---
title: "Instrumental Variables"
subtitle: "EC 425/525, Set 8"
author: "Edward Rubin"
date: "`r format(Sys.time(), '%d %B %Y')`"
output:
  xaringan::moon_reader:
    css: ['default', 'metropolis', 'metropolis-fonts', 'my-css.css']
    # self_contained: true
    nature:
      highlightStyle: github
      highlightLines: true
      countIncrementalSlides: false
---
class: inverse, middle

```{R, setup, include = F}
# devtools::install_github("dill/emoGG")
library(pacman)
p_load(
  broom, tidyverse,
  ggplot2, ggthemes, ggforce, ggridges,
  latex2exp, viridis, extrafont, gridExtra,
  kableExtra, snakecase, janitor,
  data.table, dplyr,
  lubridate, knitr,
  estimatr, here, magrittr
)
# Define pink color
red_pink <- "#e64173"
turquoise <- "#20B2AA"
orange <- "#FFA500"
red <- "#fb6107"
blue <- "#3b3b9a"
green <- "#8bb174"
grey_light <- "grey70"
grey_mid <- "grey50"
grey_dark <- "grey20"
purple <- "#6A5ACD"
slate <- "#314f4f"
# Dark slate grey: #314f4f
# Knitr options
opts_chunk$set(
  comment = "#>",
  fig.align = "center",
  fig.height = 7,
  fig.width = 10.5,
  warning = F,
  message = F
)
opts_chunk$set(dev = "svg")
options(device = function(file, width, height) {
  svg(tempfile(), width = width, height = height)
})
options(crayon.enabled = F)
options(knitr.table.format = "html")
# A blank theme for ggplot
theme_empty <- theme_bw() + theme(
  line = element_blank(),
  rect = element_blank(),
  strip.text = element_blank(),
  axis.text = element_blank(),
  plot.title = element_blank(),
  axis.title = element_blank(),
  plot.margin = structure(c(0, 0, -0.5, -1), unit = "lines", valid.unit = 3L, class = "unit"),
  legend.position = "none"
)
theme_simple <- theme_bw() + theme(
  line = element_blank(),
  panel.grid = element_blank(),
  rect = element_blank(),
  strip.text = element_blank(),
  axis.text.x = element_text(size = 18, family = "STIXGeneral"),
  axis.text.y = element_blank(),
  axis.ticks = element_blank(),
  plot.title = element_blank(),
  axis.title = element_blank(),
  # plot.margin = structure(c(0, 0, -1, -1), unit = "lines", valid.unit = 3L, class = "unit"),
  legend.position = "none"
)
theme_axes_math <- theme_void() + theme(
  text = element_text(family = "MathJax_Math"),
  axis.title = element_text(size = 22),
  axis.title.x = element_text(hjust = .95, margin = margin(0.15, 0, 0, 0, unit = "lines")),
  axis.title.y = element_text(vjust = .95, margin = margin(0, 0.15, 0, 0, unit = "lines")),
  axis.line = element_line(
    color = "grey70",
    size = 0.25,
    arrow = arrow(angle = 30, length = unit(0.15, "inches")
  )),
  plot.margin = structure(c(1, 0, 1, 0), unit = "lines", valid.unit = 3L, class = "unit"),
  legend.position = "none"
)
theme_axes_serif <- theme_void() + theme(
  text = element_text(family = "MathJax_Main"),
  axis.title = element_text(size = 22),
  axis.title.x = element_text(hjust = .95, margin = margin(0.15, 0, 0, 0, unit = "lines")),
  axis.title.y = element_text(vjust = .95, margin = margin(0, 0.15, 0, 0, unit = "lines")),
  axis.line = element_line(
    color = "grey70",
    size = 0.25,
    arrow = arrow(angle = 30, length = unit(0.15, "inches")
  )),
  plot.margin = structure(c(1, 0, 1, 0), unit = "lines", valid.unit = 3L, class = "unit"),
  legend.position = "none"
)
theme_axes <- theme_void() + theme(
  text = element_text(family = "Fira Sans Book"),
  axis.title = element_text(size = 18),
  axis.title.x = element_text(hjust = .95, margin = margin(0.15, 0, 0, 0, unit = "lines")),
  axis.title.y = element_text(vjust = .95, margin = margin(0, 0.15, 0, 0, unit = "lines")),
  axis.line = element_line(
    color = grey_light,
    size = 0.25,
    arrow = arrow(angle = 30, length = unit(0.15, "inches")
  )),
  plot.margin = structure(c(1, 0, 1, 0), unit = "lines", valid.unit = 3L, class = "unit"),
  legend.position = "none"
)
theme_set(theme_gray(base_size = 20))
# Column names for regression results
reg_columns <- c("Term", "Est.", "S.E.", "t stat.", "p-Value")
# Function for formatting p values
format_pvi <- function(pv) {
  return(ifelse(
    pv < 0.0001,
    "<0.0001",
    round(pv, 4) %>% format(scientific = F)
  ))
}
format_pv <- function(pvs) lapply(X = pvs, FUN = format_pvi) %>% unlist()
# Tidy regression results table
tidy_table <- function(x, terms, highlight_row = 1, highlight_color = "black", highlight_bold = T, digits = c(NA, 3, 3, 2, 5), title = NULL) {
  x %>%
    tidy() %>%
    select(1:5) %>%
    mutate(
      term = terms,
      p.value = p.value %>% format_pv()
    ) %>%
    kable(
      col.names = reg_columns,
      escape = F,
      digits = digits,
      caption = title
    ) %>%
    kable_styling(font_size = 20) %>%
    row_spec(1:nrow(tidy(x)), background = "white") %>%
    row_spec(highlight_row, bold = highlight_bold, color = highlight_color)
}
```

$$
\begin{align}
  \def\ci{\perp\mkern-10mu\perp}
\end{align}
$$


# Prologue

---
name: schedule

# Schedule

## Last time

Matching and propensity-score methods
- Conditional independence
- Overlap

## Today

Instrumental variables (and two-stage least squares)

## Upcoming

- Assignment due Sunday
- Proposal due Wednesday 5/22
- Midterm?
---
layout: true
# Research designs

---
class: inverse, middle
---
name: designs
## Selection on observables and/or unobservables

We've been focusing on .hi-slate[*selection-on-observables* designs], _i.e._,
$$
\begin{align}
  \left(\text{Y}_{0i},\, \text{Y}_{1i}\right) \ci \text{D}_{i}|\text{X}_{i}
\end{align}
$$
for .hi-slate[observable] variables $\text{X}_{i}$.

