ScPoEconometrics: Advanced

class: center, middle, inverse, title-slide

.title[
# ScPoEconometrics: Advanced
]
.subtitle[
## Intro and Recap 1
]
.author[
### Bluebery Planterose
]
.date[
### SciencesPo Paris 2023-01-24
]

---

layout: true

---

# Welcome to *ScPoEconometrics: Advanced*!

.pull-left[

## Today

1. Who Am I

2. This Course

3. Recap 1 of topics from intro course

]

.pull-right[

### Next time
 
* Quiz 1 (before next time)
 
* Recap 2

]

---

# Who Am I

* I'm an PhD candidate at the Paris School of Economics. Check out my [website](http://bluebery-planterose.com)!

* I work on tax evasion, climate policies, and macro topics:
    1. *Acceptability of climate policies*: who support/oppose climate policies and why?
    
    2. *Offshore real-estate in Dubai using leaked data*: how large is it, who owns it, and what does it tell us about global offshore real-estate?
    
    3. *Excess Profit Tax*: how to tax excess profits from energy firms that benefited from the war in Ukraine?

---

# This Course

## Prerequisites

* This course is the *follow-up* to [Introduction to Econometrics with R](https://github.com/ScPoEcon/ScPoEconometrics-Slides) which is taught to 2nd years.

* You are supposed to be familiar with all the econometrics material from [the slides](https://github.com/ScPoEcon/ScPoEconometrics-Slides) of that course and/or chapters 1-9 in our [textbook](https://scpoecon.github.io/ScPoEconometrics/).

* We also assume you have basic `R` working knowledge at the level of the intro course!
    * basic `data.frame` manipulation with `dplyr`
    * simple linear models with `lm`
    * basic plotting with `ggplot2`
    * Quiz 1 will try and test for that 😉, so be on top of [this chapter](https://r4ds.had.co.nz/transform.html)

---

# This Course

.pull-left[

## Grading

1. There will be ***four quizzes*** on Moodle roughly every two weeks => 40%

1. There will be ***two take home exams / case studies*** => 60%

1. There will be _no_ final exam 😅.
]

.pull-right[

## Course Materials

1. [Book](https://scpoecon.github.io/ScPoEconometrics/) chapter 10 onwards

1. The [Slides](https://github.com/ScPoEcon/Advanced-Metrics-slides)

1. The interactive [shiny apps](https://github.com/ScPoEcon/ScPoApps)

1. Quizzes on [Moodle](https://moodle.sciences-po.fr)

]

---

# Syllabus

.pull-left[

1. Intro, Recap 1
    (*Quiz 1*)

2. Recap 2
    (*Quiz 2*)

3. Intro, Difference-in-Differences

4. Tools: `Rmarkdown` and `data.table`

5. Instrumental Variables 1
    (*Quiz 3*)

6. Instrumental Variables 2
    (*Midterm exam*)
]

.pull-right[
7\. Panel Data 1

8\. Panel Data 2
    (*Quiz 4*)

9\. Discrete Outcomes

10\. Intro to Machine Learning 1

11\. Intro to Machine Learning 2

12\. Recap / Buffer
      (*Final Project*)

12\. Recap / Buffer
      (*Final Project*)
]

---

# Course Organization

---
class: separator, middle

# Recap 1

Let's get cracking! 💪

---

# Population *vs.* sample

## Models and notation

We write our (simple) population model

$$ y_i = \beta_0 + \beta_1 x_i + u_i $$

and our sample-based estimated regression model as

$$ y_i = \hat{\beta}_0 + \hat{\beta}_1 x_i + e_i $$

An estimated regression model produces estimates for each observation:

$$ \hat{y}_i = \hat{\beta}_0 + \hat{\beta}_1 x_i $$

which gives us the _best-fit_ line through our dataset.

(A lot of this set slides - in particular: pictures! - have been taken from [Ed Rubin's](https://edrub.in/index.html) outstanding material. Thanks Ed 🙏)

---
class: inverse

# Task 1: Run Simple OLS (4 minutes)
 
1. Load data [here](https://www.dropbox.com/s/wwp2cs9f0dubmhr/grade5.dta?dl=1). in `dta` format. (Hint: use `haven::read_dta("filename")` to read this format.)

1. Obtain common summary statistics for the variables `classize`, `avgmath` and `avgverb`. Hint: use the `skimr` package.

