Instrumental variables II and regression discontinuity II

# IV II & RDD II

**Session 12**

]

---

# Plan for today

.box-4.medium.sp-after-half[Treatment effects and compliance]

.box-1.medium.sp-after-half[Randomized promotion]

.box-6.medium.sp-after-half[Fuzzy regression discontinuity]

---

layout: false
name: treatment-effects
class: center middle section-title section-title-4 animated fadeIn

# Treatment effects and compliance

---

---

# Potential outcomes

---

.box-4.medium[Fundamental problem of causal inference]

.less-medium[
$$
\delta_i = Y_i^1 - Y_i^0 \quad \text{in real life is} \quad \delta_i = Y_i^1 - ???
$$
]

.box-inv-4.medium[Individual-level effects are impossible to observe!]

---

---

# Average treatment effect

.box-inv-4[Difference between average/expected value when program is on vs. expected value when program is off]

---

.box-4.medium[Every individual has a treatment/causal effect]

.box-inv-4.medium[ATE = average of all unit-level causal effects]

.box-inv-4.medium[ATE = Average effect for the whole population]

---

---

# Other versions of causal effects

.box-inv-4.medium[Average treatment on the treated]

.box-4.sp-after[ATT/TOT]

.box-inv-4.medium.sp-before[Conditional average treatment effect]

---

# Local effects

---

# LATE

.box-inv-4.medium[Local average treatment effect (LATE) = weighted ATE]

.box-4.sp-after[Narrower effect; only applies to some of the population]

.box-inv-4.medium[You can't make population-level claims with LATE]

.box-4.small[(But that can be okay!)]

---

# LATE

&nbsp;

.box-inv-4.medium.sp-after[In RDD, LATE = people in the bandwidth]

.box-inv-4.medium[In RCTs and IVs, LATE = **compliers**]

---

# Compliance

.box-4.small.sp-after[Treatment follows assignment]

.box-inv-4.less-medium[Always taker]

.box-4.small[Gets treatment regardless of assignment]
]

.box-4.small.sp-after[Rejects treatment regardless of assignment]

.box-inv-4.less-medium[Defier]

.box-4.small[Does the opposite of assignment]
]

---

---

---

# Ignoring defiers

.box-inv-4.medium[We can generally assume that defiers don't exist]

.box-4.small[In drug trials this makes sense; you can't get access to medicine without being in treatment group]

.box-4.small[In development it can make sense; in a bed net RCT, a defier assigned to treatment would have to tear down all existing bed nets out of spite]

---

# Ignoring defiers

&nbsp;

.box-inv-4.medium[Monotonicity assumption]

.box-4.small[Assignment to treatment only has an effect in one direction]

.box-4.small[Assignment to treatment can only increase—not decrease—your actual chance of treatment]

---

---

---

# More causal effects

.box-inv-4.medium[Intent to treat (ITT)]

.box-4.small.sp-after[Effect of assignment (not actual treatment!)]

.center[
<figure>
 <img src="img/12/compliance-itt.png" alt="Compliance and ITT" title="Compliance and ITT" width="90%">
</figure>
]

---

# More causal effects

.box-inv-4.medium[Complier Average Causal Effect (CACE)]

.box-4.small.sp-after[LATE for the compliers]

.center[
<figure>
 <img src="img/12/compliance-cace.png" alt="Compliance and CACE" title="Compliance and CACE" width="90%">
</figure>
]

---

# Hypothetical bed net program

.box-inv-4[An NGO distributes mosquito bed nets to help improve health by reducing malaria infection rate]

.box-inv-4[We can read everyone's minds and we know if people are always takers, never takers, or compliers]

---

# Mind reading

---

# Actual data

---

# Actual data

---

.center[
<figure>
 <img src="img/12/compliance-reversed.png" alt="Compliance with graph" title="Compliance with graph" width="90%">
</figure>
]

---

$$
`\begin{aligned}
\text{ITT}\ =\ & \color{#0D0887}{\pi_\text{compliers} \times (\text{T} - \text{C})_\text{compliers}} + \\
&\color{#B7318A}{\pi_\text{always takers} \times (\text{T} - \text{C})_\text{always takers}} + \\
&\color{#FEBA2C}{\pi_\text{never takers} \times (\text{T} - \text{C})_\text{never takers}}\\[6pt]
\text{ITT}\ =\ & \color{#0D0887}{\pi_\text{C} \text{CACE}} + \color{#B7318A}{\pi_\text{A} \text{ATACE}} + \color{#FEBA2C}{\pi_\text{N} \text{NTACE}}
\end{aligned}`
$$