--

.hi-pink[*Selection-on-unobservable* designs] replace this assumption with two new (but related) assumptions

1. $\left(\text{Y}_{0i},\, \text{Y}_{1i}\right) \perp \text{Z}_{i}$

2. $\mathop{\text{Cov}} \left( \text{Z}_{i},\, \text{D}_{i} \right) \neq 0$

---
## Selection on observables and/or unobservables

Our main goal in causal-inference minded (applied) econometrics boils down to isolating .b["good" variation] in $\text{D}_{i}$ (exogenous/as-good-as-random) from .b["bad" variation] (the part of $\text{D}_{i}$ correlated with $\text{Y}_{0i}$ and $\text{Y}_{1i}$).

--

(We want to avoid selection bias.)

--

- .hi-slate[Selection-on-observables designs] assume that we can control for all *bad variation* (selection) in $\text{D}_{i}$ through a known (observed) $\text{X}_{i}$.

--

- .hi-pink[Selection-on-unobservables designs] assume that we can extract part of the *good variation* in $\text{D}_{i}$ (generally using some $\text{Z}_{i}$) and then use this *good* part of $\text{D}_{i}$ to estimate the effect of $\text{D}_{i}$ on $\text{Y}_{i}$.
--
 We throw away the *bad variation* in $\text{D}_{i}$ (it's bad).
---
## Which route?

So set of research designs is more palatable?

--

1. There are plenty of bad applications of both sets.<br>.purple[Violated assumptions, bad controls, *etc.*]

--

1. .hi-slate[Selection on observables] assumes we know .it[everything] about selection into treatment—we can identify .it[all] of the good (or bad) variation in $\text{D}_{i}$.
--
<br>.purple[Tough in non-experimental settings. Difficult to validate in practice.]

--

1. .hi-pink[Selection on unobservables] assumes we can isolate .it[some] good/clean variation in $\text{D}_{i}$, which we then use to estimate the effect of $\text{D}_{i}$ on $\text{Y}_{i}$.
--
<br>.purple[Seems more plausible. Possible to validate. May be underpowered.]
---
layout: true
# Instrumental variables

---
name: intro
## Introduction

.attn[Instrumental variables] (IV).super[.pink[†]] is the canonical selection-on-unobservables design—isolating *good variation* in $\text{D}_{i}$ via some magical .pink[instrument] $\color{#e64173}{\text{Z}_{i}}$.

.footnote[.pink[†] For the moment, we're lumping together IV and two-stage least squares (2SLS) together—as many people do—even though they are technically different.]

--

Consider some model (structural equation)
$$
\begin{align}
  \text{Y}_{i} = \beta_0 + \beta_1 \text{D}_{i} + \varepsilon_i \tag{1}
\end{align}
$$
To guarantee consistent OLS estimates for $\beta_1$, want $\mathop{\text{Cov}} \left( \text{D}_{i},\,\varepsilon_i \right)=0$.
<br> In general, this is a heroic assumption.

--

.note[Alternative:] Estimate $\beta_1$ via instrumental variables.

---
name: defined
## Definition

For our model
$$
\begin{align}
  \text{Y}_{i} = \beta_0 + \beta_1 \text{D}_{i} + \varepsilon_i \tag{1}
\end{align}
$$

A valid .attn[instrument] is a variable $\color{#e64173}{\text{Z}_{i}}$ such that

1. $\mathop{\text{Cov}} \left( \color{#e64173}{\text{Z}_{i}},\, \text{D}_{i} \right) \neq 0$
--
<br>our .pink[instrument] correlates with treatment
--
 (so we can keep part of $\text{D}_{i}$)

--

2. $\mathop{\text{Cov}} \left( \color{#e64173}{\text{Z}_{i}},\, \varepsilon_i \right) = 0$
--
<br>our .pink[instrument] is uncorrelated with other (non- $\!\!\text{D}_{i}$) determinants of $\text{Y}_{i}$
--
, _i.e._, $\color{#e64173}{\text{Z}_{i}}$ is excludable from equation $(1)$.
--
 .attn[(exclusion restriction)]
---
name: example
## Example

Back to the returns to a college degree,
$$
\begin{align}
  \text{Income}_i = \beta_0 + \beta_1 \text{Grad}_i + \varepsilon_i
\end{align}
$$
OLS is likely biased.

--

What if that state conducts a (random) .hi-pink[lottery] for scholarships?

--

Let $\color{#e64173}{\text{Lottery}_i}$ denote an indicator for whether $i$ won a lottery scholarship..super[.pink[†]]

.footnote[.pink[†] We'll have to focus on families who were eligible/who applied.]

--

1. $\mathop{\text{Cov}} \left( \color{#e64173}{\text{Lottery}_i},\, \text{Grad}_i \right)\neq 0$ $\left( >0 \right)$ if scholarships increase grad. rates.

