Inference

# Inference
## Lecture 10
### Louis SIRUGUE
### CPES 2 - Fall 2022

---

### Quick reminder

#### 1. Regression

.pull-left[

<img src="slides_files/figure-html/unnamed-chunk-2-1.png" width="100%" style="display: block; margin: auto;" />

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

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

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

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

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

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

]

---

### Quick reminder

#### 2. Multivariate regressions

<ul>
 <li>Adding a second independent variable in the regression amounts to fitting a plane instead of a line</li>
 <ul>
 <li>Adding a third variable would fit a hyperplane of dimension 3 and so on</li>
 </ul>
</ul>

<center>Adding a continuous variable</center>

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<center>Adding a discrete variable</center>

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]

---

### Quick reminder

#### 3. Control variables

<ul>
 <li>Adding a third variable $z$ removes its potential confounding effect from the relationship between $x$ and $y$</li>
 <ul>
 <li>As we move along the $x$ axis, the third variable remains constant</li>
 </ul>
</ul>

`$$\hat{y_i} = \hat{\alpha} + \hat{\beta_1} x + \hat{\beta_2} z + \hat{\varepsilon_i}$$`

---

### Quick reminder

#### 4. Interactions

<ul>
 <li>Adding an interaction term with $z$ allows to see how the effect of $x$ on $y$ varies with $z$</li>
 <ul>
 <li>If $z$ is discrete, it amounts to regressing $y$ on $x$ separately for each $z$ group</li>
 </ul>
</ul>

`$$\hat{y_i} = \hat{\alpha} + \hat{\beta_1} x + \hat{\beta_2} z + \hat{\beta_3}(x \times z)+ \hat{\varepsilon_i}$$`

---

<h3>Today: Inference</h3>

<ul style = "margin-left:1.5cm;list-style: none">
 <li>1. Asymptotic inference</li>
 <ul style = "list-style: none">
 <li>1.1. Data generating process</li>
 <li>1.2. Standardization</li>
 <li>1.3. Confidence interval</li>
 </ul>
</ul>

<ul style = "margin-left:1.5cm;list-style: none">
 <li>2. Exact inference</li>
 <ul style = "list-style: none">
 <li>2.1. Standard error</li>
 <li>2.2. Student-t distribution</li>
 <li>2.3. Confidence interval</li>
 </ul>
</ul>

]

<ul style = "margin-left:-1cm;list-style: none">
 <li>3. Hypothesis testing</li>
 <ul style = "list-style: none">
 <li>3.1. P-value</li>
 <li>3.2. linearHypothesis()</li>
 </ul>
</ul>

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

---

<h3>Today: Inference</h3>

]

---

### 1. Asymptotic inference

#### 1.1. Data generating process

<ul>
 <li>In Part I of the course, we distinguished the empirical moments from the theoretical moments</li>
 <ul>
 <li>Like the empirical mean is a finite-sample estimation of the theoretical expected value</li>
 <li>The same principle applies to regression coefficients</li>
 </ul>
</ul>

<ul>
 <li>Take our Great Gatsby Curve for instance</li>
 <ul>
 <li></li>
 <li></li>
 </ul>
</ul>

.left-column[

<img src="slides_files/figure-html/unnamed-chunk-8-1.png" width="65%" style="display: block; margin: auto;" />
]

---

### 1. Asymptotic inference

#### 1.1. Data generating process

.left-column[

<img src="slides_files/figure-html/unnamed-chunk-9-1.png" width="65%" style="display: block; margin: auto;" />
]

---

### 1. Asymptotic inference

#### 1.1. Data generating process

<ul>
 <li>Take our Great Gatsby Curve for instance</li>
 <ul>
 <li>Had our sample of countries been a bit different, our coefficients would not be the same</li>
 <li>But they would all be estimations of a true relationship whose data-generating process is unobserved</li>
 </ul>
</ul>

.left-column[

<img src="slides_files/figure-html/unnamed-chunk-10-1.png" width="65%" style="display: block; margin: auto;" />
]

---

### 1. Asymptotic inference

#### 1.1. Data generating process

<ul>
 <li>For simplicity, let's work with a relationship whose DGP is know</li>
 <ul>
 <li>Such that we can understand how estimations from random samples behave relative to the DGP</li>
 <li>Let's generate data in R!</li>
 </ul>
</ul>

<ul>
 <li>We can use functions that output random draws from given distributions whose parameters can be chosen</li>
</ul>

&#10140; Sample size, expected value, standard deviation

```r
rnorm(n = 10, mean = 100, sd = 5)
```

```text
##  [1] 103.48482 102.78332  96.55622
##  [4]  96.46252 101.82291 103.84266
##  [7]  99.43827 104.40554 101.99053
## [10]  96.93987
```

]

&#10140; Sample size, lower bound, upper bound

```r
runif(n = 10, min = 4, max = 5)
```

```text
##  [1] 4.633493 4.213208 4.129372
##  [4] 4.478118 4.924074 4.598761
##  [7] 4.976171 4.731793 4.356727
## [10] 4.431474
```
]

