3.7 — Interaction Effects

# 3.7 — Interaction Effects
## ECON 480 • Econometrics • Fall 2021
### Ryan Safner<br> Assistant Professor of Economics <br> <a href="mailto:safner@hood.edu"><i class="fa fa-paper-plane fa-fw"></i>safner@hood.edu</a> <br> <a href="https://github.com/ryansafner/metricsF21"><i class="fa fa-github fa-fw"></i>ryansafner/metricsF21</a><br> <a href="https://metricsF21.classes.ryansafner.com"> <i class="fa fa-globe fa-fw"></i>metricsF21.classes.ryansafner.com</a><br>

---

# Outline

### [Interactions Between a Dummy and Continuous Variable](#16)
### [Interactions Between Two Dummy Variables](#69)
### [Interactions Between Two Continuous Variables](#99)

---

# Sliders and Switches

---

# Sliders and Switches

- Marginal effect of dummy variable: effect on `$Y$` of going from 0 to 1

- Marginal effect of continuous variable: effect on `$Y$` of a 1 unit change in `$X$`

---

# Interaction Effects

.smallest[
.bg-washed-green.b--dark-green.ba.bw2.br3.shadow-5.ph4.mt5[
.green[**Example**]: Consider the gender pay gap again.
]
]

.smallest[
- .hi-purple[Does experience affect wages *differently* by gender?]
  - i.e. is there an .hi[interaction effect] between gender and experience?
]
--
.smallest[
- .hi-turquoise[Note this is *NOT the same* as just asking: “do men earn more than women *with the same amount of experience*?”]

`$$\widehat{\text{wages}_i}=\beta_0+\beta_1 \, Gender_i + \beta_2 \, Experience_i$$`
]
---

# Three Types of Interactions

- Depending on the types of variables, there are 3 possible types of interaction effects

- We will look at each in turn

1. Interaction between a **dummy** and a **continuous** variable:
`$$Y_i=\beta_0+\beta_1X_i+\beta_2 D_i+\beta_3 \color{#e64173}{(X_i \times D_i)}$$`
--

2. Interaction between a **two dummy** variables:
`$$Y_i=\beta_0+\beta_1D_{1i}+\beta_2 D_{2i}+\beta_3 \color{#e64173}{(D_{1i} \times D_{2i})}$$`
--

3. Interaction between a **two continuous** variables:
`$$Y_i=\beta_0+\beta_1X_{1i}+\beta_2 X_{2i}+\beta_3 \color{#e64173}{(X_{1i} \times X_{2i})}$$`

---

# Interactions Between a Dummy and Continuous Variable

---

# Interactions: A Dummy & Continuous Variable

- Does the marginal effect of the continuous variable on `$Y$` change depending on whether the dummy is “on” or “off”?

---

# Interactions: A Dummy & Continuous Variable I

- We can model an interaction by introducing a variable that is an .hi[interaction term] capturing the interaction between two variables:

`$$Y_i=\beta_0+\beta_1X_i+\beta_2 D_i+\color{#e64173}{\beta_3(X_i \times D_i)} \quad \text{ where } D_i=\{0,1\}$$`

- `$\color{#e64173}{\beta_3}$` estimates the .hi[interaction effect] between `$X_i$` and `$D_i$` on `$Y_i$`

- What do the different coefficients `$(\beta)$`’s tell us? 
  - Again, think logically by examining each group `$(D_i=0$` or `$D_i=1)$`

---

# Interaction Effects as Two Regressions I

`$$Y_i=\beta_0+\beta_1X_i+\beta_2 D_i+\beta_3 X_i \times D_i$$`

- .red[When `\$D_i=0\$` (Control group):]

`$$\begin{align*}
\hat{Y_i}&=\hat{\beta_0}+\hat{\beta_1}X_i+\hat{\beta_2}(\color{red}{0})+\hat{\beta_3}X_i \times (\color{red}{0})\\
\hat{Y_i}& =\hat{\beta_0}+\hat{\beta_1}X_i\\
\end{align*}$$`

- .blue[When `\$D_i=1\$` (Treatment group):]

`$$\begin{align*}
\hat{Y_i}&=\hat{\beta_0}+\hat{\beta_1}X_i+\hat{\beta_2}(\color{blue}{1})+\hat{\beta_3}X_i \times (\color{blue}{1})\\
\hat{Y_i}&= (\hat{\beta_0}+\hat{\beta_2})+(\hat{\beta_1}+\hat{\beta_3})X_i\\
\end{align*}$$`

- So what we really have is *two* regression lines!

---

# Interaction Effects as Two Regressions II

<img src="3.7-slides_files/figure-html/unnamed-chunk-1-1.png" width="504" />
]

- .red[`\$D_i=0\$` group:]

`$$\color{#D7250E}{Y_i=\hat{\beta_0}+\hat{\beta_1}X_i}$$`

- .blue[`\$D_i=1\$` group:]

`$$\color{#0047AB}{Y_i=(\hat{\beta_0}+\hat{\beta_2})+(\hat{\beta_1}+\hat{\beta_3})X_i}$$`

]

---

# Interpretting Coefficients I

`$$Y_i=\beta_0+\beta_1X_i+\beta_2 D_i+\beta_3 \color{#e64173}{(X_i \times D_i)}$$`

- To interpret the coefficients, compare cases after changing `$X$` by `$\color{#e64173}{\Delta X}$`:

