# 3.9 — Logarithmic Regression — R Practice

## Set Up

To minimize confusion, I suggest creating a new R Project (e.g. regression_practice) and storing any data in that folder on your computer.

Alternatively, I have made a project in R Studio Cloud that you can use (and not worry about trading room computer limitations), with the data already inside (you will still need to assign it to an object).

### Question 1

We are returning to the speeding tickets data that we began to explore in R Practice 3.4 on Multivariate Regression and R Practice 3.7 on Dummy Variables & Interaction Effects. Download and read in (read_csv) the data below.

This data again 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
Black Dummy $$=1$$ if driver was black, $$=0$$ if not
Hispanic Dummy $$=1$$ if driver was Hispanic, $$=0$$ if not
Female Dummy $$=1$$ if driver was female, $$=0$$ if not
OutTown Dummy $$=1$$ if driver was not from local town, $$=0$$ if not
OutState Dummy $$=1$$ if driver was not from local state, $$=0$$ if not
StatePol Dummy $$=1$$ if driver was stopped by State Police, $$=0$$ if stopped by other (local)

We again want to explore who gets fines, and how much.

### Question 2

Run a regression of Amount on Age. Write out the estimated regression equation, and interpret the coefficient on Age.

### Question 3

Is the effect of Age on Amount nonlinear? Let’s run a quadratic regression.

#### Part A

Create a new variable for $$Age^2$$. Then run a quadratic regression.

#### Part B

Try running the same regression using the alternate notation: lm(Y~X+I(X^2)). This method allows you to not have to create a new variable first. Do you get the same results?

#### Part C

Write out the estimated regression equation.

#### Part D

Is this model an improvement from the linear model?Check $$R^2$$.

#### Part E

Write an equation for the marginal effect of Age on Amount.

#### Part F

Predict the marginal effect on Amount of being one year older when you are 18. How about when you are 40?

#### Part G

Our quadratic function is a $$U$$-shape. According to the model, at what age is the amount of the fine minimized?

#### Part H

Create a scatterplot between Amount and Age and add a a layer of a linear regression (as always), and an additional layer of your predicted quadratic regression curve. The regression curve, just like any regression line, is a geom_smooth() layer on top of the geom_point() layer. We will need to customize geom_smooth() to geom_smooth(method="lm", formula="y~x+I(x^2) (copy/paste this verbatim)! This is the same as a regression line (method="lm"), but we are modifying the formula to a polynomial of degree 2 (quadratic): $$y=a+bx+cx^2$$.

#### Part I

It’s quite hard to see the quadratic curve with all those data points. Redo another plot and this time, only keep the quadratic stat_smooth() layer and leave out the geom_point() layer. This will only plot the regression curve.

## Question 4

Should we use a higher-order polynomial equation? Run a cubic regression, and determine whether it is necessary.

## Question 5

Run an $$F$$-test to check if a nonlinear model is appropriate. Your null hypothesis is $$H_0: \beta_2=\beta_3=0$$ from the regression in pert (h). The command is linearHypothesis(reg_name, c("var1", "var2")) where reg_name is the name of the lm object you saved your regression in, and var1 and var2 (or more) in quotes are the names of the variables you are testing. This function requires (installing and) loading the “car” package (additional regression tools).

## Question 6

Now let’s take a look at speed (MPHover the speed limit).

### Part A

Creating new variables as necessary, run a linear-log model of Amount on MPHover. Write down the estimated regression equation, and interpret the coefficient on MPHover $$(\hat{\beta_1})$$. Make a scatterplot with the regression line.Hint: The simple geom_smooth(method="lm") is sufficient, so long as you use the right variables on the plot!

### Part B

Creating new variables as necessary, run a log-linear model of Amount on MPHover. Write down the estimated regression equation, and interpret the coefficient on MPHover $$(\hat{\beta_1})$$. Make a scatterplot with the regression line.Hint: The simple geom_smooth(method="lm") is sufficient, so long as you use the right variables on the plot!

### Part C

Creating new variables as necessary, run a log-log model of Amount on MPHover. Write down the estimated regression equation, and interpret the coefficient on MPHover $$(\hat{\beta_1})$$. Make a scatterplot with the regression line.Hint: The simple geom_smooth(method="lm") is sufficient, so long as you use the right variables on the plot!

### Part D

Which of the three log models has the best fit?Hint: Check $$R^2$$

## Question 7

Return to the quadratic model. Run a quadratic regression of Amount on Age, Age$$^2$$, MPHover, and all of the race dummy variables. Test the null hypothesis: “the race of the driver has no effect on Amount”

## Question 8

Now let’s try standardizing variables. Let’s try running a regression of Amount on Age and MPHover, but standardizing each variable.

#### Part A

Create new standardized variables for Amount, Age, and MPHover.Hint: use the scale() function inside of mutate()

#### Part B

Run a regression of standardized Amount on standardized Age and MPHover. Interpret $$\hat{\beta_1}$$ and $$\hat{\beta_2}$$ Which variable has a bigger effect on Amount?

## Question 9

Make a regression output table with huxtable of your regressions in Questions 2, 3, 4, 6a, 6b, 6c, 7 and 8.