---

### 1. Asymptotic inference

#### 1.1. Data generating process

<ul>
 <li>Consider the following data generating process:</li>
</ul>

`$$y = -2 + 0.4  \times x + \varepsilon \:\:\:\begin{cases}
x \sim \mathcal{N}(4,\,25)\\
\varepsilon \sim \mathcal{N}(0,\,1)
\end{cases}$$`

<ul>
 <li>We can randomly draw 1,000 observations
</ul>

```r
dt <- tibble(x = rnorm(1000, 4, 5),
 e = rnorm(1000, 0, 1),
 y = -2 + (.4 * x) + e)
```

<center>Check the empirical moments:</center>

```r
c(mean(dt$x), var(dt$x))
```

```
## [1]  4.127842 26.816044
```
]

```r
c(mean(dt$e), var(dt$e))
```

```
## [1] 0.009459055 1.070444754
```
]

---

### 1. Asymptotic inference

#### 1.1. Data generating process

<ul>
 <li>Because the randomly drawn sample is finite, it does not match exactly the features of the DGP:</li>
</ul>

---

### 1. Asymptotic inference

#### 1.1. Data generating process

<ul>
 <li>Same thing for the coefficients of the relationship between $x$ and $y$:</li>
</ul>

```r
lm(y ~ x, dt)
```

```
## 
## Call:
## lm(formula = y ~ x, data = dt)
## 
## Coefficients:
## (Intercept)            x  
##     -2.0044       0.4033
```

<ul>
 <li>But what would happen if we were to redo this operation many times?</li>
 <ol>
 <li>Draw a random sample from the DGP</li>
 <li>Compute the slope of the regression of $y$ on $x$</li>
 <li>Do it many times and store the coefficients</li>
</ul>

---

### 1. Asymptotic inference

#### 1.1. Data generating process

```r
#

for (i in 1:1000) {
  
#
# 
# 
  
# 
  
#
  
}
```
---

### 1. Asymptotic inference

#### 1.1. Data generating process

<ul>
 <li>We can use a loop to do that:</li>
 <ul>
 <li>First we create an empty vector</li>
 <li></li>
 <li></li>
 </ul>
</ul>

```r
beta <- c()

for (i in 1:1000) {
  
#
# 
# 
  
# 
  
#
  
}
```

---

### 1. Asymptotic inference

#### 1.1. Data generating process

<ul>
 <li>We can use a loop to do that:</li>
 <ul>
 <li>First we create an empty vector</li>
 <li>Then we put the code in a loop</li>
 <li></li>
 </ul>
</ul>

```r
beta <- c()

for (i in 1:1000) {
 
 dt_i <- tibble(x = rnorm(1000, 4, 5),
 e = rnorm(1000, 0, 1),
 y = -2 + (.4 * x) + e)
 
 reg_i <- lm(y ~ x, dt_i)
 
#
 
}
```

---

### 1. Asymptotic inference

#### 1.1. Data generating process

<ul>
 <li>We can use a loop to do that:</li>
 <ul>
 <li>First we create an empty vector</li>
 <li>Then we put the code in a loop</li>
 <li>And we fill the vector at each iteration</li>
 </ul>
</ul>

```r
beta <- c()

for (i in 1:1000) {
 
 dt_i <- tibble(x = rnorm(1000, 4, 5),
 e = rnorm(1000, 0, 1),
 y = -2 + (.4 * x) + e)
 
 reg_i <- lm(y ~ x, dt_i)
 
 beta <- c(beta, reg_i$coefficients[2])
 
}
```

---

### 1. Asymptotic inference

#### 1.1. Data generating process

<ul>
 <li>We now have 1,000 slope coefficients from 1,000 random samples of the same DGP</li>
</ul>

.left-column[
<img src="slides_files/figure-html/unnamed-chunk-24-1.png" width="95%" style="display: block; margin: auto auto auto 0;" />
]

.right.column[

<ul style = "text-align:left;">
 <li style = "margin-bottom:1cm">Some random samples give higher estimates than others</li>
 <li style = "margin-bottom:1cm">But on expectation we get the right coefficient!</li>
 <li style = "margin-bottom:1cm">The $\hat{\beta}$s actually follow a normal distribution</li>
 <li style = "margin-bottom:1cm">And at the limit their mean would converge towards $\beta$</li>
</ul>
]

---

### 1. Asymptotic inference

#### 1.2. Standardization

<ul>
 <li style = "margin-bottom:.12cm">That is crucial information because it allows to get back to something we know:</li>
 <ul>
 <li style = "margin-bottom:.12cm"></li>
 <li></li>
 <li></li>
 </ul>
</ul>
 
.left-column[
<img src="slides_files/figure-html/unnamed-chunk-25-1.png" width="80%" style="display: block; margin: auto auto auto 0;" />
]
.right-column[

`$$\hat{\beta}$$`
]

---

### 1. Asymptotic inference

#### 1.2. Standardization

<ul>
 <li>That is crucial information because it allows to get back to something we know:</li>
 <ul>
 <li style = "margin-bottom:.12cm">By subtracting $\beta$ from the distribution of $\hat{\beta}$</li>
 <li></li>
 <li></li>
 </ul>
</ul>
 
