Final Review

class: center, middle, inverse, title-slide

# Final Review
## EC 320: Introduction to Econometrics
### Kyle Raze
### Fall 2019

---

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# Prologue

---
# Housekeeping

## Final Exam

Tuesday, December 10 at 10:15am in Chapman 220.

- Room open at 10:00am.

## Problem Set 5

Due Saturday, December 7 by 11:59pm.

---
# Housekeeping

## Lab

Additional review session.

- Focus: coefficient interpretation.

## Office hours

Regular hours this week.

Next week: extended office hours on Monday.

- Times and locations TBA.

---
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# Final Exam

---
# Final Exam Topics

.green[Anything from the lectures, labs, or problem sets] .hi-green[is fair game!]

1. Midterm Topics

2. Multiple Linear Regression: Inference

3. Categorical Variables

4. Interactive Relationships

5. Nonlinear Relationships

---
# Final Exam Topics

## 1. Midterm Topics

Making predictions using fitted regression model

- *e.g.,* using a Hedonic model

Goodness of fit

Hypothesis testing

.pink[Omitted-variable bias]

- Know when omitting a variable causes bias
- Sign the bias
- Back out correlations between explanatory variables
- Provide examples of problematic omitted variables

---
# Final Exam Topics

## 2. Multiple Linear Regression: Inference

*F* tests (multiple parameters)

- State null hypothesis
- Identify restricted and unrestricted models
- Calculate *F* statistic
- Use table to find *F*.sub[*crit*]
- *F* .mono[>] *F*.sub[*crit*]?
- State conclusion of the test

*t* tests (single parameter)

**Q:** Which test should you choose?

**A:** Depends on the null hypothesis!

---
# Final Exam Topics

## 2. Multiple Linear Regression: Inference

Confidence intervals

- Formula, interpretation, and comparison of different intervals for the same coefficient

.pink[Proof:] Show that the *F* statistic formula containing *RSS* implies the *F* statistic formula containing *R*.super[*2*]

- For practice, you can also prove that the second formula implies the first

---
# Final Exam Topics

## 3. Categorical Variables

How do you interpret coefficients on binary variables?

- **Note:** Depends on the presence of interaction terms and whether the outcome variable is transformed

Dummy variable trap

What is the reference category?

- How do you back out group-specific averages from a dummy variable regression?
- How do coefficient estimates change when you change the reference category?

---
# Final Exam Topics

## 4. Interactive Relationships

How do you interpret interaction coefficients?

- Binary `$\times$` binary
- Binary `$\times$` continuous
- Continuous `$\times$` continuous

How does an interaction term change how you interpret the effect of the variable of interest on the outcome variable?

- Marginal effects (partial derivative)

---
# Final Exam Topics

## 5. Nonlinear Relationships

Identify nonlinear models

- OLS can handle nonlinear variables, but not nonlinear parameters

Transform nonlinear models

- Give OLS a chance

---
# Final Exam Topics

## 5. Nonlinear Relationships

How do you interpret coefficients in the presence of logarithmic transformations?

- Level `$Y$`, level `$X$`
- Level `$Y$`, log `$X$`
- Log `$Y$`, level `$X$`
- Log `$Y$`, log `$X$`

**Quadratic models:** interacting `$X$` with itself

- Calculate marginal effects to understand how `$X$` affects `$Y$`

---
# Final Exam Structure

## No Multiple Choice!

## No Fill-in the Blank!

## No True-False!

## Just Free Response

- 8 multi-part questions with varying numbers of points (200 points total)
- Explanations required for full credit

---
# Final Exam Protocol

## Materials

- Writing utensil
- 3-inch-by-5-inch note card
- Basic or scientific calculator (no graphing or programming capabilities)
- .hi[Nothing else]

## Procedure

- **Randomized** seating chart .red[(penalty for non-compliance)]
- 120 minutes from *"you may begin"* to *"pencils down"*
- First 30 minutes: .hi[quiet period] (no questions, no getting up)
- Last 90 minutes: ask lots of questions

---
class: inverse, middle

# Practice

---
# .mono[gapminder] Package

Data on population, GDP per capita, and life expectancy

Unit of observation: country-year

- All countries, every 5.super[th] year between 1957 and 2007

```r
p_load(gapminder)
data <- get('gapminder')
head(data)
```

```
#> # A tibble: 6 x 6
#>   country     continent  year lifeExp      pop gdpPercap
#>   <fct>       <fct>     <int>   <dbl>    <int>     <dbl>
#> 1 Afghanistan Asia       1952    28.8  8425333      779.
#> 2 Afghanistan Asia       1957    30.3  9240934      821.
#> 3 Afghanistan Asia       1962    32.0 10267083      853.
#> 4 Afghanistan Asia       1967    34.0 11537966      836.
#> 5 Afghanistan Asia       1972    36.1 13079460      740.
#> 6 Afghanistan Asia       1977    38.4 14880372      786.
```

