Categorical variables and interactions

EC 320, Set 09

Andrew Dickinson

Spring 2024

Prologue

Housekeeping

PS04:

• Due tomorrow @11:59pm

PS05:

• Posted yesterday
• Due Thursday, 05/30

Reading: (up to this point)

ItE: R, 1, 2, 3, 4, 5

MM: 1, 2

Categorical variables

Categorical Variables

Goal Make quantitative statements about qualitative information.

• e.g., race, gender, being employed, living in Oregon, etc.

Approach. Construct binary variables.

• a.k.a. dummy variables or indicator variables.
• Value equals 1 if observation is in the category or 0 if otherwise.

Regression implications.

1. Change the interpretation of the intercept.

2. Change the interpretations of the slope parameters.

Continuous Variables

Consider the relationship

\[ \text{Pay}_i = \beta_0 + \beta_1 \text{School}_i + u_i \]

where

• \(\text{Pay}_i\) is a continuous variable measuring an individual’s pay
• \(\text{School}_i\) is a continuous variable that measures years of education

Interpretation

• \(\beta_0\): \(y\)-intercept, i.e., \(\text{Pay}\) when \(\text{School} = 0\)
• \(\beta_1\): expected increase in \(\text{Pay}\) for a one-unit increase in \(\text{School}\)

Consider the relationship

\[ \text{Pay}_i = \beta_0 + \beta_1 \text{School}_i + u_i \]

Derive the slope’s interpretation.

\(\mathop{\mathbb{E}}\left[ \text{Pay} | \text{School} = \ell + 1 \right] - \mathop{\mathbb{E}}\left[ \text{Pay} | \text{School} = \ell \right]\)

\(\quad = \mathop{\mathbb{E}}\left[ \beta_0 + \beta_1 (\ell + 1) + u \right] - \mathop{\mathbb{E}}\left[ \beta_0 + \beta_1 \ell + u \right]\)

\(\quad = \left[ \beta_0 + \beta_1 (\ell + 1) \right] - \left[ \beta_0 + \beta_1 \ell \right]\)

\(\quad = \beta_0 - \beta_0 + \beta_1 \ell - \beta_1 \ell + \beta_1\) \(\: = \beta_1\).

Expected increase in pay for an additional year of schooling

Continuous Variables

Consider the relationship

\[ \text{Pay}_i = \beta_0 + \beta_1 \text{School}_i + u_i \]

Alternative derivation:

Differentiate the model with respect to schooling:

\[ \dfrac{\partial \text{Pay}}{\partial \text{School}} = \beta_1 \]

Expected increase in pay for an additional year of schooling

If we have multiple explanatory variables, e.g.,

\[ \text{Pay}_i = \beta_0 + \beta_1 \text{School}_i + \beta_2 \text{Ability}_i + u_i \]

then the interpretation changes slightly.

\(\mathop{\mathbb{E}}\left[ \text{Pay} | \text{School} = \ell + 1 \land \text{Ability} = \alpha \right] - \mathop{\mathbb{E}}\left[ \text{Pay} | \text{School} = \ell \land \text{Ability} = \alpha \right]\)

\(\quad = \mathop{\mathbb{E}}\left[ \beta_0 + \beta_1 (\ell + 1) + \beta_2 \alpha + u \right] - \mathop{\mathbb{E}}\left[ \beta_0 + \beta_1 \ell + \beta_2 \alpha + u \right]\)

\(\quad = \left[ \beta_0 + \beta_1 (\ell + 1) + \beta_2 \alpha \right] - \left[ \beta_0 + \beta_1 \ell + \beta_2 \alpha \right]\)

\(\quad = \beta_0 - \beta_0 + \beta_1 \ell - \beta_1 \ell + \beta_1 + \beta_2 \alpha - \beta_2 \alpha\) \(\: = \beta_1\)

The slope gives the expected increase in pay for an additional year of schooling, holding ability constant.