.left-column[
<img src="slides_files/figure-html/unnamed-chunk-26-1.png" width="80%" style="display: block; margin: auto auto auto 0;" />
]

---

### 1. Asymptotic inference

#### 1.2. Standardization

<ul>
 <li>That is crucial information because it allows to get back to something we know:</li>
 <ul>
 <li>By subtracting $\beta$ from the distribution of $\hat{\beta}$</li>
 <li>And dividing by the standard deviation of $\hat{\beta}$</li>
 <li></li>
 </ul>
</ul>
 
.left-column[
<img src="slides_files/figure-html/unnamed-chunk-27-1.png" width="80%" style="display: block; margin: auto auto auto 0;" />
]

---

### 1. Asymptotic inference

#### 1.2. Standardization

<ul>
 <li>That is crucial information because it allows to get back to something we know:</li>
 <ul>
 <li>By subtracting $\beta$ from the distribution of $\hat{\beta}$</li>
 <li>And dividing by the standard deviation of $\hat{\beta}$</li>
 <li>With an infinite sample we would obtain the standard normal distribution</li>
 </ul>
</ul>
 
.left-column[
<img src="slides_files/figure-html/unnamed-chunk-28-1.png" width="80%" style="display: block; margin: auto auto auto 0;" />
]

---

### 1. Asymptotic inference

#### 1.3. Confidence interval

<ul>
 <li>We can use the fact that we know the standard normal distribution:</li>
 <ul>
 <li></li>
 <li></li>
 <li></li>
 </ul>
</ul>
 
.left-column[
<img src="slides_files/figure-html/unnamed-chunk-29-1.png" width="80%" style="display: block; margin: auto auto auto 0;" />

`$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\frac{\hat{\beta}-\beta}{\text{SD}(\hat{\beta})} \sim \mathcal{N}(0, 1)$$`
]

---

### 1. Asymptotic inference

#### 1.3. Confidence interval

<ul>
 <li>We can use the fact that we know the standard normal distribution:</li>
 <ul>
 <li>That 99% of the distribution lie between $\pm$ 2.58</li>
 <li></li>
 <li></li>
 </ul>
</ul>
 
.left-column[
<img src="slides_files/figure-html/unnamed-chunk-30-1.png" width="80%" style="display: block; margin: auto auto auto 0;" />

`$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\frac{\hat{\beta}-\beta}{\text{SD}(\hat{\beta})} \sim \mathcal{N}(0, 1)$$`
`$$\text{Pr}\left[-2.58<\frac{\hat{\beta}-\beta}{\text{SD}(\hat{\beta})}<2.58\right] \approx 99\%$$`
]

---

### 1. Asymptotic inference

#### 1.3. Confidence interval

<ul>
 <li>We can use the fact that we know the standard normal distribution:</li>
 <ul>
 <li>That 99% of the distribution lie between $\pm$ 2.58</li>
 <li>That 95% of the distribution lie between $\pm$ 1.96</li>
 <li></li>
 </ul>
</ul>
 
.left-column[
<img src="slides_files/figure-html/unnamed-chunk-31-1.png" width="80%" style="display: block; margin: auto auto auto 0;" />

---

### 1. Asymptotic inference

#### 1.3. Confidence interval

<ul>
 <li>We can use the fact that we know the standard normal distribution:</li>
 <ul>
 <li>That 99% of the distribution lie between $\pm$ 2.58</li>
 <li>That 95% of the distribution lie between $\pm$ 1.96</li>
 <li>This is what allows to determine confidence intervals</li>
 </ul>
</ul>
 
.left-column[
<img src="slides_files/figure-html/unnamed-chunk-32-1.png" width="80%" style="display: block; margin: auto auto auto 0;" />

`$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\frac{\hat{\beta}-\beta}{\text{SD}(\hat{\beta})} \sim \mathcal{N}(0, 1)$$`
`$$\text{Pr}\left[-1.96<\frac{\hat{\beta}-\beta}{\text{SD}(\hat{\beta})}<1.96\right] \approx 95\%$$`
`$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\Downarrow$$`
`$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\text{Confidence interval}$$`
]

---

### 1. Asymptotic inference

#### 1.3. Confidence interval

`$$\text{Pr}\left[-1.96<\frac{\hat{\beta}-\beta}{\text{SD}(\hat{\beta})}<1.96\right] \approx 95\%$$`

--

`$$\text{Pr}\left[-1.96\times\color{SkyBlue}{\text{SD}(\hat{\beta})}<\hat{\beta}-\beta<1.96\times\color{SkyBlue}{\text{SD}(\hat{\beta})}\right] \approx 95\%$$`

--

`$$\text{Pr}\left[-1.96\times\text{SD}(\hat{\beta})\color{SkyBlue}{-\hat{\beta}}<-\beta<1.96\times\text{SD}(\hat{\beta})\color{SkyBlue}{-\hat{\beta}}\right] \approx 95\%$$`

--

`$$\text{Pr}\left[\color{SkyBlue}{+}1.96\times\text{SD}(\hat{\beta}) \color{SkyBlue}{+} \hat{\beta}\color{SkyBlue}{>}\beta\color{SkyBlue}{>}\color{SkyBlue}{-}1.96\times\text{SD}(\hat{\beta})\color{SkyBlue}{+}\hat{\beta}\right] \approx 95\%$$`

