Consider the linear regression model y = XBo + u = where y is T x 1, X is Tx k matrix of constants that are fixed in repeated samples with rank(X) k, Bo is the k x 1 parameter vector, and u~ N(0, IT) where o is an unknown positive constant. Let 3 and 62 denote the OLS estimators of Bo and of respectively. Suppose it is desired to test Ho: R30 = r against HARBO #r, where R and r are respectively a q x k matrix and a q x 1 vector of constants and rank(R) = q. Under Ho, it can be shown that F~ FT-k where (RBT r)[R(X'X)-\R]-1(R®. r) qô7 and FT-k denotes the F distribution with (q, T - k) degrees of freedom. F = - Based on this result a researcher adopts the following decision rule: Reject Ho: Ro = r at the 100a% significance level if: F > Fq,T-k(1- a) where FT-k(1a) is the 100(1- a)th percentile of the Fa,T-k distribution. (a) What would constitute a Type I error in the context of this test? (b) What is the probability of a Type I error associated with this decision rule? (c) Define mathematically the p-value of the test described above. (d) Let k = 5 and 3o, denote the ith element of o. Suppose it is desired to test Ho 0,3 + 30,5 = 2, 30,4 4. Express this null hypothesis in the form Rßor, being sure to define R and r. =
Consider the linear regression model y = XBo + u = where y is T x 1, X is Tx k matrix of constants that are fixed in repeated samples with rank(X) k, Bo is the k x 1 parameter vector, and u~ N(0, IT) where o is an unknown positive constant. Let 3 and 62 denote the OLS estimators of Bo and of respectively. Suppose it is desired to test Ho: R30 = r against HARBO #r, where R and r are respectively a q x k matrix and a q x 1 vector of constants and rank(R) = q. Under Ho, it can be shown that F~ FT-k where (RBT r)[R(X'X)-\R]-1(R®. r) qô7 and FT-k denotes the F distribution with (q, T - k) degrees of freedom. F = - Based on this result a researcher adopts the following decision rule: Reject Ho: Ro = r at the 100a% significance level if: F > Fq,T-k(1- a) where FT-k(1a) is the 100(1- a)th percentile of the Fa,T-k distribution. (a) What would constitute a Type I error in the context of this test? (b) What is the probability of a Type I error associated with this decision rule? (c) Define mathematically the p-value of the test described above. (d) Let k = 5 and 3o, denote the ith element of o. Suppose it is desired to test Ho 0,3 + 30,5 = 2, 30,4 4. Express this null hypothesis in the form Rßor, being sure to define R and r. =
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
![Consider the linear regression model
y = XBo + u
~
where y is T x 1, X is T × k matrix of constants that are fixed in repeated samples
with rank(X) = k, Bo is the kx 1 parameter vector, and u N(0, oIT) where
o is an unknown positive constant. Let 3 and 2 denote the OLS estimators
of Bo and of respectively. Suppose it is desired to test Ho: Ro r against
HARBO #r, where R and r are respectively a q × k matrix and a q × 1 vector of
constants and rank(R) q. Under Ho, it can be shown that F ~ F₁,T-k where
(R®r – r)[R(X′X)-R]-1(R®
qô7/
and FT-k denotes the F distribution with (q, T - k) degrees of freedom.
F =
-
=
r)
=
Based on this result a researcher adopts the following decision rule:
Reject Ho Ro = r at the 100a% significance level if: F > FqT-k(1 − a) where
Fa,T-k(1a) is the 100(1- a)th percentile of the Fa,T-k distribution.
(a) What would constitute a Type I error in the context of this test?
(b) What is the probability of a Type I error associated with this decision rule?
(c) Define mathematically the p-value of the test described above.
=
(d) Let k
5 and 3o, denote the ith element of o. Suppose it is desired to
test Ho 0,3 +0,5 2, 30.4 4. Express this null hypothesis in the form
R30 = r, being sure to define R and r.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6b67d426-4dcd-41ea-a5d4-47ab2d2caecd%2Fa80c1f4a-3919-43e1-8eb0-a22c729d8c10%2Fbiopnto_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Consider the linear regression model
y = XBo + u
~
where y is T x 1, X is T × k matrix of constants that are fixed in repeated samples
with rank(X) = k, Bo is the kx 1 parameter vector, and u N(0, oIT) where
o is an unknown positive constant. Let 3 and 2 denote the OLS estimators
of Bo and of respectively. Suppose it is desired to test Ho: Ro r against
HARBO #r, where R and r are respectively a q × k matrix and a q × 1 vector of
constants and rank(R) q. Under Ho, it can be shown that F ~ F₁,T-k where
(R®r – r)[R(X′X)-R]-1(R®
qô7/
and FT-k denotes the F distribution with (q, T - k) degrees of freedom.
F =
-
=
r)
=
Based on this result a researcher adopts the following decision rule:
Reject Ho Ro = r at the 100a% significance level if: F > FqT-k(1 − a) where
Fa,T-k(1a) is the 100(1- a)th percentile of the Fa,T-k distribution.
(a) What would constitute a Type I error in the context of this test?
(b) What is the probability of a Type I error associated with this decision rule?
(c) Define mathematically the p-value of the test described above.
=
(d) Let k
5 and 3o, denote the ith element of o. Suppose it is desired to
test Ho 0,3 +0,5 2, 30.4 4. Express this null hypothesis in the form
R30 = r, being sure to define R and r.
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