Econometrics

.title[
# Econometrics
]
.subtitle[
## Instrumental Variables - Applications
]
.author[
### Florian Oswald
]
.date[
### UniTo ESOMAS 2025-11-19
]

---

---

# Status

## What Did we on IV already?

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

* We used his story to motivate the IV estimator.

]

## 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.

]

---

# Prologue

## One cool `R` package a day, keeps the doctor away...

---
layout: false

# [`Rayshader`](https://www.rayshader.com)

* Can *raytrace* all kinds of elevation data.
* Can *also* make most `ggplot`s come to 3D-life!

---

# [`Rayshader`](https://www.rayshader.com)

* This is all it takes:

``` r
library(ggplot2)
library(rayshader)
#Data from Social Security administration
death = read_csv("https://www.tylermw.com/data/death.csv", skip = 1)
meltdeath = reshape2::melt(death, id.vars = "Year")

meltdeath$age = as.numeric(meltdeath$variable)

# make a ggplot
deathgg = ggplot(meltdeath) +
  geom_raster(aes(x=Year,y=age,fill=value)) +
  scale_x_continuous("Year",expand=c(0,0),breaks=seq(1900,2010,10)) +
  scale_y_continuous("Age",expand=c(0,0),breaks=seq(0,100,10),limits=c(0,100)) +
  scale_fill_viridis("Death\nProbability\nPer Year",
       trans = "log10",breaks=c(1,0.1,0.01,0.001,0.0001), 
       labels = c("1","1/10","1/100","1/1000","1/10000")) +
  ggtitle("Death Probability vs Age and Year for the USA") +
  labs(caption = "Data Source: US Dept. of Social Security")

# give it to rayshader
plot_gg(deathgg, multicore=TRUE,height=5,width=6,scale=500)
```

* Amazing, right? 🎉 🎊

---

# 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="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="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="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="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 of that year (2021).
]

* If born on 01/01/2016 they will be also and 3/4 years when *they* enter school (just one year later).

* 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 `$A$`!
]

---

# Birthdate Setup

Let's set up 2 children:

``` r
library(lubridate)
born1 = as.Date("2015-12-31")
born2 = as.Date("2016-01-01")
school1 = as.Date("2021-09-01")
school2 = as.Date("2022-09-01")
```

How much days of school if they drop out on their 16-th birthday?

---

# Birthdate Setup

Let's set up 2 children:

``` r
library(lubridate)
born1 = as.Date("2015-12-31")
born2 = as.Date("2016-01-01")
school1 = as.Date("2021-09-01")
school2 = as.Date("2022-09-01")
```

How much days of school if they drop out on their 16-th birthday?

``` r
dropout1 = born1 %m+% years(16)
dropout2 = born2 %m+% years(16)
(schooldays1 = dropout1 - school1)
```

```
## Time difference of 3773 days
```

``` r
(schooldays2 = dropout2 - school2)
```

```
## Time difference of 3409 days
```

---

# AK91 IV setup

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

]

<img src="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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---

# 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="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="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)