--

`$$\text{CI}_{95\%}: \:\hat{\beta}\pm 1.96\times\text{SD}(\hat{\beta})$$`
---

<h3>Overview</h3>

<ul style = "margin-left:1.5cm;list-style: none">
 <li>1. Asymptotic inference &#10004;</li>
 <ul style = "list-style: none">
 <li>1.1. Data generating process</li>
 <li>1.2. Standardization</li>
 <li>1.3. Confidence interval</li>
 </ul>
</ul>

]

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

---

<h3>Overview</h3>

]

---

### 2. Exact inference

#### 2.1. Standard error

<ul>
 <li>In the previous section I used phrases like "at the limit" or "with an infinite sample"</li>
 <ul>
 <li>But in practice this is not the case, so things behave slightly differently</li>
 <li>And this implies to make a few statistical adjustments to account for that</li>
 </ul>
</ul>

`$$\hat{\beta}\pm 1.96\times\color{SkyBlue}{\text{SD}(\hat{\beta})}$$`

<ul>
 <li>First we cannot measure directly the standard deviation of $\hat{\beta}$</li>
 <ul>
 <li>Indeed in practice we have only one observation of $\hat{\beta}$, not its whole distribution</li>
 <li>But like for the mean, we can compute a standard error instead</li>
 </ul>
</ul>

.pull-left[
<center>Standard deviation</center>

<center>&#10140; Measures the amount of variability, or dispersion, from the individual data values to the mean</center>
]

.pull-right[
<center>Standard error</center>

<center>&#10140; Measures how far an estimate from a given sample is likely to be from the true parameter of interest</center>
]

---

### 2. Exact inference

#### 2.1. Standard error

<ul>
 <li>We won't go through the theoretical computations together, but let's have a look at the formula:</li>
</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, decreases as:</li>
</ul>

---

### 2. Exact inference

#### 2.1. Standard error

<ul>
 <li>We won't go through the theoretical computations together, but let's have a look at the formula:</li>
</ul>

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

<ul>
 <li>Notice that the variance, and thus the standard error of our estimate, decreases as:</li>
 <ul>
 <li>The number of of observations gets bigger</li>
 </ul>
</ul>

---

### 2. Exact inference

#### 2.1. Standard error

<ul>
 <li>We won't go through the theoretical computations together, but let's have a look at the formula:</li>
</ul>

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

<ul>
 <li>Notice that the variance, and thus the standard error of our estimate, decreases as:</li>
 <ul>
 <li>The number of of observations gets bigger</li>
 <li>The number of parameters decreases</li>
 </ul>
</ul>

---

### 2. Exact inference

#### 2.1. Standard error

<ul>
 <li>We won't go through the theoretical computations together, but let's have a look at the formula:</li>
</ul>

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

<ul>
 <li>Notice that the variance, and thus the standard error of our estimate, decreases as:</li>
 <ul>
 <li>The number of of observations gets bigger</li>
 <li>The number of parameters decreases</li>
 <li>The sum of squared errors relative to the variance of $x$ decreases</li>
 </ul>
</ul>

---

### 2. Exact inference

#### 2.1. Standard error

<ul>
 <li>We won't go through the theoretical computations together, but let's have a look at the formula:</li>
</ul>

<ul>
 <li>Notice that the variance, and thus the standard error of our estimate, decreases as:</li>
 <ul>
 <li>The number of of observations gets bigger</li>
 <li>The number of parameters decreases</li>
 <li>The sum of squared errors decreases relative to the variance of $x$</li>
 </ul>
</ul>

<ul>
 <li>And as the standard error gets bigger, the confidence interval gets bigger:</li>
<ul>

`$$\hat{\beta}\pm 1.96\times\color{SkyBlue}{\text{se}(\hat{\beta})}$$`

---

### 2. Exact inference

#### 2.2. Student-t distribution

<ul>
 <li>But that's not it, remember that we took the value 1.96 from the normal distribution</li>
</ul>

`$$\hat{\beta}\pm \color{SkyBlue}{1.96}\times\text{se}(\hat{\beta})$$`
--

.left-column[
<img src="slides_files/figure-html/unnamed-chunk-33-1.png" width="80%" style="display: block; margin: auto auto auto 0;" />
]

.right-column[
<ul style = "margin-top:-.5cm;margin-left:-3.25cm;">
 <li style = "margin-bottom:.5cm">But the normal distribution is what $\frac{\hat{\beta}-\beta}{\text{SD}(\hat{\beta})}$ converges to at the limit</li>
 <li style = "margin-bottom:.5cm">In the finite world, $\frac{\hat{\beta}-\beta}{\text{se}(\hat{\beta})}$ follows a slightly flatter distribution</li>
 <li>The Student $t$ distribution, whose precise shape depends on the number of observations we have and parameters we estimate</li>
</ul>
]

---

### 2. Exact inference

#### 2.2. Student-t distribution

<ul>
 <li>The Student $t$ distribution accounts for the fact that the sample is finite</li>
 <ul>
 <li>The lower the number of degrees of freedom (#observations - #parameters) the flatter</li>
 <li>And it tends to a normal distribution as the number of degrees of freedom $\rightarrow \infty$</li>
 </ul>
</ul>

