Econometrics in R

# Econometrics in R
## Lecture 4
### Louis SIRUGUE
### M1 APE - Fall 2022

---

### What we've seen so far

#### Manipulate data with `dplyr`

```r
read.csv("ligue1.csv") 
                                                    #
                                                    #
                                                    #
                                                    #
                                                    #
                                                    #
```

```text
##     Wk Day       Date  Time          Home  xG Score xG.1          Away Attendance  ...
## 1    1 Fri 2021-08-06 21:00        Monaco 2.0   1–1  0.3        Nantes       7500  ...
## 2    1 Sat 2021-08-07 17:00          Lyon 1.4   1–1  0.8         Brest      29018  ...
## 3    1 Sat 2021-08-07 21:00        Troyes 0.8   1–2  1.2     Paris S-G      15248  ...
## 4    1 Sun 2021-08-08 13:00        Rennes 0.6   1–1  2.0          Lens      22567  ...
## 5    1 Sun 2021-08-08 15:00      Bordeaux 0.7   0–2  3.3 Clermont Foot      18748  ...
## 6    1 Sun 2021-08-08 15:00    Strasbourg 0.4   0–2  0.9        Angers      23250  ...
## 7    1 Sun 2021-08-08 15:00          Nice 0.8   0–0  0.2         Reims      18030  ...
## 8    1 Sun 2021-08-08 15:00 Saint-Étienne 2.1   1–1  1.3       Lorient      20461  ...
## 9    1 Sun 2021-08-08 17:00          Metz 0.7   3–3  1.4         Lille      15551  ...
 ...  ... ...        ...   ...           ... ...   ...  ...           ...        ...  ...                      
```

---

### What we've seen so far

#### Manipulate data with `dplyr`

```r
read.csv("ligue1.csv") %>%
  select(Home, xG, Score, xG.1, Away)               # Keep/drop certain columns
                                                    #
                                                    #
                                                    #
                                                    #
                                                    #
```

```text
##              Home  xG Score xG.1          Away
## 1          Monaco 2.0   1–1  0.3        Nantes
## 2            Lyon 1.4   1–1  0.8         Brest
## 3          Troyes 0.8   1–2  1.2     Paris S-G
## 4          Rennes 0.6   1–1  2.0          Lens
## 5        Bordeaux 0.7   0–2  3.3 Clermont Foot
## 6      Strasbourg 0.4   0–2  0.9        Angers
## 7            Nice 0.8   0–0  0.2         Reims
## 8   Saint-Étienne 2.1   1–1  1.3       Lorient
## 9            Metz 0.7   3–3  1.4         Lille
 ...             ... ...   ...  ...           ...
```

---

### What we've seen so far

#### Manipulate data with `dplyr`

```r
read.csv("ligue1.csv") %>%
  select(Home, xG, Score, xG.1, Away) %>%           # Keep/drop certain columns
  mutate(home_winner = xG > xG.1)                   # Create a new variable
                                                    #
                                                    #
                                                    #
                                                    #
```

```text
##              Home  xG Score xG.1          Away home_winner
## 1          Monaco 2.0   1–1  0.3        Nantes        TRUE
## 2            Lyon 1.4   1–1  0.8         Brest        TRUE
## 3          Troyes 0.8   1–2  1.2     Paris S-G       FALSE
## 4          Rennes 0.6   1–1  2.0          Lens       FALSE
## 5        Bordeaux 0.7   0–2  3.3 Clermont Foot       FALSE
## 6      Strasbourg 0.4   0–2  0.9        Angers       FALSE
## 7            Nice 0.8   0–0  0.2         Reims        TRUE
## 8   Saint-Étienne 2.1   1–1  1.3       Lorient        TRUE
## 9            Metz 0.7   3–3  1.4         Lille       FALSE
 ...             ... ...   ...  ...           ...         ...
```

---

### What we've seen so far

#### Manipulate data with `dplyr`

```r
read.csv("ligue1.csv") %>%
  select(Home, xG, Score, xG.1, Away) %>%           # Keep/drop certain columns
  mutate(home_winner = xG > xG.1) %>%               # Create a new variable
  filter(Home == "Rennes")                          # Keep/drop certain rows 
                                                    #
                                                    #
                                                    #
```

```text
##      Home  xG Score xG.1          Away home_winner
## 1  Rennes 0.6   1–1  2.0          Lens       FALSE
## 2  Rennes 0.9   1–0  0.5        Nantes        TRUE
## 3  Rennes 1.0   0–2  0.5         Reims        TRUE
## 4  Rennes 2.4   6–0  0.3 Clermont Foot        TRUE
## 5  Rennes 0.8   2–0  1.4     Paris S-G       FALSE
## 6  Rennes 1.5   1–0  0.6    Strasbourg        TRUE
## 7  Rennes 3.8   4–1  1.1          Lyon        TRUE
## 8  Rennes 3.1   2–0  0.7   Montpellier        TRUE
## 9  Rennes 0.8   1–2  0.6         Lille        TRUE
         ... ...   ...  ...           ...         ...
```

---

### What we've seen so far

#### Manipulate data with `dplyr`

```r
read.csv("ligue1.csv") %>%
  select(Home, xG, Score, xG.1, Away) %>%           # Keep/drop certain columns
  mutate(home_winner = xG > xG.1) %>%               # Create a new variable
  filter(Home == "Rennes") %>%                      # Keep/drop certain rows
  arrange(-xG)                                      # Sort rows
                                                    #
                                                    #
```

```text
##      Home  xG Score xG.1          Away home_winner
## 1  Rennes 3.8   4–1  1.1          Lyon        TRUE
## 2  Rennes 3.3   6–0  0.4      Bordeaux        TRUE
## 3  Rennes 3.3   6–1  0.9          Metz        TRUE
## 4  Rennes 3.1   2–0  0.7   Montpellier        TRUE
## 5  Rennes 2.7   2–0  0.3         Brest        TRUE
## 6  Rennes 2.6   4–1  0.4        Troyes        TRUE
## 7  Rennes 2.4   6–0  0.3 Clermont Foot        TRUE
## 8  Rennes 1.9   2–3  2.9        Monaco       FALSE
## 9  Rennes 1.7   2–0  0.3        Angers        TRUE
         ... ...   ...  ...           ...         ...
```

---

### What we've seen so far

#### Manipulate data with `dplyr`

```r
read.csv("ligue1.csv") %>%
  select(Home, xG, Score, xG.1, Away) %>%           # Keep/drop certain columns
  mutate(home_winner = xG > xG.1) %>%               # Create a new variable
  filter(Home == "Rennes") %>%                      # Keep/drop certain rows
  arrange(-xG) %>%                                  # Sort rows
  summarise(expected_wins = mean(home_winner),      # Aggregate into statistics
            expected_goals = sum(xG))               #
```

```
##   expected_wins expected_goals
## 1     0.8421053           36.6
```

---

### What we've seen so far

#### Plot data with `ggplot()`

```r
ggplot(read.csv("wid.csv"))                                         # Data
                                                                    #
```

---

### What we've seen so far

#### Plot data with `ggplot()`

```r
ggplot(read.csv("wid.csv"), aes(x = inc_head, y = top1)) # Data & aesthetics
 #
```
 
<img src="slides_files/figure-html/unnamed-chunk-16-1.png" width="60%" style="display: block; margin: auto;" />

---

### What we've seen so far

#### Plot data with `ggplot()`

```r
ggplot(read.csv("wid.csv"), aes(x = inc_head, y = top1)) +          # Data & aesthetics
  geom_point()                                                      # Geometry
```

---

### What we've seen so far

#### Write reports with `R markdown`

````markdown
---
title: "Starbucks"    
author: "Louis Sirugue"
output: html_document
---
````
]

---

### What we've seen so far

#### Write reports with `R markdown`

````markdown
---
title: "Starbucks"
author: "Louis Sirugue"
output: html_document
---

```{r, echo = F, message = F, warning = F}
library(ggplot2) # Load package
starbucks <- read.csv("starbucks.csv", sep = ";") # Load data
```

Nutritional values of `r nrow(starbucks)` *starbucks* beverages

````
]

---

### What we've seen so far

#### Write reports with `R markdown`

````markdown
---
title: "Starbucks"
author: "Louis Sirugue"
output: html_document
---

