#### Question Details

[answered] 11/22/2016 fmwww.edu/ec-p/data/wooldridge/fertil2.des FERTI

Please help me to solve these two questions and write the code for the second question

Homework 5

Cut and paste at the end of your submission the R code you have used in problem

2 to show your work.

1. Consider the simple regression model:

yi = 0 + 1xi + ui; for i = 1; : : : ; n;

with E(uijxi) 6= 0 and let z be a dummy instrumental variable for x, such that we

can write:

xi = 0 + 1zi + vi;

with E(uijzi) = 0 and E(vijzi) = 0.

(a) Use the result in HW2, to show that:

^0 = x0; and ^1 = x1 ???? x0;

where x0 and x1 are the sample means of x for z equal to 0 and 1 respectively.

(b) Dene:

^xi = x0 + (x1 ???? x0) zi;

show that, ^xi ???? ^

x = (x1 ???? x0) (zi ???? z), where z is the sample mean of z.

(c) Denote by n0, the number of observations for which zi = 0 and by n1 the number

of observations for which zi = 1. Show that:

Xn

i=1

(zi ???? z)2 =

n1

n

(n ???? n1) ;

and that:

Xn

i=1

(zi ???? z)(yi ???? y) =

n1

n

(n ???? n1) (y1 ???? y0) ;

where y0 and y1 are the sample means of y for z equal to 0 and 1 respectively.

(Hint: Use the fact that n = n1 + n0, and that z = n1=n).

(d) Now we regress y on ^x to obtain an estimator of 1. From the standard formula

of the slope estimator for an OLS regression and using the result in (c), show

that:

^ 1 =

y1 ???? y0

x1 ???? x0

:

This estimator is called the Wald estimator.

2. The data set fertility.csv contains information about n = 4286 women in Botswana

during 1988. This information includes number of children, years of education, age,

and religious and economic status variables. Additional information about each of the

variables in this data-set is available in the le fertility descr.pdf, which is attached

to this homework. Our policy question is to understand the eect of education (how

many years of education to undertake) on the fertitily decision of women (how many

kids to have).

(a) We rst estimate the following model using simple OLS regression:

childreni =0 + 1educi + 2agei + 3age2i

+ 4urbani + 5tvi

+6catholici + 7knowmethi + 8usemethi + ui; i = 1; : : : ; n;

where our focus is on the parameter 1. Suppose at rst that all the assumptions

for OLS are satised and estimate the model accordingly. Use heteroskedas-

ticity robust standard errors. Report the value for the estimators of the

regression coecients, their standard errors, and signicance level.

(b) Do you think that Cov(educi; ui) = 0? Explain.

(c) frsthalf is a dummy variable equal to one if the woman was born during the

rst six months of the year, and zero otherwise. Explain what is required for

frsthalf to be a valid instrumental variable for educ. Do these assumptions

seem reasonable?

(d) Estimate the rst step regression of educ on firsthalf and verify that the coef-

cient associated to firsthalf is signicant. Use heteroskedasticity robust

standard errors.

(e) Estimate the second step regression and compare the TSLS estimator of 1 with

the OLS estimator. Is the value of ^ 1 higher/lower than before? Explain.

11/22/2016 fmwww.bc.edu/ec-p/data/wooldridge/fertil2.des FERTIL2.DES

mnthborn yearborn age

electric

ceb

agefbrth children knowmeth

idlnchld heduc

agesq

urban

catholic frsthalf educ0

evermarr

Obs:

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

21.

22.

23.

24.

25.

26.

27. radio

usemeth

urbeduc tv

monthfm

spirit bicycle

yearfm

protest educ

agefm 4361 mnthborn

yearborn

age

electric

radio

tv

bicycle

educ

ceb

agefbrth

children

knowmeth

usemeth

monthfm

yearfm

agefm

idlnchld

heduc

agesq

urban

urbeduc

spirit

protest

catholic

frsthalf

educ0

evermarr http://fmwww.bc.edu/ec-p/data/wooldridge/fertil2.des month woman born

year woman born

age in years

=1 if has electricity

=1 if has radio

=1 if has tv

=1 if has bicycle

years of education

children ever born

age at first birth

number of living children

=1 if know about birth control

=1 if ever use birth control

month of first marriage

year of first marriage

age at first marriage

'ideal' number of children

husband's years of education

age^2

=1 if live in urban area

urban*educ

=1 if religion == spirit

=1 if religion == protestant

=1 if religion == catholic

=1 if mnthborn <= 6

=1 if educ == 0

=1 if ever married 1/1

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