---

### 2. Exact inference

#### 2.2. Student-t distribution

<ul>
 <li>But we know the Student $t$ distributions just as well as the standard normal distribution</li>
</ul>

--
<ul>
 <ul>
 <li>With 100 degrees of freedom, 95% of the distribution lie between $\pm$ 1.98</li>
 <li></li>
 </ul>
</ul>

<img src="slides_files/figure-html/unnamed-chunk-35-1.png" width="70%" style="display: block; margin: auto;" />
---

### 2. Exact inference

#### 2.2. Student-t distribution

<ul>
 <li>But we know the Student $t$ distributions just as well as the standard normal distribution</li>
 <ul>
 <li>With 100 degrees of freedom, 95% of the distribution lie between $\pm$ 1.98</li>
 <li>With 3,000 degrees of freedom, 90% of the distribution lie between $\pm$ 1.65</li>
 </ul>
</ul>

---

### 2. Exact inference

#### 2.2. Student-t distribution

<ul>
 <li>So instead of 1.96, we must use the value such that:</li>
 <ul>
 <li>The desired percentage of the distribution is comprised within $\pm$ that value...</li>
 <li>For a Student $t$ distribution with the relevant number of degrees of freedom</li>
 </ul>
</ul>

<ul>
 <li>We can get these values easily with the qt() function, indicating:</li>
 <ul>
 <li></li>
 <li></li>
 </ul>
</ul>

```r
qt( , )
```

---

### 2. Exact inference

#### 2.2. Student-t distribution

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

---

### 2. Exact inference

#### 2.2. Student-t distribution

<ul>
 <li>We can get these values easily with the qt() function, indicating:</li>
 <ul>
 <li>The share of the distribution below the value we're looking for (e.g., 0.975 for a 95% CI)</li>
 <li>The number of degrees of freedom of the Student $t$ distribution (e.g., 88 observations - 2 parameters)</li>
 </ul>
</ul>

```r
qt(.975, 86)
```

```
## [1] 1.987934
```

<ul>
 <li>Denote this value $t(\text{df})_{1-\frac{\alpha}{2}}$</li>
 <ul>
 <li>With $\alpha$ equal to $1 -$ the confidence level</li>
 <li>And $\text{df}$ the number of degrees of freedom</li>
 </ul>
</ul>

---

### 2. Exact inference

#### 2.3. Confidence interval
 
<ul>
 <li>The formula for the confidence interval in finite sample hence writes:</li>
</ul>

`$$\hat{\beta}\pm \color{SkyBlue}{t(\text{df})_{1-\frac{\alpha}{2}}}\times\text{se}(\hat{\beta})$$`

<ul>
 <li>The confidence interval increases as:</li>
 <ul>
 <li>The confidence level increases</li>
 <li></li>
 </ul>
</ul>

---

### 2. Exact inference

#### 2.3. Confidence interval
 
<ul>
 <li>The formula for the confidence interval in finite sample hence writes:</li>
</ul>

`$$\hat{\beta}\pm \color{SkyBlue}{t(\text{df})_{1-\frac{\alpha}{2}}}\times\text{se}(\hat{\beta})$$`

<ul>
 <li>The confidence interval increases as:</li>
 <ul>
 <li>The confidence level increases</li>
 <li>The number of degrees of freedom decreases</li>
 </ul>
</ul>

---

### Practice

#### 1) Import the `ggcurve.csv` dataset

#### 2) Regress the IGE on the Gini coefficient and store the estimated regression parameters

#### 3) Compute the 95% confidence interval of the regression slope

`$$\hat{\beta}\pm t(\text{df})_{1-\frac{\alpha}{2}}\times\text{se}(\hat{\beta})$$`

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

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

---

### Solution

#### 1) Import the `ggcurve.csv` dataset

```r
ggcurve <- read.csv("C:/User/Documents/ggcurve.csv")
```

#### 2) Regress the IGE on the Gini coefficient and store the regression slope

```r
model <- lm(ige ~ gini, ggcurve)
model
```

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

```r
alpha <- model$coefficients[1]
beta <- model$coefficients[2]
```

---

### Solution

#### 3) Compute the 95% confidence interval of the regression slope

```r
se_dat <- ggcurve %>%
 mutate(fit = alpha + gini * beta, e = ige - fit) %>%
 summarise(se = sqrt(sum(e^2)/((n()-2)*sum((gini-mean(gini))^2))))

se_dat$se
```

```
## [1] 0.2642477
```

```r
beta - se_dat$se * qt(.975, nrow(ggcurve) - 2)
```

```
##      gini 
## 0.4642511
```

```r
beta + se_dat$se * qt(.975, nrow(ggcurve) - 2)
```

```
##     gini 
## 1.566673
```

---

<h3>Overview</h3>

<ul style = "margin-left:1.5cm;list-style: none">
 <li>2. Exact inference &#10004;</li>
 <ul style = "list-style: none">
 <li>2.1. Standard error</li>
 <li>2.2. Student-t distribution</li>
 <li>2.3. Confidence interval</li>
 </ul>
</ul>