---

`$$\text{ITT}\ =\ \color{#0D0887}{\pi_\text{C} \text{CACE}} + \color{#B7318A}{\pi_\text{A} \text{ATACE}} + \color{#FEBA2C}{\pi_\text{N} \text{NTACE}}$$`
--

.box-4[Treatment received is same regardless of assignment! Being assigned to treatment doesn't influence ATs and NTs]

`$$\text{ITT}\ =\ \color{#0D0887}{\pi_\text{C} \text{CACE}} + \color{#B7318A}{\pi_\text{A} \times 0} + \color{#FEBA2C}{\pi_\text{N} \times 0}$$`

---

$$
`\begin{aligned}
\text{ITT}\ =\ & \color{#0D0887}{\pi_\text{C} \text{CACE}} + \color{#B7318A}{\pi_\text{A} \text{ATACE}} + \color{#FEBA2C}{\pi_\text{N} \text{NTACE}} \\[6pt]
=\ & \color{#0D0887}{\pi_\text{C} \text{CACE}} + \color{#B7318A}{\pi_\text{A} \times 0} + \color{#FEBA2C}{\pi_\text{N} \times 0}\\[6pt]
\text{ITT}\ =\ &\color{#0D0887}{\pi_\text{C} \text{CACE}}
\end{aligned}`
$$

---

---

# Finding the ITT

`$$\text{ITT} = (\bar{y}\ |\ \text{Treatment}) - (\bar{y}\ |\ \text{Control)}$$`

.pull-left.small-code[

```r
bed_nets |> 
  group_by(treatment) |> 
  summarize(avg = mean(health))
```

```
## # A tibble: 2 × 2
## treatment avg
## <chr> <dbl>
## 1 Control 40.9
## 2 Treatment 46.9
```
]

.pull-right.small-code[

```r
itt_model <- lm(health ~ treatment, 
 data = bed_nets)
tidy(itt_model)
```

```
## # A tibble: 2 × 2
## term estimate
## <chr> <dbl>
## 1 (Intercept) 40.9 
## 2 treatmentTreatment 5.99
```
]

---

# Finding the πC

.box-inv-4.smaller[People in treatment group who complied are a combination of Always Takers and Compliers]

$$
`\begin{aligned}
\pi_\text{A} + \pi_\text{C} =& \text{% yes in treatment; or} \\
\pi_\text{C} =& \text{% yes in treatment} - \pi_\text{A}
\end{aligned}`
$$

---

# Can we know πA?

`$$\pi_\text{C} = \text{% yes in treatment} - \pi_\text{A}$$`

.box-inv-4.small[We can assume that the proportion of Always Takers is the same across treatment and control]

.box-inv-4.small[We know how many people were in control but still used nets—that's πA!]

---

# Isolating πC

.small[
$$
`\begin{aligned}
\pi_\text{C} =& \text{% yes in treatment} - \pi_\text{A} \\
=& \text{% yes in treatment} - \text{% yes in control}
\end{aligned}`
$$
]

.pull-left.small-code.small[

```r
bed_nets |> 
  group_by(treatment, bed_net) |> 
  summarize(n = n()) |> 
  mutate(prop = n / sum(n))
```

```
## # A tibble: 4 × 4
## # Groups: treatment [2]
## treatment bed_net n prop
## <chr> <fct> <int> <dbl>
## 1 Control No bed net 808 0.805
## 2 Control Bed net 196 0.195
## 3 Treatment No bed net 388 0.390
## 4 Treatment Bed net 608 0.610
```
]

.pull-right.small-code[

```r
# pi_c = prop yes in treatment - 
# prop yes in control
pi_c <- 0.6104418 - 0.1952191
pi_c
```

```
## [1] 0.4152227
```