---
# GDP per Capita by Continent

```r
reg <- lm(gdpPercap ~ continent, data = data)
tidy(reg)
```

```
#> # A tibble: 5 x 5
#>   term              estimate std.error statistic  p.value
#>   <chr>                <dbl>     <dbl>     <dbl>    <dbl>
#> 1 (Intercept)          2194.      347.      6.33 3.21e-10
#> 2 continentAmericas    4942.      609.      8.12 8.79e-16
#> 3 continentAsia        5708.      557.     10.3  5.43e-24
#> 4 continentEurope     12276.      573.     21.4  3.06e-90
#> 5 continentOceania    16428.     1802.      9.12 2.12e-19
```

```r
nobs(reg)
```

```
#> [1] 1704
```

```r
summary(reg)$r.squared
```

```
#> [1] 0.2295766
```

---
# GDP per Capita by Continent

```r
reg2 <- lm(log(gdpPercap) ~ continent, data = data)
tidy(reg2)
```

```
#> # A tibble: 5 x 5
#>   term              estimate std.error statistic   p.value
#>   <chr>                <dbl>     <dbl>     <dbl>     <dbl>
#> 1 (Intercept)          7.25     0.0373     195.  0.       
#> 2 continentAmericas    1.37     0.0654      20.9 1.48e- 86
#> 3 continentAsia        0.823    0.0598      13.8 5.59e- 41
#> 4 continentEurope      2.10     0.0616      34.0 5.40e-194
#> 5 continentOceania     2.53     0.194       13.1 2.93e- 37
```

```r
nobs(reg2)
```

```
#> [1] 1704
```

```r
summary(reg2)$r.squared
```

```
#> [1] 0.4390461
```

---
# GDP per Capita by Continent

```r
data <- data %>% 
  mutate(continent = ifelse(as.character(continent) %in% 
                              c("Europe", "Americas", "Oceania"), 
                            "The West",
                            as.character(continent)))
reg3 <- lm(log(gdpPercap) ~ continent, data = data)
tidy(reg3)
```

```
#> # A tibble: 3 x 5
#>   term              estimate std.error statistic   p.value
#>   <chr>                <dbl>     <dbl>     <dbl>     <dbl>
#> 1 (Intercept)          7.25     0.0385     188.  0.       
#> 2 continentAsia        0.823    0.0618      13.3 1.24e- 38
#> 3 continentThe West    1.79     0.0532      33.7 8.19e-191
```

```r
summary(reg3)$r.squared
```

```
#> [1] 0.4007156
```

---
# GDP per Capita by Continent

**Null hypothesis:** Europe, the Americas, and Oceania have the same average GDP per capita.

**Setup *F* Test**

```r
# restricted model: reg3
# unrestricted model: reg2

# get r-squared from restricted and unrestricted models
r2_restrict <- summary(reg3)$r.squared
r2_unrestrict <- summary(reg2)$r.squared

# number of resrictions
n_restrict <- 2

# degrees of freedom in unrestricted model
dof <- summary(reg3)$df[2]

# significance level
alpha <- 0.01
```

---
# GDP per Capita by Continent

**Null hypothesis:** Europe, the Americas, and Oceania have the same average GDP per capita.

**Conduct *F* Test**

```r
# calculate F stat:
f_stat <- ((r2_unrestrict - r2_restrict)/n_restrict)/((1 - r2_unrestrict)/dof)

# find critical value of F:
f_crit <- qf(1-alpha, n_restrict, dof)

# reject null if:
f_stat > f_crit
```

```
#> [1] TRUE
```

