Univariate regressions

class: center, middle, inverse, title-slide

# Univariate regressions
## Lecture 8
### Louis SIRUGUE
### CPES 2 - Fall 2022

---

### Part I recap

#### Import data

```r
fb <- read.csv("C:/User/Documents/ligue1.csv", encoding = "UTF-8")
```

#### Class

```r
is.numeric("1.6180339") # What would be the output?
```

```
## [1] FALSE
```

#### Subsetting

```r
fb$Home[3]
```

```
## [1] "Troyes"
```

---

### Part I recap

#### Distributions

* The **distribution** of a variable documents all its possible values and how frequent they are

* We can describe a distribution with:
  
---

### Part I recap

#### Distributions

* The **distribution** of a variable documents all its possible values and how frequent they are

* We can describe a distribution with:
  * Its **central tendency**

---

### Part I recap

#### Distributions

* The **distribution** of a variable documents all its possible values and how frequent they are

* We can describe a distribution with:
  * Its **central tendency**
  * And its **spread**
  
---

### Part I recap

#### Central tendency

.pull-left[

* The **mean** is the sum of all values divided by the number of observations

`$$\bar{x} = \frac{1}{N}\sum_{i = 1}^Nx_i$$`

]

.pull-right[

* The **median** is the value that divides the (sorted) distribution into two groups of equal size

`$$\text{Med}(x) = \begin{cases} x[\frac{N+1}{2}] & \text{if } N \text{ is odd}\\
\frac{x[\frac{N}{2}]+x[\frac{N}{2}+1]}{2} & \text{if } N \text{ is even}
\end{cases}$$`

]

#### Spread

.pull-left[

* The **standard deviation** is square root of the average squared deviation from the mean

`$$\text{SD}(x) = \sqrt{\text{Var}(x)} = \sqrt{\frac{1}{N}\sum_{i = 1}^N(x_i-\bar{x})^2}$$`
 
]

.pull-right[

* The **interquartile range** is the difference between the maximum and the minimum value from the middle half of the distribution

$$\text{IQR} = Q_3 - Q_1 $$

]

---

### Part I recap

#### Inference

<ul> 
 <li>In Statistics, we view variables as a given realization of a data generating process</li>
 <ul>
 <li>Hence, the mean is what we call an empirical moment, which is an estimation...</li>
 <li>... of the expected value, the theoretical moment of the DGP we're interested in</li>
 </ul>
</ul>

<ul> 
 <li>To know how confident we can be in this estimation, we need to compute a confidence interval</li>
</ul>

`$$[\bar{x} - t_{n-1, \:97.5\%}\times\frac{\text{SD}(x)}{\sqrt{n}}; \:\bar{x} + t_{n-1, \:97.5\%}\times\frac{\text{SD}(x)}{\sqrt{n}}]$$`

<ul>
 <ul>
 <li>It gets larger as the variance of the distribution of $x$ increases</li>
 <li>And gets smaller as the sample size $n$ increases</li>
 </ul>
</ul>

---

### Part I recap

#### Packages

```r
library(dplyr)
```

#### Main dplyr functions

.left-column[
<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;"> Function </th>
 <th style="text-align:left;"> Meaning </th>
 </tr>
 </thead>
<tbody>
 <tr>
 <td style="text-align:left;"> mutate() </td>
 <td style="text-align:left;"> Modify or create a variable </td>
 </tr>
 <tr>
 <td style="text-align:left;"> select() </td>
 <td style="text-align:left;"> Keep a subset of variables </td>
 </tr>
 <tr>
 <td style="text-align:left;"> filter() </td>
 <td style="text-align:left;"> Keep a subset of observations </td>
 </tr>
 <tr>
 <td style="text-align:left;"> arrange() </td>
 <td style="text-align:left;"> Sort the data </td>
 </tr>
 <tr>
 <td style="text-align:left;"> group_by() </td>
 <td style="text-align:left;"> Group the data </td>
 </tr>
 <tr>
 <td style="text-align:left;"> summarise() </td>
 <td style="text-align:left;"> Summarizes variables into 1 observation per group </td>
 </tr>
</tbody>
</table>
]

.right-column[
<img style = "margin-top:0cm; margin-left:1.5cm;" src = "pipe.png" width = "180"/>
]

---

### Part I recap

#### Merge data

```r
a <- data.frame(x = c(1, 2, 3), y = c("a", "b", "c"))
b <- data.frame(x = c(4, 5, 6), y = c("d", "e", "f"))
c <- data.frame(x = 1:6, z = c("alpha", "bravo", "charlie", "delta", "echo", "foxtrot"))
```

```r
a %>% bind_rows(b) %>% left_join(c, by = "x")
```

<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:right;"> x </th>
 <th style="text-align:left;"> y </th>
 <th style="text-align:left;"> z </th>
 </tr>
 </thead>
<tbody>
 <tr>
 <td style="text-align:right;"> 1 </td>
 <td style="text-align:left;"> a </td>
 <td style="text-align:left;"> alpha </td>
 </tr>
 <tr>
 <td style="text-align:right;"> 2 </td>
 <td style="text-align:left;"> b </td>
 <td style="text-align:left;"> bravo </td>
 </tr>
 <tr>
 <td style="text-align:right;"> 3 </td>
 <td style="text-align:left;"> c </td>
 <td style="text-align:left;"> charlie </td>
 </tr>
 <tr>
 <td style="text-align:right;"> 4 </td>
 <td style="text-align:left;"> d </td>
 <td style="text-align:left;"> delta </td>
 </tr>
 <tr>
 <td style="text-align:right;"> 5 </td>
 <td style="text-align:left;"> e </td>
 <td style="text-align:left;"> echo </td>
 </tr>
 <tr>
 <td style="text-align:right;"> 6 </td>
 <td style="text-align:left;"> f </td>
 <td style="text-align:left;"> foxtrot </td>
 </tr>
</tbody>
</table>

