Multivariate regressions

class: center, middle, inverse, title-slide

# Multivariate regressions
## Lecture 9
### Louis SIRUGUE
### CPES 2 - Fall 2022

---

### Quick reminder

#### 1. Joint distribution

<center>The joint distribution shows the possible values and associated frequencies for two variables simultaneously</center>

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]

---

### Quick reminder

#### 1. Joint distribution

<center><h4> &#10140; When describing a joint distribution, we're interested in the relationship between the two variables </h4></center>

<ul>
 <li>The covariance quantifies the joint deviation of two variables from their respective mean</li>
 <ul>
 <li>It can take values from $-\infty$ to $\infty$ and depends on the unit of the data</li>
 </ul>
</ul>
 
$$ \text{Cov}(x, y) = \frac{1}{N}\sum_{i=1}^{N}(x_i − \bar{x})(y_i − \bar{y})$$

<ul>
 <li>The correlation is the covariance of two variables divided by the product of their standard deviation</li>
 <ul>
 <li>It can take values from $-1$ to $1$ and is independent from the unit of the data</li>
 </ul>
</ul>
 
 `$$\text{Corr}(x, y) = \frac{\text{Cov}(x, y)}{\text{SD}(x)\times\text{SD}(y)}$$`
 
---

### Quick reminder

#### 2. Regression

.pull-left[

<img src="slides_files/figure-html/unnamed-chunk-4-1.png" width="100%" style="display: block; margin: auto;" />

```
## 
## Call:
## lm(formula = y ~ x, data = data)
## 
## Coefficients:
## (Intercept)            x  
##    -0.09129      1.01546
```
]

.pull-right[

* This can be expressed with the **regression equation:**
 
`$$y_i = \hat{\alpha} + \hat{\beta}x_i + \hat{\varepsilon_i}$$`

* Where `$\hat{\alpha}$` is the **intercept** and `$\hat{\beta}$` the **slope** of the **line** `$\hat{y_i} = \hat{\alpha} + \hat{\beta}x_i$`, and `$\hat{\varepsilon_i}$` the **distances** between the points and the line

`$$\hat{\beta} = \frac{\text{Cov}(x_i, y_i)}{\text{Var}(x_i)}$$`

`$$\hat{\alpha} = \bar{y} - \hat{\beta} \times\bar{x}$$`

* `$\hat{\alpha}$` and `$\hat{\beta}$` minimize `$\hat{\varepsilon_i}$`

]

---

### Quick reminder

#### 3. Binary variables

.pull-left[
<center>Binary dependent variables</center>

<ul>
 <li>The fitted values can be viewed as probabilities</li>
 <ul>
 <li>$\hat{\beta}$ is the expected increase in the probability that $y = 1$ for a one unit increase in $x$</li>
 </ul>
</ul>

<ul>
 <ul>
 <li>We call that a Linear Probability Model</li>
 </ul>
</ul>

]

.pull-right[
<center>Binary independent variables</center>

<ul>
 <li>The $x$ variable should be viewed as a dummy 0/1</li>
 <ul>
 <li>$\hat{\beta}$ is the difference between the average $y$ for the group $x = 1$ and the group $x = 0$</li>
 </ul>
</ul>

<img src="slides_files/figure-html/unnamed-chunk-7-1.png" width="90%" style="display: block; margin: auto;" />
]

---
class: inverse, hide-logo

### Warm up practice

#### 1) Open the `asec.csv` data containing sex, race, weekly work hours, and annual earnings ($)

#### 2) Regress the earnings variable on the sex variable

#### 3) Check that the slope coefficient is equal to the difference between male and female average earnings

<center><h3>You've got 10 minutes!</h3></center>

---
class: inverse, hide-logo

### Solution

#### 1) Open the `asec.csv` data containing sex, race, weekly work hours, and annual earnings ($)

```r
asec <- read.csv("asec.csv")
```

