homework1

Homework Set1

Xiuli Sun

Due on:November2/3,2016

October19,2016

1.Let kids denote the number of children ever born to a woman,and let educ

denote years of education for the woman.A simple model relating fertility to years of education is

kids=β0+β1educ+u

where u is the unobserved error.

(i)What kinds of factors are contained in u?Are these likely to be correlated

with level of education?

(ii)Will a simple regression analysis uncover the ceteris paribus e?ect of edu-cation on fertility?Explain.

Note:Cateris paribus means other(relevant)factors being equal”.

2.The following table contains the ACT scores and the GP A(grade point average)

for eight college students.Grade point average is based on a four-point scale and has been rounded to one digit after the decimal.

Student GPA ACT

1 2.821

2 3.424

3 3.026

4 3.527

5 3.629

6 3.025

7 2.725

8 3.730

(i)Estimate the relationship between GP A and ACT using OLS;that is,obtain

the intercept and slope estimates in the equation

?

GP A=?β0+?β1ACT

Comment on the direction of the relationship.Does the intercept have a useful interpretation here?Explain.How much higher is the GP A predicted to be if the ACT score is increased by?ve points?

(ii)Compute the?tted values and residuals for each observation,and verify that the residuals(approximately)sum to zero.

(iii)What is the predicted value of GP A when ACT=20?

(iv)How much of the variation in GP A for these eight students is explained by ACT?Explain.

https://www.wendangku.net/doc/5a7113384.html,ing data from1988for houses sold in Andover,Massachusetts,from Kiel

and McClain(1995),the following equation relates housing price(price)to the distance from a recently built garbage incinerator(dist):

?

log(price)=9.40+0.312log(dist)

n=135R2=0.162

(i)Interpret the coe?cient on log(dist).Is the sign of this estimate what you

expect it to be?

(ii)Do you think simple regression provides an unbiased estimator of the ceteris paribus elasticity of price with respect to dist?(Think about the citys decision on where to put the incinerator.)

(iii)What other factors about a house a?ect its price?Might these be correlated with distance from the incinerator?

4.Consider the savings function

sav=β0+β1inc+u,u=√

·e

where e is a random variable with E(e)=0and V ar(e)=σ2

e .Assume that e

is independent of inc.

(i)Show that E(u|inc)=0,so that the key zero conditional mean assumption is satis?ed.[Hint:If e is independent of inc,then E(e|inc)=E(e).]

(ii)Show that V ar (u |inc )=σ2e inc ,so that the homoskedasticity Assumption

is violated.In particular,the variance of sav increases with inc .[Hint:V ar (e |inc )=V ar (e ),if e and inc are independent.]

(iii)Provide a discussion that supports the assumption that the variance of savings increases with family income.

5.Consider the standard simple regression model y =β0+β1x +u under the classical OLS assumptions.The usual OLS estimators ?β0and ?β1are unbiased for their respective population parameters.Let ?β

1be the estimator of β1obtained by assuming the intercept is zero.

(i)Find E (?β1)in terms of the x i ,β0,and β1.Verify that ?β1is unbiased for β1when the population intercept (β0)is zero.Are there other cases where ?β

1is unbiased?

(ii)Find the variance of ?β

1.(Hint:The variance does not depend on β0.)(iii)Show that V ar (?β1)≤V ar (?β1)).[Hint:For any sample of data,∑n i =1x 2i ≥∑n i =1(x i ?ˉx

)2,with strict inequality unless ˉx =0.](iv)Comment on the tradeo?between bias and variance when choosing between ?β

1and ?β1.6.Let ?β

0and ?β1be the OLS intercept and slope estimators,respectively,and let ˉu be the sample average of the errors (not the residuals!).

(i)Show that ?β1can be written as ?β1=β1+∑n i =1w i u i where w i =d i /SST x and d i =x i ?ˉx .(ii)Use part (i),along with ∑n i =1w i =0,to show that ?β1and ˉu

are uncorrelated.[Hint:You are being asked to show that E [(?β1?β1)·ˉu ]=0.]

(iii)Show that ?β0can be written as ?β0=β0+ˉu ?(?β1?β1)ˉx .

(iv)Use parts (ii)and (iii)to show that V ar (?β0)=σ2/n +σ2(ˉx )2/SST x .

https://www.wendangku.net/doc/5a7113384.html,e the data in SLEEP75.RAW from Biddle and Hamermesh (1990)to study

whether there is a tradeo?between the time spent sleeping per week and the time spent in paid work.We could use either variable as the dependent variable.For concreteness,estimate the model

sleep =β0+β1totwrk +u

where sleep is minutes spent sleeping at night per week and totwrk is total minutes worked during the week.

(i)Report your results in equation form along with the number of observations

and R2.What does the intercept in this equation mean?

(ii)If totwrk increases by2hours,by how much is sleep estimated to fall?Do you?nd this to be a large e?ect?

8.We used the data in MEAP93.RAW.Now we want to explore the relationship

between the math pass rate(math10)and spending per student(expend).

(i)Do you think each additional dollar spent has the same e?ect on the pass

rate,or does a diminishing e?ect seem more appropriate?Explain.

(ii)In the population model

math10=β0+β1log(expend)+u

argue thatβ1/10is the percentage point change in math10given a10%increase in expend.

(iii)Use the data in MEAP93.RAW to estimate the model from part(ii).Report the estimated equation in the usual way,including the sample size and R-squared.

(iv)How big is the estimated spending e?ect?Namely,if spending increases by10%,what is the estimated percentage point increase in math10?(v)One might worry that regression analysis can produce?tted values for math10that are greater than100.Why is this not much of a worry in this data set?

https://www.wendangku.net/doc/5a7113384.html,e the data in WAGE2.RAW to estimate a simple regression explaining monthly

salary(wage)in terms of IQ score(IQ).

(i)Find the average salary and average IQ in the sample.What is the sample

standard deviation of IQ?(IQ scores are standardized so that the average in the population is100with a standard deviation equal to15.)

(ii)Estimate a simple regression model where a one-point increase in IQ changes wage by a constant dollar https://www.wendangku.net/doc/5a7113384.html,e this model to?nd the predicted increase in wage for an increase in IQ of15points.Does IQ explain most of the variation in wage?

(iii)Now,estimate a model where each one-point increase in IQ has the same percentage e?ect on wage.If IQ increases by15points,what is the approximate percentage increase in predicted wage?

10.Textbook:

Problems:2.8,2.9,2.10;Computer Exercises:2.2,2.3,2.5,2.6