]

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

---

<h3>Overview</h3>

]

<ul style = "margin-left:-1cm;list-style: none">
 <li>3. Hypothesis testing</li>
 <ul style = "list-style: none">
 <li>3.1. P-value</li>
 <li>3.2. linearHypothesis()</li>
 </ul>
</ul>

]

---

### 3. Hypothesis testing

#### 3.1. P-value

<ul>
 <li>We now have the 95% confidence interval for our estimate:</li>
 <ul>
 <li>Our estimate of $\beta$ is 1.02</li>
 <li>And we are 95% sure that $\beta$ lies between 0.46 and 1.57</li>
 </ul>
</ul>

`$$0.46 \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, 1.02 \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, 1.57\\
|\underbrace{-----------}_{t(\text{df})_{1-\frac{\alpha}{2}}\times\text{se}(\hat{\beta})}\cdot\underbrace{-----------}_{t(\text{df})_{1-\frac{\alpha}{2}}\times\text{se}(\hat{\beta})}|$$`

<ul>
 <li>Note that in our confidence interval formula:</li>
 <ul>
 <li>The standard error and the relevant Student $t$ distribution are given</li>
 <li>But the confidence level $1 - \alpha$ was chosen arbitrarily</li>
 </ul>
</ul>

&#10140; Setting a higher confidence level would widen the confidence interval 
&#10140; Allowing for a lower confidence level would narrow the confidence interval

---

### 3. Hypothesis testing

#### 3.1. P-value

<ul>
 <li>So far we framed the problem as:</li>
</ul>

<center>"What are the values $\beta$ is likely to take under a given confidence level?"</center>

<ul>
 <li>But we could also think of it as:</li>
</ul>

<center>"Under which confidence level is $\beta$ is likely to take a given value?"</center>

<ul>
 <li>And this is actually a very practical way of framing the question:</li>
 <ul>
 <li>To (in)validate the predictions from a theoretical model</li>
 <li>To know under which confidence level $\beta$ is likely to be $\neq 0$ at all</li>
 </ul>
</ul>

<center>&#10140; But how to answer such questions in practice?</center>

---

### 3. Hypothesis testing

#### 3.1. P-value

<ul>
 <li>We can start from the fact that even though we do not know $\beta$, we know that:</li>
</ul>

$$\frac{\hat{\beta} - \beta}{\text{se}(\hat{\beta})} \sim t(\text{df}) $$

---

### 3. Hypothesis testing

#### 3.1. P-value

<ul>
 <li>And that in this distribution some values are quite plausible:</li>
</ul>

`$$\frac{\hat{\beta} - \color{SkyBlue}{1}}{\text{se}(\hat{\beta})}$$`

---

### 3. Hypothesis testing

#### 3.1. P-value

<ul>
 <li>And some are way less plausible:</li>
</ul>

`$$\frac{\hat{\beta} - \color{SkyBlue}{2.5}}{\text{se}(\hat{\beta})}$$`

---

### 3. Hypothesis testing

#### 3.1. P-value

<ul>
 <li>But because the distribution is continuous:</li>
 <ul>
 <li></li>
 <li></li>
 </ul>
</ul>

---

### 3. Hypothesis testing

#### 3.1. P-value

<ul>
 <li>But because the distribution is continuous:</li>
 <ul>
 <li>The probability to draw any exact value would be 0</li>
 <li></li>
 </ul>
</ul>

---

### 3. Hypothesis testing

#### 3.1. P-value

<ul>
 <li>But because the distribution is continuous:</li>
 <ul>
 <li>The probability to draw any exact value would be 0</li>
 <li>We can only compute the probability to fall below that value</li>
 </ul>
</ul>

---

### 3. Hypothesis testing

#### 3.1. P-value

<ul>
 <li>But because the distribution is continuous:</li>
 <ul>
 <li>The probability to draw any exact value would be 0</li>
 <li>Or to fall above that value if it is negative</li>
 </ul>
</ul>

---

### 3. Hypothesis testing

#### 3.1. P-value

<ul>
 <li>But generally what makes sense is to know what are the chances to fall that far from 0:</li>
 <ul>
 <li></li>
 <li></li>
 </ul>
</ul>

---

### 3. Hypothesis testing

#### 3.1. P-value

<ul>
 <li>But generally what makes sense is to know what are the chances to fall that far from 0:</li>
 <ul>
 <li>So we take 1 - the probability to fall below the absolute value</li>
 <li></li>
 </ul>
</ul>

---

### 3. Hypothesis testing

#### 3.1. P-value

---

### 3. Hypothesis testing

#### 3.1. P-value

<ul>
 <li>The resulting area is what we call a p-value</li>
 <ul>
 <li>It is the probability that $\beta$ falls at least as far from $\hat{\beta}$ as the hypothesized value</li>
 </ul>
</ul>

<ul>
 <li>Consider finding $\hat{\beta} = 4$ and a hypothesizing value of 3 for $\beta$</li>
 <ul>
 <li>A p-value of 5% indicates that there is only a 5% chance to find a $\hat{\beta} = 4$ if $\beta = 3$</li>
 <li>Below that threshold we would reject the hypothesis that $\beta = 3$ at the 95% confidence level</li>
 </ul>
</ul>