---

### Part I recap

#### Reshape data

```r
data %>% pivot_longer(c(share_tertiary, share_gdp), names_to = "Variable", values_to = "Value")
```

<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;"> country </th>
 <th style="text-align:right;"> year </th>
 <th style="text-align:left;"> Variable </th>
 <th style="text-align:right;"> Value </th>
 </tr>
 </thead>
<tbody>
 <tr>
 <td style="text-align:left;"> FRA </td>
 <td style="text-align:right;"> 2015 </td>
 <td style="text-align:left;"> share_tertiary </td>
 <td style="text-align:right;"> 44.69 </td>
 </tr>
 <tr>
 <td style="text-align:left;"> FRA </td>
 <td style="text-align:right;"> 2015 </td>
 <td style="text-align:left;"> share_gdp </td>
 <td style="text-align:right;"> 3.40 </td>
 </tr>
 <tr>
 <td style="text-align:left;"> USA </td>
 <td style="text-align:right;"> 2015 </td>
 <td style="text-align:left;"> share_tertiary </td>
 <td style="text-align:right;"> 46.52 </td>
 </tr>
 <tr>
 <td style="text-align:left;"> USA </td>
 <td style="text-align:right;"> 2015 </td>
 <td style="text-align:left;"> share_gdp </td>
 <td style="text-align:right;"> 3.21 </td>
 </tr>
</tbody>
</table>

---

### Part I recap

<center><h4> The 3 core components of the ggplot() function </h4></center>

<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;"> Component </th>
 <th style="text-align:center;"> Contribution </th>
 <th style="text-align:center;"> Implementation </th>
 </tr>
 </thead>
<tbody>
 <tr>
 <td style="text-align:left;"> Data </td>
 <td style="text-align:center;"> Underlying values </td>
 <td style="text-align:center;"> ggplot(data, | data %&gt;% ggplot(., </td>
 </tr>
 <tr>
 <td style="text-align:left;"> Mapping </td>
 <td style="text-align:center;"> Axis assignment </td>
 <td style="text-align:center;"> aes(x = V1, y = V2, ...)) </td>
 </tr>
 <tr>
 <td style="text-align:left;"> Geometry </td>
 <td style="text-align:center;"> Type of plot </td>
 <td style="text-align:center;"> + geom_point() + geom_line() + ... </td>
 </tr>
</tbody>
</table>

* Any **other element** should be added with a **`+` sign**

```r
ggplot(data, aes(x = V1, y = V2)) + 
  geom_point() + geom_line() +
  anything_else()
```

---

### Part I recap

.pull-left[

<center><h4> Main customization tools </h4></center>
<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;"> Item to customize </th>
 <th style="text-align:left;"> Main functions </th>
 </tr>
 </thead>
<tbody>
 <tr>
 <td style="text-align:left;"> Axes </td>
 <td style="text-align:left;"> scale_[x/y]_[continuous/discrete] </td>
 </tr>
 <tr>
 <td style="text-align:left;"> Baseline theme </td>
 <td style="text-align:left;"> theme_[void/minimal/.../dark]() </td>
 </tr>
 <tr>
 <td style="text-align:left;"> Annotations </td>
 <td style="text-align:left;"> geom_[[h/v]line/text](), annotate() </td>
 </tr>
 <tr>
 <td style="text-align:left;"> Theme </td>
 <td style="text-align:left;"> theme(axis.[line/ticks].[x/y] = ..., </td>
 </tr>
</tbody>
</table>
]

.pull-right[
<center><h4> Main types of geometry </h4></center>
<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;"> Geometry </th>
 <th style="text-align:center;"> Function </th>
 </tr>
 </thead>
<tbody>
 <tr>
 <td style="text-align:left;"> Bar plot </td>
 <td style="text-align:center;"> geom_bar() </td>
 </tr>
 <tr>
 <td style="text-align:left;"> Histogram </td>
 <td style="text-align:center;"> geom_histogram() </td>
 </tr>
 <tr>
 <td style="text-align:left;"> Area </td>
 <td style="text-align:center;"> geom_area() </td>
 </tr>
 <tr>
 <td style="text-align:left;"> Line </td>
 <td style="text-align:center;"> geom_line() </td>
 </tr>
 <tr>
 <td style="text-align:left;"> Density </td>
 <td style="text-align:center;"> geom_density() </td>
 </tr>
 <tr>
 <td style="text-align:left;"> Boxplot </td>
 <td style="text-align:center;"> geom_boxplot() </td>
 </tr>
 <tr>
 <td style="text-align:left;"> Violin </td>
 <td style="text-align:center;"> geom_violin() </td>
 </tr>
 <tr>
 <td style="text-align:left;"> Scatter plot </td>
 <td style="text-align:center;"> geom_point() </td>
 </tr>
</tbody>
</table>
]

