of y; on of y, on log(x)? Let Bo and B, be the OLS intercept and slope estimators, respectively, and let u be the sample average of the errors (not the residuals!). (i) wu₁, where w; = d;/SST, and d¡ = x¡ — x. w; = 0, to show that B, and u are uncorrelated. [Hint: You are Show that B, can be written as 3₁ = B₁ + Use part (i), along with being asked to show that E[(B₁-B₁) ū] = 0.] Show that B, can be written as Bo=B₁ + ū- (B₁-B₁)ñ. (iv) Use parts (ii) and (iii) to show that Var(B) = o²/n + o²(x)²/SST¸. (iii)
of y; on of y, on log(x)? Let Bo and B, be the OLS intercept and slope estimators, respectively, and let u be the sample average of the errors (not the residuals!). (i) wu₁, where w; = d;/SST, and d¡ = x¡ — x. w; = 0, to show that B, and u are uncorrelated. [Hint: You are Show that B, can be written as 3₁ = B₁ + Use part (i), along with being asked to show that E[(B₁-B₁) ū] = 0.] Show that B, can be written as Bo=B₁ + ū- (B₁-B₁)ñ. (iv) Use parts (ii) and (iii) to show that Var(B) = o²/n + o²(x)²/SST¸. (iii)
MATLAB: An Introduction with Applications
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Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
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Problem 1P
Related questions
Question
Please answer question 10
![of y on
of y, on log(x;)?
10 Let B and B, be the OLS intercept and slope estimators, respectively, and let u be the sample average
of the errors (not the residuals!).
(i)
Show that B, can be written as 3₁ = B₁ + ₁wu, where w; = d;/SST, and d¡ = x₁ - x.
(ii)
w; = 0, to show that B, and ū are uncorrelated. [Hint: You are
(iii)
(iv)
(v)
Use part (i), along with
being asked to show that E[(B₁-B₁) ·ū] = 0.]
Show that Bo can be written as Bo = Bo+ū- (B₁-B₁).
Use parts (ii) and (iii) to show that Var(B) = o²/n + o²(x)²/SST.
Do the algebra to simplify the expression in part (iv) to equation (2.58). (
[Hint: SST,/n = n¯¹Σ₁₁x²(x)²]
11 Suppose you are interested in estimating the effect of hours spent in an SAT preparation course
(hours) on total SAT score (sat). The population is all college-bound high school seniors for a par-
ticular year.
(i)
Suppose you are given a grant to run a controlled experiment. Explain how you would structure
the experiment in order to estimate the causal effect of hours on sat.
(ii)
Consider the more realistic case where students choose how much time to spend in a prepara-
tion course, and you can only randomly sample sat and hours from the population. Write the
population model as
sat Bo + B₁hours + u
where, as usual in a model with an intercept, we can assume E(u) = 0. List at least two factors
contained in u. Are these likely to have positive or negative correlation with hours?
(iii) In the equation from part (ii), what should be the sign of B, if the preparation course is effective?
(iv) In the equation from part (ii), what is the interpretation of Bo?
12 Consider the problem described at the end of Section 2-6, running a regression and only estimating an
intercept.
(i)
Given a sample {y: i = 1, 2, ..., n}, let Bo be the solution to](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fff6f9a4d-2084-42ea-abf6-3325fa2ca826%2F66689a4e-395a-4d72-a8e9-07192df526c0%2Fco0vdxa_processed.jpeg&w=3840&q=75)
Transcribed Image Text:of y on
of y, on log(x;)?
10 Let B and B, be the OLS intercept and slope estimators, respectively, and let u be the sample average
of the errors (not the residuals!).
(i)
Show that B, can be written as 3₁ = B₁ + ₁wu, where w; = d;/SST, and d¡ = x₁ - x.
(ii)
w; = 0, to show that B, and ū are uncorrelated. [Hint: You are
(iii)
(iv)
(v)
Use part (i), along with
being asked to show that E[(B₁-B₁) ·ū] = 0.]
Show that Bo can be written as Bo = Bo+ū- (B₁-B₁).
Use parts (ii) and (iii) to show that Var(B) = o²/n + o²(x)²/SST.
Do the algebra to simplify the expression in part (iv) to equation (2.58). (
[Hint: SST,/n = n¯¹Σ₁₁x²(x)²]
11 Suppose you are interested in estimating the effect of hours spent in an SAT preparation course
(hours) on total SAT score (sat). The population is all college-bound high school seniors for a par-
ticular year.
(i)
Suppose you are given a grant to run a controlled experiment. Explain how you would structure
the experiment in order to estimate the causal effect of hours on sat.
(ii)
Consider the more realistic case where students choose how much time to spend in a prepara-
tion course, and you can only randomly sample sat and hours from the population. Write the
population model as
sat Bo + B₁hours + u
where, as usual in a model with an intercept, we can assume E(u) = 0. List at least two factors
contained in u. Are these likely to have positive or negative correlation with hours?
(iii) In the equation from part (ii), what should be the sign of B, if the preparation course is effective?
(iv) In the equation from part (ii), what is the interpretation of Bo?
12 Consider the problem described at the end of Section 2-6, running a regression and only estimating an
intercept.
(i)
Given a sample {y: i = 1, 2, ..., n}, let Bo be the solution to
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