Introductory Econometrics

class: center, middle, inverse, title-slide

# Introductory Econometrics
## Lecture 18
### Louis SIRUGUE
### CPES 2 - Spring 2023

---

<h3>Today: Refresher on Introductory Econometrics</h3>

.pull-left[

<ul style = "margin-left:1cm;list-style: none">
 <li>1. Regressions with continuous variables</li>
 <ul style = "list-style: none">
 <li>1.1. Estimation</li>
 <li>1.2. Inference</li>
 </ul>
</ul>

<ul style = "margin-left:1cm;list-style: none">
 <li>2. Regressions with discrete variables</li>
 <ul style = "list-style: none">
 <li>2.1. Binary dependent variable</li>
 <li>2.2. Binary independent variable</li>
 <li>2.3. Categorical independent variable</li>
 </ul>
</ul>

]

.pull-right[

<ul style = "margin-left:-1cm;list-style: none">
 <li>3. Controls and interactions</li>
</ul>

<ul style = "margin-left:-1cm;list-style: none">
 <li>4. Interpretation</li>
</ul>

]

---

<h3>Today: Refresher on Introductory Econometrics</h3>

.pull-left[

]
 
---

### 1. Regressions with continuous variables

#### 1.1. Estimation

* 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>
 
]

---

### 1. Regressions with continuous variables

#### 1.1. Estimation

* Consider these two relationships:

.left-column[

]

.right-column[

***But a given increase in x is not associated with a same increase in y!***
 
]

---

### 1. Regressions with continuous variables

#### 1.1. Estimation

* The idea of a regression is to find the line that fits the data the best
 * Such that its slope can indicate how we expect y to change if we increase x by 1 unit
 
--

---

### 1. Regressions with continuous variables

#### 1.1. Estimation

* To do so we should minimize the distance between each point and the line

--
 
<img src="slides_files/figure-html/unnamed-chunk-5-1.png" width="90%" style="display: block; margin: auto;" />

---

### 1. Regressions with continuous variables

#### 1.1. 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 when `$x = x_i$`

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

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

]

---

### 1. Regressions with continuous variables

#### 1.1. Estimation

.pull-left[

]

.pull-right[

* Because `$\widehat{\varepsilon_i}$` is the value of 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 y-intercept
  * `$\hat{\beta}$` is the slope
  * Both are estimations of the actual `$\alpha$` and `$\beta$` of the unknown DGP
]

---

### 1. Regressions with continuous variables

#### 1.1. 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-8-1.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:** `$\hat{\alpha}$` and `$\hat{\beta}$` seem appropriate &#10140; `$\widehat{\varepsilon_i}$` are low

]

---

### 1. Regressions with continuous variables

#### 1.1. 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>

<ul>
 <li>If we replace $\widehat{\varepsilon_i}$ with $y_i -\hat{\alpha} - \hat{\beta}x_i$</li>
 <ul>
 <li>We can solve the minimization problem (see Lecture 7) to obtain:</li>
 </ul>
</ul>

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

---

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

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

<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 it is what we tune 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

---

### 1. Regressions with continuous variables

#### 1.2. Inference

<ul>
 <li>Inference refers to the fact of being able to conclude something from our estimation</li>
 <ul>
 <li>The $\hat{\beta}$ from our sample is actually an estimation of the unobserved $\beta$ of the underlying population</li>
 <li>We would like to know how reliable $\hat{\beta}$ is, how confident we are in its estimation</li>
 <li>The first step of inference is to compute the standard error of $\hat{\beta}$</li>
 </ul>
</ul>

`$$\text{se}(\hat{\beta}) = \sqrt{\widehat{\text{Var}(\hat{\beta})}} = \sqrt{\frac{\sum_{i = 1}^n\hat{\varepsilon_i}^2}{(n-\#\text{parameters})\sum_{i = 1}^n(x_i-\bar{x})^2}}$$`

<ul>
 <li>Notice that the variance, and thus the standard error of our estimate:</li>
 <ul>
 <li>Decreases as our sample gets bigger</li>
 <li>Gets larger if the points are further away from the regression line on average for a given variance of $x$</li>
 </ul>
</ul>

