ScPoEconometrics: Advanced

.title[
# ScPoEconometrics: Advanced
]
.subtitle[
## Instrumental Variables - Applications
]
.author[
### Bluebery Planterose
]
.date[
### SciencesPo Paris </br> 2023-03-07
]

---

---

# Status

## What Did we Do Last Time?

* We learned about John Snow's grand experiment in London 1850.

* We used his story to motivate the IV estimator.

* You took a quiz about some IV aspects.

]

## Today

* We'll look at further IV applications.

* We introduce an extension called *Two Stage Least Squares*.

* We will use `R` to compute the estimates.

* Finally we'll talk about *weak* instruments.

]

---

# Back to school!

---
layout: true

---

# Returns To Schooling

* [Jacob Mincer](https://en.wikipedia.org/wiki/Jacob_Mincer) was interested in this important question.

* Here's his model:

$$
\log Y_i = \alpha + \rho S_i + \beta_1 X_i + \beta_2 X_i^2 + e_i
$$
]

.pull-right[
<img src="data:image/png;base64,#05-IV2_files/figure-html/mincer-1.svg" style="display: block; margin: auto;" />
]

---

# Returns To Schooling

* He found an estimate for `$\rho$` of about 0.11,

* 11% earnings advantage for each additional year of education

* Look at the DAG. Is that a good model? Well, why would it not be? 
]

.pull-right[
<img src="data:image/png;base64,#05-IV2_files/figure-html/mincer2-1.svg" style="display: block; margin: auto;" />
]

---

# Ability Bias

* Is all else equal, after controlling for those?

* Given `$X$`, 
    * Can we find differently diligent workers out there? 
    * Can we find differently able workers?
    * Do family connections of workers vary?
]

* That's an issue, because for OLS consistency we require the orthogonality assumption 
`$$E[e_i | S_i, X_i] \neq 0$$`

* Let's introduce **ability** `$A_i$` explicitly.
]

---

# Mincer with Unobserved Ability

* Of course we can't tell them apart.

* So we defined a new unobservable factor 
`$$u_i = e_i + A_i$$`

]

<img src="data:image/png;base64,#05-IV2_files/figure-html/mincer3-1.svg" style="display: block; margin: auto;" />
]

---

# Mincer with Unobserved Ability

.pull-left[
* In terms of an equation:
`$$\log Y_i = \alpha + \rho S_i + \beta_1 X_i + \beta_2 X_i^2 + \underbrace{u_i}_{A_i + e_i}$$`

* Sometimes, this does not matter, and the OLS bias is small.

* But sometimes it does and we get it totally wrong! Example.

]

<img src="data:image/png;base64,#05-IV2_files/figure-html/mincer4-1.svg" style="display: block; margin: auto;" />
]

---

# Angrist and Krueger (1991): Birthdate is as good as Random

* Idea: 
    1. construct an IV that encodes *birth date of student*.
    1. Child born just after cutoff date will start school later!
    
* Suppose all children who reach the age of 6 by 31st of december 2021 are required to enroll in the first grade of school in september 2021.
]

.pull-right[
* If born in September 2015 (i.e. 6 years prior), will be 5 years and 3/4 by the time they start school.

* If born on the 1st of January 2016 will be 6 and 3/4 years when *they* enter school in september 2022.

* However, people can drop out of school legally on their 16-th birthday!

* So, out of people who drop out, some got more schooling than others.

* AK91 construct IV *quarter of birth* dummy: affects schooling, but not related to `$A$`!
]

---

# AK91 IV setup

* In particular: whether born in 4-th quarter or not.

]

<img src="data:image/png;base64,#05-IV2_files/figure-html/ak-mod-1.svg" style="display: block; margin: auto;" />
]

---

# AK91 Estimation: Two Stage Least Squares (2SLS)

AK91 allow us to introduce a widely used variation of our simple IV estimator: **2SLS**

1. We estimate a **first stage model** which uses only exogenous variables (like `$z$`) to explain our endgenous regressor `$s$`.

2. We then use the first stage model to *predict* values of `$s$` in what is called the **second stage** or the **reduced form** model. Performing this procedure is supposed to take out any impact of `$A$` in the correlation we observe in our data between `$s$` and `$y$`.

`\begin{align}
\text{1. Stage: }s_i &= \alpha_0 + \alpha_1 z_i + \eta_i \\ 
\text{2. Stage: }y_i &= \beta_0 + \beta_1 \hat{s}_i + u_i 
\end{align}`

**Conditions:**

1. Relevance of the IV: `$\alpha_1 \neq 0$`
1. Independence (IV assignment as good as random): `$E[\eta | z] = 0$`
1. Exogeneity (our exclusion restriction): `$E[u | z] = 0$`

---
layout: false
class: separator, middle

# Let's do Angrist and Krueger (1991)!

---
layout: true

---

# Data on birth quarter and wages

Let's load the data and look at a quick summary

```r
data("ak91", package = "masteringmetrics")
# from the modelsummary package
datasummary_skim(data.frame(ak91),histogram = TRUE)
```

