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Applied Micro-econometrics

ECON5006

Solutions to Practice problems on RDDs


Part 1: Definitions and Basic Concepts

1.   In your own words, explain why the regression discontinuity method is usually referred to “as good as a randomized experiment” .

Solution: The RDD is referred to as good as a randomized experiment” because       there exists an “optimal bandwidth” or “neighbourhood” around the discontinuity in  which observations assigned to treatment and control are statistically identical in        terms of their observable characteristics (and by assumption, are statistically identical in terms of their unobservable characteristics). Hence, the only difference between    these two groups is that half are eligible to the program (the treatment group) and the other half is not (the control group). In that sense, the RDD mimics an RCT.

2.   What is the difference between a fuzzy and a sharp regression discontinuity design (RDD)? What is the difference in the implementation of these models? (Hint: the   fuzzy RDD mimics an IV).

Solution: The difference between a fuzzy RD and a sharp RD is that in the fuzzy RD, individuals who are eligible for the treatment do not fully take-up the treatment and/or those who are not eligible for the treatment some end up benefiting from it (i.e., there is imperfect compliance); whereas in the case of the sharp RD there is perfect              compliance.

The fuzzy RD is implemented as an IV model, in which you have a first and a second stage. The first stage regresses participation to the program on the instrument, where the instrument is a dummy variable T that takes the value of 1 if the individual is       eligible for the treatment and 0 o.w. The second stage regresses the outcome of          interest on the “instrumented” participation to the program.

The sharp RD only” involves one equation that is the outcome Y regressed on the instrument T.

Both the fuzzy (first and second stage) and the sharp RD models include “weights” or “polynomials” (also sometimes referred to as the splines”) of the distance of the       running variable to the cut-off. These polynomials help absorb potential observable    (and unobservable) differences across individuals located at different points from the cut-off point.


3.   In the context of an RDD, in which of the following figures do you expect to find a “jump” around the threshold:

I.      A figure that plots the running variable against the outcome

YES!

II.      A figure that plots the running variable against an important covariate

NO!

III.      A figure that plots the running variable against the density of the observations

NO!

4.   Explain why it is often necessary to use large-scale data to estimate an RDD?

Solution: large scale data is often necessary to estimate an RDD because this model only uses the sample of observations around the threshold (i.e., those within the “optimal        bandwidth” or “optimal neighbourhood” around the discontinuity), which implies that all observations outside the optimal bandwidth (which is the majority of the data) are            excluded from the model.

So, in order to have enough power to detect statistically significant effects around the discontinuity, its better to start with a very large sample!

Part 2: Study using an RDD (*)

Interest in public health insurance in the United States is in part motivated by the relatively   poor health of children in families with low incomes. Not only are children from poor           families more likely to be born in poor health, but the relative health of these children           worsens with age. To address health disparities in child health, policy makers have increased access to medical care for children through expansions in publicly provided health insurance – i.e., Medicaid, U.S. health insurance program. These expansions aim to promote health      among children by providing timely access to health services. A major focus in preventive    care is the early diagnosis and treatment of physical and mental health needs, which also       limits pervasive health problems (and costs) in adulthood.

Despite drastic expansions in Medicaid insurance over the last 30 years, little is known about its causal effects on children’s health.

Using an exogenous source of variation in healthcare coverage, a group of researchers aimed to examine the immediate and longer-term effects of health insurance during childhood on     children’s health outcomes (i.e., mortality). To provide estimates, researchers used as             identification strategy the fact that Medicaid introduced a policy that extended eligibility to   the program for children of low-income households born after September 30, 1983. This       eligibility rule serves itself nicely to conduct an RDD that compares the outcomes of children born before and after September 30, 1983 –i.e., treated vs. untreated children.

Table 1 reports the estimated discontinuity in childhood eligibility at the September 30, 1983 cut-off, by race.1 Estimates show that, 17.13 percent of Black children gained on average just under five years of eligibility of health insurance compared to similar gains in eligibility for  8.2 percent of white children. As a result, Black children gained, on average, 0.82 years of    eligibility, which is more than twice the average years gained by white children (0.37 years).

 

To measure the effects of the policy on children’s health, policy makers use mortality records that have information on the exact date of birth, demographic characteristics, and the date and cause of mortality. These data thus allow them to follow cohorts of children –Black and white -- born before and after the cut-off and up to adulthood.

Exploiting the age eligibility cut-off for Medicaid, policy makers estimate an RDD               framework and the results are shown in Table 3. Each cell in the table reports the estimated  RD effect by race and age group using different polynomial specifications and window sizes (or bandwidths).

 

Questions:

1)  What would be a naïve specification linking changes in health insurance and individual mortality?

Yi = α + β1 * Health_Insurancei + Xi θ+ εi

where Yi represents the probability of mortality for individual i, “Health_Insurance is a dummy variable that takes the value of 1 if an individual has health insurance and 0 o.w.  X is a matrix of individual characteristics and ε is the error term.

2)  Explain why is it necessary to find an exogenous source of variation in health insurance to measure its effects on outcomes?

Solution: Because individuals with health insurance significantly differ from those   without, in ways that are observed and unobserved to the researcher.  For instance,    those with health insurance may be more likely to have underlying health conditions, be older, work in more high-risk occupations, be more risk averse, etc., and these      factors could have an effect on the health of individuals.