**Conclusion:** At least one of the continents in "The West" has a different average GDP per capita than the others.

---
class: white-slide
.center[**Association of GDP per Capita with Life Expectancy**]
<table class="table" style="font-size: 18.2px; margin-left: auto; margin-right: auto;">
 <thead>
  <tr>
   <th style="text-align:left;">  </th>
   <th style="text-align:center;"> Life Expectancy </th>
   <th style="text-align:center;"> Life Expectancy </th>
   <th style="text-align:center;"> log(Life Expectancy) </th>
   <th style="text-align:center;"> log(Life Expectancy) </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;color: #23373b !important;line-height: 110%;font-style: italic;color: black !important;"> log(GDP/Capita) </td>
   <td style="text-align:center;color: #23373b !important;line-height: 110%;font-weight: bold;"> 6.42 </td>
   <td style="text-align:center;color: #23373b !important;line-height: 110%;"> 5.69 </td>
   <td style="text-align:center;color: #23373b !important;line-height: 110%;"> 0.112 </td>
   <td style="text-align:center;color: #23373b !important;line-height: 110%;"> 0.111 </td>
  </tr>
  <tr>
   <td style="text-align:left;color: #23373b !important;color: #c2bebe !important;line-height: 110%;font-style: italic;color: black !important;">  </td>
   <td style="text-align:center;color: #23373b !important;color: #c2bebe !important;line-height: 110%;font-weight: bold;"> (0.183) </td>
   <td style="text-align:center;color: #23373b !important;color: #c2bebe !important;line-height: 110%;"> (0.325) </td>
   <td style="text-align:center;color: #23373b !important;color: #c2bebe !important;line-height: 110%;"> (0.004) </td>
   <td style="text-align:center;color: #23373b !important;color: #c2bebe !important;line-height: 110%;"> (0.006) </td>
  </tr>
  <tr>
   <td style="text-align:left;color: #23373b !important;line-height: 110%;font-style: italic;color: black !important;"> log(GDP/Capita) <d7> Americas </d7>
</td>
   <td style="text-align:center;color: #23373b !important;line-height: 110%;font-weight: bold;">  </td>
   <td style="text-align:center;color: #23373b !important;line-height: 110%;"> 4.03 </td>
   <td style="text-align:center;color: #23373b !important;line-height: 110%;">  </td>
   <td style="text-align:center;color: #23373b !important;line-height: 110%;"> 0.047 </td>
  </tr>
  <tr>
   <td style="text-align:left;color: #23373b !important;color: #c2bebe !important;line-height: 110%;font-style: italic;color: black !important;">  </td>
   <td style="text-align:center;color: #23373b !important;color: #c2bebe !important;line-height: 110%;font-weight: bold;">  </td>
   <td style="text-align:center;color: #23373b !important;color: #c2bebe !important;line-height: 110%;"> (0.674) </td>
   <td style="text-align:center;color: #23373b !important;color: #c2bebe !important;line-height: 110%;">  </td>
   <td style="text-align:center;color: #23373b !important;color: #c2bebe !important;line-height: 110%;"> (0.013) </td>
  </tr>
  <tr>
   <td style="text-align:left;color: #23373b !important;line-height: 110%;font-style: italic;color: black !important;"> log(GDP/Capita) <d7> Asia </d7>
</td>
   <td style="text-align:center;color: #23373b !important;line-height: 110%;font-weight: bold;">  </td>
   <td style="text-align:center;color: #23373b !important;line-height: 110%;"> 0.561 </td>
   <td style="text-align:center;color: #23373b !important;line-height: 110%;">  </td>
   <td style="text-align:center;color: #23373b !important;line-height: 110%;"> -0.002 </td>
  </tr>
  <tr>
   <td style="text-align:left;color: #23373b !important;color: #c2bebe !important;line-height: 110%;font-style: italic;color: black !important;">  </td>
   <td style="text-align:center;color: #23373b !important;color: #c2bebe !important;line-height: 110%;font-weight: bold;">  </td>
   <td style="text-align:center;color: #23373b !important;color: #c2bebe !important;line-height: 110%;"> (0.421) </td>
   <td style="text-align:center;color: #23373b !important;color: #c2bebe !important;line-height: 110%;">  </td>
