Regression discontinuity: review with extensions

Abstract

In treatment effect analysis, often the treatment takes a particular structure: ‘on’ if an underlying continuous variable crosses a threshold, and ‘off’ otherwise. Such a treatment occurs in various institutional settings such as a test score crossing a threshold to graduate, or income falling below a threshold to qualify for an aid. In this kind of cases, the study design is called ‘regression discontinuity (RD)’, which is popular in analyzing observational data, as long as the treatment takes the required form. This paper reviews RD to convey its essentials, and provides some extensions. First, the main RD idea based on local randomization due to an institutional/legal break is introduced. Second, treatment effects identified by RD are explored. Third, popular RD estimators are reviewed. Fourth, main specification tests are examined. Fifth, special RD topics are reviewed. Also, an empirical illustration is provided.

Appendix

1.1 Identified RD effect when continuity fails

Rewrite (2.1) as

$$\begin{aligned} E(Y|S)=\mathring{\beta }_{d}E(D|S)+\mathring{m}(S)\, \text {where} \, \mathring{m}(S)\equiv m(S)-\frac{m(0^{+})-m(0^{-})}{E(D|0^{+})-E(D|0^{-})} E(D|S). \end{aligned}$$

It holds that

$$\begin{aligned} E\{\mathring{m}(S)|0^{+}\}= & {} m(0^{+})-\frac{m(0^{+})-m(0^{-})}{ E(D|0^{+})-E(D|0^{-})}E(D|0^{+}), \\ E\{\mathring{m}(S)|0^{-}\}= & {} m(0^{-})-\frac{m(0^{+})-m(0^{-})}{ E(D|0^{+})-E(D|0^{-})}E(D|0^{-}). \end{aligned}$$

Then $E\{\mathring{m}(S)|S\}$ is continuous at $S=0$ because

$$\begin{aligned} E\{\mathring{m}(S)|0^{+}\}-E\{\mathring{m}(S)|0^{-}\}=m(0^{+})-m(0^{-})- \{m(0^{+})-m(0^{-})\}=0. \end{aligned}$$

1.2 Optimal plug-in bandwidth

The optimal bandwidth $h_{opt}$ proposed by Imbens and Kalyanaraman (2012) for SRD is

$$\begin{aligned} h_{opt}=C_{k}\left[ \frac{\hat{\sigma }^{2}(0^{-})+\hat{\sigma }^{2}(0^{+})}{ \hat{f}_{S}(0)[\{\hat{m}^{(2)}(0^{+})-\hat{m}^{(2)}(0^{-})\}^{2}+\hat{r}_{-}+ \hat{r}_{+}]}\right] ^{1/5}N^{-1/5} \end{aligned}$$

where $C_{k}$ is a kernel-dependent constant (5.40 for the uniform kernel and 3.44 for the triangular kernel $K(t)=\max (0,1-|t|)=(1-|t|)1[|t|<1]$), $\hat{f}_{S}(0)$ is an estimator for $f_{S}(0)$, $\hat{\sigma }^{2}(s)$ is an estimator for V(Y|s), $\hat{m}^{(2)}(s)$ is an estimator for the second derivative of E(Y|s), and $\hat{r}_{-}$ and $\hat{r}_{+}$ are functions of $\hat{\sigma }^{2}(0^{-})$ and $\hat{\sigma }^{2}(0^{+})$. Obtain $h_{opt}$ with the following three steps.

First, using a pilot rule-of-thumb bandwidth $h_{1}=1.84\cdot SD(S)\cdot N^{-1/5}$ for the uniform kernel, let $Q_{1i}^{-}\equiv 1[-h_{1}<S_{i}<0]$ and $Q_{1i}^{+}\equiv 1[0<S_{i}<h_{1}]$ and