`$$Y_i+\color{#e64173}{\Delta Y_i}=\beta_0+\beta_1(X_i\color{#e64173}{+\Delta X_i})\beta_2D_i+\beta_3\big((X_i\color{#e64173}{+\Delta X_i})D_i\big)$$`

- Subtracting these two equations, the difference is:

`$$\begin{align*}
	\Delta Y_i &= \beta_1 \Delta X_i + \beta_3 D_i \Delta X_i\\
	\color{#6A5ACD}{\frac{\Delta Y_i}{\Delta X_i}} &\color{#6A5ACD}{= \beta_1+\beta_3 D_i}\\
\end{align*}$$`

- .hi-purple[The effect of `\$X \rightarrow Y\$` depends on the value of `\$D_i\$`!]

- .hi[`\$\beta_3\$`: *increment* to the effect of `\$X \rightarrow Y\$` when `\$D_i=1\$` (vs. `\$D_i=0\$`)]

---

# Interpretting Coefficients II

`$$Y_i=\beta_0+\beta_1X_i+\beta_2 D_i+\beta_3 \color{#e64173}{(X_i \times D_i)}$$`

- `$\hat{\beta_0}$`: `$E[Y_i]$` for `$X_i=0$` and `$D_i=0$`

- `$\beta_1$`: Marginal effect of `$X_i \rightarrow Y_i$` for `$D_i=0$`

- `$\beta_2$`: Marginal effect on `$Y_i$` of difference between `$D_i=0$` and `$D_i=1$`

- `$\beta_3$`: The **difference** of the marginal effect of `$X_i \rightarrow Y_i$` between `$D_i=0$` and `$D_i=1$`

- This is a bit awkward, easier to think about the two regression lines:

---

# Interpretting Coefficients III

.pull-left[
.smallest[
.red[For `\$D_i=0\$` Group:  `\$\hat{Y_i}=\hat{\beta_0}+\hat{\beta_1}X_i\$`]
  - Intercept: `$\hat{\beta_0}$`
  - Slope: `$\hat{\beta_1}$`
]
]
--

.pull-right[
.smallest[
.blue[For `\$D_i=1\$` Group:  `\$\hat{Y_i}=(\hat{\beta_0}+\hat{\beta_2})+(\hat{\beta_1}+\hat{\beta_3})X_i\$`]
  - Intercept: `$\hat{\beta_0}+\hat{\beta_2}$`
  - Slope: `$\hat{\beta_1}+\hat{\beta_3}$`
]
]
--

- `$\hat{\beta_3}$`: difference in slope between groups

]

.smallest[
- How can we determine if the two lines have the same slope and/or intercept?
  - Same intercept? `$t$`-test `$H_0$`: `$\beta_2=0$`
  - Same slope? `$t$`-test `$H_0$`: `$\beta_3=0$`
]

---

# Example I

.bg-washed-green.b--dark-green.ba.bw2.br3.shadow-5.ph4.mt5[
.green[**Example**]: `$$\widehat{wage_i}=\hat{\beta_0}+\hat{\beta_1}exper_i+\hat{\beta_2}female_i+\hat{\beta_3}(exper_i \times female_i)$$`

]

- .blue[For males `\$(female=0)\$`]:
`$$\widehat{wage_i}=\hat{\beta_0}+\hat{\beta_1}exper$$`

- .pink[For females `\$(female=1)\$`]:
`$$\widehat{wage_i}=\underbrace{(\hat{\beta_0}+\hat{\beta_2})}_{\text{intercept}}+\underbrace{(\hat{\beta_1}+\hat{\beta_3})}_{\text{slope}}exper$$`

---

# Example II

```r
interaction_plot <- ggplot(data = wages)+
  aes(x = exper,
      y = wage,
      color = as.factor(Gender))+ # make factor
  geom_point(alpha = 0.5)+
  scale_y_continuous(labels=scales::dollar)+
  labs(x = "Experience (Years)",
       y = "Wage")+
  scale_color_manual(values = c("Female" = "#e64173",
                                "Male" = "#0047AB")
                     )+ # setting custom colors
  guides(color=F)+ # hide legend
  theme_slides
interaction_plot
```
]

.smallest[
- Need to make sure `color` aesthetic uses a `factor` variable
  - Can just use `as.factor()` in ggplot code
]

]

<img src="3.7-slides_files/figure-html/unnamed-chunk-2-1.png" width="504" />
]

---

# Example II

```r
interaction_plot+
* geom_smooth(method="lm")
```
]

<img src="3.7-slides_files/figure-html/unnamed-chunk-3-1.png" width="504" />
]

---

# Example II

```r
interaction_plot+
  geom_smooth(method="lm")+
* facet_wrap(~Gender)
```
]

<img src="3.7-slides_files/figure-html/unnamed-chunk-4-1.png" width="504" />
]

---

# Example Regression in R I

.smallest[
- Syntax for adding an interaction term is easy in `R`: `var1 * var2`
  - Or could just do `var1 * var2` (multiply)

```r
# both are identical in R
interaction_reg <- lm(wage ~ exper * female, data = wages)
interaction_reg <- lm(wage ~ exper + female + exper * female, data = wages)
```