#### 2) Regress the earnings variable on the sex variable

```r
lm(Earnings ~ Sex, asec)
```

```
## 
## Call:
## lm(formula = Earnings ~ Sex, data = asec)
## 
## Coefficients:
## (Intercept)      SexMale  
##       50915        21612
```

---
class: inverse, hide-logo

### Solution

#### 3) Check that the slope coefficient is equal to the difference between male and female average earnings

```r
asec %>% 
  
  # Group the data by sex
  group_by(Sex) %>%
  
  # Summarise mean earnings -> 2x2 dataset
  summarise(Mean = mean(Earnings)) %>%
  
  # Put means in columns instead of rows -> 1x2 dataset
  pivot_wider(names_from = Sex, values_from = Mean) %>%
  
  # Compute the difference in means
  mutate(Difference = Male - Female)
```

```
## # A tibble: 1 x 3
## Female Male Difference
## <dbl> <dbl> <dbl>
## 1 50915. 72527. 21612.
```

---

<h3>Today: Multivariate regressions</h3>

.pull-left[

<ul style = "margin-left:1.5cm;list-style: none">
 <li>1. Adding variables</li>
 <ul style = "list-style: none">
 <li>1.1. Continuous variables</li>
 <li>1.2. Discrete variables</li>
 </ul>
</ul>

<ul style = "margin-left:1.5cm;list-style: none">
 <li>2. Control variables</li>
 <ul style = "list-style: none">
 <li>2.1. Motivation</li>
 <li>2.2. Discrete controls</li>
 <li>2.3. Continuous controls</li>
 </ul>
</ul>

]

.pull-right[

<ul style = "margin-left:-1cm;list-style: none">
 <li>3. Interactions</li>
 <ul style = "list-style: none">
 <li>3.1. Motivation</li>
 <li>3.2. Discrete interactions</li>
 <li>3.3. Continuous interactions</li>
 </ul>
</ul>

<ul style = "margin-left:-1cm;list-style: none"><li>4. Wrap up!</li></ul>
]

---

<h3>Today: Multivariate regressions</h3>

.pull-left[

### 1. Adding variables

#### 1.1. Continuous variables

.pull-left[

* So far we focused on two-variable relationships

]

.pull-right[

* What about three variable? *(pivot the plot)*

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]
 
---

### 1. Adding variables

#### 1.1. Continuous variables

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]

.pull-right[

<ul>
 <li>In this case we must fit a plane</li>
 <ul>
 <li>It is characterized by 3 parameters</li>
 <li>And can be expressed as:</li>
 </ul>
</ul>

`$$y_i = \hat{\alpha} + \hat{\beta_1}x_{1,i} + \hat{\beta_2}x_{2,i} + \hat{\varepsilon_i}$$`

<ul>
 <li>$\hat{\alpha}$ is still the intercept</li>
 <ul>
 <li>The value of $\hat{y}$ (height) when $x_1 = x_2= 0$</li>
 </ul>
</ul>

<ul>
 <li>And now there are 2 slopes</li>
 <ul>
 <li>$\hat{\beta_1}$ along the $x_1$ axis and $\hat{\beta_2}$ along the $x_2$ axis</li>
 </ul>
 </ul>
 
]
 
---

### 1. Adding variables

#### 1.1. Continuous variables

<ul>
 <li>The same applies with more than 2 independent variables</li>
 <ul>
 <li>We would fit a hyperplane with as many dimension as $x$ variables</li>
 <li>We would obtain one intercept and one slope per $x$ variables</li>
 </ul>
</ul>

`$$y_i = \hat{\alpha} + \hat{\beta_1}x_{1, i} + \hat{\beta_2}x_{2, i} +...+ \hat{\beta_k}x_{k, i}+\hat{\varepsilon_i}$$`

<ul>
 <li>We can estimate the parameters of these hyperplanes in lm()</li>
 <ul>
 <li>Additional variables must be introduced after a + sign</li>
 </ul>
</ul>

```r
lm(ige ~ gini + third_variable, ggcurve)
```

```
## 
## Call:
## lm(formula = ige ~ gini + third_variable, data = ggcurve)
## 
## Coefficients:
##    (Intercept)            gini  third_variable  
##       -0.09536         0.98153         0.01122
```