$$\begin{aligned} \hat{\sigma }^{2}(0^{-})&\equiv \frac{1}{N_{1-}-1}\sum _{i}Q_{1i}^{-}(Y_{i}- \overline{Y}_{1}^{-})^{2}\quad \text {where}\,\,N_{1-}\equiv \sum Q_{1i}^{-}\,\,\text { and }\overline{Y}_{1}^{-}\equiv \frac{1}{N_{1-}}\sum \limits _{i}Q_{1i}^{-}Y_{i}, \\ \hat{\sigma }^{2}(0^{+})&\equiv \frac{1}{N_{1+}-1}\sum _{i}Q_{1i}^{+}(Y_{i}- \overline{Y}_{1}^{+})^{2}\quad \text {where}\,\,N_{1+}\equiv \sum Q_{1i}^{+} \,\,\text { and }\,\,\overline{Y}_{1}^{+}\equiv \frac{1}{N_{1+}}\sum \limits _{i}Q_{1i}^{+}Y_{i}, \\ \hat{f}_{S}(0)&=\frac{N_{1-}+N_{1+}}{2Nh_{1}}. \end{aligned}$$

Second, set the second pilot bandwidths:

$$\begin{aligned} h_{2}^{+}=3.56\left[ \frac{\hat{\sigma }^{2}(0^{+})}{\hat{f}_{S}(0)\{\hat{m} ^{(3)}(0)\}^{2}}\right] ^{1/7}N_{1+}^{-1/7}\,\,\text {and}\,\, h_{2}^{-}=3.56\left[ \frac{\hat{\sigma }^{2}(0^{-})}{\hat{f}_{S}(0)\{\hat{m} ^{(3)}(0)\}^{2}}\right] ^{1/7}N_{1-}^{-1/7} \end{aligned}$$

where $\hat{m}^{(3)}(0)$ for the third derivative of E(Y|0) is to be obtained as $6\hat{\gamma }_{4}$ from the LSE to

$$\begin{aligned} Y=\gamma _{0}+\gamma _{1}Z+\gamma _{2}S+\gamma _{3}S^{2}+\gamma _{4}S^{3}+error. \end{aligned}$$

Let $Q_{2i}^{-}\equiv 1[-h_{2}^{-}<S_{i}<0]$ and $Q_{2i}^{+}\equiv 1[0<S_{i}<h_{2}^{+}]$. Apply LSE to

$$\begin{aligned} Q_{2}^{-}Y=Q_{2}^{-}(\lambda _{0}^{-}+\lambda _{1}^{-}S+\lambda _{2}^{-}S^{2}+U^{-})\,\,\text {and}\,\,Q_{2}^{+}Y=Q_{2}^{+}(\lambda _{0}^{+}+\lambda _{1}^{+}S+\lambda _{2}^{+}S^{2}+U^{+}) \end{aligned}$$

to obtain $\hat{m}^{(2)}(0^{-})=2\hat{\lambda }_{2}^{-}$ and $\hat{m} ^{(2)}(0^{+})=2\hat{\lambda }_{2}^{+}$.

Finally, obtain

$$ \begin{aligned} \hat{r}_{-}=\frac{2160\cdot \hat{\sigma }^{2}(0^{-})}{N_{2-}(h_{2}^{-})^{4}}\; \& \; \hat{r}_{+}=\frac{2160\cdot \hat{\sigma }^{2}(0^{+})}{ N_{2+}(h_{2}^{+})^{4}}\quad \text {where}\,\,N_{2-}\equiv \sum _{i}Q_{2i}^{-} \; \& \; N_{2+}\equiv \sum _{i}Q_{2i}^{+} \end{aligned}$$

and then $h_{opt}$. Imbens and Kalyanaraman (2012) proposed an optimal bandwidth for FRD in their Eq. (2.4), but then noted that the above $h_{opt}$ for SRD would be adequate even for FRD, as it is based on the numerator estimation in FRD.