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  <tr>
   <th style="text-align:left;">   </th>
   <th style="text-align:right;"> Unique (#) </th>
   <th style="text-align:right;"> Missing (%) </th>
   <th style="text-align:right;"> Mean </th>
   <th style="text-align:right;"> SD </th>
   <th style="text-align:right;"> Min </th>
   <th style="text-align:right;"> Median </th>
   <th style="text-align:right;"> Max </th>
   <th style="text-align:right;">    </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> lnw </td>
   <td style="text-align:right;"> 26732 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 5.9 </td>
   <td style="text-align:right;"> 0.7 </td>
   <td style="text-align:right;"> −2.3 </td>
   <td style="text-align:right;"> 6.0 </td>
   <td style="text-align:right;"> 10.5 </td>
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   <td style="text-align:left;"> s </td>
   <td style="text-align:right;"> 21 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 12.8 </td>
   <td style="text-align:right;"> 3.3 </td>
   <td style="text-align:right;"> 0.0 </td>
   <td style="text-align:right;"> 12.0 </td>
   <td style="text-align:right;"> 20.0 </td>
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   <td style="text-align:left;"> yob </td>
   <td style="text-align:right;"> 10 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 1934.6 </td>
   <td style="text-align:right;"> 2.9 </td>
   <td style="text-align:right;"> 1930.0 </td>
   <td style="text-align:right;"> 1935.0 </td>
   <td style="text-align:right;"> 1939.0 </td>
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   <td style="text-align:left;"> qob </td>
   <td style="text-align:right;"> 4 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 2.5 </td>
   <td style="text-align:right;"> 1.1 </td>
   <td style="text-align:right;"> 1.0 </td>
   <td style="text-align:right;"> 3.0 </td>
   <td style="text-align:right;"> 4.0 </td>
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   <td style="text-align:left;"> sob </td>
   <td style="text-align:right;"> 51 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 30.7 </td>
   <td style="text-align:right;"> 14.2 </td>
   <td style="text-align:right;"> 1.0 </td>
   <td style="text-align:right;"> 34.0 </td>
   <td style="text-align:right;"> 56.0 </td>
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---

# AK91 Data Transformations

* We want to create the `q4` dummy which is `TRUE` if you are born in the 4th quarter.

* create `factor` versions of quarter and year of birth.

```r
ak91 <- mutate(ak91,
               qob_fct = factor(qob),
               q4 = as.integer(qob == "4"),
               yob_fct = factor(yob))
# get mean wage by year/quarter
ak91_age <- ak91 %>%
  group_by(qob, yob) %>%
  summarise(lnw = mean(lnw), s = mean(s)) %>%
  mutate(q4 = (qob == 4))
```

---

# AK91 Figure 1: First Stage!

Let's reproduce AK91's first figure now on education as a function of quarter of birth!

```r
ggplot(ak91_age, aes(x = yob + (qob - 1) / 4, y = s )) +
  geom_line() + 
  geom_label(mapping = aes(label = qob, color = q4)) +
  guides(label = FALSE, color = FALSE) +
  scale_x_continuous("Year of birth", breaks = 1930:1940) +
  scale_y_continuous("Years of Education", breaks = seq(12.2, 13.2, by = 0.2),
                     limits = c(12.2, 13.2)) +
  theme_bw()
```

---

# AK91 Figure 1: First Stage!

1. The 4-th quarters **did** get more education in most years!

1. There is a general trend.

]

.right-wide[
<img src="data:image/png;base64,#05-IV2_files/figure-html/ak91-dummy-1.svg" style="display: block; margin: auto;" />
]

---

# AK91 Figure 2: Impact of IV on outcome

What about earnings for those groups?

```r
ggplot(ak91_age, aes(x = yob + (qob - 1) / 4, y = lnw)) +
  geom_line() +
  geom_label(mapping = aes(label = qob, color = q4)) +
  scale_x_continuous("Year of birth", breaks = 1930:1940) +
  scale_y_continuous("Log weekly wages") +
  guides(label = FALSE, color = FALSE) +  
  theme_bw()
```

---

# AK91 Figure 2: Impact of IV on outcome

1. In general, weekly wages seem to decline somewhat over time.

]

<img src="data:image/png;base64,#05-IV2_files/figure-html/ak91-wage-1.svg" style="display: block; margin: auto;" />
]

---

# Running IV estimation in `R`

* Will use the [`iv_robust`](https://declaredesign.org/r/estimatr/reference/iv_robust.html) function from the `estimatr` package.

* *Robust*? Computes standard errors which are correcting for heteroskedasticity. [Details here.](https://declaredesign.org/r/estimatr/articles/mathematical-notes.html)
]

```r
library(estimatr)
# create a list of models
mod <- list()

# standard (biased!) OLS
mod$ols <- lm(lnw ~ s, data = ak91)

# IV: born in q4 is TRUE?
# doing IV manually in 2 stages.
mod[["1. stage"]] <- lm(s ~ q4, data = ak91)
ak91$shat         <- predict(mod[["1. stage"]])  
mod[["2. stage"]] <- lm(lnw ~ shat, data = ak91)

# run 2SLS
# doing IV all in one go
# notice the formula!
# formula = y ~ x | z
mod$`2SLS`  <- iv_robust(lnw ~ s | q4,
                         data = ak91,
                         diagnostics = TRUE)
```
]

---
count: false

# Running IV estimation in `R`

* Will use the [`iv_robust`](https://declaredesign.org/r/estimatr/reference/iv_robust.html) function from the `estimatr` package.