---

### Part I recap

.pull-left[

<center><h4> Main types of aesthetics </h4></center>

<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;"> Argument </th>
 <th style="text-align:left;"> Meaning </th>
 </tr>
 </thead>
<tbody>
 <tr>
 <td style="text-align:left;"> alpha </td>
 <td style="text-align:left;"> opacity from 0 to 1 </td>
 </tr>
 <tr>
 <td style="text-align:left;"> color </td>
 <td style="text-align:left;"> color of the geometry </td>
 </tr>
 <tr>
 <td style="text-align:left;"> fill </td>
 <td style="text-align:left;"> fill color of the geometry </td>
 </tr>
 <tr>
 <td style="text-align:left;"> size </td>
 <td style="text-align:left;"> size of the geometry </td>
 </tr>
 <tr>
 <td style="text-align:left;"> shape </td>
 <td style="text-align:left;"> shape for geometries like points </td>
 </tr>
 <tr>
 <td style="text-align:left;"> linetype </td>
 <td style="text-align:left;"> solid, dashed, dotted, etc. </td>
 </tr>
</tbody>
</table>
]

.pull-right[

<ul>
 <li>If specified in the geometry</li>
 <ul>
 <li>It will apply uniformly to every all the geometry</li>
 </ul>
</ul>

<ul>
 <li>If assigned to a variable in aes</li>
 <ul>
 <li>it will vary with the variable according to a scale documented in legend</li>
 </ul>
</ul>
]

```r
ggplot(data, aes(x = V1, y = V2, size = V3)) + 
  geom_point(color = "steelblue", alpha = .6)
```

---

### Part I recap

#### R Markdown: Three types of content

.left-column[

]

.right-column[

YAML header

Code chunks

Text
]

---

### Part I recap

#### Useful features

&#10140; **Inline code** allows to include the output of some **R code within text areas** of your report

.pull-left[
<center> <h4> Syntax </h4> </center>

```r
`paste("a", "b", sep = "-")`
```

```r
`r paste("a", "b", sep = "-")`
```

]

.pull-right[
<center> <h4> Output </h4> </center>

`paste("a", "b", sep = "-")`

a-b
 
]

&#10140; **`kable()`** for clean **html tables** and **`datatable()`** to navigate in **large tables**

```r
kable(results_table)
datatable(results_table)
```

---

### Part I recap

#### LaTeX for equations

* `$\LaTeX$` is a convenient way to display **mathematical** symbols and to structure **equations**
  * The **syntax** is mainly based on **backslashes \ and braces {}**

&nbsp;&nbsp;&nbsp;&nbsp;&#10140; What you **type** in the text area: `$x \neq \frac{\alpha \times \beta}{2}$`  
&nbsp;&nbsp;&nbsp;&nbsp;&#10140; What is **rendered** when knitting the document: `$x \neq \frac{\alpha \times \beta}{2}$`

<center>To include a LaTeX equation in R Markdown, you simply have to surround it with the $ sign</center>

.pull-left[

<h4 style = "margin-bottom:0cm;">The mean formula with one `$` on each side</h4> 
&nbsp;&nbsp;&nbsp;&nbsp;&#10140; For inline equations 
 
`$\overline{x}=\frac{1}{N}\sum_{i=1}^N x_i$`

]

.pull-right[

<h4 style = "margin-bottom:0cm;">The mean formula with two `$` on each side</h4> 
&nbsp;&nbsp;&nbsp;&nbsp;&#10140; For large/emphasized equations 
 
`$$\overline{x}=\frac{1}{N}\sum_{i=1}^N x_i$$`

]

---

<h3>Today: We start Econometrics!</h3>

.pull-left[

<ul style = "margin-left:1.5cm;list-style: none">
 <li>1. Joint distributions</li>
 <ul style = "list-style: none">
 <li>1.1. Definition</li>
 <li>1.2. Covariance</li>
 <li>1.3. Correlation</li>
 </ul>
</ul>

<ul style = "margin-left:1.5cm;list-style: none">
 <li>2. Univariate regressions</li>
 <ul style = "list-style: none">
 <li>2.1. Introduction to regressions</li>
 <li>2.2. Coefficients estimation</li>
 </ul>
</ul>

]

.pull-right[

<ul style = "margin-left:-1cm;list-style: none">
 <li>3. Binary variables</li>
 <ul style = "list-style: none">
 <li>3.1. Binary dependent variables</li>
 <li>3.2. Binary independent variables</li>
 </ul>
</ul>

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

---

<h3>Today: We start Econometrics!</h3>

.pull-left[

---

### 1. Joint distributions

#### 1.1. Definition

<ul>
 <li>The joint distribution shows the values and associated frequencies for two variables simultaneously</li>
 <ul>
 <li>Remember how the density could represent the distribution of a single variable</li>
 </ul>
</ul>

.pull-left[

<img src="slides_files/figure-html/unnamed-chunk-28-1.png" width="100%" style="display: block; margin: auto;" />
]

.pull-right[
<img src="slides_files/figure-html/unnamed-chunk-29-1.png" width="100%" style="display: block; margin: auto;" />
]

---

### 1. Joint distributions

#### 1.1. Definition

.pull-left[
<img src="slides_files/figure-html/unnamed-chunk-30-1.png" width="100%" style="display: block; margin: auto;" />
]

.pull-right[
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]

