Regression Discontinuity 1: Framework and Application

# Regression Discontinuity 1: Framework and Application
### Instructor: Yuta Toyama
### Last updated: 2021-07-15

---

---
## Introduction

-   **Regression Discontinuity Design (回帰不連続デザイン)**
    -   Exploit the discontinuous change in treatment status to estimate
        the causal effect.

-   Example:
    -   Threshold of test score for college admission
    -   Eligibility of policy due to age.
    -   Geographic boundary of two regions.

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## RD Idea

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## Course Plan and Reference

- Plan
  - Framework
  - Estimation
  - Application: Shigeoka (2014)
  - Implementation in R

- Reference
  -   Angrist and Pischke "Mostly harmless econometrics" Chapter 6
  -   R packages:
    [https://sites.google.com/site/rdpackages/rdrobust ](https://sites.google.com/site/rdpackages/rdrobust )

---

---
## Framework

- `$Y_{i}$`: observed outcome for person `$i$`
 
- Define *potential outcomes*
 - `$Y_{1i}$`: outcome for `$i$` when she is treated (treatment group)
 - `$Y_{0i}$`: outcome for `$i$` when she is not treated (control
 group)
 
- `$D_{i}$`: treatment status is deterministically determined (sharp RD
 design) `$$D_{i}=\mathbf{1}\{W_{i}\geq\bar{W}\}$$`
 - `$W_{i}$`: running variable (forcing variable).
 - Probabilistic assignment is allowed (**fuzzy RD design**)

---
## Example: Incumbent Advantage

- Consider the two-candidate elections
 - `$D_{i}$`: dummy for incumbent in the election
 - `$Y_{i}$`: whether the candidate win in the election
 - `$W_{i}:$` the vote share in the previous election.
 
- The incumbent status is defined as
 `$$D_{i}=\mathbf{1}\{W_{i}\geq0.5\}$$`
 
- Idea of RD:
 - Suppose that you won with 51%.
 - You are similar to the guy who lose at 49% (main assumption of
 RD).
 - If you focus on these people, `$D_{i}$` is as if it were randomly
 assigned.

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## Framework cont.d
- Note that `$D_{i}=\mathbf{1}\{W_{i}\geq\bar{W}\}$` implies the
 unconfoundedness `$$(Y_{1i},Y_{0i})\perp D_{i}|W_{i}$$`
 
- But the overlap assumption does not hold

`\begin{aligned}
P(D_{i}=1|W_{i}=w)=\begin{cases}
 1 & if\ w\geq\bar{W}\\
 0 & if\ w<\bar{W}
 \end{cases}
\end{aligned}`

-   To compare people with and without treatment, we need to rely on
    some sort of extrapolation around the threshold.

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## Linear approach

-   Suppose for a moment that

`\begin{aligned}
    Y_{1i} & =\rho+Y_{0i}\\
    E[Y_{0i}|W_{i}=w] & =\alpha_{0}+\beta_{0}w
\end{aligned}`

- This leads to a regression
 `$$Y_{i}=\alpha+\beta W_{i}+\rho D_{i}+\eta_{i}$$`
 - `$\rho$` is the causal effect.
 
- This approach relies on linear extrapolation. May not be good.
 - What if `$E[Y_{0i}|W_{i}=w]$` is nonlinear?

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<img src="figure_table/Figure6_1_1.png" height="630">
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## A more general approach

- Allowing for nonlinear effect of the running variable `$W_{i}$`
 `$$Y_{i}=f(W_{i})+\rho\mathbf{1}\{W_{i}\geq\bar{W}\}+\eta_{i}$$`
 
- A function `$f(\cdot)$` might be a `$p$`th order polynomial.
 `$$f(W_{i})=\beta_{1}W_{i}+\beta_{2}W_{i}^{2}+\cdots+\beta_{p}W_{i}^{p}$$`

-   nonparametric approach later.

---
## Implementation in Regression

-   Consider 
`\begin{aligned}
    E[Y_{0i}|W_{i}=w] & =f_{0}(W_{i}-\bar{W})\\
    E[Y_{1i}|W_{i}=w] & =\rho+f_{1}(W_{i}-\bar{W})
\end{aligned}`
where `$\tilde{W}_{i}=W_{i}-\bar{W}$` is a normalization.
-   Then the regression equation is
`\begin{aligned}
    Y_{i} & =\alpha+\beta_{01}\tilde{W_{i}}+\cdots+\beta_{0p}\tilde{W}_{i}^{p}\\
     & +\rho D_{i}+\beta_{1}^{*}D_{i}\tilde{W}_{i}+\cdots+\beta_{p}^{*}D_{i}\tilde{W}_{i}^{p}+\eta_{i}
\end{aligned}`

---

- When running regression, need to focus on the sample around threshold.