---

# Finding the CACE, finally!

`$$\text{CACE} = \frac{\text{ITT}}{{\pi_\text{C}}}$$`

.pull-left.small-code.small[

```r
ITT <- tidy(itt_model) |> 
 filter(term == "treatmentTreatment") |> 
 pull(estimate)
ITT
```

```
## [1] 5.987992
```

```r
pi_c
```

```
## [1] 0.4152227
```
]

.pull-right.small-code[

```r
CACE <- ITT / pi_c
CACE
```

```
## [1] 14.42116
```

.box-inv-4.small[Bed nets *cause* 14.4 point increase in health for compliers]
]

---

$$
\color{#FF4136}{\text{ITT}} = (\bar{y}\ |\ \text{Treatment}) - (\bar{y}\ |\ \text{Control)}
$$

$$
`\begin{aligned}
\color{#B10DC9}{\pi_\text{C}}\ =\ & \text{% yes in treatment} - \\
& \text{% yes in control}
\end{aligned}`
$$
]

---

# A faster way with 2SLS

.box-inv-4.medium[LATE for the compliers]

.box-4[If you use assignment to treatment as an instrument, you can find the causal effect for just compliers]

---

# CACE with 2SLS

```r
model_2sls <- iv_robust(health ~ bed_net | treatment, 
 data = bed_nets)
tidy(model_2sls)
```

```
##             term estimate std.error statistic      p.value
## 1    (Intercept) 38.12285 0.5150818  74.01320 0.000000e+00
## 2 bed_netBed net 14.42116 1.2538198  11.50178 1.086989e-29
```
]

---

layout: false
name: encouragement
class: center middle section-title section-title-1 animated fadeIn

# Promotion as an instrument

---

---

# Universal programs

.box-inv-1.medium[What if you have a program that anyone can opt in to?]

.box-1.small[Selection bias!]
]

.box-1.small[Ethics!]
]

---

# Randomized promotion

.box-inv-1.medium.sp-after[What if you *encourage* some people to participate?]

.box-inv-1.medium[What if the encouragement is randomized?]

.box-1.medium[Valid treatment/control groups?]

---

# Randomized promotion

.box-1.medium.sp-after[…but also, kind of!]

.box-inv-1.medium[Encouragement/promotion = an instrument!]

---

# Not something weird? Does that work!?

.box-inv-1.tiny.sp-after[Z → X &emsp; Cor(Z, X) ≠ 0]

.box-1.small[Promotion causes people to use the program. Yep.]
]

.pull-right.sp-after-half[
.box-inv-1.less-medium[Exclusive?]

.box-inv-1.tiny.sp-after[Z → X → Y &emsp; Z &nrarr; Y &emsp; Cor(Z, Y | X) = 0]

.box-1.small[Promotion causes outcome *only through* program? Yep.]
]

.box-inv-1.less-medium[Exogenous?]

.box-inv-1.tiny.sp-after[U &nrarr; Z &emsp; Cor(Z, U) = 0]

.box-1.small[Unobserved things that influence outcome don't also influence promotion? Yep.]

---

# Program compliance

.box-1.small[People who will always enroll in program]
]

.pull-right.sp-after[
.box-inv-1.medium[Never Takers]

.box-1.small[People who will never enroll in program]
]

.box-inv-1.medium[Compliers / Enrollers-if-Promoted]

.box-1.small[People who will enroll in the program if encouraged to]

---

# LATE for compliers

.tiny[
<table>
 <thead>
 <tr>
 <th style="text-align:center;"> id </th>
 <th style="text-align:center;"> outcome </th>
 <th style="text-align:center;"> program </th>
 <th style="text-align:center;"> promotion </th>
 </tr>
 </thead>
<tbody>
 <tr>
 <td style="text-align:center;"> 1 </td>
 <td style="text-align:center;"> 45 </td>
 <td style="text-align:center;"> TRUE </td>
 <td style="text-align:center;"> TRUE </td>
 </tr>
 <tr>
 <td style="text-align:center;"> 2 </td>
 <td style="text-align:center;"> 55 </td>
 <td style="text-align:center;"> TRUE </td>
 <td style="text-align:center;"> FALSE </td>
 </tr>
 <tr>
 <td style="text-align:center;"> 3 </td>
 <td style="text-align:center;"> 52 </td>
 <td style="text-align:center;"> FALSE </td>
 <td style="text-align:center;"> FALSE </td>
 </tr>
 <tr>
 <td style="text-align:center;"> 4 </td>
 <td style="text-align:center;"> 39 </td>
 <td style="text-align:center;"> FALSE </td>
 <td style="text-align:center;"> TRUE </td>
 </tr>
</tbody>
</table>
]

.code-small.center[

```r
iv_robust(outcome ~ program | promotion)
```
]

.box-inv-1.medium[This will show the LATE for promoted-ees!]