Continuous Variables

If we have multiple explanatory variables, e.g.,

\[ \text{Pay}_i = \beta_0 + \beta_1 \text{School}_i + \beta_2 \text{Ability}_i + u_i \]

then the interpretation changes slightly.

Alternative derivation

Differentiate the model with respect to schooling:

\[ \dfrac{\partial\text{Pay}}{\partial\text{School}} = \beta_1 \]

The slope gives the expected increase in pay for an additional year of schooling, holding ability constant.

Categorical Variables

Consider the relationship

\[ \text{Pay}_i = \beta_0 + \beta_1 \text{Female}_i + u_i \]

where \(\text{Pay}_i\) is a continuous variable measuring an individual’s pay and \(\text{Female}_i\) is a binary variable equal to \(1\) when \(i\) is female.

Interpretation of \(\beta_0\)

\(\beta_0\) is the expected \(\text{Pay}\) for males (i.e., when \(\text{Female} = 0\)):

\[ \mathop{\mathbb{E}}\left[ \text{Pay} | \text{Male} \right] = \mathop{\mathbb{E}}\left[ \beta_0 + \beta_1\times 0 + u_i \right] = \mathop{\mathbb{E}}\left[ \beta_0 + 0 + u_i \right] = \beta_0 \]

Categorical Variables

Consider the relationship

\[ \text{Pay}_i = \beta_0 + \beta_1 \text{Female}_i + u_i \]

where \(\text{Pay}_i\) is a continuous variable measuring an individual’s pay and \(\text{Female}_i\) is a binary variable equal to \(1\) when \(i\) is female.

Interpretation of \(\beta_1\)

\(\beta_1\) is the expected difference in \(\text{Pay}\) between females and males:

\(\mathop{\mathbb{E}}\left[ \text{Pay} | \text{Female} \right] - \mathop{\mathbb{E}}\left[ \text{Pay} | \text{Male} \right]\)

\(\quad = \mathop{\mathbb{E}}\left[ \beta_0 + \beta_1\times 1 + u_i \right] - \mathop{\mathbb{E}}\left[ \beta_0 + \beta_1\times 0 + u_i \right]\)

\(\quad = \mathop{\mathbb{E}}\left[ \beta_0 + \beta_1 + u_i \right] - \mathop{\mathbb{E}}\left[ \beta_0 + 0 + u_i \right]\)

\(\quad = \beta_0 + \beta_1 - \beta_0\) \(\quad = \beta_1\)

Categorical Variables

Consider the relationship

\[ \text{Pay}_i = \beta_0 + \beta_1 \text{Female}_i + u_i \]

where \(\text{Pay}_i\) is a continuous variable measuring an individual’s pay and \(\text{Female}_i\) is a binary variable equal to \(1\) when \(i\) is female.

Interpretation

\(\beta_0 + \beta_1\): is the expected \(\text{Pay}\) for females:

\(\mathop{\mathbb{E}}\left[ \text{Pay} | \text{Female} \right]\)

\(\quad = \mathop{\mathbb{E}}\left[ \beta_0 + \beta_1\times 1 + u_i \right]\)

\(\quad = \mathop{\mathbb{E}}\left[ \beta_0 + \beta_1 + u_i \right]\)

\(\quad = \beta_0 + \beta_1\)

Categorical Variables

Consider the relationship

\[ \text{Pay}_i = \beta_0 + \beta_1 \text{Female}_i + u_i \]

Interpretation

• \(\beta_0\): expected \(\text{Pay}\) for males (i.e., when \(\text{Female} = 0\))
• \(\beta_1\): expected difference in \(\text{Pay}\) between females and males
• \(\beta_0 + \beta_1\): expected \(\text{Pay}\) for females
• Males are the reference group

Categorical Variables

Consider the relationship

\[ \text{Pay}_i = \beta_0 + \beta_1 \text{Female}_i + u_i \]

Note. If there are no other variables to condition on, then \(\hat{\beta}_1\) equals the difference in group means, e.g., \(\bar{X}_\text{Female} - \bar{X}_\text{Male}\).