---

### 1. Adding variables

#### 1.2. Discrete variables

<ul>
 <li>So far we've been working with binary categorical variables:</li>
 <ul>
 <li>Accepted vs. Rejected, Male vs. Female</li>
 <li>But what about discrete variables with more than two categories?</li>
 </ul>
</ul>
 
--

.pull-left[

* Take for instance the race variable:

```r
asec %>% 
  group_by(Race) %>% 
  tally()
```

```
## # A tibble: 3 x 2
## Race n
## <chr> <int>
## 1 Black 6835
## 2 Other 6950
## 3 White 50551
```

]

.pull-right[

<center>How can we use this variable</center> 
<center>as an independent variable</center> 
<center>in our regression framework?</center>

]

---

### 1. Adding variables

#### 1.2. Discrete variables

<ul>
 <li>Remember how we converted our 2-category variable into 1 dummy variable</li>
 <ul>
 <li>We can convert an n-category variable into n-1 dummy variables</li>
 </ul>
</ul>

--

.pull-left[

<table class="table table-hover table-condensed" style="width: auto !important; margin-left: auto; margin-right: auto;">
<caption> </caption>
 <thead>
 <tr>
 <th style="text-align:left;"> Sex </th>
 <th style="text-align:right;"> Male </th>
 <th style="text-align:left;"> </th>
 <th style="text-align:left;"> Race </th>
 <th style="text-align:right;"> Black </th>
 <th style="text-align:right;"> Other </th>
 </tr>
 </thead>
<tbody>
 <tr>
 <td style="text-align:left;"> Female </td>
 <td style="text-align:right;"> 0 </td>
 <td style="text-align:left;width: 3em; "> </td>
 <td style="text-align:left;"> White </td>
 <td style="text-align:right;"> 0 </td>
 <td style="text-align:right;"> 0 </td>
 </tr>
 <tr>
 <td style="text-align:left;"> Female </td>
 <td style="text-align:right;"> 0 </td>
 <td style="text-align:left;width: 3em; "> </td>
 <td style="text-align:left;"> White </td>
 <td style="text-align:right;"> 0 </td>
 <td style="text-align:right;"> 0 </td>
 </tr>
 <tr>
 <td style="text-align:left;"> Female </td>
 <td style="text-align:right;"> 0 </td>
 <td style="text-align:left;width: 3em; "> </td>
 <td style="text-align:left;"> Black </td>
 <td style="text-align:right;"> 1 </td>
 <td style="text-align:right;"> 0 </td>
 </tr>
 <tr>
 <td style="text-align:left;"> Male </td>
 <td style="text-align:right;"> 1 </td>
 <td style="text-align:left;width: 3em; "> </td>
 <td style="text-align:left;"> Black </td>
 <td style="text-align:right;"> 1 </td>
 <td style="text-align:right;"> 0 </td>
 </tr>
 <tr>
 <td style="text-align:left;"> Male </td>
 <td style="text-align:right;"> 1 </td>
 <td style="text-align:left;width: 3em; "> </td>
 <td style="text-align:left;"> Other </td>
 <td style="text-align:right;"> 0 </td>
 <td style="text-align:right;"> 1 </td>
 </tr>
 <tr>
 <td style="text-align:left;"> Male </td>
 <td style="text-align:right;"> 1 </td>
 <td style="text-align:left;width: 3em; "> </td>
 <td style="text-align:left;"> Other </td>
 <td style="text-align:right;"> 0 </td>
 <td style="text-align:right;"> 1 </td>
 </tr>
</tbody>
</table>
]

.pull-right[

***&#10140; But why do we omit one category every time?***

<ul>
 <li>Because it would be redundant</li>
 <li>We only need 2 dummies for 3 groups:</li>
 <ul>
 <li>White: Black = 0 & Other = 0</li>
 <li>Black: Black = 1 & Other = 0</li>
 <li>Other: Black = 0 & Other = 1</li>
 </ul>
</ul>