# 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)
```
]

---

# AK91 Results Table

.left-wide[
<table class="huxtable" data-quarto-disable-processing="true"  style="margin-left: auto; margin-right: auto;" id="tab:ms1">
<col><col><col><col><col><thead>
<tr>
<th class="huxtable-cell huxtable-header" style="border-style: solid solid solid solid; border-width: 1pt 0pt 1pt 0pt;    padding: 3.5pt 6pt 3.5pt 6pt;  font-weight: normal;"> </th><th class="huxtable-cell huxtable-header" style="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 class="huxtable-cell huxtable-header" style="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 class="huxtable-cell huxtable-header" style="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 class="huxtable-cell huxtable-header" style="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>
</thead>
<tbody>
<tr>
<td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 1pt 0pt 0pt 0pt;    padding: 3.5pt 6pt 3.5pt 6pt;">(Intercept)</td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 1pt 0pt 0pt 0pt;    padding: 3.5pt 6pt 3.5pt 6pt;">4.995***</td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 1pt 0pt 0pt 0pt;    padding: 3.5pt 6pt 3.5pt 6pt;">12.747***</td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 1pt 0pt 0pt 0pt;    padding: 3.5pt 6pt 3.5pt 6pt;">4.955***</td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 1pt 0pt 0pt 0pt;    padding: 3.5pt 6pt 3.5pt 6pt;">4.955***</td></tr>
<tr>
<td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 3.5pt 6pt 3.5pt 6pt;"></td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 3.5pt 6pt 3.5pt 6pt;">(0.004)</td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 3.5pt 6pt 3.5pt 6pt;">(0.007)</td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 3.5pt 6pt 3.5pt 6pt;">(0.381)</td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 3.5pt 6pt 3.5pt 6pt;">(0.358)</td></tr>
<tr>
<td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 3.5pt 6pt 3.5pt 6pt;">s</td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 3.5pt 6pt 3.5pt 6pt;">0.071***</td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 3.5pt 6pt 3.5pt 6pt;"></td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 3.5pt 6pt 3.5pt 6pt;"></td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 3.5pt 6pt 3.5pt 6pt;">0.074**</td></tr>
<tr>
<td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 3.5pt 6pt 3.5pt 6pt;"></td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 3.5pt 6pt 3.5pt 6pt;">(0.000)</td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 3.5pt 6pt 3.5pt 6pt;"></td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 3.5pt 6pt 3.5pt 6pt;"></td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 3.5pt 6pt 3.5pt 6pt;">(0.028)</td></tr>
<tr>
<td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 3.5pt 6pt 3.5pt 6pt;">q4</td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 3.5pt 6pt 3.5pt 6pt;"></td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 3.5pt 6pt 3.5pt 6pt;">0.092***</td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 3.5pt 6pt 3.5pt 6pt;"></td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 3.5pt 6pt 3.5pt 6pt;"></td></tr>
<tr>
<td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 3.5pt 6pt 3.5pt 6pt;"></td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 3.5pt 6pt 3.5pt 6pt;"></td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 3.5pt 6pt 3.5pt 6pt;">(0.013)</td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 3.5pt 6pt 3.5pt 6pt;"></td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 3.5pt 6pt 3.5pt 6pt;"></td></tr>
<tr>
<td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 3.5pt 6pt 3.5pt 6pt;">shat</td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 3.5pt 6pt 3.5pt 6pt;"></td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 3.5pt 6pt 3.5pt 6pt;"></td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 3.5pt 6pt 3.5pt 6pt;">0.074*</td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 3.5pt 6pt 3.5pt 6pt;"></td></tr>
<tr>
<td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 0.4pt 0pt;    padding: 3.5pt 6pt 3.5pt 6pt;"></td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 0.4pt 0pt;    padding: 3.5pt 6pt 3.5pt 6pt;"></td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 0.4pt 0pt;    padding: 3.5pt 6pt 3.5pt 6pt;"></td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 0.4pt 0pt;    padding: 3.5pt 6pt 3.5pt 6pt;">(0.030)</td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 0.4pt 0pt;    padding: 3.5pt 6pt 3.5pt 6pt;"></td></tr>
<tr>
