ECO 321 Spring 2020
Homework 5
Due April 26th 11:59pm
Cut and paste at the end of your submission the R or Stata code you have used in
problems to show your work.
1. Consider the simple regression model:
yi = β0 + β1xi + ui
, for i = 1, . . . , n,
with E(ui
|xi) 6= 0 and let z be a dummy instrumental variable for x, such that we
can write:
xi = π0 + π1zi + vi
,
with E(ui
|zi) = 0 and E(vi
|zi) = 0.
(a) Use the result in HW2, to show that:
πˆ0 = ¯x0, and ˆπ1 = ¯x1 − x¯0,
where ¯x0 and ¯x1 are the sample means of x for z equal to 0 and 1 respectively.
(b) Define:
xˆi = ¯x0 + (¯x1 − x¯0) zi
,
show that, ˆxi − ¯xˆ = (¯x1 − x¯0) (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:
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 ˆx to obtain an estimator of β1. From the standard formula
of the slope estimator for an OLS regression and using the result in (c), show
that:
This estimator is called the Wald estimator.
1
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 file fertility descr.pdf, which is attached
to this homework. Our policy question is to understand the effect of education (how
many years of education to undertake) on the fertitily decision of women (how many
kids to have).
(a) We first estimate the following model using simple OLS regression:
childreni =β0 + β1educi + β2agei + β3age2
i + β4urbani + β5tvi
+β6catholici + β7knowmethi + β8usemethi + ui
, i = 1, . . . , n,
where our focus is on the parameter β1. Suppose at first that all the assumptions
for OLS are satisfied and estimate the model accordingly. Use heteroskedasticity
robust standard errors. Report the value for the estimators of the
regression coefficients, their standard errors, and significance 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
first 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 first step regression of educ on f irsthalf and verify that the coef-
ficient associated to f irsthalf is significant. Use heteroskedasticity robust
standard errors.
(e) Estimate the second step regression and compare the TSLS estimator of β1 with
the OLS estimator. Is the value of βˆ
1 higher/lower than before? Explain.