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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