   <td style="text-align:center;color: #23373b !important;color: #c2bebe !important;line-height: 110%;"> (0.008) </td>
  </tr>
  <tr>
   <td style="text-align:left;color: #23373b !important;line-height: 110%;font-style: italic;color: black !important;"> log(GDP/Capita) <d7> Europe </d7>
</td>
   <td style="text-align:center;color: #23373b !important;line-height: 110%;font-weight: bold;">  </td>
   <td style="text-align:center;color: #23373b !important;line-height: 110%;"> 0.614 </td>
   <td style="text-align:center;color: #23373b !important;line-height: 110%;">  </td>
   <td style="text-align:center;color: #23373b !important;line-height: 110%;"> -0.019 </td>
  </tr>
  <tr>
   <td style="text-align:left;color: #23373b !important;color: #c2bebe !important;line-height: 110%;font-style: italic;color: black !important;">  </td>
   <td style="text-align:center;color: #23373b !important;color: #c2bebe !important;line-height: 110%;font-weight: bold;">  </td>
   <td style="text-align:center;color: #23373b !important;color: #c2bebe !important;line-height: 110%;"> (0.597) </td>
   <td style="text-align:center;color: #23373b !important;color: #c2bebe !important;line-height: 110%;">  </td>
   <td style="text-align:center;color: #23373b !important;color: #c2bebe !important;line-height: 110%;"> (0.012) </td>
  </tr>
  <tr>
   <td style="text-align:left;color: #23373b !important;line-height: 110%;font-style: italic;color: black !important;"> log(GDP/Capita) <d7> Oceania </d7>
</td>
   <td style="text-align:center;color: #23373b !important;line-height: 110%;font-weight: bold;">  </td>
   <td style="text-align:center;color: #23373b !important;line-height: 110%;"> 5.2 </td>
   <td style="text-align:center;color: #23373b !important;line-height: 110%;">  </td>
   <td style="text-align:center;color: #23373b !important;line-height: 110%;"> 0.035 </td>
  </tr>
  <tr>
   <td style="text-align:left;color: #23373b !important;color: #c2bebe !important;line-height: 110%;font-style: italic;color: black !important;">  </td>
   <td style="text-align:center;color: #23373b !important;color: #c2bebe !important;line-height: 110%;font-weight: bold;">  </td>
   <td style="text-align:center;color: #23373b !important;color: #c2bebe !important;line-height: 110%;"> (4.36) </td>
   <td style="text-align:center;color: #23373b !important;color: #c2bebe !important;line-height: 110%;">  </td>
   <td style="text-align:center;color: #23373b !important;color: #c2bebe !important;line-height: 110%;"> (0.084) </td>
  </tr>
  <tr>
   <td style="text-align:left;color: #23373b !important;line-height: 110%;font-style: italic;color: black !important;"> Continent Dummies? </td>
   <td style="text-align:center;color: #23373b !important;line-height: 110%;font-weight: bold;"> Yes </td>
   <td style="text-align:center;color: #23373b !important;line-height: 110%;"> Yes </td>
   <td style="text-align:center;color: #23373b !important;line-height: 110%;"> Yes </td>
   <td style="text-align:center;color: #23373b !important;line-height: 110%;"> Yes </td>
  </tr>
  <tr>
   <td style="text-align:left;color: #23373b !important;line-height: 110%;font-style: italic;color: black !important;"> Observations </td>
   <td style="text-align:center;color: #23373b !important;line-height: 110%;font-weight: bold;"> 1704 </td>
   <td style="text-align:center;color: #23373b !important;line-height: 110%;"> 1704 </td>
   <td style="text-align:center;color: #23373b !important;line-height: 110%;"> 1704 </td>
   <td style="text-align:center;color: #23373b !important;line-height: 110%;"> 1704 </td>
  </tr>
  <tr>
   <td style="text-align:left;color: #23373b !important;line-height: 110%;font-style: italic;color: black !important;"> R-Squared </td>
   <td style="text-align:center;color: #23373b !important;line-height: 110%;font-weight: bold;"> 0.704 </td>
   <td style="text-align:center;color: #23373b !important;line-height: 110%;"> 0.71 </td>
   <td style="text-align:center;color: #23373b !important;line-height: 110%;"> 0.665 </td>
   <td style="text-align:center;color: #23373b !important;line-height: 110%;"> 0.669 </td>
  </tr>
</tbody>
</table>