1. Estimate the linear model
    `$$\text{avgmath}_i = \beta_0 + \text{classize}_i x_i + u_i$$`

---
class: inverse

# Task 1: Solution

1. Load the data 
    
    ```r
    grades = haven::read_dta(file ="https://www.dropbox.com/s/wwp2cs9f0dubmhr/grade5.dta?dl=1")
    ```
1. Describe the dataset:
    
    ```r
    library(dplyr)
    grades %>% 
      select(classize,avgmath,avgverb) %>%
      skimr::skim()
    ```
1. Run OLS to estimate the relationship between class size and student achievement? 
    
    ```r
    summary(lm(formula = avgmath ~ classize, data = grades))
    ```

---
layout: true

# **Question:** Why do we care about *population vs. sample*?

---

.pull-left[

.center[**Population**]

]

.pull-right[

.center[**Population relationship**]

$$ y_i = 2.53 + 0.57 x_i + u_i $$

$$ y_i = \beta_0 + \beta_1 x_i + u_i $$

]

---

.pull-left[

.center[**Sample 1:** 30 random individuals]

]

.pull-right[

.center[

**Population relationship**
 
`$y_i = 2.53 + 0.57 x_i + u_i$`

**Sample relationship**
 
`$\hat{y}_i = 2.36 + 0.61 x_i$`

]

---
count: false

.pull-left[

.center[**Sample 2:** 30 random individuals]

]

.pull-right[

.center[

**Population relationship**
 
`$y_i = 2.53 + 0.57 x_i + u_i$`

**Sample relationship**
 
`$\hat{y}_i = 2.79 + 0.56 x_i$`

]

]
---
count: false

.pull-left[

.center[**Sample 3:** 30 random individuals]

]

.pull-right[

.center[

**Population relationship**
 
`$y_i = 2.53 + 0.57 x_i + u_i$`

**Sample relationship**
 
`$\hat{y}_i = 3.21 + 0.45 x_i$`

]

---
layout: false
class: clear, middle

Let's repeat this **10,000 times**.

(This exercise is called a (Monte Carlo) simulation.)

---
count: false

---
# Population *vs.* sample

**Question:** Why do we care about *population vs. sample*?

.pull-left[
<img src="data:image/png;base64,#recap1_files/figure-html/simulation_scatter2-1.png" style="display: block; margin: auto;" />
]

.pull-right[

- On **average**, our regression lines match the population line very nicely.

- However, **individual lines** (samples) can really miss the mark.

- Differences between individual samples and the population lead to **uncertainty** for the econometrician.

]

---
layout: false

# Population *vs.* sample

**Question:** Why do we care about *population vs. sample*?

**Answer:** Uncertainty matters.

.pull-left[

* Every random sample of data is different.

* Our (OLS) estimators are computed from those samples of data.

* If there is sampling variation, there is variation in our estimates.

]

.pull-right[

* OLS inference depends on certain assumptions.

* If violated, our estimates will be biased or imprecise.

* Or both. 😧
]

---
# Linear regression

## The estimator

We can estimate a regression line in `R` (`lm(y ~ x, my_data)`) and `stata` (`reg y x`). But where do these estimates come from?

A few slides back:

> $$ \hat{y}_i = \hat{\beta}_0 + \hat{\beta}_1 x_i $$
> which gives us the *best-fit* line through our dataset.

But what do we mean by "best-fit line"?

---
layout: false

# Being the "best"

**Question:** What do we mean by *best-fit line*?

**Answers:**

- In general (econometrics), *best-fit line* means the line that minimizes the sum of squared errors (SSE):

.center[

`$\text{SSE} = \sum_{i = 1}^{n} e_i^2\quad$` where `$\quad e_i = y_i - \hat{y}_i$`

]

- Ordinary **least squares** (**OLS**) minimizes the sum of the squared errors.
- Based upon a set of (mostly palatable) assumptions, OLS
  - Is unbiased (and consistent)
  - Is the *best* (minimum variance) linear unbiased estimator (BLUE)

---
layout: true
# OLS *vs.* other lines/estimators

---

Let's consider the dataset we previously generated.

---
count: false

For any line `$\left(\hat{y} = \hat{\beta}_0 + \hat{\beta}_1 x\right)$`

---
count: false

For any line `$\left(\hat{y} = \hat{\beta}_0 + \hat{\beta}_1 x\right)$`, we can calculate errors: `$e_i = y_i - \hat{y}_i$`

---
count: false

For any line `$\left(\hat{y} = \hat{\beta}_0 + \hat{\beta}_1 x\right)$`, we can calculate errors: `$e_i = y_i - \hat{y}_i$`

---
count: false

For any line `$\left(\hat{y} = \hat{\beta}_0 + \hat{\beta}_1 x\right)$`, we can calculate errors: `$e_i = y_i - \hat{y}_i$`

---
count: false

SSE squares the errors `$\left(\sum e_i^2\right)$`: bigger errors get bigger penalties.