<ul>
 </li>$\hat{\alpha}$ is the expected $\hat{y}$ when $x_k=0 \:\forall k$</li>
 <ul>
 <li>Thus is does the job for the omitted groups!</li>
 <li>This group is called the reference group</li>
 <li>$\hat{\beta_k}$ are interpreted relative to that group</li>
 </ul>
</ul>

]

---

### 1. Adding variables

#### 1.2. Discrete variables

.pull-left[
<center>2-category variable</center>

]

.pull-right[

<center>3-category variable</center>

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]

---

### 1. Adding variables

#### 1.2. Discrete variables

* This plane can be expressed as:
 
`$$\text{Earnings}_i = \hat{\alpha} + \hat{\beta_1} 1\{\text{Race}_i = \text{Other}\} + \hat{\beta_2} 1\{\text{Race}_i = \text{White}\} + \hat{\varepsilon_i}$$`

<ul>
 <li>And the average incomes for each group equal:</li>
 <ul>
 <li>Black: $\hat{\alpha} + 0\hat{\beta_1} + 0\hat{\beta_2} = \hat{\alpha}$</li>
 <li>Other: $\hat{\alpha} + 1\hat{\beta_1} + 0\hat{\beta_2} = \hat{\alpha} + \hat{\beta_1}$</li>
 <li>White: $\hat{\alpha} + 0\hat{\beta_1} + 1\hat{\beta_2} = \hat{\alpha} + \hat{\beta_2}$</li>
 </ul>
</ul>

.pull-left[

```
## 
## Call:
## lm(formula = Earnings ~ Race, data = asec)
## 
## Coefficients:
## (Intercept)    RaceOther    RaceWhite  
##       50577        17477        12303
```
]

.pull-right[

<table class="table table-hover table-condensed" style="width: auto !important; margin-left: auto; margin-right: auto;">
<caption>Average by group</caption>
 <thead>
 <tr>
 <th style="text-align:left;"> Race </th>
 <th style="text-align:right;"> Mean earnings </th>
 </tr>
 </thead>
<tbody>
 <tr>
 <td style="text-align:left;"> Black </td>
 <td style="text-align:right;"> 50577.49 </td>
 </tr>
 <tr>
 <td style="text-align:left;"> Other </td>
 <td style="text-align:right;"> 68054.63 </td>
 </tr>
 <tr>
 <td style="text-align:left;"> White </td>
 <td style="text-align:right;"> 62880.49 </td>
 </tr>
</tbody>
</table>
]

---

### 1. Adding variables

#### 1.2. Discrete variables

<ul>
 <li>By default, lm() sorts categories by alphabetical order</li>
 <ul>
 <li>So every coefficient should be interpreted relative to the group which is first alphabetically</li>
 </ul>
</ul>

<ul>
 <li>But usually this is not the most intuitive</li>
 <ul>
 <li>You may want everything to be relative to the majority group</li>
 <li>Or to any group that has reasons to be the reference</li>
 </ul>
</ul>

<ul>
 <li>The relevel() function allows you to change the reference category</li>
 <ul>
 <li>But it works only on factor variables</li>
 </ul>
</ul>

.pull-left[

```r
asec <- asec %>% 
 mutate(Race_fct = relevel(as.factor(Race), 
 "White"))

lm(Earnings ~ Race_fct, asec)
```
]

.pull-right[

```
## 
## Call:
## lm(formula = Earnings ~ Race_fct, data = asec)
## 
## Coefficients:
##   (Intercept)  Race_fctBlack  Race_fctOther  
##         62880         -12303           5174
```
]