<td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0.4pt 0pt 0pt 0pt;    padding: 3.5pt 6pt 3.5pt 6pt;">R2</td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0.4pt 0pt 0pt 0pt;    padding: 3.5pt 6pt 3.5pt 6pt;">0.117</td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0.4pt 0pt 0pt 0pt;    padding: 3.5pt 6pt 3.5pt 6pt;">0.000</td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0.4pt 0pt 0pt 0pt;    padding: 3.5pt 6pt 3.5pt 6pt;">0.000</td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0.4pt 0pt 0pt 0pt;    padding: 3.5pt 6pt 3.5pt 6pt;">0.117</td></tr>
<tr>
<td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 0.8pt 0pt;    padding: 3.5pt 6pt 3.5pt 6pt;">RMSE</td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 0.8pt 0pt;    padding: 3.5pt 6pt 3.5pt 6pt;">0.64</td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 0.8pt 0pt;    padding: 3.5pt 6pt 3.5pt 6pt;">3.28</td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 0.8pt 0pt;    padding: 3.5pt 6pt 3.5pt 6pt;">0.68</td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 0.8pt 0pt;    padding: 3.5pt 6pt 3.5pt 6pt;">0.64</td></tr>
<tr>
<td class="huxtable-cell" colspan="5" style="border-style: solid solid solid solid; border-width: 0.8pt 0pt 1pt 0pt;    padding: 3.5pt 6pt 3.5pt 6pt;">+ p < 0.1, * p < 0.05, ** p < 0.01, *** p < 0.001</td></tr>
</tbody>
</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" data-quarto-disable-processing="true"  style="margin-left: auto; margin-right: auto;">
<col><col><col><col><col><thead>
<tr>
<th class="huxtable-cell huxtable-header" style="border-style: solid solid solid solid; border-width: 1pt 0pt 1pt 0pt;    padding: 4pt 6pt 4pt 6pt;"> </th><th class="huxtable-cell huxtable-header" style="border-style: solid solid solid solid; border-width: 1pt 0pt 1pt 0pt;    padding: 4pt 6pt 4pt 6pt;">ols</th><th class="huxtable-cell huxtable-header" style="border-style: solid solid solid solid; border-width: 1pt 0pt 1pt 0pt;    padding: 4pt 6pt 4pt 6pt;">1. stage</th><th class="huxtable-cell huxtable-header" style="border-style: solid solid solid solid; border-width: 1pt 0pt 1pt 0pt;    padding: 4pt 6pt 4pt 6pt;">2. stage</th><th class="huxtable-cell huxtable-header" style="border-style: solid solid solid solid; border-width: 1pt 0pt 1pt 0pt;    padding: 4pt 6pt 4pt 6pt;">2SLS</th></tr>
</thead>
<tbody>
<tr>
<td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 1pt 0pt 0pt 0pt;    padding: 4pt 6pt 4pt 6pt;">(Intercept)</td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 1pt 0pt 0pt 0pt;    padding: 4pt 6pt 4pt 6pt;">4.995***</td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 1pt 0pt 0pt 0pt;    padding: 4pt 6pt 4pt 6pt;">12.747***</td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 1pt 0pt 0pt 0pt;    padding: 4pt 6pt 4pt 6pt;">4.955***</td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 1pt 0pt 0pt 0pt;    padding: 4pt 6pt 4pt 6pt;">4.955***</td></tr>
<tr>
<td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 4pt 6pt 4pt 6pt;"></td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 4pt 6pt 4pt 6pt;">(0.004)</td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 4pt 6pt 4pt 6pt;">(0.007)</td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 4pt 6pt 4pt 6pt;">(0.381)</td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 4pt 6pt 4pt 6pt;">(0.358)</td></tr>
<tr>
<td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 4pt 6pt 4pt 6pt;">s</td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 4pt 6pt 4pt 6pt;">0.071***</td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 4pt 6pt 4pt 6pt;"></td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 4pt 6pt 4pt 6pt;"></td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 4pt 6pt 4pt 6pt;">0.074**</td></tr>
<tr>
<td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 4pt 6pt 4pt 6pt;"></td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 4pt 6pt 4pt 6pt;">(0.000)</td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 4pt 6pt 4pt 6pt;"></td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 4pt 6pt 4pt 6pt;"></td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 4pt 6pt 4pt 6pt;">(0.028)</td></tr>
<tr>
<td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 0.4pt 0pt;    padding: 4pt 6pt 4pt 6pt;">q4</td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 0.4pt 0pt;    padding: 4pt 6pt 4pt 6pt;"></td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 0.4pt 0pt;    padding: 4pt 6pt 4pt 6pt;">0.092***</td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 0.4pt 0pt;    padding: 4pt 6pt 4pt 6pt;"></td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 0.4pt 0pt;    padding: 4pt 6pt 4pt 6pt;"></td></tr>
