Econ7103 Homework 3 Solved

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.

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Stata or Python

1. Suppose that for a home i, you think the underlying relationship between electricity use and predictor variables is yi = eαδdiziγeηi 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.

1. (a)  Show that ln(yi) = α + ln(δ)di + γln(zi) + ηi
2. (b)  What is the intuitive interpretation of δ?
3. (c)  Show that ∆yi = δ−1 y . What is the intuitive interpretation of ∆y ? ∆di δdii ∆di
4. (d)  Show that ∂yi = γ yi . What is the intuitive interpretation of ∂yi when zi is the size of the home ∂zi zi ∂zi

   in square feet?

5. (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 z and d 􏰂dyi and ∆y 􏰃. Bootstrap the 95% confidence intervals of the coefficient i idzi ∆di

   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.

6. (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.

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