---

### 1. Joint distributions

#### 1.2. Covariance

<ul>
 <li>When describing a single distribution, we're interested in its spread and central tendency</li>
 <li>When describing a joint distribution, we're interested in the relationship between the two variables</li>
 <ul>
 <li>This can be characterized by the covariance</li>
 </ul>
</ul>
 
--

$$ \text{Cov}(x, y) = \frac{1}{N}\sum_{i=1}^{N}(x_i − \bar{x})(y_i − \bar{y}) $$

.pull-left[

<img src="slides_files/figure-html/unnamed-chunk-32-1.png" width="97%" style="display: block; margin: auto;" />
]

.pull-right[

<center>If y tends to be large relative to its mean when x is large relative to its mean, their covariance is positive</center>

<center>Conversely, if one tends to be large when the other tends to be low, the covariance is negative</center>
]

---

### 1. Joint distributions

#### 1.2. Covariance

---

### 1. Joint distributions

#### 1.2. Covariance

`$$\begin{align}
    \text{Cov}(X, a) = & 0\\[1.5em]
    \text{Cov}(X, X) = & \text{Var}(X)\\[1.5em]
    \text{Cov}(X, Y) = & \text{Cov}(Y, X)\\[1.5em]
    \text{Cov}(aX, bY) = & ab\text{Cov}(X, Y)\\[1.5em]
    \text{Cov}(X + a, Y + b) = & \text{Cov}(X, Y)\\[1.5em]
    \text{Cov}(aX + bY, cW + dZ) = & ac\text{Cov}(X, W) +   ad\text{Cov}(X, Z) + \\
    & bc\text{Cov}(Y, W) +  bd\text{Cov}(Y, Z) 
\end{align}$$`
---

### 1. Joint distributions

#### 1.3. Correlation

<ul>
 <li>One disadvantage of the covariance is that is it not standardized</li>
 <ul>
 <li>You cannot directly compare the covariance of two pairs of completely different variables</li>
 <li>Given distance variables will have a larger covariance in centimeters than in meters</li>
 </ul>
</ul>

--
 
<center>&#10140; Theoretically the covariance can take values from $-\infty$ to $+\infty$</center>

<ul>
 <li>To net out the covariance from the unit of the data, we can divide it by $\text{SD}(x)\times\text{SD}(y)$</li>
 <ul>
 <li>We call this standardized measure the correlation</li>
 <li>Correlations coefficients are comparable because they are independent from the unit of the data</li>
 </ul>
</ul>

`$$\text{Corr}(x, y) = \frac{\text{Cov}(x, y)}{\text{SD}(x)\times\text{SD}(y)}$$`

<center>&#10140; The correlation coefficient is bounded between values from $-1$ to $1$</center>

---

### 1. Joint distributions

#### 1.3. Correlation

---

### 1. Joint distributions

<center>&#10140; But a same correlation can hide very different relationships</center>

---

### 1. Joint distributions

<center>&#10140; Covariance and correlation in R</center>

```r
x <- c(50, 70, 60, 80, 60)
y <- c(10, 30, 20, 30, 40)
```

* The covariance can be obtain with the function `cov()`

```r
cov(x, y)
```

```
## [1] 70
```

* The correlation can be obtain with the function `cor()`

```r
cor(x, y)
```

```
## [1] 0.5384615
```

---

<h3>Overview</h3>

.pull-left[

<ul style = "margin-left:1.5cm;list-style: none">
 <li>1. Joint distributions &#10004;</li>
 <ul style = "list-style: none">
 <li>1.1. Definition</li>
 <li>1.2. Covariance</li>
 <li>1.3. Correlation</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. Univariate regressions

#### 2.1. Introduction to regressions

.pull-left[

* Consider the following dataset

```r
ggcurve <- read.csv("ggcurve.csv")
kable(head(ggcurve, 5), "First 5 rows")
```

<table class="table table-hover table-condensed" style="width: auto !important; margin-left: auto; margin-right: auto;">
<caption>First 5 rows</caption>
 <thead>
 <tr>
 <th style="text-align:left;"> country </th>
 <th style="text-align:right;"> ige </th>
 <th style="text-align:right;"> gini </th>
 </tr>
 </thead>
<tbody>
 <tr>
 <td style="text-align:left;"> Denmark </td>
 <td style="text-align:right;"> 0.15 </td>
 <td style="text-align:right;"> 0.38 </td>
 </tr>
 <tr>
 <td style="text-align:left;"> Norway </td>
 <td style="text-align:right;"> 0.17 </td>
 <td style="text-align:right;"> 0.33 </td>
 </tr>
 <tr>
 <td style="text-align:left;"> Finland </td>
 <td style="text-align:right;"> 0.18 </td>
 <td style="text-align:right;"> 0.38 </td>
 </tr>
 <tr>
 <td style="text-align:left;"> Canada </td>
 <td style="text-align:right;"> 0.19 </td>
 <td style="text-align:right;"> 0.46 </td>
 </tr>
 <tr>
 <td style="text-align:left;"> Australia </td>
 <td style="text-align:right;"> 0.26 </td>
 <td style="text-align:right;"> 0.44 </td>
 </tr>
</tbody>
</table>
]

.pull-right[
 
 
 The data contains 2 variables at the country level:
 
 <ul style = "margin-left:-.5cm;list-style: none">
 <li>1. IGE: Intergenerational elasticity, which captures
 </ul>
 
 the % average increase in child income for
 a 1% increase in parental income
 
 
 