---

### 1. Adding variables

#### 1.2. Discrete variables

<ul>
 <li>The factor class is made for variables whose values indicate different groups</li>
 <ul>
 <li>Values are just arbitrary group classifiers</li>
 </ul>
</ul>

```r
individuals <- as.factor(c(1, 2, 3, 4, 5))
individuals[1]
```

```
## [1] 1
## Levels: 1 2 3 4 5
```

<ul>
 <li>With factors, R understands that the different values do not mean anything</li>
 <ul>
 <li>And applying standard operations to factors does not make sense</li>
 </ul>
</ul>

```r
individuals * 2
```

```
## Warning in Ops.factor(individuals, 2): '*' not meaningful for factors
```

```
## [1] NA NA NA NA NA
```

---

### 1. Adding variables

#### 1.2. Discrete variables

* What you can also do is create the dummies yourself:

```r
asec <- asec %>%
 mutate(Black = as.numeric(Race == "Black"),
 Other = as.numeric(Race == "Other"))
```

```r
lm(Earnings ~ Black + Other, asec)
```

```
## 
## Call:
## lm(formula = Earnings ~ Black + Other, data = asec)
## 
## Coefficients:
## (Intercept)        Black        Other  
##       62880       -12303         5174
```

<center>&#10140; This might be the safest option</center>

---

### 1. Adding variables

#### 1.2. Discrete variables

* But a categorical variable must not be introduced as numeric in lm()

```r
asec <- asec %>%
 mutate(num_cat = case_when(Race == "White" ~ 0,
 Race == "Black" ~ 1,
 Race == "Other" ~ 2))
```

```r
lm(Earnings ~ num_cat, asec)
```

```
## 
## Call:
## lm(formula = Earnings ~ num_cat, data = asec)
## 
## Coefficients:
## (Intercept)      num_cat  
##     62093.8        119.6
```

<center>&#10140; lm() used our categorical variable as a continuous variable</center>

---

### 1. Adding variables

#### 1.2. Discrete variables

* Use the factor class

```r
asec <- asec %>%
 mutate(fac_cat = as.factor(num_cat))
```

```r
lm(Earnings ~ fac_cat, asec)
```

```
## 
## Call:
## lm(formula = Earnings ~ fac_cat, data = asec)
## 
## Coefficients:
## (Intercept)     fac_cat1     fac_cat2  
##       62880       -12303         5174
```

<center>&#10140; Converting all your categorical variables into factors is also a safe option</center>

---

<h3>Overview</h3>

.pull-left[

<ul style = "margin-left:1.5cm;list-style: none">
 <li>1. Adding variables &#10004;</li>
 <ul style = "list-style: none">
 <li>1.1. Continuous variables</li>
 <li>1.2. Discrete variables</li>
 </ul>
</ul>

]

.pull-right[

<ul style = "margin-left:-1cm;list-style: none"><li>4. Wrap up!</li></ul>
]

---

<h3>Overview</h3>

.pull-left[

]

---

### 2. Control variables

#### 2.1. Motivation

<ul>
 <li>But why would we include additional variables in our regressions?</li>
 <ul>
 <li>The main reason is to control for potential confounders</li>
 </ul>
</ul>

<ul>
 <li>Consider estimating the relationship between income and exposure air pollution in the Paris region</li>
</ul>

`$$\text{Pollution}_i = \hat{\alpha_1} + \hat{\beta_1} \text{Income}_i + \hat{\varepsilon_i}$$`

<ul>
 <li>You would probably expect that $\hat{\beta_1} < 0$</li>
 <ul>
 <li>Meaning that higher income earners live in less polluted areas</li>
 <li>But the closer from Paris the higher the rents and the closer the ring-road</li>
 <li>This phenomenon might counteract this effect and pull $\hat{\beta_1}$ towards 0</li>
 </ul>
</ul>

<ul>
 <li>But how to remove the impact that distance from Paris has on the relationship?</li>
 <ul>
 <li>Including it in the regression would make the corresponding coefficient absorb the confounding effect</li>
 <li>In that case we would call distance a control variable</li>
 </ul>
</ul>

