notice that we can drop x from the sample covariance.]
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