Description[^1] | Methods | Variance estimate | Degrees of freedom correction | Proof of Lemma 1 | References

Introduction | Sharp RD | Model | Plots | Inference based on local polynomial estimates | Bias-aware confidence intervals | Automatic bandwidth choice | Choice of curvature parameter | Inference when running variable is discrete | Fuzzy RD | Choice of curvature parameters | Extensions | Covariates | Aggregated data and weighted regression | Clustering | Specification testing | Optimal weights under Taylor smoothness class | Inference at a point | Diagnostics: leverage and effective observations | References

Summary | Examples | Collinear share matrix | Default way of dealing with collinear sectors | Other solutions | Extensions to multiple shifters and multiple endogenous variables | OLS | IV with a single endogenous regressor and multiple shift-share instruments | IV with multiple endogenous variables | References

Standard errors | Methods | Oracle standard errors | Overlap sample | Wald and LM tests | The score evaluated at the unrestricted estimator $\hat | Derivations | Finally, we show \eqref{eq:overlap_oracle}. The derivative of the moment condition\eqref{eq:moment_overlap_weights} with respect to $\pi_{k}=p_{k}$ (assumingcorrect specification of the propensity score) is given by\begin{equation*}-E[\lambda\frac{X_{ik}}{p_{k}^{2}(W_{i})} (\mu_{k}-\alpha^{\text{CW}}{k})\dot{p}{k}(W_{i})],\end{equation*}where we write $\lambda$ for $\lcw(W_{i})$. Since $p_{k}$ is a projection, byProposition 4 in @newey94ecta, the influence function for$\hat{\alpha}^{\text{CW}}{k}$ is given by\begin{multline*}\frac{1}{E[\lambda]}\left( \lambda \frac{X{ik}}{p_{k}(W_{i})}(Y_{i}-\alpha^{\text{CW}}{k})-\frac{\lambda}{p{k}(W_{i})}(\mu_{k}(W_{i})-\alpha^{\text{CW}}{k})(X{ik}-p_{k}(W_{i}))\right)\ | References

Setup | Example | References