--

2. $\mathop{\text{Cov}} \left(\color{#e64173}{\text{Lottery}_i},\, \varepsilon_i\right) = 0$ since the lottery is randomized.
---
layout: true
# Instrument variables
## The IV estimator

The IV estimator for our model
$$
\begin{align}
  \text{Y}_{i} = \beta_0 + \beta_1 \text{D}_{i} + \varepsilon_i \tag{1}
\end{align}
$$
with (valid) instrument $\color{#e64173}{\text{Z}_{i}}$ is
$$
\begin{align}
   \hat{\beta}_\text{IV} = \left(\text{Z}'\text{D}\right)^{-1} \left( \text{Z}'\text{Y}\right)
\end{align}
$$

---
name: iv-estimator

--

If you have no covariates, then
$$
\begin{align}
  \hat{\beta}_\text{IV} = \dfrac{\mathop{\text{Cov}} \left( \color{#e64173}{\text{Z}_{i}},\, \text{Y}_{i}\right)}{\mathop{\text{Cov}} \left( \color{#e64173}{\text{Z}_{i}},\, \text{D}_{i} \right)}
\end{align}
$$
---

If you have additional (exogenous) covariates $\text{X}_i$, then
$$
\begin{align}
  \text{Z} &= \begin{bmatrix}\color{#e64173}{\text{Z}_{i}} & \text{X}_{i}\end{bmatrix}
  \\[0.5em]
  \text{D} &= \begin{bmatrix}\color{#e64173}{\text{D}_{i}} & \text{X}_{i}\end{bmatrix}
\end{align}
$$
---
layout: true
# Instrumental variables

---
## Proof: Consistency

With a valid instrument $\text{Z}_{i}$, $\hat{\beta}_\text{IV}$ is a consistent estiamtor for $\beta_1$ in
$$
\begin{align}
  \text{Y}_{i} = \beta_0 + \beta_1 \text{X}_{i} + \varepsilon_i \tag{1}
\end{align}
$$

$\mathop{\text{plim}}\left( \hat{\beta}_{IV} \right)$

--
.pad-left[
$= \mathop{\text{plim}}\left( \left(\text{Z}'\text{D}\right)^{-1} \left( \text{Z}'\text{Y}\right) \right)$
]

--
.pad-left[
$= \mathop{\text{plim}}\left( \left(\text{Z}'\text{D}\right)^{-1} \left( \text{Z}'\text{D} \beta + \text{Z}'\varepsilon\right) \right)$
]

--
.pad-left[
$= \mathop{\text{plim}}\left( \left(\text{Z}'\text{D}\right)^{-1} \left( \text{Z}'\text{D}\right) \beta\right) + \mathop{\text{plim}}\left(\dfrac{1}{N} \text{Z}'\text{D}\right)^{-1} \mathop{\text{plim}}\left( \dfrac{1}{N} \text{Z}'\varepsilon\right)$
]

--
.pad-left[
$=\beta$  .pink[✔]
]
---
layout: true
# Two-stage least squares

---
class: inverse, middle
---
name: setup
## Setup

You'll commonly see IV implemented as a two-stage process known as<br>.attn[two-stage least squares] (2SLS).

--

.attn[First stage] Estimate the effect of the instrument $\color{#e64173}{\text{Z}_{i}}$ on our endogenous variable $\text{D}_{i}$ and (predetermined) covariates $\text{X}_{i}$. Save $\color{#6A5ACD}{\widehat{\text{D}}_{i}}$.

$$
\begin{align}
  \text{D}_{i} = \gamma_1 \color{#e64173}{\text{Z}_{i}} + \gamma_2 \text{X}_{i} + u_i
\end{align}
$$

--

.attn[Second stage] Estimate model we wanted—but only using the variation in $\text{D}_{i}$ that correlates with $\color{#e64173}{\text{Z}_{i}}$, _i.e._, $\color{#6A5ACD}{\widehat{\text{D}}_{i}}$.

$$
\begin{align}
  \text{Y}_{i} = \beta_1 \color{#6A5ACD}{\widehat{\text{D}}_{i}} + \beta_2 \text{X}_{i} + \varepsilon_i
\end{align}
$$

.note[Note] The controls $\text{X}_{i}$ must match in the first and second stages.

---
## IV estimation

This two-step procedure, with a valid instrument, produces an estimator $\hat{\beta}_1$ that is consistent for $\beta_1$.

$$
\begin{align}
  \hat{\beta}_\text{2SLS} &= \left( \text{D}' \text{P}_{\text{Z}} \text{D} \right)^{-1} \left( \text{D}' \text{P}_{\text{Z}} \text{Y} \right)
  \\[0.3em]
  \text{P}_{\text{Z}} &= \text{Z} \left( \text{Z}'\text{Z} \right)^{-1} \text{Z}'
\end{align}
$$

where $\text{D}$ is a matrix of our treatment and predetermined covariates $\left( \text{X}_{i} \right)$ and $Z$ is a matrix of our instrument and our predetermined covariates.

---
## IV estimation

Important notes

- The controls $\left( \text{X}_{i} \right)$ must match in the first and second stages.

- If you have exactly .hi-slate[one instrument] and exactly .hi-slate[one endogenous variable], then 2SLS and IV are identical.

- Your second-stage standard errors are not correct.

---
name: reduced-form
## The reduced form

In addition to the regressions within the two stages of 2SLS
1. $\text{D}_{i} = \gamma_1 \color{#e64173}{\text{Z}_{i}} + \gamma_2 \text{X}_{i} + u_i$
2. $\text{Y}_{i} = \beta_1 \color{#6A5ACD}{\widehat{\text{D}}_{i}} + \beta_2 \text{X}_{i} + \varepsilon_i$

there is a third important and related regression: the reduced form.

--

The .attn[reduced form] regresses the outcome $\text{Y}_{i}$ (LHS of the second stage) on our instrument $\color{#e64173}{\text{Z}_{i}}$ and covariates $\text{X}_{i}$ (RHS of the first stage).
$$
\begin{align}
   \text{Y}_{i} = \pi_1 \color{#e64173}{\text{Z}_{i}} + \pi_2 \text{X}_{i} + u_i
\end{align}
$$
--
Thus, the reduced form provides a consistent estimate of the causal effect of our instrument on the outcome.
---
## The reduced form, continued

While the reduced form estimates the causal effect of the instrument on our outcome, we're often actually interested in the effect of *treatment* $\left( \text{D}_{i} \right)$.

--

That said, the reduced form is still incredibly helpful/important:

- Clarifies your source of identifying variation.
--

- Does not suffer from *weak instruments* problems.
--

- Only requires $\mathop{\text{Cov}} \left( \text{Z}_{i},\, \varepsilon_i \right) = 0$.
--

- Offers insights into your estimates
--

$$
\begin{align}
  \widehat{\beta}_{1}^\text{2SLS} = \dfrac{\widehat{\pi}_{1}}{\widehat{\gamma}_{1}}
\end{align}
$$
when you have exactly one instrument.