 <ul style = "margin-left:-.5cm;list-style: none">
 <li>2. Gini: Gini index of income inequality between 
 </ul>
 
 0: everybody has the same income
 1: a single individual has all the income
 
]

---

### 2. Univariate regressions

#### 2.1. Introduction to regressions

* To investigate the **relationship** between these two variables we can start with a **scatterplot**

```r
ggplot(ggcurve , aes(x = gini, y = ige, label = country)) + geom_text()
```
<img src="slides_files/figure-html/unnamed-chunk-41-1.png" width="60%" style="display: block; margin: auto;" />

---

### 2. Univariate regressions

#### 2.1. Introduction to regressions

<ul>
 <li>We see that the two variables are positively correlated with each other:</li>
 <ul>
 <li>When one tends to be high relative to its mean, the other as well</li>
 <li>When one tends to be low relative to its mean, the other as well</li>
 </ul>
</ul>
 
--

```r
cor(ggcurve$gini, ggcurve$ige)
```

```
## [1] 0.6517277
```

<ul>
 <li>The correlation coefficient is equal to .65</li>
 <ul>
 <li>Remember that the correlation can take values from -1 to 1</li>
 <li>Here the correlation is indeed positive and fairly strong</li>
 </ul>
</ul>

<ul>
 <li>But how useful is this for real-life applications? We may want more practical information:</li>
 <ul>
 <li>Like by how much $y$ is expected to increase for a given change in $x$</li>
 <li>This is of particular interest for economists and policy makers</li>
 </ul>
</ul>

---

### 2. Univariate regressions

#### 2.1. Introduction to regressions

* Consider these two relationships :

.left-column[

]

.right-column[

&#10140; One is less noisy but flatter
 

 
&#10140; One is noisier but steeper

<h4>Both have a correlation of .75</h4>
 
]

---

### 2. Univariate regressions

#### 2.1. Introduction to regressions

* Consider these two relationships :

.left-column[

]

.right-column[

<center>But a given increase in $x$</center>
<center>is not associated with</center>
<center>a same increase in $y$!</center>
]

---

### 2. Univariate regressions

#### 2.1. Introduction to regressions

<ul>
 <li>Knowing that income inequality is negatively correlated with intergenerational mobility is one thing</li>
</ul>

<ul>
 <li>But how much more intergenerational mobility could we expect for a given reduction in inequality?</li>
 <ul>
 <li>We need to characterize the "steepness" of the relationship!</li>
 </ul>
</ul>

<ul>
 <li>It is usually the type of questions we're interested in:</li>
 <ul>
 <li>How much more should I expect to earn for an additional year of education?</li>
 <li>By how many years would life expectancy be expected to decrease for a given increase in air pollution?</li>
 <li>By how much would test scores increase for a given decrease in the number of students per teacher?</li>
 </ul>
</ul>

* And once again, this is typically what is of interest for policymakers

<center><h4>&#10140; But how to compute this expected change in $y$ for a given change of $x$?</h4></center>

---

### 2. Univariate regressions

#### 2.2. Coefficients estimation

<ul>
 <li>The idea is to find the line that fits the data the best</li>
 <ul>
 <li>Such that its slope can indicate how we expect y to change if we increase x by 1 unit</li>
 </ul>
</ul>
 
--

---

### 2. Univariate regressions

#### 2.2. Coefficients estimation

* But how do we find that line?
 
--
 
<img src="slides_files/figure-html/unnamed-chunk-46-1.png" width="100%" style="display: block; margin: auto;" />

---

### 2. Univariate regressions

#### 2.2. Coefficients estimation

* We try to minimize the distance between each point and our line
 
<img src="slides_files/figure-html/unnamed-chunk-47-1.png" width="100%" style="display: block; margin: auto;" />

---

### 2. Univariate regressions

#### 2.2. Coefficients estimation
 
.pull-left[

Take for instance the 20th observation: Peru

]

.pull-right[

And consider the following **notations**:

* We denote `$y_i$` the ige of the `$i^{\text{th}}$` country
  
  * We denote `$x_i$` the gini of the `$i^{\text{th}}$` country
  
  * We denote `$\widehat{y_i}$` the value of the `$y$` coordinate of our line for `$x = x_i$`

<center>&#10140; The distance between the $i^{\text{th}}$ y value and the line is $y_i - \widehat{y_i}$</center>

* We label that distance `$\widehat{\varepsilon_i}$`

]

---

### 2. Univariate regressions

#### 2.2. Coefficients estimation

.pull-left[

]

.pull-right[

* `$\widehat{\varepsilon_i}$` being the distance between a point `$y_i$` and its corresponding value on the line `$\widehat{y_i}$`, we can write:

`$$y_i = \widehat{y_i} + \widehat{\varepsilon_i}$$`

* And because `$\widehat{y_i}$` is a **straight line**, it can be expressed as
 
`$$\widehat{y_i} = \hat{\alpha} + \hat{\beta}x_i$$`

* Where:
  * `$\hat{\alpha}$` is the **intercept**
  * `$\hat{\beta}$` is the **slope**
]

---

### 2. Univariate regressions

#### 2.2. Coefficients estimation

* **Combining** these two **definitions** yields the equation:

`$$y_i = \hat{\alpha} + \hat{\beta}x_i + \widehat{\varepsilon_i} \begin{cases}  y_i = \widehat{y_i} + \widehat{\varepsilon_i}& \text{Definition of distance}\\
\widehat{y_i} = \hat{\alpha} + \hat{\beta}x_i & \text{Definition of the line}
\end{cases}$$`

* Depending on the values of `$\hat{\alpha}$` and `$\hat{\beta}$`, the value of every `$\widehat{\varepsilon_i}$` will change

.left-column[
<img src="slides_files/figure-html/unnamed-chunk-50-1.png" width="90%" style="display: block; margin: auto auto auto 0;" />

]

.right-column[

**Attempt 1:** `$\hat{\alpha}$` is too high and `$\hat{\beta}$` is 
too low &#10140; $\widehat{\varepsilon_i}$ are large
 
**Attempt 2:** `$\hat{\alpha}$` is too low and `$\hat{\beta}$` is 
too high &#10140; $\widehat{\varepsilon_i}$ are large
 
**Attempt 3:** both `$\hat{\alpha}$` and `$\hat{\beta}$` seem 
right &#10140; $\widehat{\varepsilon_i}$ are low

]

---

### 2. Univariate regressions

#### 2.2. Coefficients estimation

* We want to find the values of `$\hat{\alpha}$` and `$\hat{\beta}$` that **minimize** the overall **distance** between the points and the line

`$$\min_{\hat{\alpha}, \hat{\beta}}\sum_{i=1}^{n}\widehat{\varepsilon_i}^2$$`
<ul>
 <ul>
 <li>Note that we square $\widehat{\varepsilon_i}$ to avoid that its positive and negative values compensate</li>
 <li>This method is what we call Ordinary Least Squares (OLS)</li>
 </ul>
</ul>

* To solve this **optimization problem**, we need to express `$\widehat{\varepsilon_i}$` it in terms of alpha `$\hat{\alpha}$` and `$\hat{\beta}$`

`$$y_i = \hat{\alpha} + \hat{\beta}x_i + \widehat{\varepsilon_i}$$` 
`$$\Longleftrightarrow$$`
`$$\widehat{\varepsilon_i} = y_i -\hat{\alpha} - \hat{\beta}x_i$$`

---

### 2. Univariate regressions

#### 2.2. Coefficients estimation

* And our minimization problem writes

`$$\min_{\hat{\alpha}, \hat{\beta}}\sum_{i=1}^{n}(y_i -\hat{\alpha} - \hat{\beta}x_i)^2$$`

$$
`\begin{align}
\frac{\partial}{\partial\hat{\alpha}} = 0 & \:\: \Longleftrightarrow \:\: -2\sum_{i=1}^n(y_i - \hat{\alpha} - \hat{\beta}x_i) = 0 \\
\frac{\partial}{\partial\hat{\beta}}  = 0 & \:\: \Longleftrightarrow \:\: -2x_i\sum_{i=1}^n(y_i - \hat{\alpha} - \hat{\beta}x_i) = 0
\end{align}`
$$

* Rearranging the first equation yields
 
`$$\sum_{i=1}^ny_i - n\hat{\alpha} - \sum_{i=1}^n\hat{\beta}x_i = 0 \:\: \Longleftrightarrow \:\: \hat{\alpha} =\bar{y} - \hat{\beta}\bar{x}$$`

---

### 2. Univariate regressions

#### 2.2. Coefficients estimation

* Replacing `$\hat{\alpha}$` in the second equation by its new expression writes

`$$-2x_i\sum_{i=1}^n(y_i - \hat{\alpha} - \hat{\beta}x_i) = 0  \:\: \Longleftrightarrow \:\: -2x_i\sum_{i=1}^n\left[y_i - (\bar{y} - \hat{\beta}\bar{x}) - \hat{\beta}x_i\right] = 0$$`

* And by rearranging the terms we obtain
 
`$$\hat{\beta} = \frac{\sum_{i = 1}^n(x_i-\bar{x})(y_i-\bar{y})}{\sum_{i = 1}^n(x_i-\bar{x})^2}$$`

* Notice that multiplying the nominator and the denominator by `$1/n$` yields:
  
`$$\hat{\beta} = \frac{\text{Cov}(x_i, y_i)}{\text{Var}(x_i)} \:\:\:\:\:\:\:\:\: ; \:\:\:\:\:\:\:\:\: \hat{\alpha} = \bar{y} - \frac{\text{Cov}(x_i, y_i)}{\text{Var}(x_i)} \times\bar{x}$$`

---
class: inverse, hide-logo

### Practice

#### 1) Import `ggcurve.csv` and compute the `$\hat{\alpha}$` and `$\hat{\beta}$` coefficients of that equation:
 
`$$\text{IGE}_i = \hat{\alpha} + \hat{\beta}\times\text{gini}_i + \widehat{\varepsilon_i}$$`

#### 2) Create a new variable in the dataset for `$\widehat{\text{IGE}}$`

#### 3) Plot your results (scatter plot + line)

*Hints: You can use different y variables for different geometries by specifying the mapping within the geometry function:*

<center>geom_point(aes(y = y))</center>

`$$\hat{\beta} = \frac{\text{Cov}(x_i, y_i)}{\text{Var}(x_i)} \:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: \hat{\alpha} = \bar{y} - \frac{\text{Cov}(x_i, y_i)}{\text{Var}(x_i)} \times\bar{x}$$`