<tr>
<td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0.4pt 0pt 0pt 0pt;    padding: 4pt 6pt 4pt 6pt;"></td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0.4pt 0pt 0pt 0pt;    padding: 4pt 6pt 4pt 6pt;"></td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0.4pt 0pt 0pt 0pt;    padding: 4pt 6pt 4pt 6pt;">(0.013)</td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0.4pt 0pt 0pt 0pt;    padding: 4pt 6pt 4pt 6pt;"></td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0.4pt 0pt 0pt 0pt;    padding: 4pt 6pt 4pt 6pt;"></td></tr>
<tr>
<td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 4pt 6pt 4pt 6pt;">shat</td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 4pt 6pt 4pt 6pt;"></td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 4pt 6pt 4pt 6pt;"></td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 4pt 6pt 4pt 6pt;">0.074*</td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 4pt 6pt 4pt 6pt;"></td></tr>
<tr>
<td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 1pt 0pt;    padding: 4pt 6pt 4pt 6pt;"></td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 1pt 0pt;    padding: 4pt 6pt 4pt 6pt;"></td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 1pt 0pt;    padding: 4pt 6pt 4pt 6pt;"></td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 1pt 0pt;    padding: 4pt 6pt 4pt 6pt;">(0.030)</td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 1pt 0pt;    padding: 4pt 6pt 4pt 6pt;"></td></tr>
<tr>
<td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 1pt 0pt 0pt 0pt;    padding: 4pt 6pt 4pt 6pt;">R2</td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 1pt 0pt 0pt 0pt;    padding: 4pt 6pt 4pt 6pt;">0.117</td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 1pt 0pt 0pt 0pt;    padding: 4pt 6pt 4pt 6pt;">0.000</td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 1pt 0pt 0pt 0pt;    padding: 4pt 6pt 4pt 6pt;">0.000</td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 1pt 0pt 0pt 0pt;    padding: 4pt 6pt 4pt 6pt;">0.117</td></tr>
<tr>
<td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 0.8pt 0pt;    padding: 4pt 6pt 4pt 6pt;">RMSE</td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 0.8pt 0pt;    padding: 4pt 6pt 4pt 6pt;">0.64</td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 0.8pt 0pt;    padding: 4pt 6pt 4pt 6pt;">3.28</td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 0.8pt 0pt;    padding: 4pt 6pt 4pt 6pt;">0.68</td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 0.8pt 0pt;    padding: 4pt 6pt 4pt 6pt;">0.64</td></tr>
<tr>
<td class="huxtable-cell" colspan="5" style="border-style: solid solid solid solid; border-width: 0.8pt 0pt 1pt 0pt;    padding: 4pt 6pt 4pt 6pt;">+ p < 0.1, * p < 0.05, ** p < 0.01, *** p < 0.001</td></tr>
</tbody>
</table>

---

# Additional Control Variables

.left-wide[
<table class="huxtable" data-quarto-disable-processing="true"  style="margin-left: auto; margin-right: auto;">
<col><col><col><col><col><col><thead>
<tr>
<th class="huxtable-cell huxtable-header" style="border-style: solid solid solid solid; border-width: 1pt 0pt 1pt 0pt;    padding: 4pt 6pt 4pt 6pt;"> </th><th class="huxtable-cell huxtable-header" style="border-style: solid solid solid solid; border-width: 1pt 0pt 1pt 0pt;    padding: 4pt 6pt 4pt 6pt;">ols</th><th class="huxtable-cell huxtable-header" style="border-style: solid solid solid solid; border-width: 1pt 0pt 1pt 0pt;    padding: 4pt 6pt 4pt 6pt;">2SLS</th><th class="huxtable-cell huxtable-header" style="border-style: solid solid solid solid; border-width: 1pt 0pt 1pt 0pt;    padding: 4pt 6pt 4pt 6pt;">ols_yr</th><th class="huxtable-cell huxtable-header" style="border-style: solid solid solid solid; border-width: 1pt 0pt 1pt 0pt;    padding: 4pt 6pt 4pt 6pt;">2SLS_yr</th><th class="huxtable-cell huxtable-header" style="border-style: solid solid solid solid; border-width: 1pt 0pt 1pt 0pt;    padding: 4pt 6pt 4pt 6pt;">2SLS_all</th></tr>
</thead>
<tbody>
<tr>
<td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 1pt 0pt 0pt 0pt;    padding: 4pt 6pt 4pt 6pt;">(Intercept)</td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 1pt 0pt 0pt 0pt;    padding: 4pt 6pt 4pt 6pt;">5.00***</td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 1pt 0pt 0pt 0pt;    padding: 4pt 6pt 4pt 6pt;">4.96***</td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 1pt 0pt 0pt 0pt;    padding: 4pt 6pt 4pt 6pt;">5.02***</td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 1pt 0pt 0pt 0pt;    padding: 4pt 6pt 4pt 6pt;">4.97***</td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 1pt 0pt 0pt 0pt;    padding: 4pt 6pt 4pt 6pt;">4.59***</td></tr>
