## 24.1 Estimation and Inference

### 24.1.1 Local Randomization-based

Additional Assumption: Local Randomization approach assumes that inside the chosen window $$W = [c-w, c+w]$$ are assigned to treatment as good as random:

1. Joint probability distribution of scores for units inside the chosen window $$W$$ is known
2. Potential outcomes are not affected by value of the score

This approach is stronger than the Continuity-based because we assume the regressions are continuously at $$c$$ and unaffected by the running variable within window $$W$$

Because we can choose the window $$W$$ (within which random assignment is plausible), the sample size can typically be small.

To choose the window $$W$$, we can base on either

1. where the pre-treatment covariate-balance is observed
2. independent tests between outcome and score
3. domain knowledge

To make inference, we can either use

• (Fisher) randomization inference

• (Neyman) design-based

### 24.1.2 Continuity-based

• also known as the local polynomial method

• as the name suggests, global polynomial regression is not recommended (because of lack of robustness, and over-fitting and Runge’s phenomenon)

Step to estimate local polynomial regression

1. Choose polynomial order and weighting scheme
2. Choose bandwidth that has optimal MSE or coverage error
3. Estimate the parameter of interest
4. Examine robust bias-correct inference