---
class: white-slide
.center[**Association of GDP per Capita with Life Expectancy**]
<table class="table" style="font-size: 18.2px; margin-left: auto; margin-right: auto;">
 <thead>
  <tr>
   <th style="text-align:left;">  </th>
   <th style="text-align:center;"> Life Expectancy </th>
   <th style="text-align:center;"> Life Expectancy </th>
   <th style="text-align:center;"> log(Life Expectancy) </th>
   <th style="text-align:center;"> log(Life Expectancy) </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;color: #23373b !important;line-height: 110%;font-style: italic;color: black !important;"> log(GDP/Capita) </td>
   <td style="text-align:center;color: #23373b !important;line-height: 110%;"> 6.42 </td>
   <td style="text-align:center;color: #23373b !important;line-height: 110%;font-weight: bold;"> 5.69 </td>
   <td style="text-align:center;color: #23373b !important;line-height: 110%;"> 0.112 </td>
   <td style="text-align:center;color: #23373b !important;line-height: 110%;"> 0.111 </td>
  </tr>
  <tr>
   <td style="text-align:left;color: #23373b !important;color: #c2bebe !important;line-height: 110%;font-style: italic;color: black !important;">  </td>
   <td style="text-align:center;color: #23373b !important;color: #c2bebe !important;line-height: 110%;"> (0.183) </td>
   <td style="text-align:center;color: #23373b !important;color: #c2bebe !important;line-height: 110%;font-weight: bold;"> (0.325) </td>
   <td style="text-align:center;color: #23373b !important;color: #c2bebe !important;line-height: 110%;"> (0.004) </td>
   <td style="text-align:center;color: #23373b !important;color: #c2bebe !important;line-height: 110%;"> (0.006) </td>
  </tr>
  <tr>
   <td style="text-align:left;color: #23373b !important;line-height: 110%;font-style: italic;color: black !important;"> log(GDP/Capita) <d7> Americas </d7>
</td>
   <td style="text-align:center;color: #23373b !important;line-height: 110%;">  </td>
   <td style="text-align:center;color: #23373b !important;line-height: 110%;font-weight: bold;"> 4.03 </td>
   <td style="text-align:center;color: #23373b !important;line-height: 110%;">  </td>
   <td style="text-align:center;color: #23373b !important;line-height: 110%;"> 0.047 </td>
  </tr>
  <tr>
   <td style="text-align:left;color: #23373b !important;color: #c2bebe !important;line-height: 110%;font-style: italic;color: black !important;">  </td>
   <td style="text-align:center;color: #23373b !important;color: #c2bebe !important;line-height: 110%;">  </td>
   <td style="text-align:center;color: #23373b !important;color: #c2bebe !important;line-height: 110%;font-weight: bold;"> (0.674) </td>
   <td style="text-align:center;color: #23373b !important;color: #c2bebe !important;line-height: 110%;">  </td>
   <td style="text-align:center;color: #23373b !important;color: #c2bebe !important;line-height: 110%;"> (0.013) </td>
  </tr>
  <tr>
   <td style="text-align:left;color: #23373b !important;line-height: 110%;font-style: italic;color: black !important;"> log(GDP/Capita) <d7> Asia </d7>
</td>
   <td style="text-align:center;color: #23373b !important;line-height: 110%;">  </td>
   <td style="text-align:center;color: #23373b !important;line-height: 110%;font-weight: bold;"> 0.561 </td>
   <td style="text-align:center;color: #23373b !important;line-height: 110%;">  </td>
   <td style="text-align:center;color: #23373b !important;line-height: 110%;"> -0.002 </td>
  </tr>
  <tr>
   <td style="text-align:left;color: #23373b !important;color: #c2bebe !important;line-height: 110%;font-style: italic;color: black !important;">  </td>
   <td style="text-align:center;color: #23373b !important;color: #c2bebe !important;line-height: 110%;">  </td>
   <td style="text-align:center;color: #23373b !important;color: #c2bebe !important;line-height: 110%;font-weight: bold;"> (0.421) </td>
   <td style="text-align:center;color: #23373b !important;color: #c2bebe !important;line-height: 110%;">  </td>
   <td style="text-align:center;color: #23373b !important;color: #c2bebe !important;line-height: 110%;"> (0.008) </td>
  </tr>
  <tr>
   <td style="text-align:left;color: #23373b !important;line-height: 110%;font-style: italic;color: black !important;"> log(GDP/Capita) <d7> Europe </d7>
</td>
   <td style="text-align:center;color: #23373b !important;line-height: 110%;">  </td>
   <td style="text-align:center;color: #23373b !important;line-height: 110%;font-weight: bold;"> 0.614 </td>
   <td style="text-align:center;color: #23373b !important;line-height: 110%;">  </td>
   <td style="text-align:center;color: #23373b !important;line-height: 110%;"> -0.019 </td>
  </tr>
  <tr>
   <td style="text-align:left;color: #23373b !important;color: #c2bebe !important;line-height: 110%;font-style: italic;color: black !important;">  </td>
   <td style="text-align:center;color: #23373b !important;color: #c2bebe !important;line-height: 110%;">  </td>
   <td style="text-align:center;color: #23373b !important;color: #c2bebe !important;line-height: 110%;font-weight: bold;"> (0.597) </td>
   <td style="text-align:center;color: #23373b !important;color: #c2bebe !important;line-height: 110%;">  </td>
   <td style="text-align:center;color: #23373b !important;color: #c2bebe !important;line-height: 110%;"> (0.012) </td>
  </tr>
  <tr>
   <td style="text-align:left;color: #23373b !important;line-height: 110%;font-style: italic;color: black !important;"> log(GDP/Capita) <d7> Oceania </d7>
</td>
   <td style="text-align:center;color: #23373b !important;line-height: 110%;">  </td>
   <td style="text-align:center;color: #23373b !important;line-height: 110%;font-weight: bold;"> 5.2 </td>
   <td style="text-align:center;color: #23373b !important;line-height: 110%;">  </td>
   <td style="text-align:center;color: #23373b !important;line-height: 110%;"> 0.035 </td>
  </tr>
  <tr>
   <td style="text-align:left;color: #23373b !important;color: #c2bebe !important;line-height: 110%;font-style: italic;color: black !important;">  </td>
   <td style="text-align:center;color: #23373b !important;color: #c2bebe !important;line-height: 110%;">  </td>
   <td style="text-align:center;color: #23373b !important;color: #c2bebe !important;line-height: 110%;font-weight: bold;"> (4.36) </td>
   <td style="text-align:center;color: #23373b !important;color: #c2bebe !important;line-height: 110%;">  </td>
   <td style="text-align:center;color: #23373b !important;color: #c2bebe !important;line-height: 110%;"> (0.084) </td>
  </tr>
  <tr>
   <td style="text-align:left;color: #23373b !important;line-height: 110%;font-style: italic;color: black !important;"> Continent Dummies? </td>
   <td style="text-align:center;color: #23373b !important;line-height: 110%;"> Yes </td>
   <td style="text-align:center;color: #23373b !important;line-height: 110%;font-weight: bold;"> Yes </td>
   <td style="text-align:center;color: #23373b !important;line-height: 110%;"> Yes </td>
   <td style="text-align:center;color: #23373b !important;line-height: 110%;"> Yes </td>
  </tr>
  <tr>
   <td style="text-align:left;color: #23373b !important;line-height: 110%;font-style: italic;color: black !important;"> Observations </td>
   <td style="text-align:center;color: #23373b !important;line-height: 110%;"> 1704 </td>
   <td style="text-align:center;color: #23373b !important;line-height: 110%;font-weight: bold;"> 1704 </td>
   <td style="text-align:center;color: #23373b !important;line-height: 110%;"> 1704 </td>
   <td style="text-align:center;color: #23373b !important;line-height: 110%;"> 1704 </td>
  </tr>
  <tr>
   <td style="text-align:left;color: #23373b !important;line-height: 110%;font-style: italic;color: black !important;"> R-Squared </td>
   <td style="text-align:center;color: #23373b !important;line-height: 110%;"> 0.704 </td>
   <td style="text-align:center;color: #23373b !important;line-height: 110%;font-weight: bold;"> 0.71 </td>
   <td style="text-align:center;color: #23373b !important;line-height: 110%;"> 0.665 </td>
   <td style="text-align:center;color: #23373b !important;line-height: 110%;"> 0.669 </td>
  </tr>
</tbody>
</table>