- How close the sample should be to the threshold can be taken care by statistical procedure.

---

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## Effects of the minimum age drinking law

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<img src="figure_table/MMfig42.png" height="600">
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class: title-slide-section, center, middle
name: logistics
# Validation of Assumptions

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## Validation of Assumptions for RD

- Key assumption 1: STUVA. No spill over of treatment across threshold.
  - See next slide.

- Key assumption 2: Continuity of potential outcomes at the threshold.

---
## Violation of STUVA

---
## Continuity of

- The key assumptions : Both `$E[Y_{1i}|W_{i}=w]$` and
 `$E[Y_{0i}|W_{i}=w]$` are continuous at the threshold `$w=\bar{W}$`.
 
- This is not directly testable because we cannot observe `$Y_{1i}$`
 below the threshold.
 
- There are two common approaches that support this assumption:
 1. Covariate test
 2. Density test (no bunching in the running variable).

---
## Covariate Test

- The underlying idea of RDD: Comparing outcomes right above and right
 below `$\bar{W}$` provides a comparison of treated and control agents
 who are similar due to the assumed continuity in conditional
 distributions.
 
- If this is a valid comparison, then we would expect that covariates
 `$X$` also change smoothly as we pass through the threshold.

---
 
- Run the RDD on the covariate `$X$`.
 
- If we found the discontinuity, it suggests that the conditional
 expectation of `$Y$` on `$W$` may not be continuous either.
 
- If `$X$` has a direct effect on `$Y$`, the discontinuity in `$E[Y_{i}|W]$`
 at `$\bar{W}$` will confound the treatment effect.
 
- Example:
 - `$Y$` hours worked,
 - `$D$`: older-than-65 discounts,
 - `$W$`: age, `$X$`: social security benefit (non-work income)

---
## Density Test, or No Bunching

- Manipulation if agents know about the institutional details
 - If schools scoring lower than `$w = 50$` on
 standardized tests get labeled as dysfunctional, we might see 
 many schools to be right above 50
 
- In this case, we observe **bunching** around the threshold.
 - Agents are "manipulating" treatment assignment around the
 threshold.
 - Density of `$W_{i}$` is discontinuous at `$\bar{W}$`
 
- We would expect that `$E[Y_{1i}|W_{i}=w]$` would be also
 discontinuous.
 
- McCrary (2008) suggests a test of the null hypothesis that the
 density of `$W_{i}$` is continuous at `$\bar{W}$`.

---
## Digression: Bunching Analysis (集積点分析)

-   Bunching itself is an interesting economic phenomenon. It can be used to analyze a different question.

---
## Example: Ito and Sallee (2018, REStat)

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## Empirical Paper: Health Demand

- "The Effect of Patient Cost Sharing on Utilization, Health, and
  Risk Protection" by Hitoshi Shigeoka 2014 AER'

---
## Policy Issue: Medical Expenditure

- Medical expenditures are rising.
 - due to an aging population and coverage expansion
 - acute fiscal challenge to governments!
 
- Current expenditure on health (to GDP) in 2018 according to OECD
 Health Statistics 2019
 - U.S.A. (16.9%), Switzerland (12.2%), Germany (11.2%), France
 (11.2%), Sweden (11.0%), Japan (10.9%)\...
 
- One main strategy is higher patient cost sharing, that is, requiring
 patients to pay a larger share of the cost of care.
 
- Question: how does patient cost sharing affect
 - utilization (demand elasticity)?
 - health?
 - risk protection (out-of-pocket expenditures)?

---
## Background and Cross-sectional Data

- All Japanese citizens are mandatorily covered by health insurance.
 
- Use a sharp reduction in cost sharing for patients aged over 70 in
 Japan.
 
- The sources are the Patient Survey and the Comprehensive Survey of
 Living Conditions (CSLC). 1984-2008.
 
- Advantages
 - There are no confounding factors at age 70. We can isolate the
 effect of patient cost sharing.
 - Medical providers do not have incentive to differentiate prices
 by the patients' insurance type.
 - We can separate inpatient and outpatient.

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## Cost Sharing and Out-of-Pocket Medical Expenditure

- In sum, the proportion is 30% for `$<$` 69 and 10% for 70 `$\leq$`.
 
- Out-of-pocket medical expenditure for impatient admissions can reach
 27% for a 69-year-old.
 