---
name: reduced-intuition
## The reduced form, intuition

This expression for the 2SLS (and IV) estimator can be very helpful.
$$
\begin{align}
  \widehat{\beta}_{1}^\text{2SLS} = \dfrac{\color{#6A5ACD}{\widehat{\pi}_{1}}}{\color{#20B2AA}{\widehat{\gamma}_{1}}} = \dfrac{\color{#6A5ACD}{\text{Reduced-form estimate}}}{\color{#20B2AA}{\text{First-stage estimate}}}
\end{align}
$$

--

What's the interpretation/intuition?

--

Back to our example: $\widehat{\beta}_1 =$ est. effect of college graduation on income.

--

$\color{#6A5ACD}{\widehat{\pi}_1}$ gives the estimated causal effect of the scholarship lottery on income
--
, but what share of lottery winners graduate? We need to rescale if $<$ 100%.

--

$\color{#20B2AA}{\widehat{\gamma}_1}$ estimates the effect of winning the scholarship lottery on graduation
--
—the share of winners who graduated due to winning.
--
 We can scale with $\color{#20B2AA}{\widehat{\gamma}_1}$!
---
name: reduced-example
## The reduced form, example

To see why this scaling makes sense, imagine that 50% of lottery winners graduate from college due to the lottery, _i.e._, $\color{#20B2AA}{\widehat{\gamma}_1 =}$ .turquoise[0.50]..super[.pink[†]]

.footnote[.pink[†] Imagine none of the applicants would have graduated otherwise]

--

Our reduced-form estimate of $\color{#6A5ACD}{\widehat{\pi}_1=}$ .purple[$5,000] says that lottery winners make $5,000 more than the control group, on average.

--

However, half of the winners did not graduate, so $\color{#6A5ACD}{\widehat{\pi}_1}$ "underestimates" the effect of college graduation by combining graduates by nongraduates.

--

Thus, we want to double $\color{#6A5ACD}{\widehat{\pi}_1}$, _i.e._, divide by $\color{#20B2AA}{\widehat{\gamma}_1}$:
$\color{#6A5ACD}{\widehat{\pi}_1}/\color{#20B2AA}{\widehat{\gamma}_1}$ = .turquoise[$5,000]/.purple[0.5] = $10,000.
---
name: reduced-derivation

.qa[Q] How do we get this magical expression? $\left( \widehat{\beta}_1^\text{IV} = \dfrac{\widehat{\pi}_1}{\widehat{\gamma}_1} \right)$

--

## Derivation

--

$\widehat{\beta}_1^\text{IV} = \left( \text{Z}'\text{D} \right)^{-1} \left( \text{Z}'\text{Y} \right)$

--

$\color{#ffffff}{\widehat{\beta}_1^\text{IV}} = \left( \widetilde{\text{Z}}'\widetilde{\text{D}} \right)^{-1} \left( \widetilde{\text{Z}}'\text{Y} \right)$   applying FWL to reduce $\text{D}$ and $\text{Z}$ to vectors.

--

$\color{#ffffff}{\widehat{\beta}_1^\text{IV}} = \dfrac{\mathop{\text{Cov}} \left( \widetilde{\text{Z}}_{i},\, \text{Y}_{i} \right)}{\mathop{\text{Cov}} \left( \widetilde{\text{Z}}_i,\, \widetilde{\text{D}}_{i} \right)}$
--
 $= \dfrac{\mathop{\text{Cov}} \left( \widetilde{\text{Z}}_{i},\, \text{Y}_{i} \right)/\mathop{\text{Var}} \left( \widetilde{\text{Z}}_i \right)}{\mathop{\text{Cov}} \left( \widetilde{\text{Z}}_i,\, \widetilde{\text{D}}_{i} \right)/\mathop{\text{Var}} \left( \widetilde{\text{Z}}_i \right)}$

--

$\color{#ffffff}{\widehat{\beta}_1^\text{IV}} = \dfrac{\widehat{\pi}_1}{\widehat{\gamma}_1}$  .pink[✔]
---
layout: false
class: clear, middle

Let's push a bit deeper into IV's mechanics and intuition.
---
layout: true
# IV: Mechanics and intuition

---
name: iv-intuition

## Setup

In this section, we'll use medical trials as a working example..super[.pink[†]]

.footnote[.pink[†] Credit/thanks go to [Michael Anderson](https://are.berkeley.edu/~mlanderson/ARE_Website/Home.html) for this example—and much of these notes.]

--

We are interested in the regression model for the effect of some treatment (_e.g._, blood-pressure medication) on medical outcome $\text{Y}_{i}$
--
$$
\begin{align}
  \text{Y}_{i} = \beta_0 + \beta_1 \text{D}_{i} + \varepsilon_i
\end{align}
$$
$\text{D}_{i}$ indicates whether $i$ *takes* the treatment (medication). $\varepsilon_i$ captures all other factors that affect $\text{Y}_{i}$.
--
 Or in potential-outcomes framework:

$$
\begin{align}
  \text{Y}_{i} &= \text{Y}_{1i} \text{D}_{i} + \text{Y}_{0i} (1-\text{D}_{i}) \\
  \text{Y}_{0i} &= \beta_0 + \varepsilon_i \\
  \text{Y}_{1i} &= \text{Y}_{0i} + \beta_1
\end{align}
$$
---
## Research design

.note[Goal] .hi-slate[Estimate the effect of blood-pressure medication] on blood pressure.

--

.note[Challenge] .hi-slate[Selection bias:] Even if treatment reduces blood pressure, selection bias will fights against the estimated effect.

--

.note[Solution] .hi-slate[Randomized medical trial:] Ask randomly chosen individuals in treatment group to take the pill. Control individual get placebo (or nothing).

--

.note[Analysis 1] .attn[Intention to treat] (.attn[ITT]): $\widehat{\beta}_1^\text{ITT} = \overline{\text{Y}}_\text{Trt} - \overline{\text{Y}}_\text{Ctrl}$

--

.note[ITT problem] .attn[Bias from noncompliance:] People don't always follow rules.
<br>*E.g.*, treated folks who don't take pills; control folks who take pills.