---
count: false

The OLS estimate is the combination of `$\hat{\beta}_0$` and `$\hat{\beta}_1$` that minimize SSE.

---
layout: false
class: middle

```r
ScPoApps::launchApp("reg_simple")
```

---
layout: true
# OLS

## Formally

---

In simple linear regression, the OLS estimator comes from choosing the `$\hat{\beta}_0$` and `$\hat{\beta}_1$` that minimize the sum of squared errors (SSE), _i.e._,

$$ \min_{\hat{\beta}_0,\, \hat{\beta}_1} \text{SSE} $$

but we already know `$\text{SSE} = \sum_i e_i^2$`. Now use the definitions of `$e_i$` and `$\hat{y}$`.

$$
`\begin{aligned}
  e_i^2 &= \left( y_i - \hat{y}_i \right)^2 = \left( y_i - \hat{\beta}_0 - \hat{\beta}_1 x_i \right)^2 \\
  &= y_i^2 - 2 y_i \hat{\beta}_0 - 2 y_i \hat{\beta}_1 x_i + \hat{\beta}_0^2 + 2 \hat{\beta}_0 \hat{\beta}_1 x_i + \hat{\beta}_1^2 x_i^2
\end{aligned}`
$$

**Recall:** Minimizing a multivariate function requires (**1**) first derivatives equal zero (the *1.super[st]-order conditions*) and (**2**) second-order conditions (concavity).

---
layout: false

# OLS

## Interactively

```r
ScPoApps::launchApp("SSR_cone")
```

---
# OLS

## Interactively

We skipped the maths.

We now have the OLS estimators for the slope

$$ \hat{\beta}_1 = \dfrac{\sum_i (x_i - \overline{x})(y_i - \overline{y})}{\sum_i (x_i - \overline{x})^2} $$

and the intercept

$$ \hat{\beta}_0 = \overline{y} - \hat{\beta}_1 \overline{x} $$

Remember that *those* two formulae are amongst the very few ones from the intro course that you should know by heart! ❤️

We now turn to the assumptions and (implied) properties of OLS.

---
layout: true
# OLS: Assumptions and properties

---

**Question:** What properties might we care about for an estimator?

**Tangent:** Let's review statistical properties first.

---

**Refresher:** Density functions

Recall that we use **probability density functions** (PDFs) to describe the probability a **continuous random variable** takes on a range of values. (The total area = 1.)

These PDFs characterize probability distributions, and the most common/famous/popular distributions get names (_e.g._, normal, *t*, Gamma).

Here is the definition of a *PDF* `$f_X$`for a *continuous* RV `$X$`:

$$
\Pr[a \leq X \leq b] \equiv \int_a^b f_X (x) dx
$$

---

**Refresher:** Density functions

The probability a standard normal random variable takes on a value between -2 and 0: `$\mathop{\text{P}}\left(-2 \leq X \leq 0\right) = 0.48$`

---

**Refresher:** Density functions

The probability a standard normal random variable takes on a value between -1.96 and 1.96: `$\mathop{\text{P}}\left(-1.96 \leq X \leq 1.96\right) = 0.95$`

---

**Refresher:** Density functions

The probability a standard normal random variable takes on a value beyond 2: `$\mathop{\text{P}}\left(X > 2\right) = 0.023$`

---

Imagine we are trying to estimate an unknown parameter `$\beta$`, and we know the distributions of three competing estimators. Which one would we want? How would we decide?

---

**Question:** What properties might we care about for an estimator?

**Answer one: Bias.**

On average (after *many* samples), does the estimator tend toward the correct value?

**More formally:** Does the mean of estimator's distribution equal the parameter it estimates?

$$ \mathop{\text{Bias}}_\beta \left( \hat{\beta} \right) = \mathop{\boldsymbol{E}}\left[ \hat{\beta} \right] - \beta $$

---

**Answer one: Bias.**

.pull-left[

**Unbiased estimator:** `$\mathop{\boldsymbol{E}}\left[ \hat{\beta} \right] = \beta$`

]

.pull-right[

**Biased estimator:** `$\mathop{\boldsymbol{E}}\left[ \hat{\beta} \right] \neq \beta$`

]

---

**Answer two: Variance.**

The central tendencies (means) of competing distributions are not the only things that matter. We also care about the **variance** of an estimator.

$$ \mathop{\text{Var}} \left( \hat{\beta} \right) = \mathop{\boldsymbol{E}}\left[ \left( \hat{\beta} - \mathop{\boldsymbol{E}}\left[ \hat{\beta} \right] \right)^2 \right] $$

Lower variance estimators mean we get estimates closer to the mean in each sample.