```{r, echo = F, message = F, warning = F}
library(ggplot2) # Load package
starbucks <- read.csv("starbucks.csv", sep = ";") # Load data
```

Nutritional values of `r nrow(starbucks)` *starbucks* beverages

```{r, fig.height = 4}
ggplot(starbucks, aes(x = Calories, y = Cholesterol, 
                      size = Trans.Fat, color = Sugars)) +
  geom_point(alpha = .3) + theme_minimal(base_size = 14) +
  scale_color_gradient(low = "green", high = "red") 
```
````
]

---

<h3>Today: Econometrics in R!</h3>

.pull-left[
<ul style = "margin-left:1.5cm;list-style: none">
 <li>1. Regressions</li>
 <ul style = "list-style: none">
 <li>1.1. On continuous variables</li>
 <li>1.2. On binary variables</li>
 <li>1.3. On categorical variables</li>
 </ul>
</ul>

<ul style = "margin-left:1.5cm;list-style: none">
 <li>2. Case study</li>
 <ul style = "list-style: none">
 <li>2.1. Variable transformation</li>
 <li>2.2. Functional form</li>
 <li>2.3. Control variables</li>
 <li>2.4. Interactions</li>
 </ul>
</ul>
]

<ul style = "margin-left:-1cm;list-style: none">
 <li>3. Inference</li>
 <ul style = "list-style: none">
 <li>3.1. Hypothesis testing</li>
 <li>3.2. Confidence intervals</li>
 </ul>
</ul>

<ul style = "margin-left:-1cm;list-style: none">
 <li>4. Report and export results</li>
<ul style = "list-style: none">
 <li>4.1. Regression tables</li>
 <li>4.2. Plot coefficients</li>
 </ul>
</ul>

]
---

<h3>Today: Econometrics in R!</h3>

---

### 1. Regressions

#### 1.1. On continuous variables

<ul>
 <li>For this part we're going to work with the 'Great Gatsby Curve'</li>
 <ul>
 <li>It refers to the positive relationship between inequality and intergenerational income persistence</li>
 <li>The term was coined by Alan Krueger based on the research of Miles Corak</li>
 </ul>
</ul>

```r
ggcurve <- read.csv("ggcurve.csv")
str(ggcurve)
```

```
## 'data.frame':	22 obs. of  3 variables:
##  $ country: chr  "Denmark" "Norway" "Finland" "Canada" ...
##  $ ige    : num  0.15 0.17 0.18 0.19 0.26 0.27 0.29 0.32 0.34 0.4 ...
##  $ gini   : num  0.378 0.325 0.378 0.463 0.439 ...
```

<ul>
 <li>For 22 countries we have the following variables</li>
 <ul>
 <li>ige: The intergenerational income elasticity, the higher the closer child income to parent income</li>
 <li>gini: The Gini coefficient of income inequality, the higher the more concentrated the income</li>
 </ul>
</ul>
 
---

### 1. Regressions

#### 1.1. On continuous variables

<ul>
 <li>You must already be quite familiar with univariate regressions $y = \alpha + \beta x + \varepsilon$</li>
</ul>

---

### 1. Regressions

#### 1.1. On continuous variables

---

### 1. Regressions

#### 1.1. On continuous variables

<ul>
 <li>You must already be quite familiar with univariate regressions $y = \alpha + \beta x + \varepsilon$</li>
 <ul>
 <li>We're looking for the line $\hat{y_i} = \hat{\alpha} + \hat{\beta} x_i$ that mimimizes distance to the data points $\sum\varepsilon_i^2$</li>
 <li>Such that for a one unit increase in $x$</li>
 </ul>
</ul>

---

### 1. Regressions

#### 1.1. On continuous variables

---

### 1. Regressions

#### 1.1. On continuous variables

<ul>
 <li>In R we can estimate a regression model using the lm() command (for Linear Model)</li>
 <ol>
 <li></li>
 <li></li>
 </ol>
</ul>

```r
lm( , )
```

---

### 1. Regressions

#### 1.1. On continuous variables

<ul>
 <li>In R we can estimate a regression model using the lm() command (for Linear Model)</li>
 <ol>
 <li>The first argument is the formula, written as y ~ x</li>
 <li></li>
 </ol>
</ul>

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

---

### 1. Regressions

#### 1.1. On continuous variables

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

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

<ul>
 <li>This is great but the output is a bit minimalistic</li>
 <ul>
 <li>To get a more exhaustive description of our regression we can apply the summary() function to lm()</li>
 </ul>
</ul>

```r
ggmodel <- lm(ige ~ gini, ggcurve) %>% summary()
```

---

### 1. Regressions

#### 1.1. On continuous variables

```r
ggmodel
```

```
## 
## Call:
## lm(formula = ige ~ gini, data = ggcurve)
## 
## Residuals:
##       Min        1Q    Median        3Q       Max 
## -0.188991 -0.088238 -0.000855  0.047284  0.252310 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)   
## (Intercept) -0.09129    0.12870  -0.709  0.48631   
## gini         1.01546    0.26425   3.843  0.00102 **
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.1159 on 20 degrees of freedom
## Multiple R-squared:  0.4247,	Adjusted R-squared:  0.396 
## F-statistic: 14.77 on 1 and 20 DF,  p-value: 0.001016
```

---

### 1. Regressions

#### 1.1. On continuous variables

* It gives back the **command** 
 
.left-column[

```text
## 
## Call:
## lm(formula = ige ~ gini, data = ggcurve)
## 
## 
## 
## 
## 
## 
## 
## 
## 
## 
## 
## 
## 
## 
## 
```
]

---

### 1. Regressions

#### 1.1. On continuous variables

* A description of the **distribution** of **residuals**
 
.left-column[

```text
## 
## Call:
## lm(formula = ige ~ gini, data = ggcurve)
## 
## Residuals:
##       Min        1Q    Median        3Q       Max 
## -0.188991 -0.088238 -0.000855  0.047284  0.252310 
## 
## 
##
## 
##
## 
## 
## 
## 
## 
## 
```
]

---

### 1. Regressions

#### 1.1. On continuous variables

* **Coefficients** along with their **standard error**, **t-value**, and **p-value**
 
.left-column[

---

### 1. Regressions

#### 1.1. On continuous variables

* **Significance** thresholds with symbols
 
.left-column[

```text
## 
## Call:
## lm(formula = ige ~ gini, data = ggcurve)
## 
## Residuals:
##       Min        1Q    Median        3Q       Max 
## -0.188991 -0.088238 -0.000855  0.047284  0.252310 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)   
## (Intercept) -0.09129    0.12870  -0.709  0.48631   
## gini         1.01546    0.26425   3.843  0.00102 **
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## 
## 
## 
```
]

---

### 1. Regressions

#### 1.1. On continuous variables

* The **residual standard error** `$\sqrt{\sum{(y_i-\hat{y_i})^2}/\text{df}}$` and **degrees of freedom**

.right-column[

&#x1F804; Command 

&#x1F804; Residuals distribution 

&#x1F804; Coefs, s.e., t-/p-values 

&#x1F804; Significance 

&#x1F804; Residual s.e. & df. 
]

---

### 1. Regressions

#### 1.1. On continuous variables

* The `$R^2$` and **adjusted** `$R^2$`
 
.left-column[

.right-column[
&#x1F804; Command 

&#x1F804; Residuals distribution 

&#x1F804; Coefs, s.e., t-/p-values 

&#x1F804; Significance 

&#x1F804; Residual s.e. & df. 
&#x1F804; R2 & adjusted R2 
]

---

### 1. Regressions

#### 1.1. On continuous variables

<ul><li>The result of an F-test ($H_0: \beta_k = 0 \:\: \forall k$)</li></ul>
 
.left-column[

---

### 1. Regressions

#### 1.1. On continuous variables

* All these elements are then easily accessible using the `$` operator

```r
str(ggmodel, give.attr = F)
```

```
## List of 11
##  $ call         : language lm(formula = ige ~ gini, data = ggcurve)
##  $ terms        :Classes 'terms', 'formula'  language ige ~ gini
##  $ residuals    : Named num [1:22] -0.1427 -0.0687 -0.1125 -0.189 -0.094 ...
##  $ coefficients : num [1:2, 1:4] -0.0913 1.0155 0.1287 0.2642 -0.7093 ...
##  $ aliased      : Named logi [1:2] FALSE FALSE
##  $ sigma        : num 0.116
##  $ df           : int [1:3] 2 20 2
##  $ r.squared    : num 0.425
##  $ adj.r.squared: num 0.396
##  $ fstatistic   : Named num [1:3] 14.8 1 20
##  $ cov.unscaled : num [1:2, 1:2] 1.23 -2.48 -2.48 5.19
```

