Question Details

Homework 5
Cut and paste at the end of your submission the R code you have used in problem
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.

usemeth

urbeduc tv

monthfm

spirit bicycle

yearfm

protest educ

agefm 4361 mnthborn

yearborn

age

electric

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 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 &lt;= 6

=1 if educ == 0

=1 if ever married 1/1

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