---
class: white-slide
.center[**Association of GDP per Capita with Life Expectancy**]
<table class="table" style="font-size: 18.2px; margin-left: auto; margin-right: auto;">
 <thead>
  <tr>
   <th style="text-align:left;">  </th>
   <th style="text-align:center;"> Life Expectancy </th>
   <th style="text-align:center;"> Life Expectancy </th>
   <th style="text-align:center;"> log(Life Expectancy) </th>
   <th style="text-align:center;"> log(Life Expectancy) </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;color: #23373b !important;line-height: 110%;font-style: italic;color: black !important;"> log(GDP/Capita) </td>
   <td style="text-align:center;color: #23373b !important;line-height: 110%;"> 6.42 </td>
   <td style="text-align:center;color: #23373b !important;line-height: 110%;"> 5.69 </td>
   <td style="text-align:center;color: #23373b !important;line-height: 110%;font-weight: bold;"> 0.112 </td>
   <td style="text-align:center;color: #23373b !important;line-height: 110%;"> 0.111 </td>
  </tr>
  <tr>
   <td style="text-align:left;color: #23373b !important;color: #c2bebe !important;line-height: 110%;font-style: italic;color: black !important;">  </td>
   <td style="text-align:center;color: #23373b !important;color: #c2bebe !important;line-height: 110%;"> (0.183) </td>
   <td style="text-align:center;color: #23373b !important;color: #c2bebe !important;line-height: 110%;"> (0.325) </td>
   <td style="text-align:center;color: #23373b !important;color: #c2bebe !important;line-height: 110%;font-weight: bold;"> (0.004) </td>
   <td style="text-align:center;color: #23373b !important;color: #c2bebe !important;line-height: 110%;"> (0.006) </td>
  </tr>
  <tr>
   <td style="text-align:left;color: #23373b !important;line-height: 110%;font-style: italic;color: black !important;"> log(GDP/Capita) <d7> Americas </d7>
</td>
   <td style="text-align:center;color: #23373b !important;line-height: 110%;">  </td>
   <td style="text-align:center;color: #23373b !important;line-height: 110%;"> 4.03 </td>
   <td style="text-align:center;color: #23373b !important;line-height: 110%;font-weight: bold;">  </td>
   <td style="text-align:center;color: #23373b !important;line-height: 110%;"> 0.047 </td>
  </tr>
  <tr>
   <td style="text-align:left;color: #23373b !important;color: #c2bebe !important;line-height: 110%;font-style: italic;color: black !important;">  </td>
   <td style="text-align:center;color: #23373b !important;color: #c2bebe !important;line-height: 110%;">  </td>
   <td style="text-align:center;color: #23373b !important;color: #c2bebe !important;line-height: 110%;"> (0.674) </td>
   <td style="text-align:center;color: #23373b !important;color: #c2bebe !important;line-height: 110%;font-weight: bold;">  </td>
   <td style="text-align:center;color: #23373b !important;color: #c2bebe !important;line-height: 110%;"> (0.013) </td>
  </tr>
  <tr>
   <td style="text-align:left;color: #23373b !important;line-height: 110%;font-style: italic;color: black !important;"> log(GDP/Capita) <d7> Asia </d7>
</td>
   <td style="text-align:center;color: #23373b !important;line-height: 110%;">  </td>
   <td style="text-align:center;color: #23373b !important;line-height: 110%;"> 0.561 </td>
   <td style="text-align:center;color: #23373b !important;line-height: 110%;font-weight: bold;">  </td>
   <td style="text-align:center;color: #23373b !important;line-height: 110%;"> -0.002 </td>
  </tr>
  <tr>
   <td style="text-align:left;color: #23373b !important;color: #c2bebe !important;line-height: 110%;font-style: italic;color: black !important;">  </td>
   <td style="text-align:center;color: #23373b !important;color: #c2bebe !important;line-height: 110%;">  </td>
   <td style="text-align:center;color: #23373b !important;color: #c2bebe !important;line-height: 110%;"> (0.421) </td>
   <td style="text-align:center;color: #23373b !important;color: #c2bebe !important;line-height: 110%;font-weight: bold;">  </td>
   <td style="text-align:center;color: #23373b !important;color: #c2bebe !important;line-height: 110%;"> (0.008) </td>