---

### 1. Regressions with continuous variables

#### 1.2. Inference

* The magnitude of the standard error gives an indication of the precision of our estimate:
 * The larger the estimate relative to its standard error, the more precise the estimate

* But standard errors are not easily interpretable by themselves
 * A more direct way to get a sense of the precision for inference is to construct a confidence interval

<center>&#10140; Instead of saying that our estimation $\hat{\beta}$ is equal to 1.02, we would like to say that we are 95% sure that the actual $\beta$ lies between two given values</center>

* To obtain a confidence interval we can use the fact that under specific conditions (that you're gonna see next year) it is possible to derive how this object is distributed:

`$$\hat{t} \equiv \frac{\hat{\beta} - \beta}{\text{se}(\hat{\beta})}$$`

---

### 1. Regressions with continuous variables

#### 1.2. Inference

* Theory shows that `$\hat{t} \equiv \frac{\hat{\beta} - \beta}{\text{se}(\hat{\beta})}$` follows a Student t distribution whose number of degrees of freedom is equal to `$n$` (in our case 22 countries) minus the number of parameters estimated in the model (in our case 2: `$\alpha$` and `$\beta$`)

---

### 1. Regressions with continuous variables

#### 1.2. Inference

* Denote `$t_{97.5\%}$` the value such that 97.5% of the distribution is below that value
  * Then 95% of the distribution lies between `$-t_{97.5\%}$` and `$t_{97.5\%}$`

---

### 1. Regressions with continuous variables

#### 1.2. Inference

* Because we know that `$\hat{t} \equiv \frac{\hat{\beta} - \beta}{\text{se}(\hat{\beta})}$` follows this distribution, we know that it has a 95% chance to fall within the two values `$-t_{97.5\%}$` and `$t_{97.5\%}$`

--
 
`$$\text{Pr}\left[-t_{97.5\%}\leq\frac{\hat{\beta} - \beta}{\text{se}(\hat{\beta})}\leq t_{97.5\%}\right] = 95\%$$`

* Rearranging the terms yields:

`$$\text{Pr}\left[\hat{\beta} - t_{97.5\%}\times\text{se}(\hat{\beta})\leq \beta \leq\hat{\beta} + t_{97.5\%}\times\text{se}(\hat{\beta})\right] = 95\%$$`

.left-column[

* Thus, we can say that there is a 95% chance for `$\beta$` to be within
 
$$\hat{\beta} \pm t_{97.5\%}\times\text{se}(\hat{\beta}) $$

]

.right-column[

* To get `$t_{97.5\%}$` with 20 df:

```r
qt(.975, 20)
```

]

---

### 1. Regressions with continuous variables

#### 1.2. Inference

* ***Confidence intervals*** are very effective to get a sense of the precision of our estimates and of the **range of values the true parameters could reasonably take**

* But the ***p-value*** is what we tend to ultimately focus on, it is the **% chance that our estimation of the true parameter is different from a given value (generally 0) just coincidentally**

<ul>
 <li>Confidence intervals and p-values are tightly linked</li>
 <ul>
 <li>If there is a 4% chance that a parameter equal to 2 is different from 0, I know that the 95% confidence interval will start above 0 but quite close, and stop a bit before 4</li>
 <li>If a 95% confidence interval is bounded by 4 and 5, I know the the p-value will be way below 5%</li>
 </ul>
</ul>

<ul>
 <li>But these two indicators are complementary to easily get the full picture:</li>
 <ul>
 <li>With a p-value we can easily know how sure we are that the parameter is different from a given value, but it is difficult to get a sense of the set of values the parameters can reasonably take</li>
 <li>With the confidence interval it is the opposite</li>
 </ul>
</ul>

---

### 1. Regressions with continuous variables

#### 1.2. Inference

<ul>
 <li>P-val. computation: The principle is the same as for standard errors but the reasoning is reversed</li>
 <ul>
 <li>For confidence intervals: we want to know among which values the parameter has a given percentage chance to fall into</li>
 <li>For p-value: we want to know with which percentage chance 0 is out of the set of values that the parameter could reasonably take</li>
 </ul>
</ul>

<ul>
 <li>Vocabulary: We talk about significance level</li>
 <ul>
 <li> When $\text{P-value} \leq .05$, we say that the estimate is significant(ly different from 0) at the 5% level</li>
 <li> When the p-value is greater than a given threshold of acceptability, we say that the estimate is not significant</li>
 </ul>
</ul>