3)  Using the exogenous source of variation in public health insurance described above, write a model that helps identify the causal effect of the program on individuals’     health outcomes.

Solution: To identify the causal effect of the public health insurance program on the outcome, a researcher could estimate the following RDD model where the running   variable is the birth month:

Yca = α + β1 * T +f(c) + Xca θ + εca

where Yca represents the mortality rate for cohort c at age a, T is an indicator for         whether a cohort was born after September 30, 1983,f(c) is a smooth function of birth months, X is a matrix of individual characteristics, and ε is the error term. The            coefficient of interest is β1, which represents the causal effect of being born after  September 30, 1983 on the mortality rate.

The central question for implementation is how to model the function of birth month   cohortf(c). Since a priori a researcher does not know what functional form best fits” the model, he/she should test different specifications: e.g., linear splines, quadratic      splines, cubic, etc.

The model should be estimated using observations that lie within the “optimal            bandwidth” .  To determine the optimal bandwidth, a researcher should employ the      econometric test developed by Calonico et al., (2014) or by Imbens and Kalynaraman (2012) [Stata codes have been developed for this purpose and they are widely used by applied economists], and in addition, as a robustness check, re-estimate the model

using slightly larger or slightly smaller optimal bandwidths to check for the stability of the coefficient of interest.

To identify the effect of the policy by racial groups, the model could be estimated    across different samples  e.g., sample 1 only includes white individuals, sample 2   only includes African American individuals, sample 3 only includes Hispanics,        sample 4 only includes Chinese, etc. The coefficient β1 obtained from regressing the model across these different samples would provide the effect of being born after     September 30, 1983 on the mortality rate on that specific racial group.

4)  What are the necessary assumptions required for this model to provide valid estimates?

For the RDD to provide arguably causal evidence, the following assumptions need to hold:

1)  Continuity in treatment assignment: There needs to be a clear rule of treatment assignment that is consistently applied across the whole range of values in the running variable.  For instance, a clear rule of treatment assignment is:

When W >= c, treatment is always 1, when W < c treatment is always 0. where W is the running variable and c is the threshold.

2)  Unconfoundedness: After conditioning on W, a unit is either assigned to treatment or to control, and there is no further variation in treatment exposure.

3)  Other covariates that may influence the outcome (e.g., sex, race/ethnicity, etc.) do not change discontinuously at the threshold.

4)  There is NO manipulation in the running variable.

5)  What would be the different types of figures that researchers need to show to provide evidence that the empirical design is valid? Describe each of these figures and what  you expect to find.

The main figures in an RDD are:

Figure 1: shows the running variable against the probability of treatment.

•   This figure should show a jump” in the probability of treatment around the  discontinuity. The magnitude of the jump” is equal to the % of compliers. If the jump is < 100%, then it means that there is imperfect compliance.

Figure 2: shows the running variable against the covariate(s). If there are 10 covariates, then you should show 10 figures, one for each covariate.

•   This(these) figure(s) should NOT show a jump” in the relationship between the running variable and the covariates. If there is a “jump” in any of the       figures it means that there may be sorting of individuals into or out of the     treatment in terms of this particular characteristic that shows the jump” .

Figure 3: shows the running variable against the density (or distribution) of the observations.

•   This figure should NOT show a jump” .  If there is a jump” around the       discontinuity, this may be suggestive of manipulation in the running variable which is a major source of bias in the treatment effect.

Figure 4: shows the running variable against the outcome.

•   If there is an actual treatment effect (i.e., β 1 is different from 0), then there    should be a jump” around the threshold. The magnitude of the jump is        equivalent to the causal effect of the treatment (in the case of a sharp RD) or equivalent to the intent-to-treat effect of the treatment (in the case of a fuzzy RD).

6)  Describe the results shown in Table 3. What is the main finding? Are there any heterogenous effects by race or by age?

Each cell in Table 3 shows estimates of β1, the effect of being born after Sept 30th,   1983 on mortality rates by child’s age and race. The results are estimated using        different model specifications that vary by the functional form of the polynomials or splines (linear, quadratic, cubic) and by the bandwidths used (4-year bandwidths, 3- year bandwidths, and 2-year bandwidths).

While the table shows many coefficients, there are a few patterns that stand out.

•   For instance, Black children born after Sept 30th, 1983 were less likely to die by ages 15- 18 years as a result of the health insurance program, compared to other Black children of similar ages but who were born just before Sept 30th, 1983.

•   The coefficients derived from the specification using the four-year observation window are negative (suggesting a decline in mortality) and statistically           significant under all four functional form specifications.

•   If we consider the coefficient -0.443 in column 1 for Black children, ages 15-  18, under the linear specification, this coefficient suggest that children who      were randomly covered by the health insurance program based on the date of  birth, experienced a -0.443 change in the probability of dying, which, with       respect to the baseline mean (shown in the bottom row) of 2.32, would imply a change of -0.443/2.32 *100% = - 19% with respect to the outcome mean.

•   For the most part, the coefficient estimates are robust to alternative window sizes, indicating similarly sized and significant declines.

•   There is also little evidence that the health insurance policy improved       mortality rates for young (ages 4-7) or old (ages 19-23) African American children.

•   There is no evidence of a change in mortality rates for white children. In fact, the point estimates for white children are much smaller in absolute terms than those for Black children, and they switch from positive to negative across       specifications.