Problem 1. [SW 14.6] In Exercise 14.5(b) (which is copied below for your reference), again, Liam does not know the value of , but has access to a random sample Y₁, Yio- Liam has decided to predict Y₁₁ using Y/2 instead of Y. Note: This is an out-of-sample prediction scenario: Y₁₁ is not part of the estimation sample Y₁Y₁o. It can be a useful exercise to figure out where in the calculation it makes a difference. (a) Compute the bias of the prediction. (b) Compute the mean of the prediction error. (e) Compute the variance of the prediction error. (d) Compute the MSPE of the prediction. (e) Does Liam's prediction Y/2 produce a prediction with a lower MSPE than Olivia's Y prediction? (f) Suppose μ = 10 (instead of μ=2). Does Liam's prediction Y/2 produce a prediction with a lower MSPE than Olivia's Y prediction? (g) In a realistic setting, the value of μ is unknown. What advice would you give someone who is deciding between using Y and Y/2?
Problem 1. [SW 14.6] In Exercise 14.5(b) (which is copied below for your reference), again, Liam does not know the value of , but has access to a random sample Y₁, Yio- Liam has decided to predict Y₁₁ using Y/2 instead of Y. Note: This is an out-of-sample prediction scenario: Y₁₁ is not part of the estimation sample Y₁Y₁o. It can be a useful exercise to figure out where in the calculation it makes a difference. (a) Compute the bias of the prediction. (b) Compute the mean of the prediction error. (e) Compute the variance of the prediction error. (d) Compute the MSPE of the prediction. (e) Does Liam's prediction Y/2 produce a prediction with a lower MSPE than Olivia's Y prediction? (f) Suppose μ = 10 (instead of μ=2). Does Liam's prediction Y/2 produce a prediction with a lower MSPE than Olivia's Y prediction? (g) In a realistic setting, the value of μ is unknown. What advice would you give someone who is deciding between using Y and Y/2?
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter1: Functions
Section1.2: The Least Square Line
Problem 5E
Related questions
Question
Parts, E, F, and G.
![Problem 1. [SW 14.6] In Exercise 14.5(b) (which is copied below for your reference), again, Liam
does not know the value of , but has access to a random sample Y₁,...,Y₁o. Liam has decided
to predict Y₁1 using Y/2 instead of Y. Note: This is an out-of-sample prediction scenario: Y₁1 is
not part of the estimation sample Y₁Y1o. It can be a useful exercise to figure out where in the
calculation it makes a difference.
(a) Compute the bias of the prediction.
(b) Compute the mean of the prediction error.
(c) Compute the variance of the prediction error.
(d) Compute the MSPE of the prediction.
(e) Does Liam's prediction Y/2 produce a prediction with a lower MSPE than Olivia's Y prediction?
(f) Suppose μ = 10 (instead of μ = 2). Does Liam's prediction Y/2 produce a prediction with a
lower MSPE than Olivia's Y prediction?
(g) In a realistic setting, the value of 4 is unknown. What advice would you give someone who is
deciding between using Y and Y/2?
For reference, here is Exercise 14.5. (It is not part of this problem set, but it may be helpful
for you to do this ezercise before attempting 14.6.) Y is a random variable with mean = 2 and
variance o² = 25.
(a) Suppose Emma knows the value of u.
(i) What is the best (lowest MSPE) prediction of Y that Emma can make? That is, what is
the oracle prediction of Y?
(ii) What is the MSPE of Emma's prediction?
(b) Suppose Olivia does not know the value of but has access to a random sample of size n = 10
from the same population (represented by variables Y₁., Y₁o). Let Y denote the sample mean
from this random sample. Olivia wants to predict Y, which is not part of her sample Y₁.... Y10-
Let us denote Y as Y₁ to emphasize this. Olivia has decided to predict the value of Y₁ using
Y.
(i) Show that Olivia's prediction error can be decomposed as Y₁₁-Y = (Y₁₁-μ) - (Y-μ),
where Y₁1-μ is the prediction error of the oracle predictor and u-Y is the error associated
with using Y as an estimate of μ.
(ii) Show that (Y₁1-μ) has a mean of 0, that (Y-μ) has a mean of 0,
and that Y₁1 - Y has a mean of 0.
(iii) Show that Y₁1 - μ and Y - μ are uncorrelated.
(iv) Show that the MSPE of Y is](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F61ff295b-afd4-43c3-8ed2-2cc17b9c2249%2F57df3986-e6b9-4123-bc49-ff56e5aa7a7a%2Fie9rz0p_processed.png&w=3840&q=75)
Transcribed Image Text:Problem 1. [SW 14.6] In Exercise 14.5(b) (which is copied below for your reference), again, Liam
does not know the value of , but has access to a random sample Y₁,...,Y₁o. Liam has decided
to predict Y₁1 using Y/2 instead of Y. Note: This is an out-of-sample prediction scenario: Y₁1 is
not part of the estimation sample Y₁Y1o. It can be a useful exercise to figure out where in the
calculation it makes a difference.
(a) Compute the bias of the prediction.
(b) Compute the mean of the prediction error.
(c) Compute the variance of the prediction error.
(d) Compute the MSPE of the prediction.
(e) Does Liam's prediction Y/2 produce a prediction with a lower MSPE than Olivia's Y prediction?
(f) Suppose μ = 10 (instead of μ = 2). Does Liam's prediction Y/2 produce a prediction with a
lower MSPE than Olivia's Y prediction?
(g) In a realistic setting, the value of 4 is unknown. What advice would you give someone who is
deciding between using Y and Y/2?
For reference, here is Exercise 14.5. (It is not part of this problem set, but it may be helpful
for you to do this ezercise before attempting 14.6.) Y is a random variable with mean = 2 and
variance o² = 25.
(a) Suppose Emma knows the value of u.
(i) What is the best (lowest MSPE) prediction of Y that Emma can make? That is, what is
the oracle prediction of Y?
(ii) What is the MSPE of Emma's prediction?
(b) Suppose Olivia does not know the value of but has access to a random sample of size n = 10
from the same population (represented by variables Y₁., Y₁o). Let Y denote the sample mean
from this random sample. Olivia wants to predict Y, which is not part of her sample Y₁.... Y10-
Let us denote Y as Y₁ to emphasize this. Olivia has decided to predict the value of Y₁ using
Y.
(i) Show that Olivia's prediction error can be decomposed as Y₁₁-Y = (Y₁₁-μ) - (Y-μ),
where Y₁1-μ is the prediction error of the oracle predictor and u-Y is the error associated
with using Y as an estimate of μ.
(ii) Show that (Y₁1-μ) has a mean of 0, that (Y-μ) has a mean of 0,
and that Y₁1 - Y has a mean of 0.
(iii) Show that Y₁1 - μ and Y - μ are uncorrelated.
(iv) Show that the MSPE of Y is
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 4 steps
Recommended textbooks for you
Calculus For The Life Sciences
Calculus
ISBN:
9780321964038
Author:
GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:
Pearson Addison Wesley,
Glencoe Algebra 1, Student Edition, 9780079039897…
Algebra
ISBN:
9780079039897
Author:
Carter
Publisher:
McGraw Hill
Calculus For The Life Sciences
Calculus
ISBN:
9780321964038
Author:
GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:
Pearson Addison Wesley,
Glencoe Algebra 1, Student Edition, 9780079039897…
Algebra
ISBN:
9780079039897
Author:
Carter
Publisher:
McGraw Hill