Note₂. The holding all other variables constant interpretation also applies for categorical variables in multiple regression settings.

Categorical Variables

\(Y_i = \beta_0 + \beta_1 X_i + u_i\) for binary variable \(X_i = \{\color{#434C5E}{0}, \, {\color{#B48EAD}{1}}\}\)

Categorical Variables

\(Y_i = \beta_0 + \beta_1 X_i + u_i\) for binary variable \(X_i = \{\color{#434C5E}{0}, \, {\color{#B48EAD}{1}}\}\)

Categorical Variables

\(Y_i = \beta_0 + \beta_1 X_i + u_i\) for binary variable \(X_i = \{\color{#434C5E}{0}, \, {\color{#B48EAD}{1}}\}\)

Multiple Regression

\(Y_i = \beta_0 + \beta_1 X_{1i} + \beta_2 X_{2i} + u_i \quad\) \(X_1\) is continuous \(\quad X_2\) is categorical

Multiple Regression

The intercept and categorical variable \(X_2\) control for the groups’ means.

Multiple Regression

With groups’ means removed:

Multiple Regression

\(\hat{\beta}_1\) estimates the relationship between \(Y\) and \(X_1\) after controlling for \(X_2\).

Multiple Regression

Another way to think about it:

Omitted variables

Omitted Variables

Omitted Variables

Omitted Variables

Math Score

Explanatory variable	1	2
Intercept	-84.84	-6.34
	(18.57)	(15.00)
log(Spend)	-1.52	11.34
	(2.18)	(1.77)
Lunch		-0.47
		(0.01)

Data from 1823 elementary schools in Michigan

• Math Score: average fourth grade state math test scores.
• log(Spend): the natural logarithm of spending per pupil.
• Lunch: percentage of student eligible for free or reduced-price lunch.

Omitted Variables

Math Score

Explanatory variable	1	2
Intercept	-84.84	-6.34
	(18.57)	(15.00)
log(Spend)	-1.52	11.34
	(2.18)	(1.77)
Lunch		-0.47
		(0.01)

Data from 1823 elementary schools in Michigan

• Math Score: average fourth grade state math test scores.
• log(Spend): the natural logarithm of spending per pupil.
• Lunch: percentage of student eligible for free or reduced-price lunch.

Omitted-Variable Bias

Model 01: \(Y_i = \beta_0 + \beta_1 X_{1i} + u_i\).

Model 02 \(Y_i = \beta_0 + \beta_1 X_{1i} + \beta_2 X_{2i} + v_i\)

Estimating Model 01 (without \(X_2\)) yields omitted-variable bias:

\[ \color{#B48EAD}{\text{Bias} = \beta_2 \frac{\mathop{\text{Cov}}(X_{1i}, X_{2i})}{\mathop{\text{Var}}(X_{1i})}} \]

The sign of the bias depends on

1. The correlation between \(X_2\) and \(Y\), i.e., \(\beta_2\).

2. The correlation between \(X_1\) and \(X_2\), i.e., \(\mathop{\text{Cov}}(X_{1i}, X_{2i})\).

Omitted variable bias

OVB arises when we omit a variable, \(X_k\) that

1. Affects the outcome variable \(Y\), \(\beta_k \neq 0\)

2. Correlates with an explanatory variable \(X_j\), \(Cov(X_j, X_k) \neq 0\),

Biases OLS estimator of \(\beta_j\).

If we omit \(X_k\), then the formula for the bias it creates in \(\hat{\beta}_j\) is…

\[ \color{#B48EAD}{\text{Bias} = \beta_2 \frac{\mathop{\text{Cov}}(X_{1i}, X_{2i})}{\mathop{\text{Var}}(X_{1i})}} \]

Omitted variable bias

Ex. Imagine a population model for the amount individual \(i\) gets paid

\[ \text{Pay}_i = \beta_0 + \beta_1 \text{School}_i + \beta_2 \text{Male}_i + u_i \]

where \(\text{School}_i\) gives \(i\)’s years of schooling and \(\text{Male}_i\) denotes an indicator variable for whether individual \(i\) is male.