<ul>
 <li>In practice: Usually in Economics we use the 5% threshold</li>
 <ul>
 <li>But this is arbitrary, in other fields the benchmark p-value is different</li>
 <li>With this threshold we're wrong once in 20 times</li>
 </ul>
</ul>

---

<h3>Overview</h3>

.pull-left[

<ul style = "margin-left:1cm;list-style: none">
 <li>1. Regressions with continuous variables &#10004;</li>
 <ul style = "list-style: none">
 <li>1.1. Estimation</li>
 <li>1.2. Inference</li>
 </ul>
</ul>

]

.pull-right[

<ul style = "margin-left:-1cm;list-style: none">
 <li>3. Controls and interactions</li>
</ul>

<ul style = "margin-left:-1cm;list-style: none">
 <li>4. Interpretation</li>
</ul>

]
 
---

<h3>Overview</h3>

.pull-left[

]

---

### 2. Regressions with discrete variables

#### 2.1. Binary dependent variable

<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>
 
--

---

### 2. Regressions with discrete variables

#### 2.1. Binary dependent variable

<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>

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

---

### 2. Regressions with discrete variables

#### 2.1. Binary dependent variable

<ul>
 <li>The fitted values can be viewed as the probability to be accepted for a given grade</li>
</ul>

<ul>
 <ul>
 <li>The slope is thus by how much the probability of being accepted would increase 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>

---

### 2. Regressions with discrete variables

#### 2.1. Binary dependent variable

<ul>
 <li>But 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>

---

### 2. Regressions with discrete variables

#### 2.2. Binary independent variable

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

<ul>
 <li>So instead of</li>
 <ul>
 <li>having 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 have a binary independent variable</li>
 </ul>
</ul>
 
`$$\text{Height}_i = \hat{\alpha} + \hat{\beta} \times 1\{x_i = \text{Male}\} + \hat{\varepsilon_i}$$`

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

---
 
### 2. Regressions with discrete variables

#### 2.2. Binary independent variable

<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>Here is the traditionnal scatter plot representation</li>
 </ul>
</ul>

---
 
### 2. Regressions with discrete variables

#### 2.2. Binary independent variable

<ul>
 <li>Replacing 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>

---
 
### 2. Regressions with discrete variables

#### 2.2. Binary independent variable

* Let's have a look at the regression results and at the summary statistics of both distributions:

.pull-left[

```
## 
## ========================================
## Dependent variable: 
## ---------------------------
## Height 
## ----------------------------------------
## SexMale 9.5*** 
## (0.6) 
## 
## Constant 165.0*** 
## (0.4) 
## 
## ----------------------------------------
## Observations 1,000 
## R2 0.2 
## ========================================
## Note: *p<0.1; **p<0.05; ***p<0.01
```
]

.pull-right[

<table class="table table-hover table-condensed" style="width: auto !important; margin-left: auto; margin-right: auto;">
<caption>Height summary statistics by sex</caption>
 <thead>
 <tr>
 <th style="text-align:left;"> Sex </th>
 <th style="text-align:right;"> Min </th>
 <th style="text-align:right;"> Q1 </th>
 <th style="text-align:right;"> Med </th>
 <th style="text-align:right;"> Mean </th>
 <th style="text-align:right;"> Q3 </th>
 <th style="text-align:right;"> Max </th>
 </tr>
 </thead>
<tbody>
 <tr>
 <td style="text-align:left;"> Female </td>
 <td style="text-align:right;"> 135.9 </td>
 <td style="text-align:right;"> 158.8 </td>
 <td style="text-align:right;"> 164.6 </td>
 <td style="text-align:right;"> 165.0 </td>
 <td style="text-align:right;"> 170.9 </td>
 <td style="text-align:right;"> 194.7 </td>
 </tr>
 <tr>
 <td style="text-align:left;"> Male </td>
 <td style="text-align:right;"> 145.2 </td>
 <td style="text-align:right;"> 168.3 </td>
 <td style="text-align:right;"> 174.4 </td>
 <td style="text-align:right;"> 174.5 </td>
 <td style="text-align:right;"> 180.6 </td>
 <td style="text-align:right;"> 202.6 </td>
 </tr>
</tbody>
</table>