---

### 1. Regressions

#### 1.1. On continuous variables

* Take the **coefficients** for instance

```r
ggmodel$coefficients
```

```
##                Estimate Std. Error    t value    Pr(>|t|)
## (Intercept) -0.09129311  0.1287045 -0.7093234 0.486311455
## gini         1.01546204  0.2642477  3.8428420 0.001015706
```

* We can **subset** this matrix like we would do with a regular `data.frame`

```r
ggmodel$coefficients[2, 1]
```

```
## [1] 1.015462
```

```r
ggmodel$coefficients[, "Pr(>|t|)"]
```

```
## (Intercept)        gini 
## 0.486311455 0.001015706
```

---

### 1. Regressions

#### 1.1. On continuous variables

* We can also easily **plot** the **distribution** of our **residuals**

```r
ggplot(data.frame(x = ggmodel$residuals), aes(x = x)) +
  geom_density() + geom_vline(xintercept = 0, linetype = "dashed")
```

--
 
<img src="slides_files/figure-html/unnamed-chunk-43-1.png" width="60%" style="display: block; margin: auto;" />

---

### 1. Regressions

#### 1.1. On continuous variables

* Note that ggplot() has a dedicated **geometry for fitted values:** `geom_smooth()`

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

---

### Practice

<center>Check that lm() works fine by computing the $R^2$ manually</center>

#### 1) Start by creating a variable for `$\hat{\beta}$`, then for `$\hat{\alpha}$`, `$\hat{y_i}$`, and `$\hat{\varepsilon_i}$`.

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

*You're gonna need the `cov()` and `var()` functions*

#### 2) Summarise the data into only `$\hat{\alpha}$`, `$\hat{\beta}$`, and the `$R^2$`

`$$R^2 = 1 - \frac{\sum(y_i-\hat{y_i})^2}{\sum(y_i-\bar{y_i})^2}$$`

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

---

### Solution

```r
ggcurve %>% 
    mutate(beta = cov(gini, ige) / var(gini)) 
#
#
#
#
#
#
```

```text
##           country  ige      gini     beta 
## 1         Denmark 0.15 0.3781796 1.015462 
## 2          Norway 0.17 0.3250102 1.015462 
## 3         Finland 0.18 0.3779868 1.015462 
## 4          Canada 0.19 0.4631237 1.015462 
## 5       Australia 0.26 0.4385511 1.015462 
## 6          Sweden 0.27 0.3582480 1.015462 
## 7     New Zealand 0.29 0.5039373 1.015462 
## 8         Germany 0.32 0.4377010 1.015462 
## 9           Japan 0.34 0.5198383 1.015462 
## 10          Spain 0.40 0.4826243 1.015462 
## 11         France 0.41 0.4495654 1.015462 
## ..            ...  ...       ...      ... 
```

---

### Solution

```r
ggcurve %>% 
    mutate(beta = cov(gini, ige) / var(gini),
           alpha = mean(ige) - beta * mean(gini))
#
#
#
#
#
```

```text
##           country  ige      gini     beta       alpha 
## 1         Denmark 0.15 0.3781796 1.015462 -0.09129311 
## 2          Norway 0.17 0.3250102 1.015462 -0.09129311 
## 3         Finland 0.18 0.3779868 1.015462 -0.09129311 
## 4          Canada 0.19 0.4631237 1.015462 -0.09129311 
## 5       Australia 0.26 0.4385511 1.015462 -0.09129311 
## 6          Sweden 0.27 0.3582480 1.015462 -0.09129311 
## 7     New Zealand 0.29 0.5039373 1.015462 -0.09129311 
## 8         Germany 0.32 0.4377010 1.015462 -0.09129311 
## 9           Japan 0.34 0.5198383 1.015462 -0.09129311 
## 10          Spain 0.40 0.4826243 1.015462 -0.09129311 
## 11         France 0.41 0.4495654 1.015462 -0.09129311 
## ..            ...  ...       ...      ...         ... 
```

---

### Solution

```r
ggcurve %>% 
    mutate(beta = cov(gini, ige) / var(gini),
           alpha = mean(ige) - beta * mean(gini),
           fit = alpha + beta * gini)  
#
#
#
#
```

```text
##           country  ige      gini     beta       alpha       fit 
## 1         Denmark 0.15 0.3781796 1.015462 -0.09129311 0.2927339 
## 2          Norway 0.17 0.3250102 1.015462 -0.09129311 0.2387424 
## 3         Finland 0.18 0.3779868 1.015462 -0.09129311 0.2925381 
## 4          Canada 0.19 0.4631237 1.015462 -0.09129311 0.3789915 
## 5       Australia 0.26 0.4385511 1.015462 -0.09129311 0.3540389 
## 6          Sweden 0.27 0.3582480 1.015462 -0.09129311 0.2724942 
## 7     New Zealand 0.29 0.5039373 1.015462 -0.09129311 0.4204361 
## 8         Germany 0.32 0.4377010 1.015462 -0.09129311 0.3531757 
## 9           Japan 0.34 0.5198383 1.015462 -0.09129311 0.4365829 
## 10          Spain 0.40 0.4826243 1.015462 -0.09129311 0.3987935
## 11         France 0.41 0.4495654 1.015462 -0.09129311 0.3652234
## ..            ...  ...       ...      ...         ...       ...
```

---

### Solution

```r
ggcurve %>% 
    mutate(beta = cov(gini, ige) / var(gini),
           alpha = mean(ige) - beta * mean(gini),
           fit = alpha + beta * gini,
           residuals = ige - fit)
#
#
#
```

```text
##           country  ige      gini     beta       alpha       fit     residuals
## 1         Denmark 0.15 0.3781796 1.015462 -0.09129311 0.2927339 -0.1427339244
## 2          Norway 0.17 0.3250102 1.015462 -0.09129311 0.2387424 -0.0687424324
## 3         Finland 0.18 0.3779868 1.015462 -0.09129311 0.2925381 -0.1125381050
## 4          Canada 0.19 0.4631237 1.015462 -0.09129311 0.3789915 -0.1889914561
## 5       Australia 0.26 0.4385511 1.015462 -0.09129311 0.3540389 -0.0940388831
## 6          Sweden 0.27 0.3582480 1.015462 -0.09129311 0.2724942 -0.0024941569
## 7     New Zealand 0.29 0.5039373 1.015462 -0.09129311 0.4204361 -0.1304360777
## 8         Germany 0.32 0.4377010 1.015462 -0.09129311 0.3531757 -0.0331756674
## 9           Japan 0.34 0.5198383 1.015462 -0.09129311 0.4365829 -0.0965829037
## 10          Spain 0.40 0.4826243 1.015462 -0.09129311 0.3987935  0.0012065012
## 11         France 0.41 0.4495654 1.015462 -0.09129311 0.3652234  0.0447765627
## ..            ...  ...       ...      ...         ...       ...           ...
```

---

### Solution

```r
ggcurve %>% 
    mutate(beta = cov(gini, ige) / var(gini),
           alpha = mean(ige) - beta * mean(gini),
           fit = alpha + beta * gini,
           residuals = ige - fit) %>%
    summarise(alpha = alpha[1],
              beta = beta[1],
              r2 = 1 - sum(residuals^2)/sum((ige - mean(ige))^2))
```

```text
##         alpha     beta       r2
## 1 -0.09129311 1.015462 0.424749
```

```r
ggmodel$coefficients[, "Estimate"]
```

```
## (Intercept)        gini 
## -0.09129311  1.01546204
```

```r
ggmodel$r.squared
```

```
## [1] 0.424749
```

---

### 1. Regressions

#### 1.2. On binary variables

* Now consider that we want to know the **relationship** between **not** being a **European country** and the **ige**

```r
ggcurve$country
```

```
##  [1] "Denmark"        "Norway"         "Finland"        "Canada"        
##  [5] "Australia"      "Sweden"         "New Zealand"    "Germany"       
##  [9] "Japan"          "Spain"          "France"         "Singapore"     
## [13] "Pakistan"       "Switzerland"    "United States"  "Argentina"     
## [17] "Italy"          "United Kingdom" "Chile"          "Brazil"        
## [21] "China"          "Peru"
```

--

```r
europe <- c("Denmark", "Norway", "Finland", "Sweden", "Germany", "Spain", 
 "France", "Switzerland", "Italy", "United Kingdom")
```

--

```r
ggcurve <- ggcurve %>%
 mutate(continent = ifelse(country %in% europe, "Europe", "Other"))
```

---

### 1. Regressions

#### 1.2. On binary variables

* Can we just **regress** ige **on** continent even though it's a **character** variable?