Interpretation

• \(\beta_1\): returns to an additional year of schooling (ceteris paribus)
• \(\beta_2\): premium for being male (ceteris paribus)

If \(\beta_2 > 0\), then there is discrimination against women.

Omitted variable bias

Ex. From the population model

\[ \text{Pay}_i = \beta_0 + \beta_1 \text{School}_i + \beta_2 \text{Male}_i + u_i \]

An analyst focuses on the relationship between pay and schooling, i.e.,

\[ \text{Pay}_i = \beta_0 + \beta_1 \text{School}_i + \left(\beta_2 \text{Male}_i + u_i\right) \] \[ \text{Pay}_i = \beta_0 + \beta_1 \text{School}_i + \varepsilon_i \]

where \(\varepsilon_i = \beta_2 \text{Male}_i + u_i\).

Omitted variable bias

We assumed exogeniety to show that OLS is unbiased.

Even if \(\mathop{\mathbb{E}}\left[ u | X \right] = 0\), it is not necessarily true that \(\mathop{\mathbb{E}}\left[ \varepsilon | X \right] = 0\)

• If \(\beta_2 \neq 0\), then it is false

Specifically, if

\[ \mathop{\mathbb{E}}\left[ \varepsilon | \text{Male} = 1 \right] = \beta_2 + \mathop{\mathbb{E}}\left[ u | \text{Male} = 1 \right] \neq 0 \]

Then, OLS is biased

Omitted variable bias

Let’s try to see this result graphically.

The true population model:

\[ \text{Pay}_i = 20 + 0.5 \times \text{School}_i + 10 \times \text{Male}_i + u_i \]

The regression model that suffers from omitted-variable bias:

\[ \text{Pay}_i = \hat{\beta}_0 + \hat{\beta}_1 \times \text{School}_i + e_i \]

Suppose that women, on average, receive more schooling than men.

Omitted variable bias

True model: \(\text{Pay}_i = 20 + 0.5 \times \text{School}_i + 10 \times \text{Male}_i + u_i\)

Omitted variable bias

Biased regression: \(\widehat{\text{Pay}}_i = 31.3 + -0.9 \times \text{School}_i\)

Omitted variable bias

Recalling the omitted variable: Gender (female and male)

Omitted variable bias

Recalling the omitted variable: Gender (female and male)

Omitted variable bias

Unbiased regression: \(\widehat{\text{Pay}}_i = 20.9 + 0.4 \times \text{School}_i + 9.1 \times \text{Male}_i\)

Ex. Card (1995).

Ex. Card (1995)

Education is not randomly assigned across the population, it is a choice. “Depending on how these choices are made, measured earnings differences between workers with different levels of schooling may over-state or under-state the true return to education.”

Card (1995) uses geographic information to causally identify the impact of education earnings by comparing young men who grew up near higher education institutions to those who did not:

Findings suggests the greatest earnings increases are among poor men, suggesting that the presence of a local college lowers the costs/raises the perceived benefits of education.

Ex. Card (1995) Abstract

Although schooling and earnings are highly correlated, social scientists have argued for decades over the causal effect of education. This paper explores the use of college proximity as an exogenous determinant of schooling. An examination… reveals that men who grew up in local labor markets with a nearby college have significantly higher education and significantly higher earnings than other men. The education and earnings gains are concentrated among men with poorly- educated parents – men who would otherwise stop schooling at relatively low levels.

Ex. Card (1995) Data

Variable	Description
id	Person identifier
nearc4	=1 if near 4 yr college, 1966
educ	Years of schooling, 1976
age	Age in years
fatheduc	Father’s schooling
motheduc	Mother’s schooling
weight	NLS sampling weight, 1976
black	=1 if black
south	=1 if in south, 1976
wage	Hourly wage in cents, 1976
IQ	IQ score
libcrd14	=1 if lib. card in home at 14

Regress wages on an indicator for proximity to a four year institution

lm(wage ~ nearc4, data = wage_data) %>% tidy()

# A tibble: 2 × 5
  term        estimate std.error statistic  p.value
  <chr>          <dbl>     <dbl>     <dbl>    <dbl>
1 (Intercept)    516.       8.39     61.5  0       
2 nearc4          89.2     10.2       8.77 2.85e-18

Q₁: What is the reference category?