--

.note[Analysis 2] .hi-slate[IV!]
--
 Instrument medication $\text{D}_{i}$ with intention to treat $\text{Z}_{i}$.
---
## The IV solution

First question: Is $\text{Z}_{i}$ a valid instrument for $\text{D}_{i}$?

--

1. $\mathop{\text{Cov}} \left( \text{Z}_{i},\, \varepsilon_i \right) = 0$ as $\text{Z}_{i}$ was randomly assigned (exclusion restriction).

--

1. $\mathop{\text{Cov}} \left( \text{Z}_{i},\, \text{D}_{i} \right)\neq 0$ if assignment to treatment changes the likelihood you take the pills (first stage).

--

∴ $\text{Z}_{i}$ is a valid instrument for $\text{D}_{i}$ and IV consistently estimates $\beta_1$.
---
name: iv-noncompliance
## Noncompliance

.attn[Noncompliant] individuals do not abide by their treatment assignment.

--

Let's see how IV "solves" this problems.

--

First, assume noncompliance only affects treated individuals—*i.e.*, treated folks sometimes don't take their pills; control folks never take pills.

---
## Noncompliance, continued

The .hi-slate[first stage] recovers the share of treatment individuals who take the pill
$$
\begin{align}
  \text{D}_{i} = \gamma_1 \text{Z}_{i} + u_i
\end{align}
$$
--
*i.e.*, if 50% of treated individuals take the medication, $\widehat{\gamma} =$ 0.50.

--

The .hi-slate[reduced form] estimates the *ITT*
$$
\begin{align}
  \text{Y}_{i} = \pi_1 \text{Z}_{i} + v_i
\end{align}
$$
--
which we know IV rescales using the first stage
$$
\begin{align}
  \widehat{\beta}_{1}^\text{IV} = \dfrac{\widehat{\pi}_1}{\widehat{\gamma}_1} = \dfrac{\widehat{\pi}_1}{0.50} =
  2 \times \widehat{\pi}_1
\end{align}
$$
---
name: iv-rescale
## Noncompliance, continued

IV solves the noncompliance issue by rescaling by the rate of compliance.

--

If everyone perfectly complies, then $\widehat{\gamma}_1 = 1$ and $\widehat{\beta}_{1}^\text{IV} = \widehat{\pi}_1/1 = \widehat{\beta}_{1}^\text{ITT}$.

--

.ex[Further example] $N_\text{Trt}$ = 10; trt. compliance = 50%; ctrl. compliance = 100%.

$\overline{\text{Y}}_\text{Trt} = \dfrac{5 (\beta_0 + \beta_1) + 5 (\beta_0)}{10} = \beta_0 + \dfrac{\beta_1}{2}$
--
 and $\overline{\text{Y}}_\text{Ctrl} = \beta_0$.

--

So our reduced-form estimate (the ITT) is $\widehat{\gamma}_1 = \dfrac{\beta_1}{2}$ (half the true effect).

--

IV consistently estimates $\beta_1$ via rescaling the ITT by the rate of compliance
$$
\begin{align}
  \widehat{\beta}_1^\text{IV} = \dfrac{\pi}{\gamma} = \dfrac{\beta_1/2}{1/2} = \beta_1
\end{align}
$$
---
## Takeaways

Main points

1. IV .b[rescales] .pink[the causal effect of] $\color{#e64173}{\text{Z}_{i}}$ .pink[on] $\color{#e64173}{\text{Y}_{i}}$ by .purple[the causal effect of] $\color{#6A5ACD}{\text{Z}_{i}}$ .purple[on] $\color{#6A5ACD}{\text{D}_{i}}$.

--

1. IV .b[does not] compare treated compliers to untreated compliers.
--
<br>Such a comparison/estimator would re-introduce selection bias.
---
layout: true
class: clear, middle

---
name: het

Thus far, we assumed homogeneous treatment effects.

.qa[Q] What happens .b[when treatment effects are heterogeneous]?
---

.qa[A] Let's recall what our instruments are doing (with Venn diagrams!).

.note[Credit] [Glen Waddell](http://www.glenwaddell.com) introduced me to IV via Venn.