* *Robust*? Computes standard errors which are correcting for heteroskedasticity. [Details here.](https://declaredesign.org/r/estimatr/articles/mathematical-notes.html)

* Notice the `predict` to get `$\hat{s}$`.
]

```r
library(estimatr)
# create a list of models
mod <- list()

# standard (biased!) OLS
mod$ols <- lm(lnw ~ s, data = ak91)

# IV: born in q4 is TRUE?
# doing IV manually in 2 stages.
mod[["1. stage"]] <- lm(s ~ q4, data = ak91)
*ak91$shat         <- predict(mod[["1. stage"]])
mod[["2. stage"]] <- lm(lnw ~ shat, data = ak91)

---

# AK91 Results Table

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<th style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 1pt 0pt 1pt 0pt;    padding: 3.5pt 6pt 3.5pt 6pt; font-weight: normal;"> </th><th style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 1pt 0pt 1pt 0pt;    padding: 3.5pt 6pt 3.5pt 6pt; font-weight: normal;">ols</th><th style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 1pt 0pt 1pt 0pt;    padding: 3.5pt 6pt 3.5pt 6pt; font-weight: normal;">1. stage</th><th style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 1pt 0pt 1pt 0pt;    padding: 3.5pt 6pt 3.5pt 6pt; font-weight: normal;">2. stage</th><th style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 1pt 0pt 1pt 0pt;    padding: 3.5pt 6pt 3.5pt 6pt; font-weight: normal;">2SLS</th></tr>
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<tr>
<td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 3.5pt 6pt 3.5pt 6pt; font-weight: normal;"></td><td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 3.5pt 6pt 3.5pt 6pt; font-weight: normal;">(0.004)</td><td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 3.5pt 6pt 3.5pt 6pt; font-weight: normal;">(0.007)</td><td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 3.5pt 6pt 3.5pt 6pt; font-weight: normal;">(0.381)</td><td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 3.5pt 6pt 3.5pt 6pt; font-weight: normal;">(0.358)</td></tr>
<tr>
<td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 3.5pt 6pt 3.5pt 6pt; font-weight: normal;">s</td><td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 3.5pt 6pt 3.5pt 6pt; font-weight: normal;">0.071***</td><td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 3.5pt 6pt 3.5pt 6pt; font-weight: normal;"></td><td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 3.5pt 6pt 3.5pt 6pt; font-weight: normal;"></td><td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 3.5pt 6pt 3.5pt 6pt; font-weight: normal;">0.074**</td></tr>
<tr>
<td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 3.5pt 6pt 3.5pt 6pt; font-weight: normal;"></td><td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 3.5pt 6pt 3.5pt 6pt; font-weight: normal;">(0.000)</td><td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 3.5pt 6pt 3.5pt 6pt; font-weight: normal;"></td><td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 3.5pt 6pt 3.5pt 6pt; font-weight: normal;"></td><td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 3.5pt 6pt 3.5pt 6pt; font-weight: normal;">(0.028)</td></tr>
<tr>
<td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 3.5pt 6pt 3.5pt 6pt; font-weight: normal;">q4</td><td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 3.5pt 6pt 3.5pt 6pt; font-weight: normal;"></td><td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 3.5pt 6pt 3.5pt 6pt; font-weight: normal;">0.092***</td><td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 3.5pt 6pt 3.5pt 6pt; font-weight: normal;"></td><td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 3.5pt 6pt 3.5pt 6pt; font-weight: normal;"></td></tr>
<tr>
<td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 3.5pt 6pt 3.5pt 6pt; font-weight: normal;"></td><td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 3.5pt 6pt 3.5pt 6pt; font-weight: normal;"></td><td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 3.5pt 6pt 3.5pt 6pt; font-weight: normal;">(0.013)</td><td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 3.5pt 6pt 3.5pt 6pt; font-weight: normal;"></td><td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 3.5pt 6pt 3.5pt 6pt; font-weight: normal;"></td></tr>
<tr>
<td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 3.5pt 6pt 3.5pt 6pt; font-weight: normal;">shat</td><td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 3.5pt 6pt 3.5pt 6pt; font-weight: normal;"></td><td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 3.5pt 6pt 3.5pt 6pt; font-weight: normal;"></td><td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 3.5pt 6pt 3.5pt 6pt; font-weight: normal;">0.074*</td><td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 3.5pt 6pt 3.5pt 6pt; font-weight: normal;"></td></tr>
<tr>
<td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0.4pt 0pt;    padding: 3.5pt 6pt 3.5pt 6pt; font-weight: normal;"></td><td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0.4pt 0pt;    padding: 3.5pt 6pt 3.5pt 6pt; font-weight: normal;"></td><td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0.4pt 0pt;    padding: 3.5pt 6pt 3.5pt 6pt; font-weight: normal;"></td><td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0.4pt 0pt;    padding: 3.5pt 6pt 3.5pt 6pt; font-weight: normal;">(0.030)</td><td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0.4pt 0pt;    padding: 3.5pt 6pt 3.5pt 6pt; font-weight: normal;"></td></tr>
<tr>
<td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0.4pt 0pt 0pt 0pt;    padding: 3.5pt 6pt 3.5pt 6pt; font-weight: normal;">R2</td><td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0.4pt 0pt 0pt 0pt;    padding: 3.5pt 6pt 3.5pt 6pt; font-weight: normal;">0.117</td><td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0.4pt 0pt 0pt 0pt;    padding: 3.5pt 6pt 3.5pt 6pt; font-weight: normal;">0.000</td><td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0.4pt 0pt 0pt 0pt;    padding: 3.5pt 6pt 3.5pt 6pt; font-weight: normal;">0.000</td><td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0.4pt 0pt 0pt 0pt;    padding: 3.5pt 6pt 3.5pt 6pt; font-weight: normal;">0.117</td></tr>
<tr>
<td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 3.5pt 6pt 3.5pt 6pt; font-weight: normal;">RMSE</td><td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 3.5pt 6pt 3.5pt 6pt; font-weight: normal;">0.64</td><td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 3.5pt 6pt 3.5pt 6pt; font-weight: normal;">3.28</td><td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 3.5pt 6pt 3.5pt 6pt; font-weight: normal;">0.68</td><td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 3.5pt 6pt 3.5pt 6pt; font-weight: normal;">0.64</td></tr>
<tr>
<td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0.8pt 0pt;    padding: 3.5pt 6pt 3.5pt 6pt; font-weight: normal;">1. Stage F:</td><td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0.8pt 0pt;    padding: 3.5pt 6pt 3.5pt 6pt; font-weight: normal;"></td><td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0.8pt 0pt;    padding: 3.5pt 6pt 3.5pt 6pt; font-weight: normal;"></td><td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0.8pt 0pt;    padding: 3.5pt 6pt 3.5pt 6pt; font-weight: normal;"></td><td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0.8pt 0pt;    padding: 3.5pt 6pt 3.5pt 6pt; font-weight: normal;">48.990</td></tr>
<tr>
<td colspan="5" style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0.8pt 0pt 1pt 0pt;    padding: 3.5pt 6pt 3.5pt 6pt; font-weight: normal;">+ p < 0.1, * p < 0.05, ** p < 0.01, *** p < 0.001</td></tr>
</table>

]

1. First Stage: IV `q4` is statistically significant, but small effect: born in q4 has 0.092 years of educ. `$R^2$` is 0%! But F-stat is large. 😅

1. Second stage has same point estimate as `2SLS` but different std error (2. stage one is wrong)

]

---

# Remember the F-Statistic?

* We encountered this before: it's useful to test restricted vs unrestricted models against each other.

* Here, we are interested whether our instruments are *jointly* significant. Of course, with only one IV, that's not more informative than the t-stat of that IV.

* This F-Stat compares the predictive power of the first stage with and without the IVs. If they have very similar predictive power, the F-stat will be low, and we will not be able to reject the H0 that our IVs are **jointly insignificant** in the first stage model. 😞

---

# Additional Control Variables

* We saw a clear time trend in education earlier.

* There are also business-cycle fluctuations in earnings

* We should somehow control for different time periods.

* Also, we can use more than one IV! Here is how:

---

# Additional Control Variables