&#10140; The `$\hat{\alpha}$` coefficient is equal to the expected value of `$y$` when `$x = 0$`, i.e., to the average height for females

&#10140; The `$\hat{\beta}$` coefficient is equal to expected increase in `$y$` when going from `$x = 0$` to `$x = 1$`, i.e., to the difference between male and female average height

]

---
 
### 2. Regressions with discrete variables

#### 2.2. Binary independent variable

* Let's think of it in terms of a regression model:
 
`$$\text{Height}_i = \hat{\alpha} + \hat{\beta} \times 1\{x_i = \text{Male}\} + \hat{\varepsilon_i}$$`

* We now have `$\hat{\alpha}$` and `$\hat{\beta}$`:
 
`$$\text{Height}_i = 165.0 + 9.8 \times 1\{x_i = \text{Male}\} + \hat{\varepsilon_i}$$`

* The fitted values write:

`$$\widehat{\text{Height}_i} =  165.0 + 9.8 \times 1\{x_i = \text{Male}\}$$`

.pull-left[

* When the dummy equals 0 (females):
 
`$$\begin{align}
\widehat{\text{Height}_i} & = 165.0 + 9.8 \times 0\\
&= 165.0 =\overline{\text{Height}_{\left[x_i = \text{Female}\right]}}
\end{align}$$`

]

.pull-right[

* When the dummy equals 1 (males):
 
`$$\begin{align}\widehat{\text{Height}_i} & = 165.0 + 9.8 \times 1\\
&= 174.8 =\overline{\text{Height}_{\left[x_i = \text{Male}\right]}}\end{align}$$`

]

---
 
### 2. Regressions with discrete variables

#### 2.3. Categorical independent variable

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

* Take for instance the race variable:

```r
asec_2020 <- read.csv("asec_2020.csv")
kable(asec_2020 %>% group_by(Race) %>% summarise(N = n()) %>% t(),
 caption = "Distribution of the Race categorical variable")
```

<table class="table table-hover table-condensed" style="width: auto !important; margin-left: auto; margin-right: auto;">
<caption>Distribution of the Race categorical variable</caption>
<tbody>
 <tr>
 <td style="text-align:left;"> Race </td>
 <td style="text-align:left;"> Asian </td>
 <td style="text-align:left;"> Black </td>
 <td style="text-align:left;"> Other </td>
 <td style="text-align:left;"> White </td>
 </tr>
 <tr>
 <td style="text-align:left;"> N </td>
 <td style="text-align:left;"> 4528 </td>
 <td style="text-align:left;"> 6835 </td>
 <td style="text-align:left;"> 2422 </td>
 <td style="text-align:left;"> 50551 </td>
 </tr>
</tbody>
</table>

<center>&#10140; How can we use this variable as an independent variable in our regression framework?</center>

---
 
### 2. Regressions with discrete variables

#### 2.3. Categorical independent variable

* Just as we converted our `$2$`-category variable into `$1$` dummy variable, we can convert an `$n$`-category variable into `$n-1$` dummy variables:
 
--

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

.pull-right[

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

* Females are observations for which Male equals 0
 * Asians are observations for which Black, Other, and White each equals 0

&#10140; Females and Asians are ***reference categories***

* The coefficient associated with the Male dummy was interpreted ***relative*** to females
 * The coefficients associated with the Black, Other, and White dummies will be interpreted ***relative*** to Asians

]

---

### 2. Regressions with discrete variables

#### 2.3. Categorical independent variable

* Thus, regressing earnings on the race categorical variable amounts to estimate the equation:
 
`$$\text{Earnings}_i = \hat{\alpha} + \hat{\beta_1} 1\{\text{Race}_i = \text{Black}\} + \hat{\beta_2} 1\{\text{Race}_i = \text{Other}\} + \hat{\beta_3} 1\{\text{Race}_i = \text{White}\} + \hat{\varepsilon_i}$$`

* And if we compare the regression results to the average earnings by group:

.left-column[

```r
summary(lm(Earnings ~ Race, asec_2020))$coefficients
```

```
##              Estimate Std. Error   t value     Pr(>|t|)
## (Intercept)  77990.78   1149.552  67.84449 0.000000e+00
## RaceBlack   -27413.29   1482.197 -18.49503 3.571079e-76
## RaceOther   -28512.08   1947.305 -14.64181 1.819073e-48
## RaceWhite   -15110.29   1199.933 -12.59262 2.559272e-36
```

<ul><ul>
<li>$\alpha$ is still the average earnings for the reference category </li>
<li>coefficient are still relative to the reference category</li>
</ul></ul>
]

.right-column[
<table class="table table-hover table-condensed" style="width: auto !important; margin-left: auto; margin-right: auto;">
<caption>Mean earnings by race</caption>
 <thead>
 <tr>
 <th style="text-align:left;"> Race </th>
 <th style="text-align:right;"> Mean earnings </th>
 </tr>
 </thead>
<tbody>
 <tr>
 <td style="text-align:left;"> Asian </td>
 <td style="text-align:right;"> 77990.78 </td>
 </tr>
 <tr>
 <td style="text-align:left;"> Black </td>
 <td style="text-align:right;"> 50577.49 </td>
 </tr>
 <tr>
 <td style="text-align:left;"> Other </td>
 <td style="text-align:right;"> 49478.70 </td>
 </tr>
 <tr>
 <td style="text-align:left;"> White </td>
 <td style="text-align:right;"> 62880.49 </td>
 </tr>
</tbody>
</table>
]

---

### 2. Regressions with discrete variables

#### 2.3. Categorical independent variable

* As you can see from the previous regression results, by default R sorts categories by alphabetical order:

* But oftentimes we would prefer the reference category to be the majority group
  * In R we can use the `relevel()` function to change the reference category of a factor

```r
summary(lm(Earnings ~ relevel(as.factor(Race), "White"), asec_2020))$coefficients[, c(1, 2, 4)]
```

```
##                                         Estimate Std. Error     Pr(>|t|)
## (Intercept)                             62880.49   344.0464 0.000000e+00
## relevel(as.factor(Race), "White")Asian  15110.29  1199.9326 2.559272e-36
## relevel(as.factor(Race), "White")Black -12302.99   996.8981 5.947231e-35
## relevel(as.factor(Race), "White")Other -13401.79  1609.0045 8.294160e-17
```

---

<h3>Overview</h3>

.pull-left[

<ul style = "margin-left:1cm;list-style: none">
 <li>2. Regressions with discrete variables &#10004;</li>
 <ul style = "list-style: none">
 <li>2.1. Binary dependent variable</li>
 <li>2.2. Binary independent variable</li>
 <li>2.3. Categorical independent variable</li>
 </ul>
</ul>

]

.pull-right[

<ul style = "margin-left:-1cm;list-style: none">
 <li>3. Controls and interactions</li>
</ul>

<ul style = "margin-left:-1cm;list-style: none">
 <li>4. Interpretation</li>
</ul>

] 
---

<h3>Overview</h3>

.pull-left[

]

.pull-right[

<ul style = "margin-left:-1cm;list-style: none">
 <li>3. Controls and interactions</li>
</ul>
]

---

### 3. Controls and interactions

<ul>
 <li>We can add a third variable z in the regression for two reasons:</li>
 <ul>
 <li>Controlling for z allows to net out the relationship between x and y from how they both relate to z</li>
 <li>Interacting x with z allows to estimate how the relationship between x and y varies with z</li>
 </ul>
</ul>

<ul>
 <li>Consider the following fictitious dataset at the household level</li>
 <ul>
 <li>Household annual income</li>
 <li>Number of children in the household</li>
 <li>Parents' education level</li>
 </ul>
</ul>

.pull-left[

```r
data <- read.csv("household_data.csv")
head(data, 7) # fictitious data
```

```
## Income Children Education
## 1 20 1 < Highschool
## 2 10 1 < Highschool
## 3 10 2 < Highschool
## 4 15 0 < Highschool
## 5 15 1 < Highschool
## 6 20 0 < Highschool
## 7 15 2 Highschool
```
]

.pull-right[
<img src="slides_files/figure-html/unnamed-chunk-27-1.png" width="85%" style="display: block; margin: auto;" />
]