- However, for 70, it would be reduced to 8.6%.
 
- We need to take the stop-loss into account.

---

---

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## Identification Strategy

- Standard RD designs.
 
- Basic estimation equation for the CSLC is
 `$$Y_{iat}=f(a)+\beta Post70_{iat}+X_{iat}^{\prime}\gamma+\varepsilon_{iat}.$$`
 - `$Y_{iat}$`: a measure of morbidity or out-of-pocket medical
 expenditure
 - `$f(a)$`: a smooth function of age.
 - `$X_{iat}$`: a set of individual covariates
 - `$Post70_{iat}$`: `$=0$` if individual `$i$` is over 70.
 
- Patient Survey/mortality data represents individuals who are present
 in the medical institutions/deceased.
 
- As in Card, Dobkin, and Maestas (2004), basic estimation equation
 for the Patient Survey and mortality data is
 `$$\log(Y_{at})=f(a)+\beta Post70_{at}+\mu_{at}.$$`

---
## Results: Outpatient Visits

- 10.3% increase in overall visits. The implied elasticity is
 `$-0.18$`.
 
- Sharp drop in the duration from the last visit by one day.
 
- The effect is heterogeneous across institutions, genders, and
 diagnoses.

---

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## Results: Inpatient Admissions

- Left: 8.2% increase in overall admissions. The implied elasticity is
 `$-0.16$`.
 
- Right: Surge (increase by 12.0%) in admissions with surgery.
 
- From robustness checks, the implied elasticity is around `$-0.2$`.

---

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##  Benefits: Health Outcomes

- We cannot find significant discontinuity in mortality.
 
- This result is expected because health is stock (Grossman 1972).
 
- There is no discontinuity in morbidity (self-reported health).
 
- The available health measures here are limited, so we would
 underestimate the benefit.

---

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## Benefits: Risk Reduction

- Another benefit is a lower risk of unexpected out-of-pocket medical
 spending.
 
- We use a nonparametric estimator for quantile treatment effects.
 
- Patients at the right tail of the distribution in particular are
 substantially benefited.

---

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## Discussion

- Price Elasticities
 - We cannot distinguish own- from cross-price effects.
 - However, for some diagnosis groups, cross-price effects should
 be nearly zero.
 - The overall effect of the price change for the groups is an
 approximately 10 percent increase in visits.
 
- Cost-Benefit Analysis
 - Imposing many assumptions, we speculate that the welfare gain of
 risk protection from lower patient cost sharing is comparable to
 the total social cost.
 - We cannot include welfare gains from health improvements.

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class: title-slide-section, center, middle
name: logistics
# Appendix: Formal Analysis

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## Formal Identification Analysis

-   Key: continuity assumptions: Both `$E[Y_{1i}|W_{i}=w]$` and
    `$E[Y_{0i}|W_{i}=w]$` are continuous at the threshold `$w=\bar{W}$`.
    -   This is not directly testable assumption (because we cannot
        observe `$Y_{1i}$` below the threshold).
    -   Will discuss several validating approaches.

-   To see how this works, notice that

`\begin{aligned}
    E[Y_{i}|W_{i}=w]= & E[Y_{0i}|W_{i}=w]\\
     & +\mathbf{1}\{w\geq\bar{W}\}\left(E[Y_{1i}|W_{i}=w]-E[Y_{0i}|W_{i}=w]\right)
\end{aligned}`

---
 
- Taking the limit of `$w$` to `$\bar{W}$` from above and below

`\begin{aligned}
    \lim_{w\uparrow\bar{W}}E[Y_{i}|W_{i} & =w]=\lim_{w\uparrow\bar{W}}E[Y_{0i}|W_{i}=w]=E[Y_{0i}|W_{i}=\bar{W}]\\
    \lim_{w\downarrow\bar{W}}E[Y_{i}|W_{i} & =w]=\lim_{w\downarrow\bar{W}}E[Y_{1i}|W_{i}=w]=E[Y_{1i}|W_{i}=\bar{W}]
\end{aligned}`
- Notice that we use continuity in the second equalities!

---
 
- Remember that

-   So, we have
    `$$E[Y_{1i}-Y_{0i}|W_{i}=\bar{W}]=\lim_{w\downarrow\bar{W}}E[Y_{i}|W_{i}=w]-\lim_{w\uparrow\bar{W}}E[Y_{i}|W_{i}=w]$$`
    -   LHS: Average treatment effect at the threshold
    -   RHS: We can observe from the data.
        -   Conditional expectation near the threshold.