---
name: venn

```{R, venn_iv, echo = F, fig.height = 7.5}
# Colors (order: x1, x2, x3, y, z)
venn_colors <- c(purple, red, "grey60", orange, red_pink)
# Line types (order: x1, x2, x3, y, z)
venn_lines <- c("solid", "dotted", "dotted", "solid", "solid")
# Locations of circles
venn_df <- tibble(
  x  = c( 0.0,   -0.5,    1.5,   -1.0, -1.4),
  y  = c( 0.0,   -2.5,   -1.8,    2.0, -2.6),
  r  = c( 1.9,    1.5,    1.5,    1.3,  1.3),
  l  = c( "Y", "X[1]", "X[2]", "X[3]",  "Z"),
  xl = c( 0.0,    0.7,    1.6,   -1.0, -2.9),
  yl = c( 0.0,   -3.8,   -1.9,    2.2, -2.6)
)
# Venn
ggplot(data = venn_df, aes(x0 = x, y0 = y, r = r, fill = l, color = l)) +
geom_circle(aes(linetype = l), alpha = 0.3, size = 0.75) +
theme_void() +
theme(legend.position = "none") +
scale_fill_manual(values = venn_colors) +
scale_color_manual(values = venn_colors) +
scale_linetype_manual(values = venn_lines) +
geom_text(aes(x = xl, y = yl, label = l), size = 9, family = "Fira Sans Book", parse = T) +
annotate(
  x = -5.5, y = 3.3,
  geom = "text", label = "Figure 1", size = 10, family = "Fira Sans Book", hjust = 0
) +
xlim(-5.5, 4.5) +
ylim(-4.2, 3.4) +
coord_equal()
```
---
```{R, venn-endog, echo = F, fig.height = 7.5}
# Change locations of circles
venn_df %>%
mutate(
  x = x +   c(0, 0, 0, 0, 0),
  xl = xl + c(0, 0, 0, 0, 0),
  y = y +   c(0, 0, 0, 0, 1),
  yl = yl + c(0, 0, 0, 0, 1)
) %>%
# Venn
ggplot(data = ., aes(x0 = x, y0 = y, r = r, fill = l, color = l)) +
geom_circle(aes(linetype = l), alpha = 0.3, size = 0.75) +
theme_void() +
theme(legend.position = "none") +
scale_fill_manual(values = venn_colors) +
scale_color_manual(values = venn_colors) +
scale_linetype_manual(values = venn_lines) +
geom_text(aes(x = xl, y = yl, label = l), size = 9, family = "Fira Sans Book", parse = T) +
annotate(
  x = -5.5, y = 3.3,
  geom = "text", label = "Figure 2", size = 10, family = "Fira Sans Book", hjust = 0
) +
xlim(-5.5, 4.5) +
ylim(-4.2, 3.4) +
coord_equal()
```
---
```{R, venn-irrelevant, echo = F, fig.height = 7.5}
# Change locations of circles
venn_df %>%
mutate(
  x = x +   c(0, 0, 0, 0,-1),
  xl = xl + c(0, 0, 0, 0,-1),
  y = y +   c(0, 0, 0, 0, 2.3),
  yl = yl + c(0, 0, 0, 0, 2.3)
) %>%
# Venn
ggplot(data = ., aes(x0 = x, y0 = y, r = r, fill = l, color = l)) +
geom_circle(aes(linetype = l), alpha = 0.3, size = 0.75) +
theme_void() +
theme(legend.position = "none") +
scale_fill_manual(values = venn_colors) +
scale_color_manual(values = venn_colors) +
scale_linetype_manual(values = venn_lines) +
geom_text(aes(x = xl, y = yl, label = l), size = 9, family = "Fira Sans Book", parse = T) +
annotate(
  x = -5.5, y = 3.3,
  geom = "text", label = "Figure 3", size = 10, family = "Fira Sans Book", hjust = 0
) +
xlim(-5.5, 4.5) +
ylim(-4.2, 3.4) +
coord_equal()
```
---
```{R, venn-iv-endog2, echo = F, fig.height = 7.5}
# Change locations of circles
venn_df %>%
mutate(
  x = x +   c(0,    0,   0, 0,    2),
  xl = xl + c(0, -2.4, 0.8, 0,  4.6),
  y = y +   c(0,    0,   0, 0,    0),
  yl = yl + c(0,    0,   0, 0, -1.1)
) %>%
# Venn
ggplot(data = ., aes(x0 = x, y0 = y, r = r, fill = l, color = l)) +
geom_circle(aes(linetype = l), alpha = 0.3, size = 0.75) +
theme_void() +
theme(legend.position = "none") +
scale_fill_manual(values = venn_colors) +
scale_color_manual(values = venn_colors) +
scale_linetype_manual(values = venn_lines) +
geom_text(aes(x = xl, y = yl, label = l), size = 9, family = "Fira Sans Book", parse = T) +
annotate(
  x = -5.5, y = 3.3,
  geom = "text", label = "Figure 4", size = 10, family = "Fira Sans Book", hjust = 0
) +
xlim(-5.5, 4.5) +
ylim(-4.2, 3.4) +
coord_equal()
```
---
```{R, venn-iv-endog1, echo = F, fig.height = 7.5}
# Venn
ggplot(data = venn_df, aes(x0 = x, y0 = y, r = r, fill = l, color = l)) +
geom_circle(aes(linetype = l), alpha = 0.3, size = 0.75) +
theme_void() +
theme(legend.position = "none") +
scale_fill_manual(values = venn_colors) +
scale_color_manual(values = venn_colors) +
scale_linetype_manual(values = venn_lines) +
geom_text(aes(x = xl, y = yl, label = l), size = 9, family = "Fira Sans Book", parse = T) +
annotate(
  x = -5.5, y = 3.3,
  geom = "text", label = "Figure 1", size = 10, family = "Fira Sans Book", hjust = 0
) +
xlim(-5.5, 4.5) +
ylim(-4.2, 3.4) +
coord_equal()
```

---
layout: true
# IV + heterogeneity

---
## Recap

Throughout the course, we've discussed two concepts of treatment effects.

--

1. .attn[Average treatment effect] (.attn[ATE]) The average treatment effect for an individual randomly drawn from our sample.

--

1. .attn[Treatment on the treated] (.attn[TOT]) The average treatment effect for a .it.hi-slate[treated] individual
randomly drawn from our sample.

--

When we assume homogeneous/constant treatment effects, ATE = TOT.

--

.qa[Q] If treatment effects vary, then what do IV and 2SLS estimate?

--

.qa[A] Not ATE.
--
 And not TOT.
--
 They estimate the LATE..super[.pink[†]]

 .footnote[
 .pink[†] See [Angrist, Imbens, and Rubin (1996)](https://www.jstor.org/stable/2291629).
 ]
---
## The LATE

IV generally estimates the .attn[LATE]—the .attn[Local Average Treatment Effect].

--

.note[Recall] IV "works" by isolating variation in $\text{D}_{i}$ induced by our instrument $\text{Z}_{i}$.

--

In other words: IV focuses on the individuals whose $\text{D}_{i}$ changes due to $\text{Z}_{i}$.

Angrist, Imbens, and Rubin (1996) call these folks .attn[compliers].

--

However, *compliers* are only one of four possible groups.

.col-left[
1. .attn[Compliers] $\text{D}_{i} = 1$ iff $\text{Z}_{i}=1$.
1. .attn[Always-takers] $\text{D}_{i} = 1$ $\forall \text{Z}_{i}$.
1. .attn[Never-takers] $\text{D}_{i} = 0$ $\forall \text{Z}_{i}$.
1. .attn[Defiers] $\text{D}_{i} = 1$ iff $\text{Z}_{i}=0$.
]
--
.col-right[
Only take pills .hi-slate[when treated].
<br>.hi-slate[Always] take pills.
<br>.hi-slate[Never] take pills.
<br>Only take pills .hi-slate[when untreated].
]
---
## The LATE

Because IV only uses variation in $\text{D}_{i}$ that correlates with $\text{Z}_{i}$, IV mechanically drops *always-takers* and *never-takers*.

--

Most IV derivations/applications assume away the existence of *defiers*.