<tr>
<td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 4pt 6pt 4pt 6pt;"></td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 4pt 6pt 4pt 6pt;">(0.00)</td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 4pt 6pt 4pt 6pt;">(0.36)</td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 4pt 6pt 4pt 6pt;">(0.01)</td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 4pt 6pt 4pt 6pt;">(0.35)</td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 4pt 6pt 4pt 6pt;">(0.25)</td></tr>
<tr>
<td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 4pt 6pt 4pt 6pt;">s</td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 4pt 6pt 4pt 6pt;">0.07***</td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 4pt 6pt 4pt 6pt;">0.07**</td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 4pt 6pt 4pt 6pt;">0.07***</td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 4pt 6pt 4pt 6pt;">0.08**</td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 4pt 6pt 4pt 6pt;">0.11***</td></tr>
<tr>
<td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 1pt 0pt;    padding: 4pt 6pt 4pt 6pt;"></td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 1pt 0pt;    padding: 4pt 6pt 4pt 6pt;">(0.00)</td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 1pt 0pt;    padding: 4pt 6pt 4pt 6pt;">(0.03)</td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 1pt 0pt;    padding: 4pt 6pt 4pt 6pt;">(0.00)</td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 1pt 0pt;    padding: 4pt 6pt 4pt 6pt;">(0.03)</td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 1pt 0pt;    padding: 4pt 6pt 4pt 6pt;">(0.02)</td></tr>
<tr>
<td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 1pt 0pt 0.4pt 0pt;    padding: 4pt 6pt 4pt 6pt;">R2</td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 1pt 0pt 0.4pt 0pt;    padding: 4pt 6pt 4pt 6pt;">0.117</td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 1pt 0pt 0.4pt 0pt;    padding: 4pt 6pt 4pt 6pt;">0.117</td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 1pt 0pt 0.4pt 0pt;    padding: 4pt 6pt 4pt 6pt;">0.118</td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 1pt 0pt 0.4pt 0pt;    padding: 4pt 6pt 4pt 6pt;">0.117</td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 1pt 0pt 0.4pt 0pt;    padding: 4pt 6pt 4pt 6pt;">0.091</td></tr>
<tr>
<td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0.4pt 0pt 0pt 0pt;    padding: 4pt 6pt 4pt 6pt;">RMSE</td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0.4pt 0pt 0pt 0pt;    padding: 4pt 6pt 4pt 6pt;">0.64</td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0.4pt 0pt 0pt 0pt;    padding: 4pt 6pt 4pt 6pt;">0.64</td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0.4pt 0pt 0pt 0pt;    padding: 4pt 6pt 4pt 6pt;">0.64</td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0.4pt 0pt 0pt 0pt;    padding: 4pt 6pt 4pt 6pt;">0.64</td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0.4pt 0pt 0pt 0pt;    padding: 4pt 6pt 4pt 6pt;">0.65</td></tr>
<tr>
<td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 4pt 6pt 4pt 6pt;">Instruments</td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 4pt 6pt 4pt 6pt;">none</td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 4pt 6pt 4pt 6pt;">Q4</td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 4pt 6pt 4pt 6pt;">none</td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 4pt 6pt 4pt 6pt;">Q4</td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 4pt 6pt 4pt 6pt;">All Quarters</td></tr>
<tr>
<td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 0.8pt 0pt;    padding: 4pt 6pt 4pt 6pt;">Year of birth</td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 0.8pt 0pt;    padding: 4pt 6pt 4pt 6pt;">no</td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 0.8pt 0pt;    padding: 4pt 6pt 4pt 6pt;">no</td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 0.8pt 0pt;    padding: 4pt 6pt 4pt 6pt;">yes</td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 0.8pt 0pt;    padding: 4pt 6pt 4pt 6pt;">yes</td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 0.8pt 0pt;    padding: 4pt 6pt 4pt 6pt;">yes</td></tr>
<tr>
<td class="huxtable-cell" colspan="6" style="border-style: solid solid solid solid; border-width: 0.8pt 0pt 1pt 0pt;    padding: 4pt 6pt 4pt 6pt;">+ p < 0.1, * p < 0.05, ** p < 0.01, *** p < 0.001</td></tr>
</tbody>
</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)
```