```r
# we keep adding to our `mod` list:
mod$ols_yr  <- update(mod$ols, . ~ . + yob_fct)  #  previous OLS model
# add exogenous vars on both sides of the `|` !
mod[["2SLS_yr"]] <- estimatr::iv_robust(lnw ~ s  + yob_fct | q4 + yob_fct, data = ak91, diagnostics = TRUE )  
# use all quarters as IVs
mod[["2SLS_all"]] <- estimatr::iv_robust(lnw ~ s  + yob_fct | qob_fct + yob_fct, data = ak91, diagnostics = TRUE  )
```

<table class="huxtable" style="border-collapse: collapse; border: 0px; margin-bottom: 2em; margin-top: 2em; ; margin-left: auto; margin-right: auto;  " id="tab:unnamed-chunk-7">
<col><col><col><col><col><col><tr>
<th style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 1pt 0pt 1pt 0pt;    padding: 4pt 6pt 4pt 6pt; font-weight: bold;"> </th><th style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 1pt 0pt 1pt 0pt;    padding: 4pt 6pt 4pt 6pt; font-weight: bold;">ols</th><th style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 1pt 0pt 1pt 0pt;    padding: 4pt 6pt 4pt 6pt; font-weight: bold;">2SLS</th><th style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 1pt 0pt 1pt 0pt;    padding: 4pt 6pt 4pt 6pt; font-weight: bold;">ols_yr</th><th style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 1pt 0pt 1pt 0pt;    padding: 4pt 6pt 4pt 6pt; font-weight: bold;">2SLS_yr</th><th style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 1pt 0pt 1pt 0pt;    padding: 4pt 6pt 4pt 6pt; font-weight: bold;">2SLS_all</th></tr>
<tr>
<td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 1pt 0pt 0pt 0pt;    padding: 4pt 6pt 4pt 6pt; font-weight: normal;">(Intercept)</td><td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 1pt 0pt 0pt 0pt;    padding: 4pt 6pt 4pt 6pt; font-weight: normal;">4.995</td><td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 1pt 0pt 0pt 0pt;    padding: 4pt 6pt 4pt 6pt; font-weight: normal;">4.955</td><td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 1pt 0pt 0pt 0pt;    padding: 4pt 6pt 4pt 6pt; font-weight: normal;">5.017</td><td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 1pt 0pt 0pt 0pt;    padding: 4pt 6pt 4pt 6pt; font-weight: normal;">4.966</td><td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 1pt 0pt 0pt 0pt;    padding: 4pt 6pt 4pt 6pt; font-weight: normal;">4.592</td></tr>
<tr>
<td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 4pt 6pt 4pt 6pt; font-weight: normal;"></td><td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 4pt 6pt 4pt 6pt; font-weight: normal;">(0.004)</td><td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 4pt 6pt 4pt 6pt; font-weight: normal;">(0.358)</td><td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 4pt 6pt 4pt 6pt; font-weight: normal;">(0.005)</td><td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 4pt 6pt 4pt 6pt; font-weight: normal;">(0.354)</td><td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 4pt 6pt 4pt 6pt; font-weight: normal;">(0.251)</td></tr>
<tr>
<td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 4pt 6pt 4pt 6pt; font-weight: normal;">s</td><td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 4pt 6pt 4pt 6pt; font-weight: normal;">0.071</td><td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 4pt 6pt 4pt 6pt; font-weight: normal;">0.074</td><td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 4pt 6pt 4pt 6pt; font-weight: normal;">0.071</td><td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 4pt 6pt 4pt 6pt; font-weight: normal;">0.075</td><td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 4pt 6pt 4pt 6pt; font-weight: normal;">0.105</td></tr>
<tr>
<td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 1pt 0pt;    padding: 4pt 6pt 4pt 6pt; font-weight: normal;"></td><td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 1pt 0pt;    padding: 4pt 6pt 4pt 6pt; font-weight: normal;">(0.000)</td><td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 1pt 0pt;    padding: 4pt 6pt 4pt 6pt; font-weight: normal;">(0.028)</td><td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 1pt 0pt;    padding: 4pt 6pt 4pt 6pt; font-weight: normal;">(0.000)</td><td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 1pt 0pt;    padding: 4pt 6pt 4pt 6pt; font-weight: normal;">(0.028)</td><td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 1pt 0pt;    padding: 4pt 6pt 4pt 6pt; font-weight: normal;">(0.020)</td></tr>
<tr>
<td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 1pt 0pt 0.4pt 0pt;    padding: 4pt 6pt 4pt 6pt; font-weight: normal;">R2</td><td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 1pt 0pt 0.4pt 0pt;    padding: 4pt 6pt 4pt 6pt; font-weight: normal;">0.117</td><td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 1pt 0pt 0.4pt 0pt;    padding: 4pt 6pt 4pt 6pt; font-weight: normal;">0.117</td><td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 1pt 0pt 0.4pt 0pt;    padding: 4pt 6pt 4pt 6pt; font-weight: normal;">0.118</td><td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 1pt 0pt 0.4pt 0pt;    padding: 4pt 6pt 4pt 6pt; font-weight: normal;">0.117</td><td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 1pt 0pt 0.4pt 0pt;    padding: 4pt 6pt 4pt 6pt; font-weight: normal;">0.091</td></tr>
<tr>
<td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0.4pt 0pt 0pt 0pt;    padding: 4pt 6pt 4pt 6pt; font-weight: normal;">RMSE</td><td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0.4pt 0pt 0pt 0pt;    padding: 4pt 6pt 4pt 6pt; font-weight: normal;">0.64</td><td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0.4pt 0pt 0pt 0pt;    padding: 4pt 6pt 4pt 6pt; font-weight: normal;">0.64</td><td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0.4pt 0pt 0pt 0pt;    padding: 4pt 6pt 4pt 6pt; font-weight: normal;">0.64</td><td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0.4pt 0pt 0pt 0pt;    padding: 4pt 6pt 4pt 6pt; font-weight: normal;">0.64</td><td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0.4pt 0pt 0pt 0pt;    padding: 4pt 6pt 4pt 6pt; font-weight: normal;">0.65</td></tr>
<tr>
<td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 4pt 6pt 4pt 6pt; font-weight: normal;">1. Stage F:</td><td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 4pt 6pt 4pt 6pt; font-weight: normal;"></td><td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 4pt 6pt 4pt 6pt; font-weight: normal;">48.990</td><td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 4pt 6pt 4pt 6pt; font-weight: normal;"></td><td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 4pt 6pt 4pt 6pt; font-weight: normal;">47.731</td><td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 4pt 6pt 4pt 6pt; font-weight: normal;">32.323</td></tr>
<tr>