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

```
## 
## Call:
## lm(formula = ige ~ continent, data = ggcurve)
## 
## Coefficients:
##    (Intercept)  continentOther  
##         0.3360          0.1065
```

* Seems like we can!
 
--
 

 
<center>&#10140; But what's actually going on?</center>

---

### 1. Regressions

#### 1.2. On binary variables

* R implicitely **converts** character variables into **binary** variables

---

### 1. Regressions

#### 1.2. On binary variables

* Such that just **like** in the **continuous** case...

---

### 1. Regressions

#### 1.2. On binary variables

* ... we're looking at a **1-unit increase** in `$x$` *(i.e., swiching from 0 (Europe) to 1 (Other))*

---

### 1. Regressions

#### 1.2. On binary variables

* As you know the **fit** is necessarily going through the **mean of each category**
 
<img src="slides_files/figure-html/unnamed-chunk-66-1.png" width="40%" style="display: block; margin: auto;" />

---

### 1. Regressions

#### 1.2. On binary variables

* Such that `$\hat{\alpha}$` is the **mean of the reference group** 
 
<img src="slides_files/figure-html/unnamed-chunk-67-1.png" width="40%" style="display: block; margin: auto;" />

---

### 1. Regressions

#### 1.2. On binary variables

* And `$\hat{\beta}$` is the **difference in means**
 

 
<img src="slides_files/figure-html/unnamed-chunk-68-1.png" width="40%" style="display: block; margin: auto;" />

---

### 1. Regressions

#### 1.2. On binary variables

* We can verify that easily:

--
 
.pull-left[

```r
ggcurve %>% 
  group_by(continent) %>%
  summarise(ybar = mean(ige)) %>%
  mutate(relative = ybar - ybar[1])
```

```
## # A tibble: 2 x 3
## continent ybar relative
## <chr> <dbl> <dbl>
## 1 Europe 0.336 0 
## 2 Other 0.442 0.106
```
]

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

```
## 
## Call:
## lm(formula = ige ~ continent, data = ggcurve)
## 
## Coefficients:
##    (Intercept)  continentOther  
##         0.3360          0.1065
```
]

<center>&#10140; And what about discrete variables with more than 2 categories?</center>

---

### 1. Regressions

#### 1.3. On categorical variables

<ul>
 <li>Let's work on the <a href = "https://www.census.gov/data/datasets/time-series/demo/cps/cps-asec.html">2020 Annual Social and Economic (ASEC) Supplement</a> to the US CPS</li>
 <ul>
 <li>Here is an extract on 64,336 working individuals with positive earnings</li>
 <li>For which I kept only 4 variables:</li>
 </ul>
</ul>

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

```
## 'data.frame':	64336 obs. of  4 variables:
##  $ Sex     : chr  "Female" "Male" "Female" "Male" ...
##  $ Earnings: int  52500 34000 40000 8424 58000 42000 55000 28000 200 25000 ...
##  $ Race    : chr  "White" "White" "White" "White" ...
##  $ Hours   : int  40 40 44 21 60 40 40 40 20 40 ...
```

* Let's say we want to regress earnings on Race

```r
unique(asec$Race)
```

```
## [1] "White" "Other" "Black"
```

---

### 1. Regressions

#### 1.3. On categorical variables

<ul>
 <li>Just like the 2-category variable was equivalent to 1 dummy variable</li>
 <ul>
 <li>An n-category variable is equivalent to n-1 dummy variables</li>
 </ul>
</ul>

<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;"> Continent </th>
 <th style="text-align:right;"> Other </th>
 <th style="text-align:left;"> </th>
 <th style="text-align:left;"> Race </th>
 <th style="text-align:right;"> Other </th>
 <th style="text-align:right;"> White </th>
 </tr>
 </thead>
<tbody>
 <tr>
 <td style="text-align:left;"> Europe </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;"> 0 </td>
 <td style="text-align:right;"> 0 </td>
 </tr>
 <tr>
 <td style="text-align:left;"> Europe </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;"> 0 </td>
 <td style="text-align:right;"> 0 </td>
 </tr>
 <tr>
 <td style="text-align:left;"> Europe </td>
 <td style="text-align:right;"> 0 </td>
 <td style="text-align:left;width: 3em; "> </td>
 <td style="text-align:left;"> Other </td>
 <td style="text-align:right;"> 1 </td>
 <td style="text-align:right;"> 0 </td>
 </tr>
 <tr>
 <td style="text-align:left;"> Other </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;"> 1 </td>
 <td style="text-align:right;"> 0 </td>
 </tr>
 <tr>
 <td style="text-align:left;"> Other </td>
 <td style="text-align:right;"> 1 </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;"> 1 </td>
 </tr>
 <tr>
 <td style="text-align:left;"> Other </td>
 <td style="text-align:right;"> 1 </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;"> 1 </td>
 </tr>
</tbody>
</table>
]

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

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

]

* Instead of 1 `$x$` axis we're going to have n-1 `$x$` axes

---

### 1. Regressions

#### 1.3. On categorical variables

]

.pull-right[

<center>3-category variable</center>
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]

---

### 1. Regressions

#### 1.3. On categorical variables

<ul>
 <li>Once again the constant is the average $y$ for the reference category</li>
 <ul>
 <li>And the slopes are the relative differences in means</li>
 </ul>
</ul>

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

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

```r
asec %>%
  group_by(Race) %>%
  summarise(ybar = mean(Earnings)) %>%
  mutate(relative = ybar - ybar[1])
```

```
## # A tibble: 3 x 3
## Race ybar relative
## <chr> <dbl> <dbl>
## 1 Black 50577. 0 
## 2 Other 68055. 17477.
## 3 White 62880. 12303.
```

]

<ul>
 <li>Note that R always sorts character variables by alphabetical order</li>
 <ul>
 <li>But it would be more natural to interpret relative to the majority group</li>
 <li>Then how to change the reference category in the regression?</li>
 </ul>
</ul>

---

### 1. Regressions

#### 1.3. On categorical variables

<ul>
 <li>The function to set the reference category is relevel()</li>
 <ul>
 <li>The first argument is the vector to indicate the reference of</li>
 <li>The second argument is the value of the reference group</li>
 </ul>
</ul>

```r
lm(Earnings ~ relevel(Race, "White"), asec)
```

```
## Error in relevel.default(Race, "White"): 'relevel' only for (unordered) factors
```

* Oops! I need to introduce you a new class of R objects first: `factors`

```r
as.factor(asec$Race) %>% 
  levels()
```

```
## [1] "Black" "Other" "White"
```
]

```r
as.factor(asec$Race) %>% 
  relevel("White") %>% 
  levels()
```

```
## [1] "White" "Black" "Other"
```
]

---

### 1. Regressions

#### 1.3. On categorical 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
```

---

### Practice

#### 1) Open the data from the World Inequality Database we used in lecture 2

#### 2) Regress the income share of the top 1% on the year variable

#### 3) Redo the same regression after having converted the year variable as a `factor`

<center><h3>You've got 5 minutes!</h3></center>

---

### Solution

#### 1) Open the data from the World Inequality Database we used in lecture 2

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

#### 2) Regress the income share of the top 1% on the year variable

```r
lm(top1 ~ year, wid)
```

```
## 
## Call:
## lm(formula = top1 ~ year, data = wid)
## 
## Coefficients:
## (Intercept)         year  
##   2.242e-01   -3.131e-05
```

---

### Solution

#### 3) Redo the same regression after having converted the year variable as a `factor`