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

---
class: inverse, hide-logo

### Solution

#### 1) Import `ggcurve.csv` and compute the `$\hat{\alpha}$` and `$\hat{\beta}$` coefficients of that equation:

```r
# Read the data
ggcurve <- read.csv("ggcurve.csv")
# Compute beta
beta <- cov(ggcurve$gini, ggcurve$ige) / var(ggcurve$gini)
# Compute alpha
alpha <- mean(ggcurve$ige) - (beta * mean(ggcurve$gini))
```

```r
c(alpha, beta)
```

```
## [1] -0.09129311  1.01546204
```

#### 2) Create a new variable in the dataset for `$\widehat{\text{IGE}}$`

```r
ggcurve <- ggcurve %>%
 mutate(fit = alpha + beta * gini)
```

---
class: inverse, hide-logo

### Solution

#### 3) Plot your results (scatter plot + line)

```r
ggplot(ggcurve, aes(x = gini)) + 
  geom_point(aes(y = ige)) + geom_line(aes(y = fit))
```
 
--

---

### 2. Univariate regressions

#### 2.2. Coefficients estimation

* As usual there are functions to do that in R
 
--
 
.pull-left[

<ul>
 <li>lm() to estimate regression coefficients</li>
 <li>It has two main arguments:</li>
 <ul>
 <li>Formula: written as y ~ x</li>
 <li>Data: where y and x are</li>
 </ul>
</ul>

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

```
## 
## Call:
## lm(formula = ige ~ gini, data = ggcurve)
## 
## Coefficients:
## (Intercept)         gini  
##    -0.09129      1.01546
```

]

.pull-right[

<ul>
 <li>geom_smooth() to plot the fit</li>
</ul>

```r
ggplot(ggcurve, aes(x = gini, y = ige)) + 
  geom_point() + 
  geom_smooth(method = "lm", formula = y ~ x)
```

]

---
class: inverse, hide-logo

<center><h3> Vocabulary </h3></center>

* This equation we're working on is called a regression model
 
`$$y_i = \hat{\alpha} + \hat{\beta}x_i + \widehat{\varepsilon_i}$$`

<ul>
 <ul>
 <li> We say that we regress $y$ on $x$ to find the coefficients $\hat{\alpha}$ and $\hat{\beta}$ that characterize the regression line</li>
 <li> We often call $\hat{\alpha}$ and $\hat{\beta}$ parameters of the regression because we tune them to fit our model to the data</li>
 </ul>
</ul>

<ul>
 <li>We also have different names for the $x$ and $y$ variables</li>
 <ul>
 <li> $y$ is called the dependent or explained variable
 <li> $x$ is called the independent or explanatory variable
 </ul>
</ul>

* We call `$\widehat{\varepsilon_i}$` the residuals because it is what is left after we fitted the data the best we could

* And `$\hat{y_i} = \hat{\alpha} + \hat{\beta}x_i$`, i.e., the value on the regression line for a given `$x_i$` are called the fitted values

---

<h3>Overview</h3>

.pull-left[

<ul style = "margin-left:1.5cm;list-style: none">
 <li>2. Univariate regressions &#10004;</li>
 <ul style = "list-style: none">
 <li>2.1. Introduction to regressions</li>
 <li>2.2. Coefficients estimation</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. Binary variables

#### 3.1. Binary dependent variables

<ul>
 <li>So far we've considered only continuous variables in our regression models</li>
 <ul>
 <li>But what if our dependent variable is discrete?</li>
 </ul>
</ul>

<ul>
 <li>Consider that we have data on candidates to a job:</li>
 <ul>
 <li>Their Baccalauréat grade (/20) </li>
 <li>Whether they got accepted</li>
 </ul>
</ul>
 
--

---

### 3. Binary variables

#### 3.1. Binary dependent variables

<ul>
 <li>Even if the outcome variable is binary we can regress it on the grade variable</li>
 <ul>
 <li>We can convert it into a dummy variable, a variable taking either the value 0 or 1</li>
 <li>Here consider a dummy variable taking the value 1 if the person was accepted</li>
 </ul>
</ul>

`$$1\{y_i = \text{Accepted}\} = \hat{\alpha} + \hat{\beta} \times \text{Grade}_i + \hat{\varepsilon_i}$$`

.left-column[

]

.right-column[

<center><h4> &#10140; How would you interpret the beta coefficient from this regression?</h4></center>

]

---

### 3. Binary variables

#### 3.1. Binary dependent variables

<ul>
 <li>The fitted values can be viewed as the probability to be accepted for a given grade</li>
 <ul>
 <li>$\hat{\beta}$ is thus by how much this probability would vary on expectation for a 1 point increase in the grade</li>
 <li>That's why we call OLS regression models with a binary outcome Linear Probability Models</li>
 </ul>
</ul>

---

### 3. Binary variables

#### 3.1. Binary dependent variables

<ul>
 <li>But what kind of problems could we encounter with such models?</li>
 <ul>
 <li>What would be the $\hat{\alpha}$ coefficient here?</li>
 <li>And what's the probability to be accepted for a grade of 18?</li>
 </ul>
</ul>

---

### 3. Binary variables

#### 3.1. Binary dependent variables

<ul>
 <li>With an LPM you can end up with "probabilities" that are lower than 0 and greater than 1</li>
 <ul>
 <li>Interpretation is only valid for values of x sufficiently close to the mean</li>
 <li>Keep that in mind and be careful when interpreting the results of an LPM</li>
 </ul>
</ul>