Q₂: Interpret the coefficients.

Q₃: Suppose instead we run¹

lm(wage ~ farc4, data = wage_data

How does interpretation change?

Men who did not grow up near a 4 year institution

---

\(\beta_0\): On average, men who did not grow up near a 4-year institution earn 516 cents per hour.

\(\widehat{\beta}_{\text{nearc4}}\): On average, men who grew up near a 4 year institution earn 89.2 cents more per hour than men who did not.

---

\(\beta_0\): On average, men who grow up near a 4-year institution earn \(\widehat{\beta}_0\) cents per hour.

\(\widehat{\beta}_{\text{farc4}}\): On average, men who grew up far away from a 4 year institution earn \(\widehat{\beta}_{farc4}\) cents less per hour than men who did.

1. where farc4 is an indicator for living far from 4-year institutions

Regress wages on an indicator for proximity to a four year institution and race

lm(wage ~ nearc4 + black, data = wage_data) %>% tidy()

# A tibble: 3 × 5
  term        estimate std.error statistic  p.value
  <chr>          <dbl>     <dbl>     <dbl>    <dbl>
1 (Intercept)    562.       8.65     65.0  0       
2 nearc4          78.1      9.84      7.94 2.73e-15
3 black         -162.      10.8     -15.0  7.16e-49

Q₁: Reference category?

Q₂: Interpretation?

Non-black men who did not grow up near a 4-year institution

---

\(\beta_0\): On average, non-black men who grew up far from a 4-year institution earn 562 cents per hour.

\(\widehat{\beta}_{\text{nearc4}}\): On average, men who grew up near a 4-year institution earn 78.1 cents more per hour than men who did not.

\(\widehat{\beta}_{\text{black}}\): On average, black men earn 162 cents less per hour than non-black men, holding proximity to a 4-year institution constant.

Regress wages on an indicator for proximity to a four year institution and race

lm(wage ~ nearc4 + nonblack, data = wage_data) %>% tidy()

# A tibble: 3 × 5
  term        estimate std.error statistic   p.value
  <chr>          <dbl>     <dbl>     <dbl>     <dbl>
1 (Intercept)    400.      11.2      35.6  1.30e-231
2 nearc4          78.1      9.84      7.94 2.73e- 15
3 nonblack       162.      10.8      15.0  7.16e- 49

Q₁: Reference category?

Q₂: Interpretation?

Black men who did not grow up near a 4-year institution

---

\(\beta_0\): On average, black men who did not grow up near a 4-year institution earn 400 cents per hour.

\(\widehat{\beta}_{\text{nearc4}}\): On average, men who grew up near a 4-year institution earn 78.1 cents more per hour than men who did not.

\(\widehat{\beta}_{\text{black}}\): On average, nonblack men earn 162 cents more per hour than non-black men, holding proximity to a 4-year institution constant.

Notice:

• The coefficient on nearc4 is the same in both models
• The interpretation of the coefficient on black changes—magnitude is the same

lm(wage ~ nearc4 + black, data = wage_data) %>% 
  tidy()

# A tibble: 3 × 5
  term        estimate std.error statistic  p.value
  <chr>          <dbl>     <dbl>     <dbl>    <dbl>
1 (Intercept)    562.       8.65     65.0  0       
2 nearc4          78.1      9.84      7.94 2.73e-15
3 black         -162.      10.8     -15.0  7.16e-49

lm(wage ~ nearc4 + nonblack, data = wage_data) %>% 
  tidy()

