ECO 321Homework 

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:

π?0 = ˉx0, and ?π1 = ˉx1 ? xˉ0,

where ˉx0 and ˉx1 are the sample means of x for z equal to 0 and 1 respectively.

(b) Define:

x?i = ˉx0 + (ˉx1 ? xˉ0) zi

,

show that, ?xi ? ˉx? = (ˉx1 ? xˉ0) (zi ? zˉ), where ˉz 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 ˉy0 and ˉy1 are the sample means of y for z equal to 0 and 1 respectively.

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

(d) Now we regress y on ?x 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 β?

1 higher/lower than before? Explain.