--

Thus, IV estimates a treatment effect .hi-slate[using only *compliers*].

--

Hence the "local" in *local average treatment effect*.
---
name: late-ex
## The LATE: Medical-trial example

Imagine treatment works for some $\left( \beta_{1,i} < 0 \right)$ and not for others $\left( \beta_{1,j} = 0 \right)$.

Suppose individuals know their response to blood-pressure medication.

--
- $\beta_{1,i}<0$ individuals always take the pill.

--
- $\beta_{1,j}=0$ individuals only take the pill when treated.

--

Then our compliers will be individuals for whom $\beta_{1,j}=0$.

--

Thus, IV's LATE will indicate no treatment effect $\left( \widehat{\beta}_1^\text{IV} = 0 \right)$.

---
## The LATE

.qa[Q] So is IV actually inconsistent?

--

.qa[A] It depends what you are trying to estimate (and how you interpret it).

IV doesn't estimate the ATE or TOT, so it would be inconsistent for them..super[.pink[†]]

.footnote[
.pink[†] Just as the TOT is not consistent for the ATE.
]

--

IV estimates the *local* average treatment effect.

--

.note[Takeaway] Because IV identifies off of compliers, it estimates an average treatment effect for these individuals (who *comply* with the instrument).

--

.note[Takeaway.sub[2]] Different instruments have different LATEs.
---
name: monotonicity
## Monotonicity

We've already written down the two classical IV/2SLS assumptions

- .note[First stage:] $\mathop{\text{Cov}} \left( \text{Z}_{i},\, \text{D}_{i} \right) > 0$
- .note[Exclusion restriction:] $\mathop{\text{Cov}} \left( \text{Z}_{i},\, \varepsilon_{i} \right) = 0$

but we need a third assumption to get ensure IV's complier-based LATE interpretation.

--

- .attn[Monotonicity] (.attn[Uniformity]).attn[:] $\text{D}_{i}(z)\geq \text{D}_{i}(z')$ or $\text{D}_{i}(z)\leq \text{D}_{i}(z') \enspace \forall i$
<br> [Heckman](chrome-extension://oemmndcbldboiebfnladdacbdfmadadm/http://jenni.uchicago.edu/papers/koop2006/koop2-IV_ho_2006-09-25a_mms.pdf): *Uniformity* of responses *across persons.*
<br> [Imbens and Angrist (1994)](https://www.jstor.org/stable/2951620): Instrument has monotone effect on $\text{D}_{i}$.
---
## Monotonicity

If "defiers" exist, then monotonicity/uniformity is violated.

--

In this case, the IV estimand is

$$
\begin{align}
  \dfrac{\tau_{c} \mathop{\text{Pr}}\left(\text{complier}\right) - \tau_{d} \mathop{\text{Pr}}\left(\text{defier}\right)}{ \mathop{\text{Pr}}\left(\text{complier}\right) -  \mathop{\text{Pr}}\left(\text{defier}\right)}
\end{align}
$$

which is not bound between $\tau_{c}$ and $\tau_{d}$.

--

.ex[Example] $\tau_c=$ 1 and $\tau_d=$ 2. $\mathop{\text{Pr}}\left(\text{complier}\right)=$ 2/3 and $\mathop{\text{Pr}}\left(\text{defier}\right)=$ 1/3.

--

Then the "LATE" is 0..super[.pink[†]]

.footnote[
.pink[†] Some people would instead say that there is no LATE when you violate monotonicity.
]

---
layout: false
class: clear, middle

Until now, we've focused on using a single instrument.

The 2SLS estimator accomodates multiple instruments..super[.pink[†]]

.footnote[
.pink[†] Whether you can find multiple valid instruments is another question.
]
---
layout: true
# Multiple instruments

---
class: inverse, middle
name: multi-inst
---
## Motivation

.qa[Q] Why include multiple instruments?

--

.qa[A] Multiple instruments can capture more variation in $\text{D}_{i}$ (efficiency).

--

Using terminology from the *system-of-equations* literature,

- one instrument for one endogenous variable: .attn[just identified]
- multiple instruments for one endogenous variable: .attn[over identified]

---
## In practice

With (valid) instruments $\text{Z}_{1i}$ and $\text{Z}_{2i}$, or first stage becomes
$$
\begin{align}
  \text{D}_{i} = \gamma_0 + \gamma_1 \text{Z}_{1i} + \gamma_2 \text{Z}_{2i} + \gamma_3 \text{X}_{i} + u_i
\end{align}
$$

--

while our second stage is still
$$
\begin{align}
  \text{Y}_{i} = \beta_0 + \beta_1 \widehat{\text{D}}_{i} + \beta_2 \text{X}_{i} + v_i
\end{align}
$$
---
layout: true
# Multiple instruments
## Example: Quarter of birth

---
name: multi-ex

Back to our quest to estimate the returns to education.

--

[Angrist and Krueger (1991)](https://www.jstor.org/stable/2937954) proposed *quarter of birth* as a set of instruments for years of schooling.

--

Accordingly, their first stage looks something like.super[.pink[†]]

.footnote[
.pink[†] We need to drop one of the quarter-of-birth indicators to avoid perfect collinearity.
]

$$
\begin{align}
  \text{Schooling}_i = \gamma_0 &+ \gamma_1 \mathbb{I}(\text{Born Q1})_{i} + \gamma_2 \mathbb{I}(\text{Born Q2})_{i}
  \\&+ \gamma_3 \mathbb{I}(\text{Born Q3})_{i} + \gamma_4 \mathbb{I}(\text{Born Q4})_{i}
  \\&+ \gamma_5 \text{X}_{i} + u_{i}
\end{align}
$$


---
.qa[Q] Is quarter of birth a valid instrument?

--

.qa[Q1] Why would quarter of birth affect schooling? (.note[First stage])

--

.qa[A1] Students cannot drop out of school until a certain age, and quarter of birth affects your age at the time you begin school.

--

.ex[Example] Some states require students to stay in school until they are 16.
- Students who start school at age .hi-slate[6] drop out after .hi-slate[10] years of schooling.
- Students who start school at age .hi-slate[5] drop out after .hi-slate[11] years of schooling.

