Regression Discontinuity Extensions

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# Regression Discontinuity Extensions
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### Ian McCarthy | Emory University
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### Econ 771, Fall 2022
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# RD with Discrete Running Variable

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# Discrete Running Variables

- With lots of mass points, not a problem
- With sufficiently few mass points...
- Standard smoothness/continuity argument no longer applies
- Need to assume that observations are as good as randomly assigned above/below threshold (Local Randomization)

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# Local randomization versus continuity

- Often used as heuristic device in continuous running variable RD
- Requires stronger assumptions than continuity-based RD
    - Continuity-based RD allows for running variable to directly affect potential outcomes
    - Local randomization instead assumes `$x_{i}$` is randomly generated and independent of `$Y_{i}(1)$` and `$Y_{i}(0)$`:
    
`$$\small E[Y_{i} | x_{i} \geq c] - E[Y_{i} | x_{i}<c] = E[Y_{i}(1) | x_{i}\geq c] - E[Y_{i}(0)|x_{i}<c] = E[Y_{i}(1)] - E[Y_{i}(0)]$$`

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# Local randomization versus continuity

- Practical rephrasing...regression of `$y$` on `$x$` should have zero slope (pretty strong)
- "Regression functions are constant in the entire region where the score is randomly assigned"
- Possible to weaken this and impose assumption on some transformed potential outcome, for known transformation

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# Local randomization and inference

- Large sample approximation may be invalid if number of observations in bandwidth is small
- Turn to randomization-based inference (`rdrandinf`)

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# Local randomization and bandwidth

- Can still select "optimal" bandwidth or window around the threshold, but harder to do
- Cattaneo et al (2020) suggest using covariates known to be correlated with the outcome and the running variable
- Find largest window where this correlation is no different from 0 (`rdwinselect`)

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# Local randomization and bandwidth

With few mass points

- Minimum window is "interval between two consecutive mass points where the treatment status changes from zero to one"
- Only need to use two mass points
- Actual value of cutoff is meaningless since running variable is discrete

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# Discrete running variable in practice

- `rdrobust` has some adjustments for mass points and unique values
- `rdrandinf` for randomization inference with discrete running variable
- `rdwinselect` for optimal window (using covariate argument previously)
- Kolesar and Rothe (2018), `RDHonest`
    - Approximates bias due to larger-than-ideal bandwidth
    - Assume maximum bias, construct adjusted CI

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# Regression Kink

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# RD as a kink

Some discontinuities are just in the slope rather than the intercept:

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# RD as a kink

- Highly applicable
- Challenging to implement as it requires a lot of data
    - Slopes are hard to disentangle in raw data
    - Ganong and Jager (2020) JASA offer a possible test
- Implemented with `rdrobust` and `deriv=1` option
- Fuzzy Kink RD...with `rdrobust` and `deriv=1` and `fuzzy=t` options

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# RD with Multiple Cutoffs

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# Different types of mutliple cutoffs

- Noncumulative: Cutoff values differ across groups, but each group has only one cutoff (e.g., test scores with different thresholds across schools)
- Cumulative: one common score system for each person, but different treatments along the running variable (e.g., treatment 1 at age 60 and treatment 2 at age 65)
- Multiple scores: Units receive two scores and treatment is assigned based on a boundary for both scores (e.g., geographic boundary with latitude and longitude)
- Implemented with `rdmulti` in Stata and R

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# Noncumulative

- Nothing new here
- Just apply standard RD to pooled sample with re-centered running variable, `$\tilde{x} = x - c$`

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# Cumulative

- Receive treatment 1 if `$x_{i} < c_{1}$`, treatment 2 if `$c_{1} \leq x_{i} < c_{2}$`, etc.
- Problematic if person `$i$` is in multiple bandwidths
- Choose bandwidths to ensure non-overlapping populations around each threshold
- Can pool the estimates once non-overlapping windows are ensured

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# Multiple scores

- Applies to "geographic" RD

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# Bias and Undersmoothing

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# Asymptotic distribution

From Cattaneo et al. (2020):
`$$\text{MSE}(\hat{\tau}_{SRD}) = \text{Bias}^{2}(\hat{\tau}_{SRD}) + \text{Var}(\hat{\tau}_{SRD}) = (h^{2(p+1)}\mathcal{B})^{2} + \frac{1}{nh}\mathcal{V}$$`
- `$p$` is the polynomial degree of the local linear estimator ($p=1$ for linear)
- `$\mathcal{B}$` is the leading bias term of an expansion 
- `$\mathcal{V}$` is the leading variance term

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# Asymptotic distribution

Choice of `$h$` that minimizes this MSE (conditional on `$p$` and the kernel) is:
`$$h_{MSE} = \left(\frac{\mathcal{V}}{2(p+1)\mathcal{B}^{2}}\right)^{1/(2p+3)}n^{-1/(2p+3)}$$`

- for `$p = 1$`, `$o(h_{MSE}) = n^{-1/5}$`
- but choosing `$h_{MSE} \propto  n^{-1/5}$` does not lead to zero bias because bias reduces to zero at slower rate

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# Asymptotic distribution

- Calonico, Cattaneo, and Titunik (2015) argues for plug-in estimator for bias
- Idea: 
    - Estimate local RD estimate with quadratic terms
    - Use second derivative to approximate bias for the local linear RD
    - Adjust resulting local linear RD
- Calonico, Cattaneo, and Farrell (2020) argues for CE-optimal bandwidths