<div data-pagedtable="false">
  <script data-pagedtable-source type="application/json">
{"columns":[{"label":["term"],"name":[1],"type":["chr"],"align":["left"]},{"label":["estimate"],"name":[2],"type":["dbl"],"align":["right"]},{"label":["std.error"],"name":[3],"type":["dbl"],"align":["right"]},{"label":["statistic"],"name":[4],"type":["dbl"],"align":["right"]},{"label":["p.value"],"name":[5],"type":["dbl"],"align":["right"]}],"data":[{"1":"(Intercept)","2":"6.15827549","3":"0.34167408","4":"18.023830","5":"7.998534e-57"},{"1":"exper","2":"0.05360476","3":"0.01543716","4":"3.472450","5":"5.585255e-04"},{"1":"female","2":"-1.54654677","3":"0.48186030","4":"-3.209534","5":"1.411253e-03"},{"1":"exper:female","2":"-0.05506989","3":"0.02217496","4":"-2.483427","5":"1.332533e-02"}],"options":{"columns":{"min":{},"max":[10]},"rows":{"min":[10],"max":[10]},"pages":{}}}
  </script>
</div>
]

---

# Example Regression in R III

```r
library(huxtable)
huxreg(interaction_reg,
       coefs = c("Constant" = "(Intercept)",
                 "Experience" = "exper",
                 "Female" = "female",
                 "Experience * Female" = "exper:female"),
       statistics = c("N" = "nobs",
                      "R-Squared" = "r.squared",
                      "SER" = "sigma"),
       number_format = 2)
```

]
]

.quitesmall[
<table class="huxtable" style="border-collapse: collapse; border: 0px; margin-bottom: 2em; margin-top: 2em; ; margin-left: auto; margin-right: auto;  " id="tab:unnamed-chunk-7">
<col><col><tr>
<th style="vertical-align: top; text-align: center; white-space: normal; border-style: solid solid solid solid; border-width: 0.8pt 0pt 0pt 0pt;    padding: 6pt 6pt 6pt 6pt; font-weight: normal;"></th><th style="vertical-align: top; text-align: center; white-space: normal; border-style: solid solid solid solid; border-width: 0.8pt 0pt 0.4pt 0pt;    padding: 6pt 6pt 6pt 6pt; font-weight: normal;">(1)</th></tr>
<tr>
<th style="vertical-align: top; text-align: left; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">Constant</th><td style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0.4pt 0pt 0pt 0pt;    padding: 6pt 6pt 6pt 6pt; font-weight: normal;">6.16 ***</td></tr>
<tr>
<th style="vertical-align: top; text-align: left; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;"></th><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">(0.34)   </td></tr>
<tr>
<th style="vertical-align: top; text-align: left; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">Experience</th><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">0.05 ***</td></tr>
<tr>
<th style="vertical-align: top; text-align: left; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;"></th><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">(0.02)   </td></tr>
<tr>
<th style="vertical-align: top; text-align: left; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">Female</th><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">-1.55 ** </td></tr>
<tr>
<th style="vertical-align: top; text-align: left; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;"></th><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">(0.48)   </td></tr>
<tr>
<th style="vertical-align: top; text-align: left; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">Experience * Female</th><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">-0.06 *  </td></tr>
<tr>
<th style="vertical-align: top; text-align: left; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;"></th><td style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0.4pt 0pt;    padding: 6pt 6pt 6pt 6pt; font-weight: normal;">(0.02)   </td></tr>
<tr>
<th style="vertical-align: top; text-align: left; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">N</th><td style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0.4pt 0pt 0pt 0pt;    padding: 6pt 6pt 6pt 6pt; font-weight: normal;">526       </td></tr>
<tr>
<th style="vertical-align: top; text-align: left; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">R-Squared</th><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">0.14    </td></tr>
<tr>
<th style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0.8pt 0pt;    padding: 6pt 6pt 6pt 6pt; font-weight: normal;">SER</th><td style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0.8pt 0pt;    padding: 6pt 6pt 6pt 6pt; font-weight: normal;">3.44    </td></tr>
<tr>
<th colspan="2" style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0.8pt 0pt 0pt 0pt;    padding: 6pt 6pt 6pt 6pt; font-weight: normal;"> *** p < 0.001;  ** p < 0.01;  * p < 0.05.</th></tr>
</table>

]
]

---

# Example Regression in R: Interpretting Coefficients

`$$\widehat{wage_i}=6.16+0.05 \, Experience_i - 1.55 \, Female_i - 0.06 \, (Experience_i \times Female_i)$$`

- `$\hat{\beta_0}$`:

---

# Example Regression in R: Interpretting Coefficients

`$$\widehat{wage_i}=6.16+0.05 \, Experience_i - 1.55 \, Female_i - 0.06 \, (Experience_i \times Female_i)$$`

- `$\hat{\beta_0}$`: **Men** with 0 years of experience earn 6.16

- `$\hat{\beta_1}$`:

---

# Example Regression in R: Interpretting Coefficients

`$$\widehat{wage_i}=6.16+0.05 \, Experience_i - 1.55 \, Female_i - 0.06 \, (Experience_i \times Female_i)$$`

- `$\hat{\beta_0}$`: **Men** with 0 years of experience earn 6.16

- `$\hat{\beta_1}$`: For every additional year of experience, **men** earn $0.05

- `$\hat{\beta_2}$`:

