2. Consider the linear regression model Y = Bot Bi Xe tr t=1,2,...,n, where X₁0 is a non-stochastic variable and u, is a random error term such that E(u) =0 for all t. (a) Suppose that u = 0₁44-1 +0₂04-2+1, where ₁ and ₂ are known parameters and e, are random variables such that E() = 0 for all t, var(₁)=²> 0 for all t, and cov(t, ₂) = 0 for all t = 8. (i) What are the statistical properties of the ordinary least squares (OLS) estimators of 30 and 3₁? Explain. (ii) Is it possible to find estimators of 5 and 3, which are more efficient than the OLS estimators? If so, explain how to estimate 3 and 3₁ effi- ciently. Include in your answer a careful explanation of why your proposed estimation method would produce efficient estimates. (iii) Is a conventional OLS-based t-test for the hypothesis that 3₁ = 0 valid? If not, explain how to carry out a valid test of the hypothesis that 3₁ = 0. (iv) What do the hypotheses Ho: 01 02 = 0 and H₁: 010 and/or 02/0 imply about the regression errors u? Explain in detail how to test Ho against H, using the Breusch-Godfrey procedure. (b) Indicate whether you agree or disagree with each of the following statements and provide a brief justification for your answer. (i) "If the hypothesis that ₁ = 2 = 0 is rejected, it is advisable to use OLS with heteroskedasticity-consistent standard errors". (ii) "If the hypothesis that o₁ = 2 = 0 is rejected, a natural response is not to use an estimator other than OLS but to change the specification of the model". (iii) "If ₁ = 2 = 0 and var(t)=o² exp(X₁), o and 3₁ should be estimated by OLS in the model (Y₁/X₂) = (1/X₂) + B₁ + ₂". (iv) "OLS estimates of the coefficients of the model Y₁ = 30 +³₁X₁ + ₂ are accurate even if the true model is Y₁ = Bo + B₁X₁ + B₂X² + €₁".
2. Consider the linear regression model Y = Bot Bi Xe tr t=1,2,...,n, where X₁0 is a non-stochastic variable and u, is a random error term such that E(u) =0 for all t. (a) Suppose that u = 0₁44-1 +0₂04-2+1, where ₁ and ₂ are known parameters and e, are random variables such that E() = 0 for all t, var(₁)=²> 0 for all t, and cov(t, ₂) = 0 for all t = 8. (i) What are the statistical properties of the ordinary least squares (OLS) estimators of 30 and 3₁? Explain. (ii) Is it possible to find estimators of 5 and 3, which are more efficient than the OLS estimators? If so, explain how to estimate 3 and 3₁ effi- ciently. Include in your answer a careful explanation of why your proposed estimation method would produce efficient estimates. (iii) Is a conventional OLS-based t-test for the hypothesis that 3₁ = 0 valid? If not, explain how to carry out a valid test of the hypothesis that 3₁ = 0. (iv) What do the hypotheses Ho: 01 02 = 0 and H₁: 010 and/or 02/0 imply about the regression errors u? Explain in detail how to test Ho against H, using the Breusch-Godfrey procedure. (b) Indicate whether you agree or disagree with each of the following statements and provide a brief justification for your answer. (i) "If the hypothesis that ₁ = 2 = 0 is rejected, it is advisable to use OLS with heteroskedasticity-consistent standard errors". (ii) "If the hypothesis that o₁ = 2 = 0 is rejected, a natural response is not to use an estimator other than OLS but to change the specification of the model". (iii) "If ₁ = 2 = 0 and var(t)=o² exp(X₁), o and 3₁ should be estimated by OLS in the model (Y₁/X₂) = (1/X₂) + B₁ + ₂". (iv) "OLS estimates of the coefficients of the model Y₁ = 30 +³₁X₁ + ₂ are accurate even if the true model is Y₁ = Bo + B₁X₁ + B₂X² + €₁".
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
Related questions
Question
How would you answer and workout section a
Transcribed Image Text:2. Consider the linear regression model
Y₁ = Bo + Bi X₁ + U.
t=1,2,..., n.
where X, 0 is a non-stochastic variable and u, is a random error term such that
E(u) =0 for all t.
(a) Suppose that u = 0₁1-1+0₂-2+, where 1 and 2 are known parameters
and e, are random variables such that E() = 0 for all t, var(t) = o² > 0 for
all t, and cov(et, Es) = 0 for all t = s.
(i) What are the statistical properties of the ordinary least squares (OLS)
estimators of 30 and 3₁? Explain.
(ii) Is it possible to find estimators of 50 and 3₁ which are more efficient
than the OLS estimators? If so, explain how to estimate 3 and 3₁ effi-
ciently. Include in your answer a careful explanation of why your proposed
estimation method would produce efficient estimates.
(iii) Is a conventional OLS-based t-test for the hypothesis that 3₁ = 0 valid?
If not, explain how to carry out a valid test of the hypothesis that 3₁ = 0.
(iv) What do the hypotheses Ho: 1 = 2 = 0 and H₁: 01 70 and/or 02 #0
imply about the regression errors u? Explain in detail how to test Ho
against H, using the Breusch-Godfrey procedure.
(b) Indicate whether you agree or disagree with each of the following statements
and provide a brief justification for your answer.
(i) "If the hypothesis that 1 = 2 = 0 is rejected, it is advisable to use OLS
with heteroskedasticity-consistent standard errors".
(ii) "If the hypothesis that 01 = 2 = 0 is rejected, a natural response is not
to use an estimator other than OLS but to change the specification of the
model".
(iii) "If 01 = 02 = 0 and var(e) = o² exp(X), 3o and 31 should be estimated
by OLS in the model (Y/X₂) = o(1/X₁) +³₁ +e₂".
(iv) “OLS estimates of the coefficients of the model Y₁ = Bo + B₁ X₂ + u are
accurate even if the true model is Y₁ = Bo + B₁X + B₂X² + st".
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