<table class="huxtable" data-quarto-disable-processing="true"  style="margin-left: auto; margin-right: auto;">
<col><col><col><col><thead>
<tr>
<th class="huxtable-cell huxtable-header" style="border-style: solid solid solid solid; border-width: 1pt 0pt 1pt 0pt;    padding: 5pt 6pt 5pt 6pt;"> </th><th class="huxtable-cell huxtable-header" style="border-style: solid solid solid solid; border-width: 1pt 0pt 1pt 0pt;    padding: 5pt 6pt 5pt 6pt;">OLS</th><th class="huxtable-cell huxtable-header" style="border-style: solid solid solid solid; border-width: 1pt 0pt 1pt 0pt;    padding: 5pt 6pt 5pt 6pt;">First Stage</th><th class="huxtable-cell huxtable-header" style="border-style: solid solid solid solid; border-width: 1pt 0pt 1pt 0pt;    padding: 5pt 6pt 5pt 6pt;">IV</th></tr>
</thead>
<tbody>
<tr>
<td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 1pt 0pt 0pt 0pt;    padding: 5pt 6pt 5pt 6pt;">(Intercept)</td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 1pt 0pt 0pt 0pt;    padding: 5pt 6pt 5pt 6pt;">-0.185</td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 1pt 0pt 0pt 0pt;    padding: 5pt 6pt 5pt 6pt;">10.237</td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 1pt 0pt 0pt 0pt;    padding: 5pt 6pt 5pt 6pt;">0.441</td></tr>
<tr>
<td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 5pt 6pt 5pt 6pt;"></td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 5pt 6pt 5pt 6pt;">(0.185)</td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 5pt 6pt 5pt 6pt;">(0.276)</td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 5pt 6pt 5pt 6pt;">(0.467)</td></tr>
<tr>
<td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 5pt 6pt 5pt 6pt;">educ</td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 5pt 6pt 5pt 6pt;">0.109</td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 5pt 6pt 5pt 6pt;"></td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 5pt 6pt 5pt 6pt;">0.059</td></tr>
<tr>
<td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 5pt 6pt 5pt 6pt;"></td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 5pt 6pt 5pt 6pt;">(0.014)</td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 5pt 6pt 5pt 6pt;"></td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 5pt 6pt 5pt 6pt;">(0.037)</td></tr>
<tr>
<td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 5pt 6pt 5pt 6pt;">fatheduc</td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 5pt 6pt 5pt 6pt;"></td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 5pt 6pt 5pt 6pt;">0.269</td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 5pt 6pt 5pt 6pt;"></td></tr>
<tr>
<td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 0.4pt 0pt;    padding: 5pt 6pt 5pt 6pt;"></td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 0.4pt 0pt;    padding: 5pt 6pt 5pt 6pt;"></td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 0.4pt 0pt;    padding: 5pt 6pt 5pt 6pt;">(0.029)</td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 0.4pt 0pt;    padding: 5pt 6pt 5pt 6pt;"></td></tr>
<tr>
<td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0.4pt 0pt 0pt 0pt;    padding: 5pt 6pt 5pt 6pt;">Num.Obs.</td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0.4pt 0pt 0pt 0pt;    padding: 5pt 6pt 5pt 6pt;">428</td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0.4pt 0pt 0pt 0pt;    padding: 5pt 6pt 5pt 6pt;">428</td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0.4pt 0pt 0pt 0pt;    padding: 5pt 6pt 5pt 6pt;">428</td></tr>
<tr>
<td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 1pt 0pt;    padding: 5pt 6pt 5pt 6pt;">R2</td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 1pt 0pt;    padding: 5pt 6pt 5pt 6pt;">0.118</td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 1pt 0pt;    padding: 5pt 6pt 5pt 6pt;">0.173</td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 1pt 0pt;    padding: 5pt 6pt 5pt 6pt;">0.093</td></tr>
</tbody>
</table>

