4.1 — Multivariate OLS Estimators — Practice

Quarto file R Studio Cloud


Answer Key (html) Answer Key (.qmd)

Download and read in (read_csv) the data below.

This data comes from a paper by Makowsky and Strattman (2009) that we will examine later. Even though state law sets a formula for tickets based on how fast a person was driving, police officers in practice often deviate from that formula. This dataset includes information on all traffic stops. An amount for the fine is given only for observations in which the police officer decided to assess a fine. There are a number of variables in this dataset, but the one’s we’ll look at are:

Variable Description
Amount Amount of fine (in dollars) assessed for speeding
Age Age of speeding driver (in years)
MPHover Miles per hour over the speed limit

We want to explore who gets fines, and how much. We’ll come back to the other variables (which are categorical) in this dataset in later lessons.

Question 1

How does the age of a driver affect the amount of the fine? Make a scatterplot of the Amount of the fine (y) and the driver’s Age (x) along with a regression line.

Question 2

Next, we’ll want to find the correlation between Amount and Age. Do this first.

Then notice that it won’t work. This is because there are a lot of NAs (missing data) for Amount (if tried to get the mean() of Amount, it would do the same thing.

You can verify the NAs with:

data %>% # use your named dataframe!
  select(Amount) %>%

# OR
# data %>% count(Amount) # but this has a lot of rows!

In order to run a correlation, we need to drop or ignore all of the NAs. You could filter() the data:

# this would OVERWRITE data
data <- data %>%
  filter(!is.na(Amount)) # remove all NAs

Or, if you don’t want to change your data, the cor() command allows you to set use = "pairwise.complete.obs" as an argument.

Question 3

We want to estimate the following model:

\[\widehat{\text{Amount}_i}= \hat{\beta_0}+\hat{\beta_1}\text{Age}_i\]

Run a regression, and save it as an object. Then get a summary() of it.

Part A

Write out the estimated regression equation.

Part B

What is \(\hat{\beta_0}\) for this model? What does it mean in the context of our question?

Part C

What is \(\hat{\beta_1}\) for this model? What does it mean in the context of our question?

Part D

What is the marginal effect of Age on Amount?

Question 4

Redo question 3 with the broom package. Try out tidy() and glance(). This is just to keep you versatile!

Question 5

How big would the difference in expected fine be for two drivers, one 18 years old and one 40 years old?

Question 6

Now run the regression again, controlling for speed (MPHover).

Part A

Write the new regression equation.

Part B

What is the marginal effect of Age on Amount? What happened to it, compared to Question 3D?

Part C

What is the marginal effect of MPHover on Amount?

Part D

What is \(\hat{\beta_0}\) for our model, and what does it mean in the context of our question?

Part E

What is the adjusted \(\bar{R}^2\)? What does it mean?

Question 7

Now suppose both the 18 year old and the 40 year old each went 10 MPH over the speed limit. How big would the difference in expected fine be for the two drivers?

Question 8

What is the difference in expected fine between two 18 year-olds, one who went 10 MPH over, and one who went 30 MPH over?

Question 9

Use the modelsummary package’s modelsummary() command to make a regression table of your two regressions: the one from question 3, and the one from question 6.

Question 10

Are our two independent variables multicollinear? Do younger people tend to drive faster?

Part A

Get the correlation between Age and MPHover.

Part B

Make a scatterplot of MPHover (y) on Age (x).

Part C

Run an auxiliary regression of MPHover on Age.

Part D

Interpret the coefficient on Age from this regression.

Part E

Look at your regression table in question 10. What happened to the standard error on Age? Why (consider the formula for variance of \(\hat{\beta_1})\)?

Part F

Calculate the Variance Inflation Factor (VIF) using the car package’s vif() command. Run it on your regression object saved from Question 7.

Part G

Calculate the VIF manually, using what you learned in this question.

Question 11

Let’s now think about the omitted variable bias. Suppose the “true” model is the one we ran from Question 7.

Part A

Do you suppose that MPHover fits the two criteria for omitted variable bias?

Part B

Look at the regression we ran in Question 4. Consider this the “omitted” regression, where we left out MPHover. Does our estimate of the marginal effect of Age on Amount overstate or understate the true marginal effect?

Part C

Use the “true” model (Question 7), the “omitted” regression (Question 3), and our “auxiliary” regression (Question 10) to identify each of the following parameters that describe our biased estimate of the marginal effect of Age on Amount:


See the notation I used in class.

Part D

From your answer in part C, how large is the omitted variable bias from leaving out MPHover?

Question 12

Make a coefficient plot of your coefficients from the regression in Question 6. The package modelsummary (which you will need to install and load) has a great command modelplot() to do this on your regression object.