<ul>
 <li>Notice that in this example, the 95% confidence interval of $\hat{\beta}$ would not include the value 3</li>
 <ul>
 <li>With a hypothesized value equal to the bound of a confidence interval the p-value would equal 1 - the corresponding confidence level</li>
 <li>So a p-value lower than $\alpha$ means that the hypothesized value is outside the $(1-\alpha)$% confidence interval</li>
 </ul>
</ul>

<center>&#10140; Let's go through a formal example with our data</center>

---

### 3. Hypothesis testing

#### 3.1. P-value

<ul>
 <li>Can we reject at the 95% confidence level that $\beta = 0$?</li>
</ul>

```r
beta
```

```
##     gini 
## 1.015462
```

<ul>
 <li>We should start by hypothesizing that $\beta = 0$</li>
 <ul>
 <li>This is what we call the "null hypothesis" $H_0$</li>
 </ul>
</ul>

`$$H_0: \beta = 0$$`

<center>Under $H_0$:</center>

`$$\frac{\hat{\beta} - 0}{\text{se}(\hat{\beta})} \sim t(\text{df})$$`

---

### 3. Hypothesis testing

#### 3.1. P-value

<ul>
 <li>We should find the area below $(\hat{\beta} - 0)/\text{se}(\hat{\beta})$ in a Student $t$ distribution we the right number of $\text{df}$</li>
 <ul>
 <li>$(\hat{\beta} - 0)/\text{se}(\hat{\beta})$ is what we call the $t$-stat</li>
 </ul>
</ul>

```r
(beta - 0) / se_dat$se
```

```
##     gini 
## 3.842842
```

<ul>
 <li>While qt() gave us the value for a certain probability, pt() gives the the probability for a given value:</li>
 <ul>
 <li></li>
 <li></li>
 </ul>
</ul>

```r
pt( , )
```

---

### 3. Hypothesis testing

#### 3.1. P-value

```r
(beta - 0) / se_dat$se
```

```
##     gini 
## 3.842842
```

<ul>
 <li>While qt() gave us the value for a certain probability, pt() gives the the probability for a given value:</li>
 <ul>
 <li>Put in the t-stat</li>
 <li></li>
 </ul>
</ul>

```r
pt((beta - 0) / se_dat$se, )
```

---

### 3. Hypothesis testing

#### 3.1. P-value

```r
(beta - 0) / se_dat$se
```

```
##     gini 
## 3.842842
```

```r
pt((beta - 0) / se_dat$se, nrow(ggcurve) - 2)
```

```
##      gini 
## 0.9994921
```

---

### 3. Hypothesis testing

#### 3.1. P-value

<ul>
 <li>We must then:</li>
 <ul>
 <li>Take 1 - this probability (area above the t-stat)</li>
 <li></li>
 </ul>
</ul>

```r
1 - pt(abs((beta - 0) / se_dat$se), nrow(ggcurve) - 2)
```

```
##         gini 
## 0.0005078528
```

---

### 3. Hypothesis testing

#### 3.1. P-value

<ul>
 <li>We must then:</li>
 <ul>
 <li>Take 1 - this probability (area above the t-stat)</li>
 <li>And multiply it by 2 (consider the absolute distance and not the signed distance)</li>
 </ul>
</ul>

```r
2 * (1 - pt(abs((beta - 0) / se_dat$se), nrow(ggcurve) - 2))
```

```
##        gini 
## 0.001015706
```

<ul>
 <li>The p-value is lower than 1%:</li>
 <ul>
 <li>We can reject at the 99% confidence level that $\beta = 0$</li>
 <li>In that case we say that $\hat{\beta}$ is significantly different from 0 at the 1% significance level</li>
 </ul>
</ul>

<ul>
 <li>But the p-value is greater than 0.1%:</li>
 <ul>
 <li>We cannot reject at the 99.9% confidence level that $\beta = 0$</li>
 <li>In that case we say that $\hat{\beta}$ is not significantly different from 0 at the 0.1% significance level</li>
 </ul>
</ul>

---

### 3. Hypothesis testing

#### 3.1. P-value

<ul>
 <li>By default, the summary() function tests whether or not each coefficient is significantly different from 0</li>
 <ul>
 <li></li>
 </ul>
</ul>

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

---

### 3. Hypothesis testing

#### 3.1. P-value

<ul>
 <li>By default, the summary() function tests whether or not each coefficient is significantly different from 0</li>
 <ul>
 <li>You can extract the information from the $coefficient attribute of the output</li>
 </ul>
</ul>

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

<ul>
 <li>For each coefficient it indicates:</li>
 <ul>
 <li>The standard error</li>
 <li>The $t$-stat $(H_0:\beta=0)$</li>
 <li>The p-value $(H_0:\beta=0)$</li>
 </ul>
</ul>

<ul>
 <li>The output of the summary() function is great to have a quick overview of the model:</li>
</ul>