---
count: false

**Answer two: Variance.**

---

**Answer one: Bias.**

**Answer two: Variance.**

**Subtlety:** The bias-variance tradeoff.

Should we be willing to take a bit of bias to reduce the variance?

In econometrics, we generally stick with unbiased (or consistent) estimators. But other disciplines (especially computer science) think a bit more about this tradeoff.

---
layout: false

# The bias-variance tradeoff.

---
# OLS: Assumptions and properties

## Properties

As you might have guessed by now,

- OLS is **unbiased**.
- OLS has the **minimum variance** of all unbiased linear estimators.

---
# OLS: Assumptions and properties

## Properties

But... these (very nice) properties depend upon a set of assumptions:

1. The population relationship is linear in parameters with an additive disturbance.

2. Our `$X$` variable is **exogenous**, _i.e._, `$\mathop{\boldsymbol{E}}\left[ u |X \right] = 0$`.

3. The `$X$` variable has variation. And if there are multiple explanatory variables, they are not perfectly collinear.

4. The population disturbances `$u_i$` are independently and identically distributed as normal random variables with mean zero `$\left( \mathop{\boldsymbol{E}}\left[ u \right] = 0 \right)$` and variance `$\sigma^2$` (_i.e._,  `$\mathop{\boldsymbol{E}}\left[ u^2 \right] = \sigma^2$`). Independently distributed and mean zero jointly imply `$\mathop{\boldsymbol{E}}\left[ u_i u_j \right] = 0$` for any `$i\neq j$`.

---
# OLS: Assumptions and properties

## Assumptions

Different assumptions guarantee different properties:

- Assumptions (1), (2), and (3) make OLS unbiased.
- Assumption (4) gives us an unbiased estimator for the variance of our OLS estimator.

We will discuss solutions to **violations of these assumptions**. See also our discussion [in the book](https://scpoecon.github.io/ScPoEconometrics/std-errors.html#class-reg)

- Non-linear relationships in our parameters/disturbances (or misspecification).
- Disturbances that are not identically distributed and/or not independent.
- Violations of exogeneity (especially omitted-variable bias).

---
# OLS: Assumptions and properties

## Conditional expectation

For many applications, our most important assumption is **exogeneity**, _i.e._,
$$
`\begin{align}
  \mathop{E}\left[ u | X \right] = 0
\end{align}`
$$
but what does it actually mean?

One way to think about this definition:

> For *any* value of `$X$`, the mean of the residuals must be zero.

- _E.g._, `$\mathop{E}\left[ u | X=1 \right]=0$` *and* `$\mathop{E}\left[ u | X=100 \right]=0$`

- _E.g._, `$\mathop{E}\left[ u | X_2=\text{Female} \right]=0$` *and* `$\mathop{E}\left[ u | X_2=\text{Male} \right]=0$`

- Notice: `$\mathop{E}\left[ u | X \right]=0$` is more restrictive than `$\mathop{E}\left[ u \right]=0$`
---
layout: false
class: clear, middle

Graphically...
---
exclude: true

---
class: clear

Valid exogeneity, _i.e._, `$\mathop{E}\left[ u | X \right] = 0$`

<img src="data:image/png;base64,#recap1_files/figure-html/ex_good_exog-1.svg" style="display: block; margin: auto;" />
---
class: clear

Invalid exogeneity, _i.e._, `$\mathop{E}\left[ u | X \right] \neq 0$`

---
layout: false
class: title-slide-final, middle
background-image: url(data:image/png;base64,#../../img/logo/ScPo-econ.png)
background-size: 250px
background-position: 9% 19%

# END

| | |
| :--------------------------------------------------------------------------------------------------------- | :-------------------------------- |
| <a href="mailto:bluebery.planterose@sciencespo.fr">.ScPored[] | bluebery.planterose@sciencespo.fr |
| <a href="https://github.com/ScPoEcon/Advanced-Metrics-slides">.ScPored[] | Original Slides from Florian Oswald |
| <a href="https://scpoecon.github.io/ScPoEconometrics">.ScPored[] | Book |
| <a href="http://twitter.com/ScPoEcon">.ScPored[] | @ScPoEcon |
| <a href="http://github.com/ScPoEcon">.ScPored[] | @ScPoEcon |