```r
lm(top1 ~ year, wid %>% mutate(year = as.factor(year)))
```

```
## 
## Call:
## lm(formula = top1 ~ year, data = wid %>% mutate(year = as.factor(year)))
## 
## Coefficients:
## (Intercept)     year2011     year2012     year2013     year2014     year2015  
##   0.1621494   -0.0011366   -0.0026891   -0.0013401   -0.0004469   -0.0008683  
##    year2016     year2017     year2018     year2019  
##  -0.0001450   -0.0004857   -0.0015236   -0.0018488
```

---

### 1. Regressions

#### 1.3. On categorical variables

<ul>
 <li>Another option to include categorical variables is to one hot encode the data</li>
 <ul>
 <li>It simply means converting the discrete variables into dummies such that everything is numeric</li>
 </ul>
</ul>

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

* Then we can use the `+` symbol to include n-1 categories in the regression

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

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

### 1. Regressions

#### 1.3. On categorical variables

<ul>
 <li>But do we really need to omit one category?</li>
 <ul>
 <li>As we are in the multivariate case let's move from $\hat{\beta} = \frac{\text{Cov}(x, y)}{\text{Var}(x)}$</li>
 <li>To $\hat{\beta} = (X'X)^{-1}X'y$</li>
 </ul>
</ul>

```r
y <- as.matrix(asec$Earnings)

X <- asec %>% 
 mutate(constant = 1) %>% 
 select(constant, White, Black, Other) %>% 
 as.matrix()
```

```r
dim(X)
```

```
## [1] 64336     4
```

]
.pull-right[

```r
dim(t(X))
```

```
## [1] 4 64336
```
]
]
.right-column[

```r
dim(t(X) %*% X)
```

```
## [1] 4 4
```
]

---

### 1. Regressions

#### 1.3. On categorical variables

* Because of perfect multicollinearity it will not be possible to invert `$X'X$`

```r
solve(t(X) %*% X)
```

```text
## Error in solve.default(t(X) %*% X): system is computationally singular
```

* We have to remove one category

```r
X <- asec %>% mutate(constant = 1) %>% 
* select(constant, Black, Other) %>% 
 as.matrix()
 
solve(t(X) %*% X) %*% (t(X) %*% y)
```

```
##                [,1]
## constant  62880.488
## Black    -12302.993
## Other      5174.141
```
]

* Or to remove the constant

```r
X <- asec %>% #mutate(constant = 1) %>% 
* select(White, Black, Other) %>% 
 as.matrix()
 
solve(t(X) %*% X) %*% (t(X) %*% y)
```

```
##           [,1]
## White 62880.49
## Black 50577.49
## Other 68054.63
```
]

---

### 1. Regressions

#### 1.3. On categorical variables

* Note that you can remove the constant in lm() by adding `- 1` in the formula

```r
lm(Earnings ~ White + Black + Other - 1, asec)
```

```text
## Coefficients:
## White  Black  Other  
## 62880  50577  68055
```

* And that it would drop a category anyway

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

```text
## Coefficients:
## (Intercept)        White        Black        Other  
##       68055        -5174       -17477           NA
```

]

.right-column[

<center>But it's not because multicollinearity does not break lm() that you should not pay attention to it!</center>
]

---

<h3>Overview</h3>

.pull-left[
<ul style = "margin-left:1.5cm;list-style: none">
 <li>1. Regressions &#10004;</li>
 <ul style = "list-style: none">
 <li>1.1. On continuous variables</li>
 <li>1.2. On binary variables</li>
 <li>1.3. On categorical variables</li>
 </ul>
</ul>

]

---

<h3>Overview</h3>

---

### 2. Case study

#### 2.1. Variable transformation

* Imagine you want to estimate the **relationship** between **weekly hours** of work and **annual earnings**

`$$\text{Earnings}_i = \alpha + \beta \text{Hours}_i + \varepsilon_i$$`

* We can **estimate it in R** using the Annual Social and Economic Supplement to the US CPS

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

```text
## Coefficients:
## (Intercept)        Hours  
##      -20039         2078
```

* Are we done?
 
--

---

### 2. Case study

#### 2.1. Variable transformation

```r
ggplot(asec, aes(x = Hours, y = Earnings)) +
  geom_point() + geom_smooth(method = "lm")
```

.left-column[
<img src="slides_files/figure-html/unnamed-chunk-108-1.png" width="90%" style="display: block; margin: auto;" />
]

<center>&#10140; Not very satisfactory</center>

<ul>
 <li>The joint distribution does not seem adequate on the the y dimension</li>
</ul>

<ul>
 <li>Let's take a look at the earnings distribution</li>
</ul>
]

---

### 2. Case study

#### 2.1. Variable transformation

```r
ggplot(asec, aes(x = Earnings)) +
  geom_density() 
```

.left-column[
<img src="slides_files/figure-html/unnamed-chunk-110-1.png" width="90%" style="display: block; margin: auto;" />
]
--

<center>It's clearly log-normal</center>
</ul>

<center>&#10140; Let's plot the log of it</center>
]

---

### 2. Case study

#### 2.1. Variable transformation

```r
ggplot(asec, aes(x = log(Earnings))) +
  geom_density() 
```

.left-column[
<img src="slides_files/figure-html/unnamed-chunk-112-1.png" width="90%" style="display: block; margin: auto;" />
]
--

<center>Better!</center>

<center>&#10140; Let's update the scatterplot</center>
]

---

### 2. Case study

#### 2.2. Functional form

```r
ggplot(asec, aes(x = Hours, y = log(Earnings))) +
  geom_point(alpha = .1) + geom_smooth(method = "lm")
```

]

<center>Definitely better!</center>

<center>&#10140; But something still feels off</center>
]

---

### 2. Case study

#### 2.2. Functional form

```r
ggplot(asec, aes(x = Hours, y = log(Earnings))) +
  geom_point(alpha = .1) + geom_smooth(method = "lm", formula = y ~ poly(x, 2))
```

]

<center>&#10140; There we go</center>
]

---

### 2. Case study

#### 2.2. Functional form

* We'd better rewrite our model as:
 
`$$\log(\text{Earnings}_i) = \alpha + \beta_1 \text{Hours}_i + \beta_2 \text{Hours}^2_i + \varepsilon_i$$`

* Create the new variables

```r
asec <- asec %>%
 mutate(lEarnings = log(Earnings),
 sqHours = Hours^2)
```

* And run the regression

```r
lm(lEarnings ~ Hours + sqHours, asec)
```

```text
## Coefficients:
## (Intercept)        Hours      sqHours  
##   7.3637848    0.1192609   -0.0008804
```

]

<center>Isn't there something we could/should add in our regression?</center>
]

---

### 2. Case study

#### 2.3. Control variables

<ul>
 <li>This positive relationship could be driven by a third variable</li>
 <ul>
 <li>Males tend both to work part time less often and to earn more</li>
 <li>The higher the hours, the higher the probability to be a male, the higher the expected earnings</li>
 </ul>
</ul>

<ul>
 <li>Let's control for that in the regression</li>
</ul>

`$$\log(\text{Earnings}_i) = \alpha + \beta_1 \text{Hours}_i + \beta_2 \text{Hours}^2_i + \beta_3 \text{Male}_i + \varepsilon_i$$`

--

.pull-left[
<ul>
 <li>Such that Males and Females would have</li>
 <ul>
 <li>The same slope</li>
 <li>Different intercepts</li>
 </ul>
</ul>
]

.pull-right[
<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;"> </th>
 <th style="text-align:center;"> $\text{Male} = 0$ </th>
 <th style="text-align:center;"> $\text{Male} = 1$ </th>
 </tr>
 </thead>
<tbody>
 <tr>
 <td style="text-align:left;"> Intercept </td>
 <td style="text-align:center;"> $\alpha$ </td>
 <td style="text-align:center;"> $\alpha + \beta_3$ </td>
 </tr>
 <tr>
 <td style="text-align:left;"> Slope </td>
 <td style="text-align:center;"> $\beta_1 + 2\beta_2\text{Hours}$ </td>
 <td style="text-align:center;"> $\beta_1 + 2\beta_2\text{Hours}$ </td>
 </tr>
</tbody>
</table>
]

---

### 2. Case study

#### 2.3. Control variables

.pull-left[

<img src="slides_files/figure-html/unnamed-chunk-122-1.png" width="100%" style="display: block; margin: auto;" />
]
 
.pull-right[

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]

---

### 2. Case study

#### 2.3. Control variables

* Once again we can include a control variable using the `+` symbol

```r
lm(lEarnings ~ Hours + sqHours + Sex, asec)
```

```text
## Coefficients:
## (Intercept)        Hours      sqHours      SexMale  
##   7.3356057    0.1177514   -0.0008806    0.1700972
```

--