---
If students must begin school in calendar year in which they turn 6
- December birthdates: begin school at 5.75; drop out with 10.25 yrs.
- January birthdates: begin school at 6.75; drop out with 9.25 yrs.

--

For some group, quarter of birth may affect the number of years in school.
---

It turns out that the first stage is also pretty weak in this setting.

.attn[Weak instruments] can cause several problems for 2SLS/IV:

--

1. Our estimator is a ratio of the reduced form and the first stage, so a weak first stage can blow up reduced-form estimates (amplifying reduced-form noise/bias).

--

2. Many weak instruments lead to a finite-sample issue in which 2SLS is biased toward OLS—our first stage is essentially overfitting.

--

What about our other requirements for a valid instrument?

---
.qa[Q2] Is quarter of birth uncorrelated with $\varepsilon_i$ (.note[excludable])?

--

.qa[A2] While quarter of birth may be fairly arbitrary for some families, other families might time births.

If these birth timers differ from other couples along other dimensions (_e.g._, income or education), then quarter of birth may correlate with $\varepsilon_i$.
---
.qa[Q3] Is the effect monotone?

--

.qa[A3] Some.super[.pink[†]] argue that monotonicity may be violated in this setting.

.footnote[
.pink[†] _E.g._, [Aliprantis (2012)](https://journals.sagepub.com/doi/abs/10.3102/1076998610396885)
]

--

Consider December births.

--

- Original idea: December birthdates will start school at age 5.7, inducing more years of education before 16.

--

- *Redshirting* idea: Parents hold back December kids so they can be older (_i.e._, 6.7), inducing fewer years of education before 16.

---
layout: true
# 2SLS and .mono[R]

---
name: 2sls-r

## `estimatr`

You can implement 2SLS/IV in many ways in .mono[R].

Today: `esitmatr` and `iv_robust()`.

--

Specifically, we give `iv_robust()` the relationship that we want separted from the instrument by `|`

--
, *e.g.*,

```{R, iv-data, include = F}
# Set seed
set.seed(12345)
# Sample size
n <- 1e2
# Define our variance-covariance matrix (D, ε, Z)
Σ <- matrix(data = c(1, 0.3, 0.3, 0.3, 1, 0, 0.3, 0, 1), ncol = 3)
# Our vector of means (D, ε, Z)
μ = c(10, 0, 3)
# Draw n observations; convert to tibble
sample_df <- MASS::mvrnorm(n = n, mu = μ, Sigma = Σ) %>% as_tibble()
# Name variables
names(sample_df) <- c("D", "ε", "Z")
# Calculate Y
sample_df %<>% mutate(Y = 7 + 1 * D + ε)
```

```{R, r-iv1}
# Estimate 2SLS
iv_robust(Y ~ D | Z, data = sample_df, se_type = "classical") %>%
  tidy() %>% select(1:5)
```
---

## Now in two stages!

Of course, we can estimate 2SLS in two stages.

```{R, r-iv-s1}
# First stage
stage1 <- lm_robust(D ~ Z, data = sample_df, se_type = "classical")
# First-stage results
stage1 %>% tidy() %>% select(1:5)
```
---

## Second stage

We just need to add $\widehat{\text{D}}_{i}$ to our dataset.

```{R, r-iv-s2}
# Add fitted (first-stage) values to data
sample_df %<>% mutate(D_hat = stage1$fitted.values)
# Second stage
stage2 <- lm_robust(Y ~ D_hat, data = sample_df, se_type = "classical")
# Second-stage results
stage2 %>% tidy() %>% select(1:5)
```
---
## Standard errors

However, recall that our second-stage standard errors are not correct.

--

.center.hi-purple[Second-stage results]
<span style="display:block; margin-bottom:-1em;"> </span>

```{R, r-iv-2sls1, echo = F}
stage2 %>%
  tidy_table(
    terms = c("Int", "D hat"),
    highlight_bold = F
  )
```

--

.center.hi-pink[2SLS results]
<span style="display:block; margin-bottom:-1em;"> </span>
```{R, r-iv-2sls2, echo = F}
iv_robust(Y ~ D | Z, data = sample_df, se_type = "classical") %>%
  tidy_table(
    terms = c("Int", "D"),
    highlight_bold = F
  )
```






---
layout: true
# IV and 2SLS
## Conclusions

---
name: conclusions

1. IV/2SLS focus on .hi-slate[isolating some "good" variation] in $\text{D}_{i}$ via $\text{Z}_{i}$.

1. Important .hi-slate[requirements]: strong first stage, excludability, monotonicity.

1. IV and 2SLS .hi-slate[rescale the reduced form] with the first stage.

1. Estimates are .hi-slate[LATE from compliers].

1. Different instruments can produce .hi-slate[different LATEs].

1. A .hi-slate[weak first stage] can lead to problems.
---
layout: false
# Table of contents

.col-left[
### Admin
.smallest[

1. [Schedule](#schedule)
]

### Instrumental variables
.smallest[
1. [Research designs](#designs)
1. [Introduction](#intro)
1. [Definition](#defined)
1. [Example](#example)
1. [IV estimator](#iv-estimator)
]

]

.col-right[

### Two-stage least squares
.smallest[
1. [Setup](#setup)
1. [The reduced form](#reduced-form)
  - [Defined](#reduced-form)
  - [Intuition](#reduced-intuition)
  - [Example](#reduced-example)
  - [Derivation](#reduced-derivation)
1. [Intuition and mechanics](#iv-intuition)
  - [Noncompliance](#iv-noncompliance)
  - [Rescaling](#iv-rescale)
1. [Heterogeneous treatment effects](#het)
  - [Venn diagram](#venn)
  - [LATE](#late)
  - [Example](#late-ex)
  - [Monotonicity](#monotonicity)
1. [Multiple instruments](#multi-inst)
  - [Example](#multi-ex)
1. [2SLS and .mono[R]](#2sls-r)
1. [Conclusions](#conclusions)
]

]
---
exclude: true

```{R, generate pdfs, include = F, eval = T}
source("../../ScriptsR/unpause.R")
unpause("08IV.Rmd", ".", T, T)
```