<td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 4pt 6pt 4pt 6pt; font-weight: normal;">Instruments</td><td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 4pt 6pt 4pt 6pt; font-weight: normal;">none</td><td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 4pt 6pt 4pt 6pt; font-weight: normal;">Q4</td><td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 4pt 6pt 4pt 6pt; font-weight: normal;">none</td><td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 4pt 6pt 4pt 6pt; font-weight: normal;">Q4</td><td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 4pt 6pt 4pt 6pt; font-weight: normal;">All Quarters</td></tr>
<tr>
<td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 1pt 0pt;    padding: 4pt 6pt 4pt 6pt; font-weight: normal;">Year of birth</td><td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 1pt 0pt;    padding: 4pt 6pt 4pt 6pt; font-weight: normal;">no</td><td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 1pt 0pt;    padding: 4pt 6pt 4pt 6pt; font-weight: normal;">no</td><td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 1pt 0pt;    padding: 4pt 6pt 4pt 6pt; font-weight: normal;">yes</td><td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 1pt 0pt;    padding: 4pt 6pt 4pt 6pt; font-weight: normal;">yes</td><td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 1pt 0pt;    padding: 4pt 6pt 4pt 6pt; font-weight: normal;">yes</td></tr>
</table>

---

# Additional Control Variables

.left-wide[
<table class="huxtable" style="border-collapse: collapse; border: 0px; margin-bottom: 2em; margin-top: 2em; ; margin-left: auto; margin-right: auto;  " id="tab:unnamed-chunk-8">
<col><col><col><col><col><col><tr>
<th style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 1pt 0pt 1pt 0pt;    padding: 4pt 6pt 4pt 6pt; font-weight: bold;"> </th><th style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 1pt 0pt 1pt 0pt;    padding: 4pt 6pt 4pt 6pt; font-weight: bold;">ols</th><th style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 1pt 0pt 1pt 0pt;    padding: 4pt 6pt 4pt 6pt; font-weight: bold;">2SLS</th><th style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 1pt 0pt 1pt 0pt;    padding: 4pt 6pt 4pt 6pt; font-weight: bold;">ols_yr</th><th style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 1pt 0pt 1pt 0pt;    padding: 4pt 6pt 4pt 6pt; font-weight: bold;">2SLS_yr</th><th style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 1pt 0pt 1pt 0pt;    padding: 4pt 6pt 4pt 6pt; font-weight: bold;">2SLS_all</th></tr>
<tr>
<td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 1pt 0pt 0pt 0pt;    padding: 4pt 6pt 4pt 6pt; font-weight: normal;">(Intercept)</td><td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 1pt 0pt 0pt 0pt;    padding: 4pt 6pt 4pt 6pt; font-weight: normal;">4.995</td><td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 1pt 0pt 0pt 0pt;    padding: 4pt 6pt 4pt 6pt; font-weight: normal;">4.955</td><td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 1pt 0pt 0pt 0pt;    padding: 4pt 6pt 4pt 6pt; font-weight: normal;">5.017</td><td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 1pt 0pt 0pt 0pt;    padding: 4pt 6pt 4pt 6pt; font-weight: normal;">4.966</td><td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 1pt 0pt 0pt 0pt;    padding: 4pt 6pt 4pt 6pt; font-weight: normal;">4.592</td></tr>
<tr>
<td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 4pt 6pt 4pt 6pt; font-weight: normal;"></td><td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 4pt 6pt 4pt 6pt; font-weight: normal;">(0.004)</td><td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 4pt 6pt 4pt 6pt; font-weight: normal;">(0.358)</td><td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 4pt 6pt 4pt 6pt; font-weight: normal;">(0.005)</td><td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 4pt 6pt 4pt 6pt; font-weight: normal;">(0.354)</td><td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 4pt 6pt 4pt 6pt; font-weight: normal;">(0.251)</td></tr>
<tr>
<td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 4pt 6pt 4pt 6pt; font-weight: normal;">s</td><td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 4pt 6pt 4pt 6pt; font-weight: normal;">0.071</td><td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 4pt 6pt 4pt 6pt; font-weight: normal;">0.074</td><td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 4pt 6pt 4pt 6pt; font-weight: normal;">0.071</td><td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 4pt 6pt 4pt 6pt; font-weight: normal;">0.075</td><td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 4pt 6pt 4pt 6pt; font-weight: normal;">0.105</td></tr>
<tr>
<td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 1pt 0pt;    padding: 4pt 6pt 4pt 6pt; font-weight: normal;"></td><td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 1pt 0pt;    padding: 4pt 6pt 4pt 6pt; font-weight: normal;">(0.000)</td><td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 1pt 0pt;    padding: 4pt 6pt 4pt 6pt; font-weight: normal;">(0.028)</td><td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 1pt 0pt;    padding: 4pt 6pt 4pt 6pt; font-weight: normal;">(0.000)</td><td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 1pt 0pt;    padding: 4pt 6pt 4pt 6pt; font-weight: normal;">(0.028)</td><td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 1pt 0pt;    padding: 4pt 6pt 4pt 6pt; font-weight: normal;">(0.020)</td></tr>
<tr>
<td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 1pt 0pt 0.4pt 0pt;    padding: 4pt 6pt 4pt 6pt; font-weight: normal;">R2</td><td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 1pt 0pt 0.4pt 0pt;    padding: 4pt 6pt 4pt 6pt; font-weight: normal;">0.117</td><td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 1pt 0pt 0.4pt 0pt;    padding: 4pt 6pt 4pt 6pt; font-weight: normal;">0.117</td><td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 1pt 0pt 0.4pt 0pt;    padding: 4pt 6pt 4pt 6pt; font-weight: normal;">0.118</td><td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 1pt 0pt 0.4pt 0pt;    padding: 4pt 6pt 4pt 6pt; font-weight: normal;">0.117</td><td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 1pt 0pt 0.4pt 0pt;    padding: 4pt 6pt 4pt 6pt; font-weight: normal;">0.091</td></tr>
<tr>
<td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0.4pt 0pt 0pt 0pt;    padding: 4pt 6pt 4pt 6pt; font-weight: normal;">RMSE</td><td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0.4pt 0pt 0pt 0pt;    padding: 4pt 6pt 4pt 6pt; font-weight: normal;">0.64</td><td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0.4pt 0pt 0pt 0pt;    padding: 4pt 6pt 4pt 6pt; font-weight: normal;">0.64</td><td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0.4pt 0pt 0pt 0pt;    padding: 4pt 6pt 4pt 6pt; font-weight: normal;">0.64</td><td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0.4pt 0pt 0pt 0pt;    padding: 4pt 6pt 4pt 6pt; font-weight: normal;">0.64</td><td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0.4pt 0pt 0pt 0pt;    padding: 4pt 6pt 4pt 6pt; font-weight: normal;">0.65</td></tr>