  </tr>
  <tr>
   <td style="text-align:left;color: #23373b !important;line-height: 110%;font-style: italic;color: black !important;"> log(GDP/Capita) <d7> Europe </d7>
</td>
   <td style="text-align:center;color: #23373b !important;line-height: 110%;">  </td>
   <td style="text-align:center;color: #23373b !important;line-height: 110%;"> 0.614 </td>
   <td style="text-align:center;color: #23373b !important;line-height: 110%;font-weight: bold;">  </td>
   <td style="text-align:center;color: #23373b !important;line-height: 110%;"> -0.019 </td>
  </tr>
  <tr>
   <td style="text-align:left;color: #23373b !important;color: #c2bebe !important;line-height: 110%;font-style: italic;color: black !important;">  </td>
   <td style="text-align:center;color: #23373b !important;color: #c2bebe !important;line-height: 110%;">  </td>
   <td style="text-align:center;color: #23373b !important;color: #c2bebe !important;line-height: 110%;"> (0.597) </td>
   <td style="text-align:center;color: #23373b !important;color: #c2bebe !important;line-height: 110%;font-weight: bold;">  </td>
   <td style="text-align:center;color: #23373b !important;color: #c2bebe !important;line-height: 110%;"> (0.012) </td>
  </tr>
  <tr>
   <td style="text-align:left;color: #23373b !important;line-height: 110%;font-style: italic;color: black !important;"> log(GDP/Capita) <d7> Oceania </d7>
</td>
   <td style="text-align:center;color: #23373b !important;line-height: 110%;">  </td>
   <td style="text-align:center;color: #23373b !important;line-height: 110%;"> 5.2 </td>
   <td style="text-align:center;color: #23373b !important;line-height: 110%;font-weight: bold;">  </td>
   <td style="text-align:center;color: #23373b !important;line-height: 110%;"> 0.035 </td>
  </tr>
  <tr>
   <td style="text-align:left;color: #23373b !important;color: #c2bebe !important;line-height: 110%;font-style: italic;color: black !important;">  </td>
   <td style="text-align:center;color: #23373b !important;color: #c2bebe !important;line-height: 110%;">  </td>
   <td style="text-align:center;color: #23373b !important;color: #c2bebe !important;line-height: 110%;"> (4.36) </td>
   <td style="text-align:center;color: #23373b !important;color: #c2bebe !important;line-height: 110%;font-weight: bold;">  </td>
   <td style="text-align:center;color: #23373b !important;color: #c2bebe !important;line-height: 110%;"> (0.084) </td>
  </tr>
  <tr>
   <td style="text-align:left;color: #23373b !important;line-height: 110%;font-style: italic;color: black !important;"> Continent Dummies? </td>
   <td style="text-align:center;color: #23373b !important;line-height: 110%;"> Yes </td>
   <td style="text-align:center;color: #23373b !important;line-height: 110%;"> Yes </td>
   <td style="text-align:center;color: #23373b !important;line-height: 110%;font-weight: bold;"> Yes </td>
   <td style="text-align:center;color: #23373b !important;line-height: 110%;"> Yes </td>
  </tr>
  <tr>
   <td style="text-align:left;color: #23373b !important;line-height: 110%;font-style: italic;color: black !important;"> Observations </td>
   <td style="text-align:center;color: #23373b !important;line-height: 110%;"> 1704 </td>
   <td style="text-align:center;color: #23373b !important;line-height: 110%;"> 1704 </td>
   <td style="text-align:center;color: #23373b !important;line-height: 110%;font-weight: bold;"> 1704 </td>
   <td style="text-align:center;color: #23373b !important;line-height: 110%;"> 1704 </td>
  </tr>
  <tr>
   <td style="text-align:left;color: #23373b !important;line-height: 110%;font-style: italic;color: black !important;"> R-Squared </td>
   <td style="text-align:center;color: #23373b !important;line-height: 110%;"> 0.704 </td>
   <td style="text-align:center;color: #23373b !important;line-height: 110%;"> 0.71 </td>
   <td style="text-align:center;color: #23373b !important;line-height: 110%;font-weight: bold;"> 0.665 </td>
   <td style="text-align:center;color: #23373b !important;line-height: 110%;"> 0.669 </td>
  </tr>
</tbody>
</table>

