You have access to imaginary data on an energy-efficiency retrofit program in Atlanta kwh.csv (the same
as the previous homework) and you are interested in whether the program reduced energy use. In your
dataset is the following information: After recruiting the households for the program, you assigned them to
Variable
Description
electricity
kWh of electricity used by the household in the month
sqft
Square feet of the home
retrofit
= 1 if the home received a retrofit
temp
The outdoor average temperature (◦ F) during the month at the home’s location
Table 1: Variable descriptions for homework 3.
treatment and control groups. Treatment homes received the retrofits on the first of the month and control
homes did not have any work done.
1 Stata or Python
1. Suppose that for a home i, you think the underlying relationship between electricity use and predictor
variables is yi = eαδdi ziγeηi where e is Euler’s number or the base of the natural logarithm, di is a
binary variable equal to one if home i received the retrofit program, zi is a vector of the other control
variables, ηi is unobserved error, and {α, δ, γ} are parameters to estimate.
(a) Show that ln(yi) = α + ln(δ)di + γln(zi) + ηi
(b) What is the intuitive interpretation of δ?
(c) Show that
∆y
i
∆
d
i
= δ−
1
δ
di
yi . What is the intuitive interpretation of ∆∆dy
i
?
(d) Show that ∂y
i
∂zi
= γ y
i
zi
. What is the intuitive interpretation of ∂y
i
∂zi
when zi is the size of the home
in square feet?
(e) Estimate the log-transformed equation via ordinary least squares on the transformed parameters
using any algorithm you would like. Save the coefficient estimates and the average marginal effects
estimates of zi and di dy
i
dzi
and ∆∆dy
i . Bootstrap the 95% confidence intervals of the coefficient
estimates and the marginal effects estimates using 1000 sampling replications (note that each
bootstrap replication should perform both the regression and the second stage calculation of the
marginal effect). Display the results in a table with three columns (one for the variable name, one
for the coefficient estimate, and one for the marginal effect estimate). Show the 95% confidence
intervals for each estimate under each number.
(f) Graph the average marginal effects of outdoor temperature and square feet of the home with
bands for their bootstrapped confidence intervals so that they are easy to interpret and compare.
1