# A tibble: 3 × 5
  term        estimate std.error statistic   p.value
  <chr>          <dbl>     <dbl>     <dbl>     <dbl>
1 (Intercept)    400.      11.2      35.6  1.30e-231
2 nearc4          78.1      9.84      7.94 2.73e- 15
3 nonblack       162.      10.8      15.0  7.16e- 49

Ex. Card (1995)

lm(wage ~ educ + nearc4 + south + black, data = wage_data) %>% tidy()

# A tibble: 5 × 5
  term        estimate std.error statistic  p.value
  <chr>          <dbl>     <dbl>     <dbl>    <dbl>
1 (Intercept)    312.      25.1      12.5  8.47e-35
2 educ            21.5      1.73     12.4  1.02e-34
3 nearc4          47.3      9.75      4.85 1.31e- 6
4 south          -74.1      9.81     -7.56 5.52e-14
5 black          -98.5     11.3      -8.68 6.27e-18

Q₁: What is the reference category?

Q₂: Interpret the coefficients.

We considered a model where schooling has the same effect for everyone (F and M)

We will consider models that allow effects to differ by another variable (e.g., by gender: (F and M)):

Interactive relationships

Motivation

Regression coefficients describe average effects. But for whom does on average mean?

Averages can mask heterogeneous effects that differ by group or by the level of another variable.

We can use interaction terms to model heterogeneous effects, accommodating complexity and nuance by going beyond “the effect of \(X\) on \(Y\) is \(\beta_1\).”

Interaction Terms

Starting point: \(Y_i = \beta_0 + \beta_1 X_{1i} + \beta_2 X_{2i} + u_i\)

• \(X_{1i}\) is the variable of interest

• \(X_{2i}\) is a control variable

A richer model: Interactions test whether \(X_{2i}\) moderates the effect of \(X_{1i}\)

\[ Y_i = \beta_0 + \beta_1 X_{1i} + \beta_2 X_{2i} + \beta_3 X_{1i} \cdot X_{2i} + u_i \]

Interpretation: The partial derivative of \(Y_i\) with respect to \(X_{1i}\) is the marginal effect of \(X_1\) on \(Y_i\):

\[ \color{#81A1C1}{\dfrac{\partial Y}{\partial X_1} = \beta_1 + \beta_3 X_{2i}} \]

The effect of \(X_1\) depends on the level of \(X_2\) 🤯

Ex. Differential returns to education

Ex. Differential returns to education

Research question: Do the returns to education vary by race?

Consider the interactive regression model:

\[\begin{align*} \text{Wage}_i = \beta_0 &+ \beta_1 \text{Education}_i + \beta_2 \text{Black}_i \\ &+ \beta_3 \text{Education}_i \times \text{Black}_i + u_i \end{align*}\]

What is the marginal effect of an additional year of education?

\[ \dfrac{\partial \text{Wage}}{\partial \text{Education}} = \beta_1 + \beta_3 \text{Black}_i \]

Ex. Differential returns to education

What is the return to education for black workers?

lm(wage ~ educ + black + educ:black, data = wage2) %>% tidy()

# A tibble: 4 × 5
  term        estimate std.error statistic  p.value
  <chr>          <dbl>     <dbl>     <dbl>    <dbl>
1 (Intercept)    196.      82.2       2.38 1.75e- 2
2 educ            58.4      5.96      9.80 1.19e-21
3 black          321.     263.        1.22 2.23e- 1
4 educ:black     -40.7     20.7      -1.96 4.99e- 2

\[ \widehat{\left(\dfrac{\partial \text{Wage}}{\partial \text{Education}} \right)}\Bigg|_{\small \text{Black}=1} = \hat{\beta}_1 + \hat{\beta}_3 = 17.65 \]

Ex. Differential returns to education

What is the return to education for non-black workers?

lm(wage ~ educ + black + educ:black, data = wage2) %>% tidy()

# A tibble: 4 × 5
  term        estimate std.error statistic  p.value
  <chr>          <dbl>     <dbl>     <dbl>    <dbl>
1 (Intercept)    196.      82.2       2.38 1.75e- 2
2 educ            58.4      5.96      9.80 1.19e-21
3 black          321.     263.        1.22 2.23e- 1
4 educ:black     -40.7     20.7      -1.96 4.99e- 2

\[ \widehat{\left(\dfrac{\partial \text{Wage}}{\partial \text{Education}} \right)}\Bigg|_{\small \text{Black}=0} = \hat{\beta}_1 = 58.38 \]

Ex. Differential returns to education

Q: Does the return to education differ by race?