---

# Example Regression in R: Interpretting Coefficients

`$$\widehat{wage_i}=6.16+0.05 \, Experience_i - 1.55 \, Female_i - 0.06 \, (Experience_i \times Female_i)$$`

- `$\hat{\beta_0}$`: **Men** with 0 years of experience earn 6.16

- `$\hat{\beta_1}$`: For every additional year of experience, **men** earn $0.05

- `$\hat{\beta_2}$`: **Women** with 0 years of experience earn $1.55 **less than men**

- `$\hat{\beta_3}$`:

---

# Example Regression in R: Interpretting Coefficients

`$$\widehat{wage_i}=6.16+0.05 \, Experience_i - 1.55 \, Female_i - 0.06 \, (Experience_i \times Female_i)$$`

- `$\hat{\beta_0}$`: **Men** with 0 years of experience earn 6.16

- `$\hat{\beta_1}$`: For every additional year of experience, **men** earn $0.05

- `$\hat{\beta_2}$`: **Women** with 0 years of experience earn $1.55 **less than men**

- `$\hat{\beta_3}$`: **Women** earn $0.06 **less than men** for every additional year of experience

---

# Interpretting Coefficients as 2 Regressions I

`$$\widehat{wage_i}=6.16+0.05 \, Experience_i - 1.55 \, Female_i - 0.06 \, (Experience_i \times Female_i)$$`

- Men with 0 years of experience earn $6.16 on average

- For every additional year of experience, men earn $0.05 more on average

---

# Interpretting Coefficients as 2 Regressions II

`$$\widehat{wage_i}=6.16+0.05 \, Experience_i - 1.55 \, Female_i - 0.06 \, (Experience_i \times Female_i)$$`

.pink[Regression for women `\$(female=1)\$`]
`$$\begin{align*}
\widehat{wage_i}&=6.16+0.05 \, Experience_i - 1.55\color{#e64173}{(1)}-0.06 \, Experience_i \times \color{#e64173}{(1)}\\
&= (6.16-1.55)+(0.05-0.06) \, Experience_i\\
&= 4.61-0.01 \, Experience_i \\
\end{align*}$$`

- Women with 0 years of experience earn $4.61 on average

- For every additional year of experience, women earn $0.01 *less* on average

---

# Example Regression in R: Hypothesis Testing

.pull-left[
.smallest[
- Are slopes & intercepts of the 2 regressions statistically significantly different?

]
]

`$$\begin{align*}
\widehat{wage_i}&=6.16+0.05 \, Experience_i - 1.55 \, Female_i\\
&- 0.06 \, (Experience_i \times Female_i) \\ \end{align*}$$`

]

```
## # A tibble: 4 × 5
##   term         estimate std.error statistic  p.value
##   <chr>           <dbl>     <dbl>     <dbl>    <dbl>
## 1 (Intercept)    6.16      0.342      18.0  8.00e-57
## 2 exper          0.0536    0.0154      3.47 5.59e- 4
## 3 female        -1.55      0.482      -3.21 1.41e- 3
## 4 exper:female  -0.0551    0.0222     -2.48 1.33e- 2
```
]
]

---

# Example Regression in R: Hypothesis Testing

.pull-left[
.smallest[
- Are slopes & intercepts of the 2 regressions statistically significantly different?

- .purple[Are intercepts different?] `$\color{#6A5ACD}{H_0: \beta_2=0}$`
  - Difference between men vs. women for no experience?
  - Is `$\hat{\beta_2}$` significant?
  - .purple[Yes (reject)] `$\color{#6A5ACD}{H_0}$`: `$p$`-value = 0.00

]
]

`$$\begin{align*}
\widehat{wage_i}&=6.16+0.05 \, Experience_i - 1.55 \, Female_i\\
&- 0.06 \, (Experience_i \times Female_i) \\ \end{align*}$$`

]

---

# Example Regression in R: Hypothesis Testing

.pull-left[
.smallest[
- Are slopes & intercepts of the 2 regressions statistically significantly different?

- .purple[Are slopes different?] `$\color{#6A5ACD}{H_0: \beta_3=0}$`
  - Difference between men vs. women for marginal effect of experience?
  - Is `$\hat{\beta_3}$` significant?
  - .purple[Yes (reject)] `$\color{#6A5ACD}{H_0}$`: `$p$`-value = 0.01
]
]

`$$\begin{align*}
\widehat{wage_i}&=6.16+0.05 \, Experience_i - 1.55 \, Female_i\\
&- 0.06 \, (Experience_i \times Female_i) \\ \end{align*}$$`

]

---
class: inverse, center, middle

# Interactions Between Two Dummy Variables

---

# Interactions Between Two Dummy Variables

- Does the marginal effect on `$Y$` of one dummy going from “off” to “on” change depending on whether the *other* dummy is “off” or “on”?