.box-1.small[Says nothing about the effect of the program on Always Takers or Never Takers]

---

layout: false
name: fuzzy-rdd
class: center middle section-title section-title-6 animated fadeIn

# Fuzzy RDD

---

---

# Fuzzy discontinuities

---

# Fuzzy discontinuities

.box-inv-6.medium.sp-after[Fuzzy discontinuities imply noncompliance]

.box-inv-6.medium[Address noncompliance with instrumental variables]

---

# What do we use as instrument?

.box-inv-6.medium[Instrument = above/below cutoff]

.box-6.small.sp-after[i.e. what they were supposed to do]

&nbsp;

.box-6.medium.sp-before[(This is just like the CACE we just did!)]

---

# Not something weird? Does that work!?

.box-inv-6.tiny.sp-after[Z → X &emsp; Cor(Z, X) ≠ 0]

.box-6.small[Cutoff causes program? Yep.]
]

.pull-right.sp-after-half[
.box-inv-6.less-medium[Exclusive?]

.box-inv-6.tiny.sp-after[Z → X → Y &emsp; Z &nrarr; Y &emsp; Cor(Z, Y | X) = 0]

.box-6.small[Cutoff causes outcome *only through* program? Yep.]
]

.box-inv-6.less-medium[Exogenous?]

.box-inv-6.tiny.sp-after[U &nrarr; Z &emsp; Cor(Z, U) = 0]

.box-6.small[Unobserved things that influence outcome don't also influence cutoff? It's an arbitrary cutoff, so sure.]

---

# Doubly local LATE

.box-inv-6.medium[Effect is only for (1) compliers (2) near the cutoff]

---

# Parametric fuzzy RD

```r
tutoring_centered <- tutoring |> 
 mutate(entrance_centered = entrance_exam - 70,
 below_cutoff = entrance_exam <= 70)
head(tutoring_centered, 6)
```

```
## # A tibble: 6 × 6
## id entrance_exam tutoring exit_exam entrance_centered below_cutoff
## <int> <dbl> <lgl> <dbl> <dbl> <lgl> 
## 1 1 92.4 FALSE 78.1 22.4 FALSE 
## 2 2 72.8 FALSE 58.2 2.77 FALSE 
## 3 3 53.7 TRUE 62.0 -16.3 TRUE 
## 4 4 98.3 FALSE 67.5 28.3 FALSE 
## 5 5 69.7 TRUE 54.1 -0.288 TRUE 
## 6 6 68.1 TRUE 60.1 -1.93 TRUE
```
]

---

# Parametric fuzzy RD

```r
# Bandwidth ± 10
fuzzy1 <- iv_robust(
 exit_exam ~ entrance_centered + tutoring | entrance_centered + below_cutoff,
 data = filter(tutoring_centered, entrance_centered >= -10 & entrance_centered <= 10)
)

tidy(fuzzy1)
```

```
##                term   estimate  std.error statistic       p.value
## 1       (Intercept) 60.1413558 1.01765573 59.097939 9.746624e-200
## 2 entrance_centered  0.4366281 0.09929619  4.397229  1.407213e-05
## 3      tutoringTRUE  9.7410444 1.91184891  5.095091  5.384163e-07
```
]

---

# Nonparametric fuzzy RD

.box-6.small[Important! Specify actual treatment status, *not* the instrument of above/below the cutoff]

```r
rdrobust(y = tutoring$exit_exam, x = tutoring$entrance_exam, 
         c = 70, fuzzy = tutoring$tutoring) |> 
  summary()
```

```
## First-stage estimates.
## 
## =============================================================================
##         Method     Coef. Std. Err.         z     P>|z|      [ 95% C.I. ]       
## =============================================================================
##   Conventional    -0.708     0.073    -9.751     0.000    [-0.850 , -0.565]
```
]