<tr>
<td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 4pt 6pt 4pt 6pt; font-weight: normal;">1. Stage F:</td><td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 4pt 6pt 4pt 6pt; font-weight: normal;"></td><td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 4pt 6pt 4pt 6pt; font-weight: normal;">48.990</td><td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 4pt 6pt 4pt 6pt; font-weight: normal;"></td><td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 4pt 6pt 4pt 6pt; font-weight: normal;">47.731</td><td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 4pt 6pt 4pt 6pt; font-weight: normal;">32.323</td></tr>
<tr>
<td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 4pt 6pt 4pt 6pt; font-weight: normal;">Instruments</td><td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 4pt 6pt 4pt 6pt; font-weight: normal;">none</td><td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 4pt 6pt 4pt 6pt; font-weight: normal;">Q4</td><td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 4pt 6pt 4pt 6pt; font-weight: normal;">none</td><td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 4pt 6pt 4pt 6pt; font-weight: normal;">Q4</td><td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 4pt 6pt 4pt 6pt; font-weight: normal;">All Quarters</td></tr>
<tr>
<td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 1pt 0pt;    padding: 4pt 6pt 4pt 6pt; font-weight: normal;">Year of birth</td><td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 1pt 0pt;    padding: 4pt 6pt 4pt 6pt; font-weight: normal;">no</td><td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 1pt 0pt;    padding: 4pt 6pt 4pt 6pt; font-weight: normal;">no</td><td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 1pt 0pt;    padding: 4pt 6pt 4pt 6pt; font-weight: normal;">yes</td><td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 1pt 0pt;    padding: 4pt 6pt 4pt 6pt; font-weight: normal;">yes</td><td style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 1pt 0pt;    padding: 4pt 6pt 4pt 6pt; font-weight: normal;">yes</td></tr>
</table>

]

* leaves OLS mostly unchanged
* slight increase in 2SLS estimate

**Using all quarters as IV**...

* Increases precision of 2SLS estimate a lot!
* Point estimate is 10.5% now!

]

---

# AK91: Taking Stock - The Quarter of Birth (QOB) IV

* This will produce consistent estimates if
    1. The IV predicts the endogenous regressor well.
    2. The IV is as good as random / independent of OVs.
    3. Can only impact outcome through schooling.
    
* How does the QOB perform along those lines?

]

1. Plot of first stage and high F-stat offer compelling evidence for **relevance**. ✅

2. Is QOB **independent** of, say, *maternal characteristics*? Birthdays are not really random - there are birth seasons for certain socioeconomic backgrounds. highest maternal schooling give birth in second quarter. (not in 4th! ✅)

3. Exclusion: What if the youngest kids (born in Q4!) are the disadvantaged ones early on, which has long-term negative impacts? That would mean `$E[u|z] \neq 0$`! Well, with QOB the youngest ones actually do better (more schooling and higher wage)! ✅
]

---

# Mechanics of IV

## Identification and Inference

---

# IV Identification

Let's go back to our simple linear model:

$$
y = \beta_0 + \beta_1 x + u
$$

where we fear that `$Cov(x,u) \neq 0$`, `$x$` is *endogenous*.

## Conditions for IV

1. **first stage** or **relevance**: `$Cov(z,x) \neq 0$`
2. **IV exogeneity**: `$Cov(z,u) = 0$`: the IV is exogenous in the outcome equation.

---

# Valid Model (A) vs Invalid Model (B) for IV `z`

---

# IV Identification

>## Conditions for IV

>1. **first stage** or **relevance**: `$Cov(z,x) \neq 0$`
>2. **IV exogeneity**: `$Cov(z,u) = 0$`: the IV is exogenous in the outcome equation.

]

* How does this *identify* `$\beta_1$`?

* (How can we express `$\beta_1$` in terms of population moments to pin it's value down?)

]

---

# IV Identification

`\begin{align}
Cov(z,y) &= Cov(z, \beta_0 + \beta_1 x + u) \\
         &= \beta_1 Cov(z,x) + Cov(z,u) 
\end{align}`

$$
Cov(z,y) = \beta_1 Cov(z,x) 
$$
]

.pull-right[
and under condition 1. (**relevance**), we have `$Cov(z,x)\neq0$`, so that we can divide the equation through to obtain

$$
\beta_1 =  \frac{Cov(z,y)}{Cov(z,x)}.
$$
* `$\beta_1$` is *identified* via population moments `$Cov(z,y)$` and `$Cov(z,x)$`.

* We can *estimate* those moments via their *sample analogs*
]

---

# IV Estimator

Just plugging in for the population moments:

`$$\hat{\beta}_1 = \frac{\sum_{i=1}^n (z_i - \bar{z})(y_i - \bar{y})}{\sum_{i=1}^n (z_i - \bar{z})(x_i - \bar{x})}$$`

* The intercept estimate is `$\hat{\beta}_0 = \bar{y} - \hat{\beta}_1 \bar{x}$`

* Given both assumptions 1. and 2. are satisfied, we say that *the IV estimator is consistent for `$\beta_1$`*. We write

$$
\text{plim}(\hat{\beta}_1) = \beta_1
$$

in words: the *probability limit* of `$\hat{\beta}_1$` is the true `$\beta_1$`.

* If this is true, we say that this estimator is **consistent**.

---

# IV Inference

Assuming `$E(u^2|z) = \sigma^2$` the variance of the IV slope estimator is

`$$Var(\hat{\beta}_{1,IV}) = \frac{\sigma^2}{n \sigma_x^2 \rho_{x,z}^2}$$`

* `$\sigma_x^2$` is the population variance of `$x$`,

* `$\sigma^2$` the one of `$u$`, and

* `$\rho_{x,z}$` is the population correlation between `$x$` and `$z$`.

You can see 2 important things here:

1. Without the term `$\rho_{x,z}^2$`, this is **like OLS variance**.
2. As sample size `$n$` increases, the **variance decreases**.

---

# IV Variance is Always Larger than OLS Variance

* Replace `$\rho_{x,z}^2$` with `$R_{x,z}^2$`, i.e. the R-squared of a regression of `$x$` on `$z$`:

`$$Var(\hat{\beta}_{1,IV}) = \frac{\sigma^2}{n \sigma_x^2 R_{x,z}^2}$$`

1. Given `$R_{x,z}^2 < 1$` in most real life situations, we have that `$Var(\hat{\beta}_{1,IV}) > Var(\hat{\beta}_{1,OLS})$` almost certainly.

1. The higher the correlation between `$z$` and `$x$`, the closer their `$R_{x,z}^2$` is to 1. With `$R_{x,z}^2 = 1$` we get back to the OLS variance. This is no surprise, because that implies that in fact `$z = x$`.

So, if you have a valid, exogenous regressor `$x$`, you should *not* perform IV estimation using `$z$` to obtain `$\hat{\beta}$`, since your variance will be unnecessarily large.

---

# Returns to Education for Married Women

Consider the following model for married women's wages:

$$
\log wage = \beta_0 + \beta_1 educ + u
$$
Let's run an OLS on this, and then compare it to an IV estimate using *father's education*. Keep in mind that this is a valid IV `$z$` if

1. *fatheduc* and *educ* are correlated
2. *fatheduc* and `$u$` are not correlated.

---

# Returns to Education for Married Women