---

# Interactions Between Two Dummy Variables

`$$Y_i=\beta_0+\beta_1D_{1i}+\beta_2 D_{2i}+\beta_3 \color{#e64173}{(D_{1i} \times D_{2i})}$$`

- `$D_{1i}$` and `$D_{2i}$` are dummy variables

- `$\hat{\beta_1}$`: effect on `$Y$` of going from `$D_{1i}=0$` to `$D_{1i}=1$` when `$D_{2i}=0$`

- `$\hat{\beta_2}$`: effect on `$Y$` of going from `$D_{2i}=0$` to `$D_{2i}=1$` when `$D_{1i}=0$`

- `$\hat{\beta_3}$`: effect on `$Y$` of going from `$D_{1i}=0$` to `$D_{1i}=1$` when `$D_{2i}=1$`
  - *increment* to the effect of `$D_{1i}$` going from 0 to 1 when `$D_{2i}=1$` (vs. 0)

- As always, best to think logically about possibilities (when each dummy `$=0$` or `$=1)$`

---

# 2 Dummy Interaction: Interpretting Coefficients

.smaller[
`$$Y_i=\beta_0+\beta_1D_{1i}+\beta_2 D_{2i}+\beta_3 \color{#e64173}{(D_{1i} \times D_{2i})}$$`
]

.smallest[
- To interpret coefficients, compare cases:
  - Hold `$D_{2i}$` constant (set to some value `$D_{2i}=\mathbf{d_2}$`)
  - Plug in 0s or 1s for `$D_{1i}$`

`$$\begin{align*}
E(Y_i|D_{1i}&=\color{#FFA500}{0}, D_{2i}=\mathbf{d_2}) = \beta_0+\beta_2 \mathbf{d_2}\\
E(Y_i|D_{1i}&=\color{#44C1C4}{1}, D_{2i}=\mathbf{d_2}) = \beta_0+\beta_1(\color{#44C1C4}{1})+\beta_2 \mathbf{d_2}+\beta_3(\color{#44C1C4}{1})\mathbf{d_2}\\
\end{align*}$$`
]

`$$\color{#6A5ACD}{\beta_1+\beta_3 \mathbf{d_2}}$$`

- .hi-purple[The marginal effect of] `$\color{#6A5ACD}{D_{1i} \rightarrow Y_i\\}$` .hi-purple[depends on the value of] `$\color{#6A5ACD}{D_{2i}\\}$`
  - `$\color{#e64173}{\hat{\beta_3}}$` is the *increment* to the effect of `$D_1$` on `$Y$` when `$D_2$` goes from `$0$` to `$1$`
]

---

# Interactions Between 2 Dummy Variables: Example

.smallest[
.bg-washed-green.b--dark-green.ba.bw2.br3.shadow-5.ph4.mt5[
.green[**Example**]: Does the gender pay gap change if a person is married vs. single?

`$$\widehat{wage_i}=\hat{\beta_0}+\hat{\beta_1}female_i+\hat{\beta_2}married_i+\hat{\beta_3}(female_i \times married_i)$$`
]
]

.quitesmall[
- Logically, there are 4 possible combinations of `$female_i = \{\color{#0047AB}{0},\color{#e64173}{1}\}$` and `$married_i = \{\color{#FFA500}{0},\color{#44C1C4}{1}\}$`
]

.pull-left[
.quitesmall[
1) .hi-orange[Unmarried] .hi-blue[men] `$(female_i=\color{#0047AB}{0}, \, married_i=\color{#FFA500}{0})$`
`$$\widehat{wage_i}=\hat{\beta_0}$$`
]
]

---

# Interactions Between 2 Dummy Variables: Example

.smallest[
.bg-washed-green.b--dark-green.ba.bw2.br3.shadow-5.ph4.mt5[
.green[**Example**]: Does the gender pay gap change if a person is married vs. single?

`$$\widehat{wage_i}=\hat{\beta_0}+\hat{\beta_1}female_i+\hat{\beta_2}married_i+\hat{\beta_3}(female_i \times married_i)$$`
]
]

.quitesmall[
- Logically, there are 4 possible combinations of `$female_i = \{\color{#0047AB}{0},\color{#e64173}{1}\}$` and `$married_i = \{\color{#FFA500}{0},\color{#44C1C4}{1}\}$`
]

.pull-left[
.quitesmall[
1) .hi-orange[Unmarried] .hi-blue[men] `$(female_i=\color{#0047AB}{0}, \, married_i=\color{#FFA500}{0})$`
`$$\widehat{wage_i}=\hat{\beta_0}$$`

2) .hi-turquoise[Married] .hi-blue[men] `$(female_i=\color{#0047AB}{0}, \, married_i=\color{#44C1C4}{1})$`
`$$\widehat{wage_i}=\hat{\beta_0}+\hat{\beta_2}$$`
]
]

.pull-right[
.quitesmall[
3) .hi-orange[Unmarried] .hi-pink[women] `$(female_i=\color{#e64173}{1}, \, married_i=\color{#FFA500}{0})$`
`$$\widehat{wage_i}=\hat{\beta_0}+\hat{\beta_1}$$`
]
]

---

# Interactions Between 2 Dummy Variables: Example

.smallest[
.bg-washed-green.b--dark-green.ba.bw2.br3.shadow-5.ph4.mt5[
.green[**Example**]: Does the gender pay gap change if a person is married vs. single?

