Q1.3 2 Points
Suppose that a computer glitch causes you to lose all the observations for which household size is greater than 6. Will OLS still provide an unbiased estimator of if you estimate Model 1 using the remaining data? Explain.
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Q1.4 2 Points
Suppose that, contrary to our assumption above, our data was not generated as a random sample from the population. Instead, data was collected by issuing a population-wide voluntary survey that consumption < i 10 β 1
β 1
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households could choose to respond to. Will OLS still provide an unbiased estimator of if you estimate Model 1? Explain.
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Q2 Electricity Consumption and Temperature 10 Points
You are interested in whether the daily average temperature is a good predictor of electricty consumption. Specifically, you would like to estimate the following population regression model:
Model 1:
where is daily electricity consumption in day and is the average daily temperature.
You have data on the daily amount of electricity consumed over the summer in a single randomly chosen house in California. For each day, you have the following variables for the dates May 1, 2012 to Sep 30, 2012:
. sasdate: Date . kwh_daily: Daily electricity consumption (in kilowatt hours) . temp_avg: Average daily temperature Download the data set ARE106-Final-F2021-Q2.RData from Canvas and load it in RStudio using the following commands (make sure the data are in the working directory, or provide a path to the directory in which the data are stored):
load(“ARE106-Final-F2021-Q2.RData”)
β 1
log(elec ) = t β + 0 β log(temp )+ 1 t ε , t
elec t t temp t
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For this question, you will need to load (and install if you haven’t already) the following packages:
library(lmtest) # Testing Linear Regression Models library(dynlm) # Dynamic Linear Regression library(sandwich) # Robust Covariance Matrix Estimators
Before you start, RStudio has some nice packages that are designed specifically to work with time series data. To reap the benefits of these packages, we need to convert our data to two time-series objects using the following command:
elec <- ts(D[,”kwh_daily”]) temp <- ts(D[,”temp_avg”])
Q2.1 1 Point
Do you expect in Model 1 to be positive or negative? Explain.
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Q2.2 1 Point
Estimate Model 1 using ordinary least squares (OLS). Interpret your estimate of .
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β 1
β 1
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Q2.3 1 Point
Compute and interpret a 95% confidence interval for .
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Q2.4 1 Point
Suppose we have a failure of CR3. Describe what this means and what the implications are for our OLS estimate of .
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Q2.5 1.5 Points
Test the null hypothesis that there is no autocorrelation. What do you conclude? Explain how you arrived at your conclusion.
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β 1
β 1
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Q2.6 1.5 Points
Perform an ex-post correction of the estimated standard errors for your estimate of . How does this change your 95% confidence interval for ? Explain your answer.
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Q2.7 1 Point
Explain what Feasible Generalized Least Squares (FGLS) is and how it attempts to address autocorrelation.
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Q2.8 1 Point
Assume that the population regression errors follow a first-order autoregressive process. Estimate Model 1 using FGLS. Explain how you estimated Model 1.
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β 1
β 1
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Q2.9 1 Point
Do you think you’ve gotten the model right with FGLS? Explain.
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Q3 Estimating a Demand Curve 8 Points
You are interested in estimating the slope of the demand curve in following demand equation:
Demand Equation: ,
where .
denotes quantity demanded, denotes price, denotes observed factors that shift the demand curve, and denotes unobserved factors that shift the demand curve.
Suppose that the quantities and prices we are observe are determined such that the quantity supplied and demanded are equal. In other words, observed quantities are such that , where the quantity supplied is determined by the following equation:
Supply Equation: ,
where .
denotes quantity demanded, denotes price, denotes observed factors that shift the supply curve, and denotes unobserved factors that shift the supply curve.
β 1
Q = Di β P + 1 i β XD + 2 i ϵ Di
Cov(P ,ϵ ) = i Di 0
Q Di P i XD i ϵ Di
Q = Di Q = Si Q i
Q = Si α P + 1 i α XS + 2 i ϵ Si
Cov(P ,ϵ ) = i Si 0
Q Si P i XS i ϵ Si
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Using observed quantities and prices , you estimate the following population regression model:
Estimation Equation: .
Q3.1 2 Points
Explain intuitively why assumption CR5 fails for your Estimation Equation.
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Q3.2 2 Points
Use an equation to demonstrate that price is endogenously determined with quantity .
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Q3.3 2 Points
Suppose you decide to use an Instrumental Variables (IV) approach to identify the slope of the demand curve. What is a possible instrument? Explain.
Q i P i
Q = i β P + 1 i β XD + 2 i ϵ i
P i
Q i
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Q3.4 2 Points
Explain the procedure you would use to estimate your Estimation Equation using an IV approach. For example, what are your first- and second-stage regression equations?
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Q4 Returns to Education 8 Points
You are interested in estimating the causal effect of education on wages. The sample you will be working with consists of 1980 Census data on wages for 10,000 white and black men born between 1930- 1939 in the United States. The variables that we will use for our analysis are:
. wage: weekly wage, in 1980 USD . educ: years of schooling . age: age . married: =1 if individual is married . race: =1 if individual is black . QTR1: =1 if born in the first quarter (Jan-Mar) of the year . QTR2: =1 if born in the second quarter (Apr-Jun) of the year . QTR3: =1 if born in the third quarter (Jul-Sep) of the year No file chosenChoose Files
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. QTR4: =1 if born in the fourth quarter (Oct-Dec) of the year For this question, you will need to load (and install if you haven’t already) the following package:
library(AER) # Applied Econometrics with R
You can load the data in RStudio using the following commands:
load(“ARE106-Final-F2021-Q4.RData”)
Q4.1 1 Point
Consider the following regression model:
Model 1:
Estimate Model 1 using OLS and interpret your estimate of .
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Q4.2 2 Points
Do you think CR5 holds for Model 1? Explain.
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log(wage ) = i β + 0 β educ + 1 i ε . i
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Q4.3 2 Points
Suppose you are worried that Model 1 fails CR5. As a solution, you decide to control for some omitted variables in Model 1 by estimating the following regression model:
Model 2:
Estimate Model 2 using OLS. Is your estimate of in Model 2 different from your estimate of in Model 1? Explain.
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Q4.4 2 Points
Suppose you are still concerned that education is endogenous, even after adding additional control variables. In a famous study, economists Joshua Angrist and Alan Krueger proposed that the quarter of year in which a person is born is a good instrument for education. Their reasoning is as follows: individuals born in the beginning of the calendar year start school in the fall at an older age. Compulsory schooling laws generally require students to remain in school until their sixteenth or seventeenth birthday. Thus, individuals born in earlier in the year are able to drop out of school after completing less schooling than individuals born near the end of the year.
Estimate Model 2 using an instrumental variables approach, with quarter-of-birth dummy variables acting as an instrument for educ. Does your estimate of differ from your earlier estimates of ? Explain.
log(wage ) = i β + 0 β educ + 1 i β age + 2 i β age + 3 i 2 β married + 4 i β race + 5 i ε . i
β 1
β 1
β 1 β 1
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Q4.5 1 Point
Do you think your instrumental variables approach successfully estimates the causal effect of education on wages? Justify your answer.
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Q5 Attestation 1 Point
The answers above are my own answers. I did not coordinate with anyone else while completing this final exam.
Do you attest to this statement?
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Q6 Code 0 Points
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Upload any files that correspond to your R code and any outputs/figures associated with your answers above. It’s easiest to copy and paste your code and outputs from the RStudio console and paste them into a .txt file. Any figures can be saved and uploaded as .png files. Make sure to organize and label your code so that it is easy to follow which code and output corresponds to a particular question. Homework assignments that do not include their code and R output will be given a score of zero!