```r
data(mroz,package = "wooldridge")
mods = list()
mods$OLS <- lm(lwage ~ educ, data = mroz)
mods[['First Stage']] <- lm(educ ~ fatheduc, data = subset(mroz, inlf == 1))
mods$IV  <- estimatr::iv_robust(lwage ~ educ | fatheduc, data = mroz)
```

---

# IV Standard Errors

---

# IV with a Weak Instrument

* IV is consistent under given assumptions.

* However, *even if* we have only very small `$Cor(z,u)$`, we can get wrong-footed

* Small corrleation between `$x$` and `$z$` can produce **inconsistent** estimates.

$$
\text{plim}(\hat{\beta}_{1,IV}) = \beta_1 + \frac{Cor(z,u)}{Cor(z,x)} \cdot \frac{\sigma_u}{\sigma_x}
$$
]

.pull-right[
* Take `$Cor(z,u)$` is very small,
* A **weak instrument** is one with only a small absolute value for `$Cor(z,x)$` 
* This will blow up this second term in the probability limit. 
* Even with a very big sample size `$n$`, our estimator would *not* converge to the true population parameter `$\beta_1$`, because we are using a weak instrument.
]

---

# Weak Stuff

To illustrate this point, let's assume we want to look at the impact of number of packs of cigarettes smoked per day by pregnant women (*packs*) on the birthweight of their child (*bwght*):

$$
\log(bwght) = \beta_0 + \beta_1 packs + u
$$

We are worried that smoking behavior is correlated with a range of other health-related variables which are in `$u$` and which could impact the birthweight of the child. So we look for an IV. Suppose we use the price of cigarettes (*cigprice*), assuming that the price of cigarettes is uncorrelated with factors in `$u$`. Let's run the first stage of *cigprice* on *packs* and then let's show the 2SLS estimates:

---

# Weak Stuff