`$$\widehat{wage_i}=\hat{\beta_0}+\hat{\beta_1}female_i+\hat{\beta_2}married_i+\hat{\beta_3}(female_i \times married_i)$$`
]
]

.quitesmall[
- Logically, there are 4 possible combinations of `$female_i = \{\color{#0047AB}{0},\color{#e64173}{1}\}$` and `$married_i = \{\color{#FFA500}{0},\color{#44C1C4}{1}\}$`
]

.pull-left[
.quitesmall[
1) .hi-orange[Unmarried] .hi-blue[men] `$(female_i=\color{#0047AB}{0}, \, married_i=\color{#FFA500}{0})$`
`$$\widehat{wage_i}=\hat{\beta_0}$$`

2) .hi-turquoise[Married] .hi-blue[men] `$(female_i=\color{#0047AB}{0}, \, married_i=\color{#44C1C4}{1})$`
`$$\widehat{wage_i}=\hat{\beta_0}+\hat{\beta_2}$$`
]
]

3) .hi-orange[Unmarried] .hi-pink[women] `$(female_i=\color{#e64173}{1}, \, married_i=\color{#FFA500}{0})$`
`$$\widehat{wage_i}=\hat{\beta_0}+\hat{\beta_1}$$`

4) .hi-turquoise[Married] .hi[women] `$(female_i=\color{#e64173}{1}, \, married_i=\color{#44C1C4}{1})$`
`$$\widehat{wage_i}=\hat{\beta_0}+\hat{\beta_1}+\hat{\beta_2}+\hat{\beta_3}$$`

]
]

---

# Looking at the Data

```r
# get average wage for unmarried men
wages %>%
  filter(female == 0,
         married == 0) %>%
  summarize(mean = mean(wage))
```

```
##       mean
## 1 5.168023
```

```r
# get average wage for married men
wages %>%
  filter(female == 0,
         married == 1) %>%
  summarize(mean = mean(wage))
```

```
##       mean
## 1 7.983032
```
]
]
]
.pull-right[
.quitesmall[
.code60[

```r
# get average wage for unmarried women
wages %>%
  filter(female == 1,
         married == 0) %>%
  summarize(mean = mean(wage))
```

```
##       mean
## 1 4.611583
```

```r
# get average wage for married women
wages %>%
  filter(female == 1,
         married == 1) %>%
  summarize(mean = mean(wage))
```

```
##       mean
## 1 4.565909
```
]
]
]

---

# Two Dummies Interaction: Group Means

`$$\widehat{wage_i}=\hat{\beta_0}+\hat{\beta_1}female_i+\hat{\beta_2}married_i+\hat{\beta_3}(female_i \times married_i)$$`

|  | .hi-blue[Men] |.hi[Women] |
|--------|-----------|---------|
| .hi-orange[Unmarried]   | $5.17     | $4.61   |
| .hi-turquoise[Married] | $7.98     | $4.57   |

---

# Interactions Between Two Dummy Variables: In R I

```r
reg_dummies <- lm(wage ~ female + married + female:married, data = wages)
reg_dummies %>% tidy()
```

```
## # A tibble: 4 × 5
##   term           estimate std.error statistic  p.value
##   <chr>             <dbl>     <dbl>     <dbl>    <dbl>
## 1 (Intercept)       5.17      0.361     14.3  2.26e-39
## 2 female           -0.556     0.474     -1.18 2.41e- 1
## 3 married           2.82      0.436      6.45 2.53e-10
## 4 female:married   -2.86      0.608     -4.71 3.20e- 6
```
]

---

# Interactions Between Two Dummy Variables: In R II

```r
library(huxtable)
huxreg(reg_dummies,
       coefs = c("Constant" = "(Intercept)",
                 "Female" = "female",
                 "Married" = "married",
                 "Female * Married" = "female:married"),
       statistics = c("N" = "nobs",
                      "R-Squared" = "r.squared",
                      "SER" = "sigma"),
       number_format = 2)
```
]
]

.tiny[
<table class="huxtable" style="border-collapse: collapse; border: 0px; margin-bottom: 2em; margin-top: 2em; ; margin-left: auto; margin-right: auto;  " id="tab:unnamed-chunk-14">
<col><col><tr>
<th style="vertical-align: top; text-align: center; white-space: normal; border-style: solid solid solid solid; border-width: 0.8pt 0pt 0pt 0pt;    padding: 6pt 6pt 6pt 6pt; font-weight: normal;"></th><th style="vertical-align: top; text-align: center; white-space: normal; border-style: solid solid solid solid; border-width: 0.8pt 0pt 0.4pt 0pt;    padding: 6pt 6pt 6pt 6pt; font-weight: normal;">(1)</th></tr>
<tr>
<th style="vertical-align: top; text-align: left; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">Constant</th><td style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0.4pt 0pt 0pt 0pt;    padding: 6pt 6pt 6pt 6pt; font-weight: normal;">5.17 ***</td></tr>
<tr>
<th style="vertical-align: top; text-align: left; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;"></th><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">(0.36)   </td></tr>
<tr>
<th style="vertical-align: top; text-align: left; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">Female</th><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">-0.56    </td></tr>
<tr>
<th style="vertical-align: top; text-align: left; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;"></th><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">(0.47)   </td></tr>
<tr>
<th style="vertical-align: top; text-align: left; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">Married</th><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">2.82 ***</td></tr>
<tr>
<th style="vertical-align: top; text-align: left; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;"></th><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">(0.44)   </td></tr>
<tr>
<th style="vertical-align: top; text-align: left; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">Female * Married</th><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">-2.86 ***</td></tr>
<tr>
<th style="vertical-align: top; text-align: left; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;"></th><td style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0.4pt 0pt;    padding: 6pt 6pt 6pt 6pt; font-weight: normal;">(0.61)   </td></tr>
<tr>
<th style="vertical-align: top; text-align: left; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">N</th><td style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0.4pt 0pt 0pt 0pt;    padding: 6pt 6pt 6pt 6pt; font-weight: normal;">526       </td></tr>
<tr>
<th style="vertical-align: top; text-align: left; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">R-Squared</th><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">0.18    </td></tr>
<tr>
<th style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0.8pt 0pt;    padding: 6pt 6pt 6pt 6pt; font-weight: normal;">SER</th><td style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0.8pt 0pt;    padding: 6pt 6pt 6pt 6pt; font-weight: normal;">3.35    </td></tr>
<tr>
<th colspan="2" style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0.8pt 0pt 0pt 0pt;    padding: 6pt 6pt 6pt 6pt; font-weight: normal;"> *** p < 0.001;  ** p < 0.01;  * p < 0.05.</th></tr>
</table>