```r
lm(lEarnings ~ Hours + sqHours, asec)
```

```text
## Coefficients:
## (Intercept)        Hours      sqHours  
##   7.3637848    0.1192609   -0.0008804
```

<center>Actually the baseline coefficient was not that inflated</center>

---

### 2. Case study

#### 2.4. Interactions

<ul>
 <li>But what if we were to allow for different slopes?</li>
 <ul>
 <li>The relationship between hours and earnings might be heterogeneous across sex/gender</li>
 <li>This is what interactions allow to account for</li>
 </ul>
</ul>

* We simply have to include the **product** of the two variables in the model

`$$\log(\text{Earnings}_i) = \alpha + \beta_1 \text{Hours}_i + \beta_2 \text{Hours}^2_i + \beta_3 \text{Male}_i + \beta_4 \text{Hours}_i \times \text{Male}_i + \varepsilon_i$$`

--

.pull-left[
<ul>
 <li>Such that Males and Females would have</li>
 <ul>
 <li>Not only different intercepts</li>
 <li>But also different slopes</li>
 </ul>
</ul>
]

---

### 2. Case study

#### 2.4. Interactions

.pull-left[

<img src="slides_files/figure-html/unnamed-chunk-129-1.png" width="100%" style="display: block; margin: auto;" />
]
 
.pull-right[

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]

---

### 2. Case study

#### 2.4. Interactions

* Now we can use the `*` symbol for products

```r
results <- summary(lm(lEarnings ~ Hours + sqHours + Sex + Hours * Sex, asec))$coefficients
```

```r
results
```

```
##                    Estimate   Std. Error    t value     Pr(>|t|)
## (Intercept)    7.3900468002 2.319198e-02 318.646690 0.000000e+00
## Hours          0.1174998826 1.034522e-03 113.578950 0.000000e+00
## sqHours       -0.0009103523 1.321706e-05 -68.877048 0.000000e+00
## SexMale       -0.0282905673 2.753222e-02  -1.027544 3.041682e-01
## Hours:SexMale  0.0050321134 6.767013e-04   7.436240 1.048736e-13
```

What do you conclude ?

```r
results[5, 1] / results[2, 1]
```

```
## [1] 0.04282654
```

<center>&#10140; A 4% difference</center>
]

```r
results[5, 4]
```

```
## [1] 1.048736e-13
```

<center>&#10140; Which is highly significant</center>
]

---

<h3>Overview</h3>

<ul style = "margin-left:1.5cm;list-style: none">
 <li>2. Case study &#10004;</li>
 <ul style = "list-style: none">
 <li>2.1. Variable transformation</li>
 <li>2.2. Functional form</li>
 <li>2.3. Control variables</li>
 <li>2.4. Interactions</li>
 </ul>
</ul>
]

]

---

<h3>Overview</h3>

---

### 3. Inference

#### 3.1. Hypothesis testing

<ul>
 <li>According to our previous regression, $\hat{\beta_1}$ is significantly different from 0</li>
 <ul>
 <li>But let's pretend that you know that this coefficient is equal to .12 in Canada</li>
 <li>How could we test whether or not our coefficient is different from .12?</li>
 </ul>
</ul>

```r
results
```

`$$t = \frac{\hat{\beta} - .12}{\text{s.e.}(\hat{\beta})}$$`
]

```r
t <- (results[2, 1] - .12) / results[2, 2]
t
```

```
## [1] -2.416689
```
]

---

### 3. Inference

#### 3.1. Hypothesis testing

<ul>
 <li>And then we need the value of the area outside the interval $\left[-t;t\right]$</li>
 <ul>
 <li>From a Student-t distribution with the correct number of degrees of freedom (#obs - #parameters)</li>
 </ul>
</ul>

---

### 3. Inference

#### 3.1. Hypothesis testing

<ul>
 <li>You can get the area below a certain t-value with the pt() function</li>
 <ol>
 <li></li>
 <li></li>
 </ol>
</ul>

```r
pt( , )
```

---

### 3. Inference

#### 3.1. Hypothesis testing

<ul>
 <li>You can get the area below a certain t-value with the pt() function</li>
 <ol>
 <li>The first argument is the t-value</li>
 <li></li>
 </ol>
</ul>

```r
pt(t, )
```

---

### 3. Inference

#### 3.1. Hypothesis testing

<ul>
 <li>You can get the area below a certain t-value with the pt() function</li>
 <ol>
 <li>The first argument is the t-value</li>
 <li>The second argument is the number of degrees of freedom</li>
 </ol>
</ul>

```r
pt(t, nrow(asec) - nrow(results))
```
--

```
## [1] 0.007832573
```

* We just have to multiply this value by 2 to obtain the p-value

```r
2 * pt(t, nrow(asec) - nrow(results))
```

```
## [1] 0.01566515
```
]

* Had our t-stat been positive we would have needed to multiply 1 - `$t$` by 2

```r
2 * (1 - pt(abs(t), nrow(asec)-nrow(results)))
```

```
## [1] 0.01566515
```
]

---

### 3. Inference

#### 3.1. Hypothesis testing

<ul>
 <li>A very handy function for hypothesis testing is linearHypothesis() from the car package</li>
 <ul>
 <li></li>
 <li></li>
 </ul>
</ul>

```r
linearHypothesis( , )
```

---

### 3. Inference

#### 3.1. Hypothesis testing

<ul>
 <li>A very handy function for hypothesis testing is linearHypothesis() from the car package</li>
 <ul>
 <li>The first argument is the model</li>
 <li></li>
 </ul>
</ul>

```r
linearHypothesis(lm(lEarnings ~ Hours + sqHours + Sex + Hours * Sex, asec), )
```

---

### 3. Inference

#### 3.1. Hypothesis testing

```r
linearHypothesis(lm(lEarnings ~ Hours + sqHours + Sex + Hours * Sex, asec), c("Hours = .12"))
```

```
## Linear hypothesis test
## 
## Hypothesis:
## Hours = 0.12
## 
## Model 1: restricted model
## Model 2: lEarnings ~ Hours + sqHours + Sex + Hours * Sex
## 
##   Res.Df   RSS Df Sum of Sq      F  Pr(>F)  
## 1  64332 46102                              
## 2  64331 46098  1    4.1851 5.8404 0.01567 *
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
```

---

### 3. Inference

#### 3.1. Hypothesis testing

<ul>
 <li>It can be used for F tests like the one from the summary</li>
</ul>

```r
linearHypothesis(lm(lEarnings ~ Hours + sqHours + Sex + Hours * Sex, asec), 
                 c("Hours = 0", "sqHours = 0", "SexMale = 0", "Hours:SexMale = 0"))
```

---

### 3. Inference

#### 3.1. Hypothesis testing

```r
linearHypothesis(lm(lEarnings ~ Hours + sqHours + Sex + Hours * Sex, asec), 
                 c("Hours = 0", "sqHours = 0", "SexMale = 0", "Hours:SexMale = 0"))
```

```
## Linear hypothesis test
## 
## Hypothesis:
## Hours = 0
## sqHours = 0
## SexMale = 0
## Hours:SexMale = 0
## 
## Model 1: restricted model
## Model 2: lEarnings ~ Hours + sqHours + Sex + Hours * Sex
## 
## Res.Df RSS Df Sum of Sq F Pr(>F) 
## 1 64335 68395 
## 2 64331 46098 4 22297 7779.1 < 2.2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
```

---

### 3. Inference

```text
## 
## Call:
## lm(formula = lEarnings ~ Hours + sqHours + Sex + Hours * Sex, 
## data = asec)
## 
## Residuals:
## Min 1Q Median 3Q Max 
## -10.3453 -0.4299 0.0133 0.4810 4.8506 
## 
## Coefficients:
## Estimate Std. Error t value Pr(>|t|) 
## (Intercept) 7.390e+00 2.319e-02 318.647 < 2e-16 ***
## Hours 1.175e-01 1.035e-03 113.579 < 2e-16 ***
## sqHours -9.103e-04 1.322e-05 -68.877 < 2e-16 ***
## SexMale -2.829e-02 2.753e-02 -1.028 0.304 
## Hours:SexMale 5.032e-03 6.767e-04 7.436 1.05e-13 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.8465 on 64331 degrees of freedom
## Multiple R-squared: 0.326, Adjusted R-squared: 0.326 
*## F-statistic: 7779 on 4 and 64331 DF, p-value: < 2.2e-16
```

---

### 3. Inference

#### 3.2. Confidence intervals

<ul>
 <li>Now we know how to get the area below a given value of a Student $t$ distribution</li>
 <ul>
 <li>But sometimes the opposite is useful as well</li>
 <li>In particular to compute confidence intervals</li>
 </ul>
</ul>

* Indeed the confidence interval for a `$\hat{\beta}$` coefficient is given by:
 
`$$\hat{\beta} \pm t_{1-(\alpha/2), \text{df}} \times \text{s.e.}(\hat{\beta})$$`

<ul><ul><li>Where $\alpha$ denotes the desired significance level</li></ul></ul>

<ul>
 <li>qt() gives the $t$-statistic below which lies the desired share of the area of a given Student $t$ distribution</li>
 <ul>
 <li></li>
 <li></li>
 </ul>
</ul>

