RDD

Question 1

How do you reduce the mistake of mistaking non-linearity for discontinuity in RDD?

Answer 1

1) try different pth-order polynomials.
2) focus on observations near the cutoff

Question 2

Regression Discontinuity Designs (RDD)

Answer 2

-RDD is a quasi-experimental design used to study causal relationships in situations where randomized controlled trials are not feasible or ethical.
-RDD evaluates the impact of a program by defining the effect of a treatment on a population of individuals.
-Examples include admission to school affecting educational outcomes and abortion policy affecting women’s health, adolescent pregnancies, and school dropout

Question 3

Sharp RD vs. Fuzzy RD:

Answer 3

-RDD comes in two styles: sharp and fuzzy.
-Sharp RD is used when treatment status is a deterministic and discontinuous function of a covariate.
-Fuzzy RD exploits discontinuities in the probability or expected value of treatment conditional on a covariate.

Question 4

Identifying Causal Effects with RDD:

Answer 4

RDD identifies causal effects by exploiting precise knowledge of the rules determining treatment.
RDD focuses on arbitrary rules that provide good experiments in a highly rule-based world.
RDD is suitable for scenarios where researchers are interested in the causal effect of a binary or probability intervention treatment on a dependent variable.

Question 5

Applications of RDD:

Answer 5

-Lemieux and Milligan (2008) used RDD to study the incentive effects of social assistance in Quebec on the employment rate.
-RDD can be applied in various contexts, exploiting specific rules and thresholds to create experiments and estimate causal effects.

Question 6

Challenges and Considerations in RDD:

Answer 6

-RDD may face challenges in handling non-linearities in the running variable.
-Graphical inspection is crucial in RDD analysis to ensure the validity of the results.
-Researchers need to carefully choose polynomial specifications and focus on observations near the cutoff for reliable estimates.

Question 7

Fuzzy RD as Instrumental Variable:

Answer 7

-Fuzzy RD uses discontinuities in the probability or expected value of treatment as an instrumental variable.
-It defines the probability of treatment at a cutoff, with different functions on either side of the cutoff.
-Fuzzy RD naturally leads to a 2SLS estimation strategy for causal effect estimation.

Question 8

FD-RD (First Difference Regression Discontinuity):

Answer 8

-FD-RD estimates the difference in outcome variables by exploiting the longitudinal nature of information around the discontinuity.
-It involves estimating the first difference between outcome variables, assuming similarity of individuals close to the discontinuity.

Question 9

Advantages of RDD:

Answer 9

-RDD is powerful in measuring differences in small windows around the discontinuity point.
-It exploits heterogeneity in treatment, eliminating the need for a control group as in Difference-in-Differences (DiD).

Question 10

Setting Up RDD:

Answer 10

-RDD is based on the premise of a known threshold or cutoff in the assignment variable.
-The assignment variable is the running variable that determines treatment status.
-The cutoff represents a point beyond which treatment changes, creating a natural experiment.

Question 11

Cutoff Specification:

Answer 11

-Choosing an appropriate cutoff is crucial for RDD.
-The cutoff should be theoretically justified and align with the discontinuity in the treatment.
-Researchers often conduct sensitivity analyses with different cutoff specifications to ensure robustness.

Question 12

Assignment Mechanism:

Answer 12

The assignment mechanism in RDD is deterministic, where treatment status changes sharply at the cutoff.
For sharp RDD, treatment is a discontinuous function of the running variable, making it a binary switch.
For fuzzy RDD, treatment is determined by the probability of treatment, allowing for a more gradual transition

Question 13

Graphical Inspection:

Answer 13

A key step in RDD is graphically inspecting the relationship between the running variable and the outcome.
A scatter plot with the cutoff as a reference helps visualize the discontinuity.
Visual confirmation of a jump or discontinuity in the outcome around the cutoff is essential.

Question 14

Bandwidth Selection:

Answer 14

Bandwidth refers to the range of observations around the cutoff used in the analysis.
Choosing an appropriate bandwidth is important for balancing precision and bias.
Researchers often perform sensitivity analyses with different bandwidths to assess robustness.

Question 15

Common Pitfalls in RDD:

Answer 15

RDD assumes that observable and unobservable characteristics change smoothly around the cutoff.
Researchers should be cautious of confounding factors that may violate this assumption.
Small sample sizes and insufficient data density around the cutoff can lead to imprecise estimates.

Question 16

2SLS Estimation in Fuzzy RDD:

Answer 16

Fuzzy RDD naturally leads to a 2SLS (Two-Stage Least Squares) estimation strategy.
The first stage involves regressing treatment status on the running variable.
The second stage estimates the causal effect using predicted treatment status as an instrument.

Question 17

mathematical set up

Answer 17

Running Variable (X):

X represents the continuous variable that is used to determine treatment assignment.
Cutoff Point (c):

c is the predetermined cutoff point or threshold value of the running variable. It is the point at which the treatment changes.
Treatment Indicator (D):

D is a binary treatment indicator variable that takes the value 1 if X >= c (treatment group) and 0 if X < c (control group).
Outcome Variable (Y):

Y represents the outcome variable of interest.
Now, let’s set up the regression equations for the potential outcomes:

Potential Outcome for Treated Units (Y1):

For units receiving treatment (when X >= c):
Y1 = f(X) + beta1 * D + epsilon1
Potential Outcome for Control Units (Y0):

For units not receiving treatment (when X < c):
Y0 = f(X) + beta0 * D + epsilon0
Observed Outcome (Y):

The observed outcome is a combination of Y1 and Y0 based on the treatment assignment:
Y = D * Y1 + (1 - D) * Y0
Regression Model:

The regression model includes the observed outcome as a dependent variable and the treatment indicator:
Y = alpha + rho * D + X’ * gamma + epsilon
where alpha is the intercept, rho is the treatment effect, X’ is a vector of covariates, and epsilon is the error term.
In this setup, the parameter rho represents the causal effect of the treatment at the cutoff point. The main idea is to estimate rho by exploiting the discontinuity in treatment assignment at the cutoff. Researchers often estimate the local average treatment effect (LATE) around the cutoff, focusing on the individuals very close to the threshold. This provides an estimate of the treatment effect for those affected by the specific cutoff rule.