27.7 Estimation and Inference

27.7.1 Local Randomization-Based Approach

The local randomization approach additionally assumes that within the chosen window $W = [c - w, c + w]$, treatment assignment is as good as random. This requires:

1. The joint probability distribution of running variable values inside $W$ to be known.
2. Potential outcomes to be independent of the running variable within $W$.

This is a stronger assumption than continuity-based identification, as it requires that regression functions are smooth at $c$ and remain unaffected by $X_i$ within $W$.

Since researchers can choose the window $W$ (where random assignment plausibly holds), the sample size can often be small.

The selection of $W$ can be based on:

1. Pre-treatment covariate balance: Ensure covariates are similar across the threshold.
2. Independent tests: Check for independence between the outcome and the running variable.
3. Domain knowledge: Use theoretical or empirical justification for the window choice.

For inference, researchers can use:

• (Fisher) randomization inference
• (Neyman) design-based methods

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27.7.2 Continuity-Based Approach

Also known as the local polynomial regression method, this approach estimates treatment effects by fitting a polynomial model locally around the cutoff. Global polynomial regression is not recommended due to issues such as:

• Lack of robustness
• Overfitting
• Runge’s phenomenon (oscillatory behavior at boundaries)

Steps for Local Polynomial Estimation

1. Choose the polynomial order and weighting scheme
2. Select an optimal bandwidth (minimizing MSE or coverage error)
3. Estimate the parameter of interest
4. Perform robust bias-corrected inference

This method ensures that estimation remains local, capturing the treatment effect precisely at the cutoff.

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