---
class: white-slide
.center[**Association of GDP per Capita with Life Expectancy**]
<table class="table" style="font-size: 18.2px; margin-left: auto; margin-right: auto;">
 <thead>
  <tr>
   <th style="text-align:left;">  </th>
   <th style="text-align:center;"> Life Expectancy </th>
   <th style="text-align:center;"> Life Expectancy </th>
   <th style="text-align:center;"> log(Life Expectancy) </th>
   <th style="text-align:center;"> log(Life Expectancy) </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;color: #23373b !important;line-height: 110%;font-style: italic;color: black !important;"> log(GDP/Capita) </td>
   <td style="text-align:center;color: #23373b !important;line-height: 110%;"> 6.42 </td>
   <td style="text-align:center;color: #23373b !important;line-height: 110%;"> 5.69 </td>
   <td style="text-align:center;color: #23373b !important;line-height: 110%;"> 0.112 </td>
   <td style="text-align:center;color: #23373b !important;line-height: 110%;font-weight: bold;"> 0.111 </td>
  </tr>
  <tr>
   <td style="text-align:left;color: #23373b !important;color: #c2bebe !important;line-height: 110%;font-style: italic;color: black !important;">  </td>
   <td style="text-align:center;color: #23373b !important;color: #c2bebe !important;line-height: 110%;"> (0.183) </td>
   <td style="text-align:center;color: #23373b !important;color: #c2bebe !important;line-height: 110%;"> (0.325) </td>
   <td style="text-align:center;color: #23373b !important;color: #c2bebe !important;line-height: 110%;"> (0.004) </td>
   <td style="text-align:center;color: #23373b !important;color: #c2bebe !important;line-height: 110%;font-weight: bold;"> (0.006) </td>
  </tr>
  <tr>
   <td style="text-align:left;color: #23373b !important;line-height: 110%;font-style: italic;color: black !important;"> log(GDP/Capita) <d7> Americas </d7>
</td>
   <td style="text-align:center;color: #23373b !important;line-height: 110%;">  </td>
   <td style="text-align:center;color: #23373b !important;line-height: 110%;"> 4.03 </td>
   <td style="text-align:center;color: #23373b !important;line-height: 110%;">  </td>
   <td style="text-align:center;color: #23373b !important;line-height: 110%;font-weight: bold;"> 0.047 </td>
  </tr>
  <tr>
   <td style="text-align:left;color: #23373b !important;color: #c2bebe !important;line-height: 110%;font-style: italic;color: black !important;">  </td>
   <td style="text-align:center;color: #23373b !important;color: #c2bebe !important;line-height: 110%;">  </td>
   <td style="text-align:center;color: #23373b !important;color: #c2bebe !important;line-height: 110%;"> (0.674) </td>
   <td style="text-align:center;color: #23373b !important;color: #c2bebe !important;line-height: 110%;">  </td>
   <td style="text-align:center;color: #23373b !important;color: #c2bebe !important;line-height: 110%;font-weight: bold;"> (0.013) </td>
  </tr>
  <tr>
   <td style="text-align:left;color: #23373b !important;line-height: 110%;font-style: italic;color: black !important;"> log(GDP/Capita) <d7> Asia </d7>
</td>
   <td style="text-align:center;color: #23373b !important;line-height: 110%;">  </td>
   <td style="text-align:center;color: #23373b !important;line-height: 110%;"> 0.561 </td>
   <td style="text-align:center;color: #23373b !important;line-height: 110%;">  </td>
   <td style="text-align:center;color: #23373b !important;line-height: 110%;font-weight: bold;"> -0.002 </td>
  </tr>
  <tr>
   <td style="text-align:left;color: #23373b !important;color: #c2bebe !important;line-height: 110%;font-style: italic;color: black !important;">  </td>
   <td style="text-align:center;color: #23373b !important;color: #c2bebe !important;line-height: 110%;">  </td>
   <td style="text-align:center;color: #23373b !important;color: #c2bebe !important;line-height: 110%;"> (0.421) </td>
   <td style="text-align:center;color: #23373b !important;color: #c2bebe !important;line-height: 110%;">  </td>
   <td style="text-align:center;color: #23373b !important;color: #c2bebe !important;line-height: 110%;font-weight: bold;"> (0.008) </td>
  </tr>
  <tr>
   <td style="text-align:left;color: #23373b !important;line-height: 110%;font-style: italic;color: black !important;"> log(GDP/Capita) <d7> Europe </d7>
</td>
   <td style="text-align:center;color: #23373b !important;line-height: 110%;">  </td>
   <td style="text-align:center;color: #23373b !important;line-height: 110%;"> 0.614 </td>
   <td style="text-align:center;color: #23373b !important;line-height: 110%;">  </td>
   <td style="text-align:center;color: #23373b !important;line-height: 110%;font-weight: bold;"> -0.019 </td>
  </tr>
  <tr>
   <td style="text-align:left;color: #23373b !important;color: #c2bebe !important;line-height: 110%;font-style: italic;color: black !important;">  </td>
   <td style="text-align:center;color: #23373b !important;color: #c2bebe !important;line-height: 110%;">  </td>
   <td style="text-align:center;color: #23373b !important;color: #c2bebe !important;line-height: 110%;"> (0.597) </td>
   <td style="text-align:center;color: #23373b !important;color: #c2bebe !important;line-height: 110%;">  </td>
   <td style="text-align:center;color: #23373b !important;color: #c2bebe !important;line-height: 110%;font-weight: bold;"> (0.012) </td>
  </tr>
  <tr>
   <td style="text-align:left;color: #23373b !important;line-height: 110%;font-style: italic;color: black !important;"> log(GDP/Capita) <d7> Oceania </d7>
</td>
   <td style="text-align:center;color: #23373b !important;line-height: 110%;">  </td>
   <td style="text-align:center;color: #23373b !important;line-height: 110%;"> 5.2 </td>
   <td style="text-align:center;color: #23373b !important;line-height: 110%;">  </td>
   <td style="text-align:center;color: #23373b !important;line-height: 110%;font-weight: bold;"> 0.035 </td>
  </tr>
  <tr>
   <td style="text-align:left;color: #23373b !important;color: #c2bebe !important;line-height: 110%;font-style: italic;color: black !important;">  </td>
   <td style="text-align:center;color: #23373b !important;color: #c2bebe !important;line-height: 110%;">  </td>
   <td style="text-align:center;color: #23373b !important;color: #c2bebe !important;line-height: 110%;"> (4.36) </td>
   <td style="text-align:center;color: #23373b !important;color: #c2bebe !important;line-height: 110%;">  </td>
   <td style="text-align:center;color: #23373b !important;color: #c2bebe !important;line-height: 110%;font-weight: bold;"> (0.084) </td>
  </tr>
  <tr>
   <td style="text-align:left;color: #23373b !important;line-height: 110%;font-style: italic;color: black !important;"> Continent Dummies? </td>
   <td style="text-align:center;color: #23373b !important;line-height: 110%;"> Yes </td>
   <td style="text-align:center;color: #23373b !important;line-height: 110%;"> Yes </td>
   <td style="text-align:center;color: #23373b !important;line-height: 110%;"> Yes </td>
   <td style="text-align:center;color: #23373b !important;line-height: 110%;font-weight: bold;"> Yes </td>
  </tr>
  <tr>
   <td style="text-align:left;color: #23373b !important;line-height: 110%;font-style: italic;color: black !important;"> Observations </td>
   <td style="text-align:center;color: #23373b !important;line-height: 110%;"> 1704 </td>
   <td style="text-align:center;color: #23373b !important;line-height: 110%;"> 1704 </td>
   <td style="text-align:center;color: #23373b !important;line-height: 110%;"> 1704 </td>
   <td style="text-align:center;color: #23373b !important;line-height: 110%;font-weight: bold;"> 1704 </td>
  </tr>
  <tr>
   <td style="text-align:left;color: #23373b !important;line-height: 110%;font-style: italic;color: black !important;"> R-Squared </td>
   <td style="text-align:center;color: #23373b !important;line-height: 110%;"> 0.704 </td>
   <td style="text-align:center;color: #23373b !important;line-height: 110%;"> 0.71 </td>
   <td style="text-align:center;color: #23373b !important;line-height: 110%;"> 0.665 </td>
   <td style="text-align:center;color: #23373b !important;line-height: 110%;font-weight: bold;"> 0.669 </td>
  </tr>
</tbody>
</table>