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

---

### 3. Hypothesis testing

#### 3.1. P-value

```
## 
## 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
```
]

---

### 3. Hypothesis testing

#### 3.1. P-value

]

---

### 3. Hypothesis testing

#### 3.1. P-value

]

---

### 3. Hypothesis testing

#### 3.1. P-value

---

### 3. Hypothesis testing

#### 3.1. P-value

.right-column[
&#x1F804; Command 

&#x1F804; Residuals distribution 

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

&#x1F804; Significance 
]
---

### 3. Hypothesis testing

#### 3.1. P-value

]

---

### 3. Hypothesis testing

#### 3.2. linearHypothesis()

<ul>
 <li>But the linearHypothesis() function from the car package allows to easily test other hypotheses:</li>
 <ul>
 <li></li>
 <li></li>
 </ul>
</ul>

```r
linearHypothesis( , )
```

---

### 3. Hypothesis testing

#### 3.2. linearHypothesis()

<ul>
 <li>But the linearHypothesis() function from the car package allows to easily test other hypotheses:</li>
 <ul>
 <li>You must provide the model</li>
 <li></li>
 </ul>
</ul>

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

---

### 3. Hypothesis testing

#### 3.2. linearHypothesis()

```r
linearHypothesis(lm(ige ~ gini, ggcurve), "gini = 0")
```

```
## Linear hypothesis test
## 
## Hypothesis:
## gini = 0
## 
## Model 1: restricted model
## Model 2: ige ~ gini
## 
##   Res.Df     RSS Df Sum of Sq      F   Pr(>F)   
## 1     21 0.46733                                
## 2     20 0.26883  1    0.1985 14.767 0.001016 **
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
```
  
---

### 3. Hypothesis testing

#### 3.2. linearHypothesis()

<ul>
 <li>You can also test more complex hypotheses</li>
 <ul>
 <li>Like equality between coefficients</li>
 </ul>
</ul>

```r
linearHypothesis(lm(ige ~ gini, ggcurve), "gini = (Intercept)")
```

```
## Linear hypothesis test
## 
## Hypothesis:
## - (Intercept)  + gini = 0
## 
## Model 1: restricted model
## Model 2: ige ~ gini
## 
##   Res.Df     RSS Df Sum of Sq      F  Pr(>F)  
## 1     21 0.37634                              
## 2     20 0.26883  1   0.10751 7.9983 0.01039 *
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
```
  
---

### 3. Hypothesis testing

#### 3.2. linearHypothesis()

<ul>
 <li>You can also test more complex hypotheses</li>
 <ul>
 <li>Like equality between coefficients, or joint hypotheses (relying a generalization of the t-test called F-test)</li>
 </ul>
</ul>

```r
linearHypothesis(lm(ige ~ gini, ggcurve), c("gini = 0", "(Intercept) = 0"))
```

```
## Linear hypothesis test
## 
## Hypothesis:
## gini = 0
## (Intercept) = 0
## 
## Model 1: restricted model
## Model 2: ige ~ gini
## 
##   Res.Df    RSS Df Sum of Sq      F    Pr(>F)    
## 1     22 3.8841                                  
## 2     20 0.2688  2    3.6153 134.48 2.523e-12 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
```

---

<h3>Overview</h3>

]

<ul style = "margin-left:-1cm;list-style: none">
 <li>3. Hypothesis testing &#10004;</li>
 <ul style = "list-style: none">
 <li>3.1. P-value</li>
 <li>3.2. linearHypothesis()</li>
 </ul>
</ul>

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

---

### 4. Wrap up!

#### Data generating process

<ul>
 <li>In practice we estimate coefficients on a given realization of a data generating process</li>
 <ul>
 <li>So the true coefficient is unobserved</li>
 <li>But our estimation is informative on the values the true coefficient is likely to take</li>
 </ul>
</ul>

.left-column[
<img src="slides_files/figure-html/unnamed-chunk-84-1.png" width="90%" style="display: block; margin: auto auto auto 0;" />
]

---

### 4. Wrap up!

#### Confidence interval

<ul>
 <li>This allows to infer a confidence interval:</li>
</ul>

`$$\hat{\beta}\pm t(\text{df})_{1-\frac{\alpha}{2}}\times\text{se}(\hat{\beta})$$`

<ul>
 <li>Where $t(\text{df})_{1-\frac{\alpha}{2}}$ is the value from a Student $t$ distribution</li>
 <ul>
 <li>With the relevant number of degrees of freedom $\text{df}$ (n - #parameters)</li>
 <li>And the desired confidence level $1-\alpha$</li>
 </ul>
</ul>

<ul>
 <li>And where $\text{se}(\hat{\beta})$ denotes the standard error of $\hat{\beta}$:</li>
</ul>

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

---

### 4. Wrap up!

#### P-value

<ul>
 <li>It also allows to test how likely is $\beta$ to be different from a given value:</li>
 <ul>
 <li>If the p-value < 5%, we can reject that $\beta$ equals the hypothesized value at the 95% confidence level</li>
 <li>This threshold, very common in Economics, implies that we have 1 chance out of 20 to be wrong</li>
 </ul>
</ul>

```r
linearHypothesis(lm(ige ~ gini, ggcurve), "gini = 0")
```