`$$\text{Pollution}_i = \hat{\alpha_2} + \hat{\beta_2} \text{Income}_i + \hat{\beta_3} \text{Distance}_i + \hat{\epsilon_i}$$`
---

### 2. Control variables

#### 2.2. Discrete

<ul>
 <li>The most common control variable is probably sex/gender</li>
 <ul>
 <li>It may play a role in the relationship between earnings and hours worked for instance</li>
 <li>The fact that women work part time more often and earn less contribute to the relationship</li>
 <li>Just like distance did in the previous example</li>
 </ul>
</ul>

---

### 2. Control variables

#### 2.2. Discrete

---

### 2. Control variables

#### 2.2. Discrete

---

### 2. Control variables

#### 2.2. Discrete

---

### 2. Control variables

#### 2.2. Discrete

&#10140; The relationship is indeed inflated by the sex variable

.pull-left[

<ul>
 <li>Because being a male is positively correlated with both $x$ and $y$</li>
</ul>

<ul>
 <li>Controlling for sex would solve that problem by absorbing this effect</li>
</ul>

<ul>
 <li>Controlling for a discrete variable amounts to allow one intercept per category</li>
</ul>

<ul>
 <li>Giving two parallel fitted lines which are the intersections of the plane and the scatterplots</li>
</ul>

]

.pull-right[

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---

### 2. Control variables

#### 2.2. Discrete

&#10140; The relationship is indeed inflated by the sex variable

.pull-left[

<ul>
 <li>Because being a male is positively correlated with both $x$ and $y$</li>
</ul>

<ul>
 <li>Controlling for sex would solve that problem by absorbing this effect</li>
</ul>

<ul>
 <li>Controlling for a discrete variable amounts to allow one intercept per category</li>
</ul>

<ul>
 <li>Giving two parallel fitted lines which are the intersections of the plane and the scatterplots</li>
</ul>

]

.pull-right[

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]

---

### 2. Control variables

#### 2.2. Discrete

`$$\text{Earnings}_i = \hat{\alpha} + \hat{\beta_1}\text{Hours}_i  + \hat{\beta_2}1\{\text{Sex}_i = \text{Male}\} + \hat{\varepsilon_i}$$`

```
## (Intercept)       Hours     SexMale 
##  1019.34269    11.86326   200.98782
```

.left-column[
<img src="slides_files/figure-html/unnamed-chunk-40-1.png" width="85%" style="display: block; margin: auto auto auto 0;" />
]

.right-column[

<center>Graphical counterpart</center>

`$\hat{\alpha}$`: Intercept of the reference group

`$\hat{\beta_1}$`: Common slope

`$\hat{\beta_2}$`: Gap between the two lines

`$\hat{\alpha} +\hat{\beta_2}$`: Intercept of the other group  
]

---

### 2. Control variables

#### 2.2. Discrete

<ul>
 <li>We can obtain this common slope by:</li>
 <ol>
 <li>Demeaning earnings and hours by group</li>
 <li>Regressing the demeaned earnings on the hours</li>
 </ol>
</ul>

---

### 2. Control variables

#### 2.2. Discrete

<ul>
 <li>Note that once we control for third variable</li>
 <ol>
 <li>As we move along the x axis, this third variable remains constant</li>
 <li>Here, as the number of hours increases the probability to be a male does not increase anymore</li>
 </ol>
</ul>

---

### 2. Control variables

#### 2.3. Continuous

<ul>
 <li>The same idea apply when we control for continuous variables</li>
 <ul>
 <li>Including it in the regression allows to account for another dimension</li>
 <li>Such that when $x$ moves this variable remains constant</li>
 <li>This nets out the relationship between $x$ and $y$ from the potential confounding effect of this variable</li>
 <li>This is why we call it controlling for something</li>
 </ul>
</ul>

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---
class: inverse, hide-logo