```r
qt( , )
```

---

### 3. Inference

#### 3.2. Confidence intervals

* Indeed the confidence interval for a `$\hat{\beta}$` coefficient is given by:
 
`$$\hat{\beta} \pm t_{1-(\alpha/2), \text{df}} \times \text{s.e.}(\hat{\beta})$$`

<ul><ul><li>Where $\alpha$ denotes the desired significance level</li></ul></ul>

<ul>
 <li>qt() gives the $t$-statistic below which lies the desired share of the area of a given Student $t$ distribution</li>
 <ul>
 <li>The first argument is $1-(\alpha/2)$</li>
 <li></li>
 </ul>
</ul>

```r
qt(.975, )
```

---

### 3. Inference

#### 3.2. Confidence intervals

* Indeed the confidence interval for a `$\hat{\beta}$` coefficient is given by:
 
`$$\hat{\beta} \pm t_{1-(\alpha/2), \text{df}} \times \text{s.e.}(\hat{\beta})$$`

<ul><ul><li>Where $\alpha$ denotes the desired significance level</li></ul></ul>

```r
qt(.975, Inf)
```

```
## [1] 1.959964
```

---

### 3. Inference

#### 3.2. Confidence intervals

<ul>
 <li>So if we want a 97% confidence interval for our coefficients associated with hours, we feed qt() with:</li>
 <ul>
 <li></li>
 <li></li>
 </ul>
</ul>

```r
t003 <- qt( , )
```

---

### 3. Inference

#### 3.2. Confidence intervals

<ul>
 <li>So if we want a 97% confidence interval for our coefficients associated with hours, we feed qt() with:</li>
 <ul>
 <li>The share of the Student $t$ distribution below the desired $t$-stat: $1-(\alpha/2) = 1-(0.03/2) = 0.985$</li>
 <li></li>
 </ul>
</ul>

```r
t003 <- qt(.985, )
```

---

### 3. Inference

#### 3.2. Confidence intervals

```r
t003 <- qt(.985, nrow(asec) - nrow(results))
```

---

### 3. Inference

#### 3.2. Confidence intervals

```r
t003 <- qt(.985, nrow(asec) - nrow(results))
t003
```

```
## [1] 2.170139
```

--

* And we apply the formula:

```r
results[2, 1] - (results[2, 2] * t003)
```

```
## [1] 0.1152548
```
]
 
--

```r
results[2, 1] + (results[2, 2] * t003)
```

```
## [1] 0.1197449
```
]

---

<h3>Overview</h3>

<ul style = "margin-left:-1cm;list-style: none">
 <li>3. Inference &#10004;</li>
 <ul style = "list-style: none">
 <li>3.1. Hypothesis testing</li>
 <li>3.2. Confidence intervals</li>
 </ul>
</ul>

]

---

<h3>Overview</h3>

]

---

### 4. Report and export results

#### 4.1. Regression tables

<ul>
 <li>The output of the summary() function is very practical but not very convenient to report the main results</li>
 <ul>
 <li>Academic regression tables rather look like that</li>
 </ul>
</ul>

---

### 4. Report and export results

#### 4.1. Regression tables

<ul>
 <li>The huxreg() function from huxtable allows to create such tables.</li>
 <ul>
 <li></li>
 <li></li>
 <li></li>
 <li></li>
 <li></li>
 <li></li>
 </ul>
</ul>

```r
outreg <- huxreg()
#
#
#
#
#
#
#
#
```

---

### 4. Report and export results

#### 4.1. Regression tables

---

### 4. Report and export results

#### 4.1. Regression tables

---

### 4. Report and export results

#### 4.1. Regression tables

<ul>
 <li>The huxreg() function from huxtable allows to create such tables. Main arguments include:</li>
 <ul>
 <li>As many models as you want, named or not</li>
 <li>Which uncertainty statistic to display (std.error, p.value, conf.low, conf.high)</li>
 <li>Where to place the uncertainty statistic (below, same, right)</li>
 <li></li>
 <li></li>
 <li></li>
 </ul>
</ul>

---

### 4. Report and export results

#### 4.1. Regression tables

```r
outreg <- huxreg(Baseline = lm(lEarnings ~ Hours, asec),
 lm(lEarnings ~ Hours + sqHours, asec),
 lm(lEarnings ~ Hours + sqHours + Sex, asec),
 lm(lEarnings ~ Hours + sqHours + Sex + Hours * Sex, asec),
 error_format = "({std.error})",
 error_pos = "below",
 statistics = c(N = "nobs", R2 = "r.squared"))
#
#
```

---

### 4. Report and export results

#### 4.1. Regression tables

---

### 4. Report and export results

#### 4.1. Regression tables

---

### 4. Report and export results

#### 4.1. Regression tables

```text
## ─────────────────────────────────────────────────────────────────────────────────────────────────────
## Baseline (2) (3) (4) 
## ────────────────────────────────────────────────────────────────────────────────
## (Intercept) 8.604 *** 7.364 *** 7.336 *** 7.390 *** 
## (0.014) (0.022) (0.022) (0.023) 
## Hours 0.051 *** 0.119 *** 0.118 *** 0.117 *** 
## (0.000) (0.001) (0.001) (0.001) 
## sqHours -0.001 *** -0.001 *** -0.001 *** 
## (0.000) (0.000) (0.000) 
## SexMale 0.170 *** -0.028 
## (0.007) (0.028) 
## Hours:SexMale 0.005 *** 
## (0.001) 
## ────────────────────────────────────────────────────────────────────────────────
## N 64336 64336 64336 64336 
## R2 0.268 0.319 0.325 0.326 
## ─────────────────────────────────────────────────────────────────────────────────────────────────────
## Dependent variable: log annual earnings. *** p < 0.01; ** p < 0.05; * p < 0.1
```

* Then export it with `quick_[latex/html/pdf/docx](outreg, file = "path/filename.format")`

---

### 4. Report and export results

#### 4.1. Regression tables

---

### 4. Report and export results

#### 4.1. Regression tables

---

<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-169">
<col><col><col><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;">Baseline</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;">(2)</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;">(3)</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;">(4)</th></tr>
<tr>
<th style="vertical-align: top; text-align: left; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">(Intercept)</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;">8.604 ***</td><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;">7.364 ***</td><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;">7.336 ***</td><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;">7.390 ***</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;">(0.014)   </td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">(0.022)   </td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">(0.022)   </td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">(0.023)   </td></tr>
<tr>
<th style="vertical-align: top; text-align: left; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">Hours</th><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">0.051 ***</td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">0.119 ***</td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">0.118 ***</td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">0.117 ***</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;">(0.000)   </td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">(0.001)   </td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">(0.001)   </td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">(0.001)   </td></tr>
<tr>
<th style="vertical-align: top; text-align: left; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">sqHours</th><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">        </td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">-0.001 ***</td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">-0.001 ***</td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">-0.001 ***</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;">        </td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">(0.000)   </td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">(0.000)   </td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">(0.000)   </td></tr>
<tr>
<th style="vertical-align: top; text-align: left; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">SexMale</th><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">        </td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">        </td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">0.170 ***</td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">-0.028    </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;">        </td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">        </td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">(0.007)   </td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">(0.028)   </td></tr>
<tr>
<th style="vertical-align: top; text-align: left; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">Hours:SexMale</th><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">        </td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">        </td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">        </td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">0.005 ***</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;">        </td><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;">        </td><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;">        </td><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;">(0.001)   </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;">64336        </td><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;">64336        </td><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;">64336        </td><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;">64336        </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;">R2</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;">0.268    </td><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;">0.319    </td><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;">0.325    </td><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;">0.326    </td></tr>
<tr>
<th colspan="5" 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;">Dependent variable: log annual earnings. *** p < 0.01; ** p < 0.05; * p < 0.1</th></tr>
</table>

---

### 4. Report and export results

#### 4.1. Regression tables

---

### Practice

#### Use the functions `huxreg()`, `insert_row()`, and `merge_cells()` to reproduce this table:

```text
## Dependent variable: Log annual earnings 
## (1) (2) (3) (4) 
## ────────────────────────────────────────────────────
## Hours worked 0.051 *** 0.119 *** 0.118 *** 0.117 *** 
## (0.000) (0.000) (0.000) (0.000) 
## (Hours worked)² -0.001 *** -0.001 *** -0.001 *** 
## (0.000) (0.000) (0.000) 
## Male 0.170 *** -0.028 
## (0.000) (0.304) 
## Hours worked x Male 0.005 *** 
## (0.000) 
## Constant 8.604 *** 7.364 *** 7.336 *** 7.390 *** 
## (0.000) (0.000) (0.000) (0.000) 
## ────────────────────────────────────────────────────
## N 64336 64336 64336 64336 
## R2 0.268 0.319 0.325 0.326 
## ──────────────────────────────────────────────────────────────────────────
## Significance: *** p < 0.01; ** p < 0.05; * p < 0.1 
```

---

### Solution

#### Use the functions `huxreg()`, `insert_row()`, and `merge_cells()` to reproduce the table

---

### Solution

#### Use the functions `huxreg()`, `insert_row()`, and `merge_cells()` to reproduce the table

---

### Solution

#### Use the functions `huxreg()`, `insert_row()`, and `merge_cells()` to reproduce the table

---

### Solution

#### Use the functions `huxreg()`, `insert_row()`, and `merge_cells()` to reproduce the table

---

### Solution

#### Use the functions `huxreg()`, `insert_row()`, and `merge_cells()` to reproduce the table

---

### Solution

#### Use the functions `huxreg()`, `insert_row()`, and `merge_cells()` to reproduce the table

```r
huxreg(lm(lEarnings ~ Hours, asec),
       lm(lEarnings ~ Hours + sqHours, asec),
       lm(lEarnings ~ Hours + sqHours + Sex, asec),
       lm(lEarnings ~ Hours + sqHours + Sex + Hours * Sex, asec),
       error_format = "({p.value})",
       error_pos = "below",
       statistics = c(N = "nobs", R2 = "r.squared"), 
       stars = c(`***` = 0.01, `**` = 0.05, `*` = 0.1),
       note = "Significance: {stars}",
       align = "c",
       coefs = c("Hours worked" = "Hours",
                 "(Hours worked)²" = "sqHours",
                 "Male" = "SexMale",
                 "Hours worked x Male" = "Hours:SexMale",
                 "Constant" = "(Intercept)"))
#
#
```

---

### Solution

#### Use the functions `huxreg()`, `insert_row()`, and `merge_cells()` to reproduce the table

---

### Solution

#### Use the functions `huxreg()`, `insert_row()`, and `merge_cells()` to reproduce the table

---

### 4. Report and export results

#### 4.2. Plot coefficients

<ul>
 <li>It can also be useful to provide a graphical representation of the coefficients</li>
 <ul>
 <li>By now you should be able to work it around with dplyr and ggplot</li>
 <li>But there exists a shortcut</li>
 </ul>
</ul>

<ul>
 <li>The plot_summs() function from the jtools package takes regression models as inputs and plots the results</li>
 <ul>
 <li></li>
 <li></li>
 <li></li>
 <li></li>
 <li></li>
 </ul>
</ul>

```r
plot_summs()
#
#
#
#
#
```

---

### 4. Report and export results

#### 4.2. Plot coefficients

```r
plot_summs(lm(lEarnings ~ Hours + Race + Sex, asec),
           lm(lEarnings ~ Hours + Race + Sex + sqHours, asec))
#
#
#
#
```

---

### 4. Report and export results

#### 4.2. Plot coefficients

<ul>
 <li>The plot_summs() function from the jtools package takes regression models as inputs and plots the results</li>
 <ul>
 <li>First feed it with your models</li>
 <li>You can choose to omit some coefficients</li>
 <li></li>
 <li></li>
 <li></li>
 </ul>
</ul>

```r
plot_summs(lm(lEarnings ~ Hours + Race + Sex, asec),
           lm(lEarnings ~ Hours + Race + Sex + sqHours, asec),
           omit.coefs = "(Intercept)")
#
#
#
```

---

### 4. Report and export results

#### 4.2. Plot coefficients

```r
plot_summs(lm(lEarnings ~ Hours + Race + Sex, asec),
           lm(lEarnings ~ Hours + Race + Sex + sqHours, asec),
           omit.coefs = "(Intercept)",
           ci_level = 0.99)
#
#
```

---

### 4. Report and export results

#### 4.2. Plot coefficients

```r
plot_summs(lm(lEarnings ~ Hours + Race + Sex, asec),
           lm(lEarnings ~ Hours + Race + Sex + sqHours, asec),
           omit.coefs = "(Intercept)",
           ci_level = 0.99,
           colors = c("#014D64", "#00A2D9")) 
#
```

---

### 4. Report and export results

#### 4.2. Plot coefficients

---

### 4. Report and export results

#### 4.2. Plot coefficients
 
<img src="slides_files/figure-html/unnamed-chunk-185-1.png" width="76%" style="display: block; margin: auto;" />

---

<h3>Overview</h3>

<ul style = "margin-left:-1cm;list-style: none">
 <li>4. Report and export results &#10004;</li>
 <ul style = "list-style: none">
 <li>4.1. Regression tables</li>
 <li>4.2. Plot coefficients</li>
 </ul>
</ul>

]

---

### 5. Wrap up!

#### Regressions in R

```r
summary(lm(formula = ige ~ gini, data = ggcurve))
```

---

### 5. Wrap up!

#### Variable transformations, functional forms, controls, interactions

`$$\log(\text{Earnings}_i) = \alpha + \beta_1 \text{Hours}_i + \beta_2 \text{Hours}^2_i + \beta_3 \text{Male}_i + \beta_4 \text{Hours}_i \times \text{Male}_i + \varepsilon_i$$`

.pull-left[

<img src="slides_files/figure-html/unnamed-chunk-188-1.png" width="100%" style="display: block; margin: auto;" />
]
 
.pull-right[

```r
summary(
  lm(lEarnings ~ Hours + 
                 sqHours + 
                 Sex + 
                 Hours * Sex, 
     asec)
)$coefficients[, 1:2]
```

```
##                    Estimate   Std. Error
## (Intercept)    7.3900468002 2.319198e-02
## Hours          0.1174998826 1.034522e-03
## sqHours       -0.0009103523 1.321706e-05
## SexMale       -0.0282905673 2.753222e-02
## Hours:SexMale  0.0050321134 6.767013e-04
```
]

---

### 5. Wrap up!

#### Hypothesis testing

```r
linearHypothesis(lm(lEarnings ~ Hours + sqHours + Sex + Hours * Sex, asec), 
                 c("Hours = 0", "sqHours = 0", "SexMale = 0", "Hours:SexMale = 0"))
```

```text
## Linear hypothesis test
## 
## Hypothesis:
## Hours = 0
## sqHours = 0
## SexMale = 0
## Hours:SexMale = 0
## 
## Model 1: restricted model
## Model 2: lEarnings ~ Hours + sqHours + Sex + Hours * Sex
## 
## Res.Df RSS Df Sum of Sq F Pr(>F) 
## 1 64335 68395 
## 2 64331 46098 4 22297 7779.1 < 2.2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
```

---

### 5. Wrap up!

#### Report/export results

```r
huxreg(
  Baseline = lm(lEarnings ~ Hours, asec),
  lm(lEarnings ~ Hours + sqHours, asec),
  lm(lEarnings ~ Hours + sqHours + Sex, asec),
  lm(lEarnings ~ Hours + sqHours + Sex + 
                 Hours * Sex, asec),
  error_format = "({std.error})",
  error_pos = "below",
  statistics = c(N="nobs", R2="r.squared"), 
  stars = c(`***` = 0.01, 
            `**` = 0.05, 
            `*` = 0.1),
  note = paste("Dependent variable: log",
               "annual earnings. {stars}"
)
```

]