---

# 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)
```

<table class="huxtable" data-quarto-disable-processing="true"  style="margin-left: auto; margin-right: auto;">
<col><col><col><thead>
<tr>
<th class="huxtable-cell huxtable-header" style="border-style: solid solid solid solid; border-width: 1pt 0pt 1pt 0pt;    padding: 5pt 6pt 5pt 6pt;"> </th><th class="huxtable-cell huxtable-header" style="border-style: solid solid solid solid; border-width: 1pt 0pt 1pt 0pt;    padding: 5pt 6pt 5pt 6pt;">First Stage</th><th class="huxtable-cell huxtable-header" style="border-style: solid solid solid solid; border-width: 1pt 0pt 1pt 0pt;    padding: 5pt 6pt 5pt 6pt;">IV</th></tr>
</thead>
<tbody>
<tr>
<td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 1pt 0pt 0pt 0pt;    padding: 5pt 6pt 5pt 6pt;">(Intercept)</td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 1pt 0pt 0pt 0pt;    padding: 5pt 6pt 5pt 6pt;">0.067</td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 1pt 0pt 0pt 0pt;    padding: 5pt 6pt 5pt 6pt;">4.448</td></tr>
<tr>
<td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 5pt 6pt 5pt 6pt;"></td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 5pt 6pt 5pt 6pt;">(0.103)</td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 5pt 6pt 5pt 6pt;">(0.940)</td></tr>
<tr>
<td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 5pt 6pt 5pt 6pt;">cigprice</td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 5pt 6pt 5pt 6pt;">0.000</td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 5pt 6pt 5pt 6pt;"></td></tr>
<tr>
<td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 5pt 6pt 5pt 6pt;"></td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 5pt 6pt 5pt 6pt;">(0.001)</td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 5pt 6pt 5pt 6pt;"></td></tr>
<tr>
<td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 5pt 6pt 5pt 6pt;">packs</td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 5pt 6pt 5pt 6pt;"></td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 0pt 0pt;    padding: 5pt 6pt 5pt 6pt;">2.989</td></tr>
<tr>
<td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 0.4pt 0pt;    padding: 5pt 6pt 5pt 6pt;"></td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 0.4pt 0pt;    padding: 5pt 6pt 5pt 6pt;"></td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0pt 0pt 0.4pt 0pt;    padding: 5pt 6pt 5pt 6pt;">(8.996)</td></tr>
<tr>
<td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0.4pt 0pt 1pt 0pt;    padding: 5pt 6pt 5pt 6pt;">R2</td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0.4pt 0pt 1pt 0pt;    padding: 5pt 6pt 5pt 6pt;">0.000</td><td class="huxtable-cell" style="border-style: solid solid solid solid; border-width: 0.4pt 0pt 1pt 0pt;    padding: 5pt 6pt 5pt 6pt;">-23.230</td></tr>
</tbody>
</table>

---

# 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. ❌
]

---

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background-image: url(../img/logo/esomas.png)
background-size: 250px
background-position: 9% 19%

# END