F8 Regression discontinuity design

Question 1

What is the running variable and cutoff?

Answer 1

Running variable is an observed confounder that determines the treatment status at a specific value/cutoff. Typically continuous.

Question 2

What is a RDD?

Answer 2

Regression discontinuity design. Exploits a natural cutoff/threshold which distributes units into either control or treatment group.

Question 3

What does RDD estimate?

Answer 3

Local average treatment effect: LATE.

Question 4

What is the challenge with LATE?

Answer 4

The effect is limited to the individuals around the cutoff. Generalizability to the rest of the population is not possible.

Internal validity > external validity

Question 5

What is a key advantage of RDD?

Answer 5

Superior in handling unobserved confounders. It convincingly eliminates selection bias if assumptions holds.

Question 6

Was is the key assumption?

Answer 6

Continuity. Potential outcome are continous functions of the running variable and smooth passing through the cutoff.

All confounders are assumed to be continuous at the cutoff.

Treatment becomes independent of potential outcomes. D is the only variable that affects the outcome and jumps discontinuously at the cutoff.

No simultaneous treatments and the cutoff cannot be endogenous.

Question 7

What are examples of RDD?

Answer 7

Test-scores (SAT), geographical boarders, time, close elections and policy changes

Question 8

What must the assignment rule/cutoff be?

Answer 8

Known, precise and free of manipulation (if the RDD is sharp)

Question 9

Draw the DAG for RDD (both of them)

Answer 9

Squares.

D –> Y: The causal relation of interest.

X –> Y: Confounder. The running variable affect the outcome (independently from D). Out of influence under RDD

U: Unobserved confounder causing bias. Out of influence under RDD

Question 10

What is the mathematical formula?

Answer 10

Homogenous effects: Y_i = α + βx_i + δD_i + ε_i (changes only the intercept).

Heterogenous effects: Y_i = α + βx_i + δD_i + ε_i + θD_i x_i (the interaction term lets the function differ on both sides at the cutoff)

Question 11

What are the potential outcomes framework for RDD?

Answer 11

The thing with limit.

Question 12

What are the two types of RDDs?

Answer 12

Sharp RDD: Probability of treatment changes from 0 to 1 at the cutoff (deterministic). No common support (relies on extrapolation).

Fuzzy RDD: Gradual increase in probability of treatment. With a minor jump at the cutoff.

Question 13

What happens with the estimator in a fuzzy RDD?

Answer 13

It’s scaled to the probability of being treated.

Wald estimator (special case of IV estimator - binary outcome and some degree of non-compliance).

Question 14

Bandwidth

Answer 14

Narrow: Loss of statistical power
Broad: Risk of specification bias

Question 15

What is the main challenge to RDD?

Answer 15

Sorting.

If the cutoff is known then self-selection into treatment or control is possible. The continuity assumption doesn’t hold up.

Question 16

What are reasons for sorting?

Answer 16

The assignment rule is known in advance

Agents are interested in adjusting and they have time to adjust

The cutoff is endogenous to factors that affect the potential outcomes

Non-random heaping along the running variable

Question 17

How can possible sorting be inspected?

Answer 17

Covariate balance test

McCracy’s density test (power intensive - treat bin frequencies as dependent variable and running variable as independent for each group)

Question 18

What is the difference between matching and RDD?

Answer 18

Matching relies on confounders. RDD handles both observed and unobserved confounders.

Matching estimates ATE while RDD estimates LATE.

Question 19

What is extrapolation?

Answer 19

Lack of common support. We compared units with different values on the running variable.

Question 20

What are important considerations for specification of the function? Draw different examples.

Answer 20

Data could be linear or non-linear. This could lead to and effect being estimated that is due to misspecification and vice versa.

Question 21

What is the difference between parametric and nonparametric specification?

Answer 21

Parametric: Specify the functional form before hand.

Nonparametric: Data driven without prior assumptions on the functional form (local E[Y] on the running variable - could be both quadratic, linear and lowess)

Question 22

How do you avoid overfitting?

Answer 22

Start with a linear model and try a polynomium (allow one turning point).

Gelman & Imbens (2019) have shown that adding more polynomials introduce bias

Question 23

What an example of a nonparametric approach?

Answer 23

Kernel regression (weighting observations closer to cutoff higher - you sort of phase in the bandwidth)

Question 24

Should you or should you not let the functions differ for the control and treatment group?

Answer 24

Always let them differ according to Lee & Lemieux (2010)

Question 25

Should you cluster on the running variable?

Answer 25

No never. Use honest confidence intervals or robust standard errors.

Question 26

How does the continuity assumption slightly change in fuzzy RDDs?

Answer 26

The conditional expectation of the potential outcomes is changing smoothly through the cutoff

Question 27

What do you estimate with fuzzy RDD?

Answer 27

LATE for compliers (those whose treatment status changed right around the cutoff)

Question 28

What is nonrandom heaping on the running variable? And what is the solution?

Answer 28

Individuals disproportionately report certain values of a variable, often due to convenience, cognitive biases, or intentional behavior. E.g. clustering specific values/rounding like babies weight.

Solution: Donut-hole RDD.

Question 29

What is a popular design in RDD?

Answer 29

The close election: This “at the margins of a close race” is crucial because the idea is that it is at the margins of a close race that the distribution of voter preferences is the same.

Question 30

What is a regression kink design?

Answer 30

The linear trends flatten out after the cutoff (jump in the first derivative) e.g. government benefits at a threshold.