Exercise - Selection

Why to select one model or another?

Date

October 15, 2024

Instructions

This page gives both instructions and some interactive snippets. You will have to run most of your code on RStudio, on your own computer.

First, let’s load useful packages. We need the tidyverse package but also to install the wooldridge package (it contains the dataset we are interested in).

Code 
# install.packages("wooldridge")

library(tidyverse)
library(wooldridge)

Exploring the Data Set

The Data

Load the data by calling wage1 (after loading the package using library(wooldridge)).

Print the dataset. It prints in a different format than what we are used to. It is stored in a dataframe and not the tibbles we are used to. Let’s create a tibble version of this data set, using the function as_tibble(); run the following line of code:

wage <- as_tibble(wage1)

What does each variable represent? (use the help() function). How many observations are there in this data set? .

“Research” Question

Objective

We will study how to better explain/predict hourly wage, based on the set of covariates available in the data set

We want to build a model that accurately describes the variation in wage. Why can understanding how wage evolves with various covariates matter for economics research?

Summarising the Data

Before anything, let’s quickly explore the data. To have an overview of the dataset, we can use the function glimpse(wage).

What type of variable does profocc seem to be? Confirm this by exploring the documentation for the data set (the help)

The summary() function can also provide interesting information on the dataset. Call this function. What is the median number of years of educations in the data set?

What is the proportion of married individuals in this data set? 1

Distribution of Wage

We want to study the variation in wage, before anything, let’s plot its distribution as we did in the previous exercise!

# plot distrib wage

Do you notice anything weird or erroneous?

Relationship between wage and education

We can then explore its correlation with education. Which ggplot function should we use? 

# plot relationship between wage and education

What do we notice?

In addition, larger levels of education seem to be associated with hourly wages.

We may also be interested in disparities by gender. We can plot the same graph distinguishing by gender. To do so, add color = female within the aes() function: aes(x = ..., y = ..., color = female).

ggplot treats female as a continuous variable (an integer). Instead, this variable is binary. To transform it as categorical variable, we can use the as_factor() function: aes(x = ..., y = ..., color = as_factor(female)).

For a given level of education, the wage of female individuals seems to be than the wage of non-female ones.

Does the relationship between wage and education seem different for female and non-female?

Relationship Between Wage and Other Variables

Go on with this data exploration: look at the correlation between wage and experience, tenure and their squares. Can you see a clear relationship between the square of experience and wage?

Does that mean that there is no statistical relationship between these two variables?

How would you expect this relationship to be Why?

Note

You can do a lot more data exploration using graphs but also tables. We will however now turn to modeling.

Quadratics

Let’s first explore the relationship between hourly wage, level of education, experience and experience squared. Regress wage on these variables, as we did last weeks.

Tip

The variable expersq is already in the data set; you can call it directly. If you use exper^2 in your regression, what happens? To remedy this, you need to tell R that you are actually creating a new variable, using the function I(exper^2): formula = ... ~ ... + I(exper^2).

# regression

What is the point estimate for educ? . How would you interpret this coefficient?

What is the point estimate for exper? . How would you interpret this coefficient?

For how many years of experience could you easily interpret this coefficient? . The presence of which variable in the analysis can explain this difference?

Comparing two individuals with the same level of education, we expect one with 3 years of experience to earn on average more per hour than one with 2 years of experience. What is the general formula to compute this quantity?

\[\frac{\partial \widehat{wage}}{\partial exp} = \hat{\beta_2} + 2 \hat{\beta_3} exp \]

How do you interpret the sign of expersq?

There are decreasing marginal returns to experience.

Indicators

We will now explore how the relationship between level of education and wage varies by marital status. If we add a married dummy, what is the coefficient for educ?

How do we interpret it?

For a given level of education, on average, we expect married individual to earn dollars more/less per hour than non married one.

Does the relationship between level of education and wage varies by marital status?

If you plot the regression lines for both groups, they will be

Interactions

To make this relationship vary by marital status, we can interact the variable of interest, educ, with married. To interact two variables, we can use :: eg y ~ x:z.

Which variables do we need to include in the model?

educ, married, educ:married, exper and expersq

Tip

Try only having educ*married, exper and expersq in your formula. What happens to your results? What does the * do?

Comparing two married individuals with the same level of experience, we expect one with one more year of education to earn on average more per hour than the other.

Comparing two non-married individuals with the same level of experience, we expect one with one more year of education to earn on average more per hour than the other.

Plot the relationship between wage and education. Add a regression line. To do so, we need to use the function.

#plot

Let’s now plot separate lines for married vs non-married individuals. To do so, we need to add the color = married aesthetic (aes(..., color = married)).

Model selection

Let’s now dive into model selection.

\(R^2\)

What is the \(R^2\) of the regression of wage on the level of education, the level of experience and its square? 2

What is the adjusted\(-R^2\)? . Is the difference with the \(R^2\) large? Why so?

How would you interpret the \(R^2\) (or the adjusted\(-R^2\))?

Add nonwhite, tenure, female, married, numdep to the model. Does the this model explains a larger proportion of the variation in wage?

Does the coefficient for educ vary? Why?

Is the interpretation of this coefficient the same as before?

Misspecification

Let’s get back to our simple model of wage on the level of education, the level of experience and its square. Does including female affect the coefficient for educ? Why?

In the first specification, female is an . What other variables in the data set could have similar effects? Should we include them in the model?

Should we include any variables in the model? Why?

Variable selection involves arbitrating this bias-variance trade-off.

Information Criteria

To select which variables to include and not to include, a basic and usual approach is to use Information Criteria. We will choose the model that minimizes an information criterion, either the Akaike Information Criterion (AIC) or the Bayesian Information Criterion (BIC). The associated functions are AIC and BIC.

Write several models including some of the following variables, in addition of educ: exper, expersq, married, tenure, female, numdep and nonwhite.

reg_1 <- lm(...)
reg_2 <- lm(...)
reg_3 <- lm(...)
reg_4 <- lm(...)

AIC(reg_1, reg_2, reg_3, reg_4)

BIC(reg_1, reg_2, reg_3, reg_4)

Which model would you choose?

In particular, estimate the model including all the covariates, compute its AIC. Then do the same thing but excluding nonwhite. Which model would you choose?3

Footnotes

  1. This information is directly accessible through summary(wage)↩︎

  2. Run the regression and print its summary.↩︎

  3. Here the line between the two is pretty fine and the difference not particularly stark.↩︎