Conduct a two-sided \(t\)-test of the null hypothesis that the interaction coefficient equals 0 at the 5% level.

lm(wage ~ educ + black + educ:black, data = wage2) %>% tidy()

# A tibble: 4 × 5
  term        estimate std.error statistic  p.value
  <chr>          <dbl>     <dbl>     <dbl>    <dbl>
1 (Intercept)    196.      82.2       2.38 1.75e- 2
2 educ            58.4      5.96      9.80 1.19e-21
3 black          321.     263.        1.22 2.23e- 1
4 educ:black     -40.7     20.7      -1.96 4.99e- 2

p-value = 0.0499 < 0.05 \(\implies\) reject null hypothesis.

A: The return to education is significantly lower for black workers.

Ex. Differential returns to education

We can also test hypotheses about specific marginal effects.

• e.g., H₀: \(\left(\dfrac{\partial \text{Wage}}{\partial \text{Education}} \right)\Bigg|_{\small \text{Black}=1} = 0\).
• Conduct a \(t\) test or construct confidence intervals.

Problem 1: lm() output does not include \(\hat{\text{SE}}\) for the marginal effects.

Problem 2: The formula for marginal effect standard errors includes covariances between coefficient estimates. The math is messy.¹

Solution: Construct confidence intervals using the margins package.

1. Stay tuned

Ex. Differential returns to education

The margins function provides standard errors and 95% confidence intervals for each marginal effect.

pacman::p_load(margins)

reg = lm(wage ~ educ + black + educ:black, data = wage2)

margins(reg, at = list(black = 0:1)) %>% summary() %>% filter(factor == "educ")

 factor  black     AME      SE      z      p    lower   upper
   educ 0.0000 58.3773  5.9539 9.8049 0.0000  46.7079 70.0466
   educ 1.0000 17.6544 19.8941 0.8874 0.3749 -21.3373 56.6462

Ex. Differential returns to education

We can use the geom_pointrange() option in ggplot2 to plot the marginal effects with 95% confidence intervals.

Ex. Policy intervention

Ex. Policy intervention

Suppose we are interested in the relationship of some policy intervention that on health outcomes. We estimate the following model:

\[ \text{Health Score}_i = \beta_0 + \beta_1 \text{Age}_i + \beta_2 \text{Policy}_i + \beta_3 \text{Age}_i \times \text{Policy}_i + u_i \]

Where:

• \(\text{Health Score}\): is a continuous variable representing the health of individual \(i\).
• \(\text{Age}_i\): the age of individual \(i\)
• \(\text{Policy}_i\): An indicator for whether the individual \(i\) experienced the policy intervention or not.

Ex. Policy intervention

OLS output is the following:

# A tibble: 4 × 5
  term             estimate std.error statistic   p.value
  <chr>               <dbl>     <dbl>     <dbl>     <dbl>
1 (Intercept)   46.5        1.08         42.8   4.55e-228
2 income        -0.00000466 0.0000212    -0.220 8.26e-  1
3 policy        12.1        1.52          7.97  4.36e- 15
4 income:policy  0.100      0.0000297  3365.    0        

Interpretation:

• \(\beta_0\): Expected health score for an individual with income of 0 who was not exposed to the policy

• \(\beta_1\): Expected change in health for an additional dollar of income of for those not exposed to the policy.

• \(\beta_2\): Expected difference in health score between an individual of income of 0 exposed to the policy and an individual of income of 0 not exposed to the policy.

• \(\beta_3\): Expected difference in the effect of income on the health score between those exposed to the policy and those not exposed.