---

### 3. Binary variables

#### 3.2. Binary independent variables

<ul>
 <li>Now consider that we have individual data containing</li>
 <ul>
 <li>The sex</li>
 <li>The height (centimeters)</li>
 </ul>
</ul>

<ul>
 <li>So the situation is different</li>
 <ul>
 <li>We used to have a binary dependent variable:</li>
 </ul>
</ul>

`$$1\{y_i = \text{Accepted}\} = \hat{\alpha} + \hat{\beta} \times \text{Grade}_i + \hat{\varepsilon_i}$$`
 
<ul>
 <ul>
 <li>We now have a binary independent variable:</li>
 </ul>
</ul>
 
`$$\text{Height}_i = \hat{\alpha} + \hat{\beta} \times 1\{\text{Sex}_i = \text{Male}\} + \hat{\varepsilon_i}$$`

<center><h4> &#10140; How would you interpret the coefficient $\hat{\beta}$ from this regression?</h4></center>

---

### 3. Binary variables

#### 3.2. Binary independent variables

<ul>
 <li>If the sex variable was continuous it would be the expected increase in height for a "1 unit increase" in sex</li>
 <ul>
 <li>Here the "1 unit increase" is switching from 0 to 1, i.e. from female to male</li>
 <li>With that in mind, how would you interpret the coefficient $\hat{\beta}$?</li>
 </ul>
</ul>

---

### 3. Binary variables

#### 3.2. Binary independent variables

<ul>
 <li>If I replace the point geometry by the corresponding boxplots</li>
 <ul>
 <li>What this "1 unit increase" corresponds to should be clearer</li>
 <li>The coefficient $\hat{\beta}$ is actually the difference between the average height for males and females</li>
 </ul>
</ul>

---

### 3. Binary variables

#### 3.2. Binary independent variables

.pull-left[

`$\overline{\text{Height}_{\left[\text{Sex}_i = \text{Female}\right]}} = 165$`

`$\overline{\text{Height}_{\left[\text{Sex}_i = \text{Male}\right]}} = 176$`

`$$\text{Height}_i = \hat{\alpha} + \hat{\beta} \times 1\{\text{Sex}_i = \text{Male}\} + \hat{\varepsilon_i}$$`
`$$\hat{\alpha} = 165 \:\:\:\:\:\:\:\:\:\:\:\:\:\: \hat{\beta} = 11$$`

`$$\text{Height}_i = \hat{\alpha} + \hat{\beta} \times 1\{\text{Sex}_i = \text{Female}\} + \hat{\varepsilon_i}$$`
`$$\hat{\alpha} = 176 \:\:\:\:\:\:\:\:\:\:\:\:\:\: \hat{\beta} = -11$$`

]
 
.pull-right[

<img src="slides_files/figure-html/unnamed-chunk-65-1.png" width="100%" style="display: block; margin: auto;" />
]

---

### 3. Binary variables

#### 3.2. Binary independent variables

* In terms of fitted values:
 
`$$\text{Height}_i = \hat{\alpha} + \hat{\beta} \times 1\{\text{Sex}_i = \text{Male}\} + \hat{\varepsilon_i}$$`

* We now have `$\hat{\alpha}$` and `$\hat{\beta}$`:
 
`$$\text{Height}_i = 165 + 11 \times 1\{\text{Sex}_i = \text{Male}\} + \hat{\varepsilon_i}$$`

* The fitted values write:

`$$\widehat{\text{Height}_i} =  165 + 11 \times 1\{\text{Sex}_i = \text{Male}\}$$`

.pull-left[

* When the dummy equals 0 *(females)*:
 
`$$\begin{align}
\widehat{\text{Height}_i} & = 165 + 11 \times 0\\
&= 165 =\overline{\text{Height}_{\left[\text{Sex}_i = \text{Female}\right]}}
\end{align}$$`

]

.pull-right[

* When the dummy equals 1 *(males)*:
 
`$$\begin{align}\widehat{\text{Height}_i} & = 165 + 11 \times 1\\
&= 176 =\overline{\text{Height}_{\left[\text{Sex}_i = \text{Male}\right]}}\end{align}$$`

]

---

<h3>Overview</h3>

.pull-left[

]

.pull-right[

<ul style = "margin-left:-1cm;list-style: none">
 <li>3. Binary variables &#10004;</li>
 <ul style = "list-style: none">
 <li>3.1. Binary dependent variables</li>
 <li>3.2. Binary independent variables</li>
 </ul>
</ul>

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

---

### 4. Wrap up!

#### 1. Joint distribution

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

.pull-left[
<img src="slides_files/figure-html/unnamed-chunk-66-1.png" width="100%" style="display: block; margin: auto;" />
]

.pull-right[

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]

---

### 4. Wrap up!

#### 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>
 
 `$$\text{Corr}(x, y) = \frac{\text{Cov}(x, y)}{\text{SD}(x)\times\text{SD}(y)}$$`
 
---

### 4. Wrap up!

#### 2. Regression

.pull-left[

<img src="slides_files/figure-html/unnamed-chunk-68-1.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 `$\hat{\varepsilon_i}$`

]

---

### 4. Wrap up!

#### 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-71-1.png" width="90%" style="display: block; margin: auto;" />
]