]
]
---

# 2 Dummies Interaction: Interpretting Coefficients

.smallest[
`$$\widehat{wage_i}=5.17-0.56 \, female_i + 2.82 \, married_i - 2.86 \, (female_i \times married_i)$$`

|  | .hi-blue[Men] |.hi[Women] |
|--------|-----------|---------|
| .hi-orange[Unmarried]   | $5.17     | $4.61   |
| .hi-turquoise[Married] | $7.98     | $4.57   |

]

.smallest[
- Wage for .hi-turquoise[married] .hi-blue[men]: `$\hat{\beta_0}+\hat{\beta_2}=5.17+2.82=7.98$`
]

.smallest[
- Wage for .hi-orange[unmarried] .hi[women]: `$\hat{\beta_0}+\hat{\beta_1}=5.17-0.56=4.61$`

]

.smallest[
- Wage for .hi-turquoise[married] .hi[women]: `$\hat{\beta_0}+\hat{\beta_1}+\hat{\beta_2}+\hat{\beta_3}=5.17-0.56+2.82-2.86=4.57$`
]

---

# 2 Dummies Interaction: Interpretting Coefficients

.smallest[
`$$\widehat{wage_i}=5.17-0.56 \, female_i + 2.82 \, married_i - 2.86 \, (female_i \times married_i)$$`

|  | .hi-blue[Men] |.hi[Women] |
|--------|-----------|---------|
| .hi-orange[Unmarried]   | $5.17     | $4.61   |
| .hi-turquoise[Married] | $7.98     | $4.57   |

]

]

.smallest[
- `$\hat{\beta_1}$`: **Difference** in wages between .hi-blue[men] and .hi[women] who are .hi-orange[unmarried]

]

.smallest[
- `$\hat{\beta_2}$`: **Difference** in wages between .hi-turquoise[married] and .hi-orange[unmarried] .hi-blue[men]

]
--

.smallest[
- `$\hat{\beta_3}$`: **Difference** in:
  - effect of **Marriage** on wages between .hi-blue[men] and .hi[women]
  - effect of **Gender** on wages between .hi-orange[unmarried] and .hi-turquoise[married] individuals

]
---

# Interactions Between Two Continuous Variables

---

# Interactions Between Two Continuous Variables

- Does the marginal effect of `$X_1$` on `$Y$` depend on what `$X_2$` is set to?

---

# Interactions Between Two Continuous Variables

`$$Y_i=\beta_0+\beta_1X_{1i}+\beta_2 X_{2i}+\beta_3 \color{#e64173}{(X_{1i} \times X_{2i})}$$`

- To interpret coefficients, compare changes after changing `$\color{#e64173}{\Delta X_{1i}}$` (holding `$X_2$` constant):

`$$Y_i+\color{#e64173}{\Delta Y_i}=\beta_0+\beta_1(X_1+\color{#e64173}{\Delta X_{1i}})\beta_2X_{2i}+\beta_3((X_{1i}+\color{#e64173}{\Delta X_{1i}}) \times X_{2i})$$`

- Take the difference to get:

`$$\begin{align*}
\Delta Y_i &= \beta_1 \Delta X_{1i}+ \beta_3 X_{2i} \Delta X_{1i}\\
\color{#6A5ACD}{\frac{\Delta Y_i}{\Delta X_{1i}}} &= \color{#6A5ACD}{\beta_1+\beta_3 X_{2i}}\\ 	
\end{align*}$$`

- .hi-purple[The effect of `\$X_1 \rightarrow Y_i\$` depends on `\$X_2\$`]
  - `$\color{#e64173}{\beta_3}$`: *increment* to the effect of `$X_1 \rightarrow Y_i$` for every 1 unit change in `$X_2$`

---

# Continuous Variables Interaction: Example

.smallest[
.bg-washed-green.b--dark-green.ba.bw2.br3.shadow-5.ph4.mt5[
.green[**Example**]: Do education and experience interact in their determination of wages?

`$$	\widehat{wage_i}=\hat{\beta_0}+\hat{\beta_1} \, educ_i+\hat{\beta_2} \, exper_i+\hat{\beta_3}(educ_i \times exper_i)$$`

]

- Estimated effect of education on wages depends on the amount of experience (and vice versa)!