```r
data(bwght, package = "wooldridge")
mods <- list()
mods[["First Stage"]] <- lm(packs ~ cigprice, data = bwght)
mods[["IV"]] <- estimatr::iv_robust(log(bwght) ~  packs | cigprice, data = bwght, diagnostics = TRUE)
```

---

# Weak Stuff

* The first columns shows: very weak first stage. *cigprice* has zero impact on packs it seems!

* `$R^2$` is zero.

* What is we use this IV nevertheless?
]

* in the second column: very large, positive(!) impact of packs smoked on birthweight. 🤔

* Huge Standard Error though.

* An `$R^2$` of -23?!

* F-stat of first stage: 0.121. Corresponds to a p-value of 0.728 : we **cannot** reject the H0 of an insignificant first stage here *at all*.

* So: **invalid** approach. ❌
]

---

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[<i class="fa fa-paper-plane fa-fw"></i>]               | bluebery.planterose@sciencespo.fr      |
| <a href="https://github.com/ScPoEcon/ScPoEconometrics-Slides">.ScPored[<i class="fa fa-link fa-fw"></i>] | Original Slides from Florian Oswald |
| <a href="https://scpoecon.github.io/ScPoEconometrics">.ScPored[<i class="fa fa-link fa-fw"></i>] | Book |
| <a href="http://twitter.com/ScPoEcon">.ScPored[<i class="fa fa-twitter fa-fw"></i>]                          | @ScPoEcon                         |
| <a href="http://github.com/ScPoEcon">.ScPored[<i class="fa fa-github fa-fw"></i>]                          | @ScPoEcon                       |