---

### 3. Controls and interactions

<ul>
 <li>There's a clear positive relationship</li>
</ul>

```
##             Estimate Pr(>|t|)
## (Intercept)   -0.885    0.319
## Income         0.166    0.000
```

<ul>
 <ul>
 <li>But what if this relationship was driven by a third variable?</li>
 <li>Maybe it's just that more educated parents tend to earn more and to have more children</li>
 </ul>
</ul>

.pull-right[
<ul>
 <li>In this example, education is indeed positively correlated with both variables</li>
 <ul>
 <li>So at least part of the positive relationship we observe is actually due to education</li>
 <li>Controlling for education estimates the relationship by netting our the contribution of education</li>
 </ul>
</ul>
]

---

### 3. Controls and interactions

<ul>
 <li>Controlling for education does the same to the slope as recentering the graph with respect to education</li>
 <ul>
 <li>In that way, when moving along the x axis, z does not increase but remains constant</li>
 </ul>
</ul>

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

<ul>
 <li>The crosses are located at the average x and y values for each education group</li>
 <ul>
 <li>Controlling for education shifts x and y by group such that crosses superimpose</li>
 </ul>
</ul>

]

.pull-right[
<img src="slides_files/figure-html/unnamed-chunk-31-1.png" width="100%" style="display: block; margin: auto;" />

```
##                     Estimate Pr(>|t|)
## (Intercept)           -0.120    0.892
## Income                 0.064    0.196
## EducationCollege       3.456    0.015
## EducationHighschool    1.856    0.037
```
]

---

### 3. Controls and interactions

<ul>
 <li>Here when we do not control for education:</li>
</ul>

`$$Children_i = \alpha + \beta Income_i + \varepsilon_i$$`

<ul><ul>
 <li>We estimate the overall relationship (here, significantly positive)</li>
</ul></ul>

<ul>
 <li>But when we control for education:</li>
</ul>

`$$Children_i = \alpha + \beta Income_i + \gamma_1 1\{Education_i=\text{Highschool}\} + \gamma_2 1\{Education_i=\text{College}\} +\varepsilon_i$$`

<ul><ul>
 <li>We estimate the relationship net of the effect of education (here, not significant)</li>
</ul></ul>

<ul>
 <li>Interacting the two variables is going one step further:</li>
</ul>

`$$\begin{align}Children_i & = \alpha + \beta Income_i + \gamma_1 1\{Education_i=\text{Highschool}\} + \gamma_2 1\{Education_i=\text{College}\} + \\ 
& \delta_1 Income_i\times1\{Education_i=\text{Highschool}\} + \delta_2 Income_i \times 1\{Education_i=\text{College}\} + \varepsilon_i\end{align}$$`

<ul><ul>
 <li>It is not simply taking into account the fact that education may plays a role</li>
 <li>It estimates by how much the relationship between x and y varies according to z</li>
</ul></ul>

---

### 3. Controls and interactions

* Interacting income with education provides one slope per education group:

.pull-left[
<img src="slides_files/figure-html/unnamed-chunk-33-1.png" width="100%" style="display: block; margin: auto;" />
]

.pull-right[

```
##                            Estimate Pr(>|t|)
## (Intercept)                   2.333    0.225
## Income                       -0.100    0.411
## EducationCollege             -1.768    0.553
## EducationHighschool           0.596    0.819
## Income:EducationCollege       0.239    0.095
## Income:EducationHighschool    0.111    0.445
```
]

<ul>
 <li>The principle is the same when the third variable is continuous:</li>
 <ul>
 <li>Controlling nets out the slope from how the third variable enters the relationship</li>
 <li>Interacting gives by how much the slope changes on expectation when the third variable increases by 1</li>
 <li>And we can control for/interact with multiple third variables</li>
 </ul>
</ul>

---

<h3>Overview</h3>

.pull-left[

]

.pull-right[

<ul style = "margin-left:-1cm;list-style: none">
 <li>3. Controls and interactions &#10004;</li>
</ul>

<ul style = "margin-left:-1cm;list-style: none">
 <li>4. Interpretation</li>
</ul>

] 
---

class: inverse, hide-logo

### 4. Interpretation

<center>Train at interpreting coefficients from randomly drawn relationships</center>