`$$\frac{\Delta wage}{\Delta educ}=\hat{\beta_1}+\beta_3 \, exper_i$$`

`$$\frac{\Delta wage}{\Delta exper}=\hat{\beta_2}+\beta_3 \, educ_i$$`

- This is a type of nonlinearity (we will examine nonlinearities next lesson)

]

---

# Continuous Variables Interaction: In R I

```r
reg_cont <- lm(wage ~ educ + exper + educ:exper, data = wages)
reg_cont %>% tidy()
```

```
## # A tibble: 4 × 5
##   term        estimate std.error statistic  p.value
##   <chr>          <dbl>     <dbl>     <dbl>    <dbl>
## 1 (Intercept) -2.86      1.18       -2.42  1.58e- 2
## 2 educ         0.602     0.0899      6.69  5.64e-11
## 3 exper        0.0458    0.0426      1.07  2.83e- 1
## 4 educ:exper   0.00206   0.00349     0.591 5.55e- 1
```
]

---

# Continuous Variables Interaction: In R II

```r
library(huxtable)
huxreg(reg_cont,
       coefs = c("Constant" = "(Intercept)",
                 "Education" = "educ",
                 "Experience" = "exper",
                 "Education * Experience" = "educ:exper"),
       statistics = c("N" = "nobs",
                      "R-Squared" = "r.squared",
                      "SER" = "sigma"),
       number_format = 3)
```

]
]

.tiny[
<table class="huxtable" style="border-collapse: collapse; border: 0px; margin-bottom: 2em; margin-top: 2em; ; margin-left: auto; margin-right: auto;  " id="tab:unnamed-chunk-16">
<col><col><tr>
<th style="vertical-align: top; text-align: center; white-space: normal; border-style: solid solid solid solid; border-width: 0.8pt 0pt 0pt 0pt;    padding: 6pt 6pt 6pt 6pt; font-weight: normal;"></th><th style="vertical-align: top; text-align: center; white-space: normal; border-style: solid solid solid solid; border-width: 0.8pt 0pt 0.4pt 0pt;    padding: 6pt 6pt 6pt 6pt; font-weight: normal;">(1)</th></tr>
<tr>
<th style="vertical-align: top; text-align: left; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">Constant</th><td style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0.4pt 0pt 0pt 0pt;    padding: 6pt 6pt 6pt 6pt; font-weight: normal;">-2.860 *  </td></tr>
<tr>
<th style="vertical-align: top; text-align: left; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;"></th><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">(1.181)   </td></tr>
<tr>
<th style="vertical-align: top; text-align: left; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">Education</th><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">0.602 ***</td></tr>
<tr>
<th style="vertical-align: top; text-align: left; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;"></th><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">(0.090)   </td></tr>
<tr>
<th style="vertical-align: top; text-align: left; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">Experience</th><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">0.046    </td></tr>
<tr>
<th style="vertical-align: top; text-align: left; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;"></th><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">(0.043)   </td></tr>
<tr>
<th style="vertical-align: top; text-align: left; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">Education * Experience</th><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">0.002    </td></tr>
<tr>
<th style="vertical-align: top; text-align: left; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;"></th><td style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0.4pt 0pt;    padding: 6pt 6pt 6pt 6pt; font-weight: normal;">(0.003)   </td></tr>
<tr>
<th style="vertical-align: top; text-align: left; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">N</th><td style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0.4pt 0pt 0pt 0pt;    padding: 6pt 6pt 6pt 6pt; font-weight: normal;">526        </td></tr>
<tr>
<th style="vertical-align: top; text-align: left; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">R-Squared</th><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">0.226    </td></tr>
<tr>
<th style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0.8pt 0pt;    padding: 6pt 6pt 6pt 6pt; font-weight: normal;">SER</th><td style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0.8pt 0pt;    padding: 6pt 6pt 6pt 6pt; font-weight: normal;">3.259    </td></tr>
<tr>
<th colspan="2" style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0.8pt 0pt 0pt 0pt;    padding: 6pt 6pt 6pt 6pt; font-weight: normal;"> *** p < 0.001;  ** p < 0.01;  * p < 0.05.</th></tr>
</table>

]
]

---

# Continuous Variables Interaction: Marginal Effects

.smallest[
`$$\widehat{wages_i}=-2.860+0.602 \, educ_i + 0.047 \, exper_i + 0.002 \, (educ_i \times exper_i)$$`
]

| Experience | `$\displaystyle\frac{\Delta wage}{\Delta educ}=\hat{\beta_1}+\hat{\beta_3} \, exper$` |
|------------|------------------------------------------------|
| 5 years | `$0.602+0.002(5)=0.612$` |
| 10 years | `$0.602+0.002(10)=0.622$` |
| 15 years | `$0.602+0.002(15)=0.632$` |
]

- Marginal effect of education `$\rightarrow$` wages **increases** with more experience

---

# Continuous Variables Interaction: Marginal Effects

.smallest[
`$$\widehat{wages_i}=-2.860+0.602 \, educ_i + 0.047 \, exper_i + 0.002 \, (educ_i \times exper_i)$$`
]

| Education | `$\displaystyle\frac{\Delta wage}{\Delta exper}=\hat{\beta_2}+\hat{\beta_3} \, educ$` |
|------------|------------------------------------------------|
| 5 years | `$0.047+0.002(5)=0.057$` |
| 10 years | `$0.047+0.002(10)=0.067$` |
| 15 years | `$0.047+0.002(15)=0.077$` |
]

- Marginal effect of experience `$\rightarrow$` wages **increases** with more education

- If you want to estimate the marginal effects more precisely, and graph them, see the appendix in [today’s class page](/content/3.7-content)