### Practice

#### 1) Using the `asec` data, regress (yearly) earnings on (weekly) hours worked

#### 2) Regress earnings on hours worked controlling for sex

#### 3) Interpret the difference between the results from 1) and 2)

<center><h3>You've got 8 minutes!</h3></center>

---
class: inverse, hide-logo

### Solution

#### 1) Using the `asec` data, regress (yearly) earnings on (weekly) hours worked

```r
lm(Earnings ~ Hours, asec)$coefficients
```

```
## (Intercept)       Hours 
##   -20038.85     2077.79
```

#### 2) Regress earnings on hours worked controlling for sex

```r
lm(Earnings ~ Hours + Sex, asec)$coefficients
```

```
## (Intercept)       Hours     SexMale 
##  -22296.150    1953.829   13794.385
```

---
class: inverse, hide-logo

### Solution

#### 3) Interpret the difference between the results from 1) and 2)

.pull-left[
<ul><li>The slope is still positive less steep</li></ul>

<ul><ul><li>In the first regression as the number of hours increases the probability to be a male does increase as well</li></ul></ul>

<ul><ul><li>Because males tend to earn more this contributes to the positive relationship between Hours and Earnings</li></ul></ul>

<ul><ul><li>In the second regression, controlling for sex allows to maintain the probability to be a male constant along the hour axis to remove this effect</li></ul></ul>

]

---

<h3>Overview</h3>

.pull-left[

<ul style = "margin-left:1.5cm;list-style: none">
 <li>2. Control variables &#10004;</li>
 <ul style = "list-style: none">
 <li>2.1. Motivation</li>
 <li>2.2. Discrete controls</li>
 <li>2.3. Continuous controls</li>
 </ul>
</ul>

]

.pull-right[

<ul style = "margin-left:-1cm;list-style: none"><li>4. Wrap up!</li></ul>
]

---

<h3>Overview</h3>

.pull-left[

]

.pull-right[

---

### 3. Interactions

#### 3.1. Motivation

<ul>
 <li>Now we know how to remove the confounding effect of a third variable by controlling for it</li>
 <ul>
 <li>But what if the main relationship varies depending on the value of the third variable?</li>
 </ul>
</ul>

* Let's get back to the previous example

`$$\text{Pollution}_i = \hat{\alpha} + \hat{\beta_1} \text{Income}_i + \hat{\beta_2} \text{Distance}_i + \hat{\epsilon_i}$$`

<ul>
 <li>The equation imposes that the effect of income on pollution is constant: $\hat{\beta_2}$</li>
 <ul>
 <li>But what if the relationship was actually not the same close to Paris than further away?</li>
 <li>Maybe that the closer from Paris the larger the effect (higher segregation, ...)</li>
 </ul>
</ul>

<ul>
 <li>But how to capture how the relationship between income and pollution varies with distance?</li>
 <ul>
 <li>We should allow for it in the equation!</li>
 <li>By adding a term that depends both on income and distance</li>
 <li>What we use is their product, and we call that an interaction</li>
 </ul>
</ul>

`$$\text{Pollution}_i = \hat{\alpha_2} + \hat{\beta_3} \text{Income}_i + \hat{\beta_4} \text{Distance}_i+ \hat{\beta_5} (\text{Distance}_i\times\text{Income}_i) + \hat{\epsilon_i}$$`

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### 3. Interactions

#### 3.2. Discrete

<ul>
 <li>Take for instance the following relationship between household income and the number of children</li>
</ul>

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### 3. Interactions

#### 3.2. Discrete

<ul>
 <li>Take for instance the following relationship between household income and the number of children</li>
 <ul>
 <li>The level of education seems to play a role in the relationship</li>
 </ul>
</ul>

---

### 3. Interactions

#### 3.2. Discrete

---

### 3. Interactions

#### 3.2. Discrete

<ul>
 <li>This is because the relationship between income and children varies with education</li>
 <ul>
 <li>Interacting income with education allows to account for that</li>
 <li>Like controlling allows for different intercepts, interacting allows for different slopes</li>
 </ul>
</ul>

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### 3. Interactions

#### 3.2. Discrete