1.3 LLR-type score-density continuity test

Let $h_{1}$ be the interval size for the first-stage histogram of the McCrary (2008) test, and for an even number n, there are n / 2 intervals to be constructed on either side of 0. Consider left and right intervals of width $h_{1}$ around 0:

$$ \begin{aligned}{}[-jh_{1},\ -(j-1)h_{1}),\ \ j=\frac{n}{2},\frac{n}{2}-1,\ldots ,1 \quad \& \quad [(j-1)h_{1},\ jh_{1}),\ j\,=\,1,2,\ldots ,\frac{n}{2}. \end{aligned}$$

The midpoints of these intervals are

$$ \begin{aligned} G_{j}\equiv -(j-0.5)h_{1},\ \ j=\frac{n}{2},\frac{n}{2}-1,\ldots ,1 \quad \& \quad (j-0.5)h_{1},\ \ j\,=\,1,2,\ldots ,\frac{n}{2}; \end{aligned}$$

‘G’ stands for grid points and there are n midpoints. Let $R_{j}$ denote the histogram height at $G_{j}$; $(R_{j},G_{j})$, $j=1,\ldots ,n$ are to be taken as “ observations” in the second stage.

The test is based on $\ln \hat{\varphi }_{0}-\ln \hat{\psi }_{0}$ that come from the LLR intercept estimates in ($h_{2}$ is a second-stage bandwidth)

$$\begin{aligned}&\min _{\varphi _{0},\varphi _{1}}\sum _{j=1}^{n}(R_{j}-\varphi _{0}-\varphi _{1}G_{j})^{2}K(\frac{G_{j}}{h_{2}})1[0<G_{j}],\\&\min _{\psi _{0},\psi _{1}}\sum _{j=1}^{n}(R_{j}-\psi _{0}-\psi _{1}G_{j})^{2}K(\frac{ G_{j}}{h_{2}})1[G_{j}<0]. \end{aligned}$$

Specifically, the test statistic is

$$\begin{aligned} \hat{\theta }&\equiv \ln \hat{f}^{+}-\ln \hat{f}^{-}\,\,\text {where} \,\,\hat{f}^{+}\equiv \sum _{j}K\big (\frac{G_{j}}{h_{2}}\big )1[0<G_{j}]\frac{ H_{2}^{+}-H_{1}^{+}G_{j}}{H_{2}^{+}H_{0}^{+}-(H_{1}^{+})^{2}}R_{j},\\ \hat{f}^{-}&\equiv \sum _{j}K\big (\frac{G_{j}}{h_{2}}\big )1[G_{j}<0]\frac{ H_{2}^{-}-H_{1}^{-}G_{j}}{H_{2}^{-}H_{0}^{-}-(H_{1}^{-})^{2}}R_{j},\\ H_{k}^{+}&\equiv \sum _{j}K\big (\frac{G_{j}}{h_{2}}\big )1[0<G_{j}]G_{j}{}^{k}\,\,\text {and}\,\, H_{k}^{-}\equiv \sum _{j}K\big (\frac{G_{j}}{h_{2}}\big )1[G_{j}<0]G_{j}{}^{k}. \end{aligned}$$

As for the asymptotic distribution, the proposition in McCrary (2008), p.702) is that, for the triangular kernel, under $h_{2}\rightarrow 0$, $Nh_{2}\rightarrow \infty$, $h_{1}/h_{2}\rightarrow 0$, and $h_{2}^{2}\sqrt{ Nh_{2}}\rightarrow B\in [0,\infty )$,

$$\begin{aligned} \sqrt{Nh_{2}}(\hat{\theta }-\theta )\leadsto N\left\{ \frac{B}{20}\left( \frac{ -f^{+\prime \prime }}{f^{+}}-\frac{-f^{-^{\prime \prime }}}{f^{-}}\right) ,\ \frac{ 24}{5}\left( \frac{1}{f^{+}}+\frac{1}{f^{-}}\right) \right\} \end{aligned}$$

where $f^{+}\equiv f_{S}(0^{+}),$ $f^{-}\equiv f_{S}(0^{-})$, $\theta \equiv \ln f^{+}-\ln f^{-}$ and ‘$^{\prime \prime }$’ denotes the second derivative. With under-smoothing $h_{2}^{2}\sqrt{Nh_{2}}\rightarrow 0$, the asymptotic bias can be ignored. In practice, adopt the under-smoothing to ignore the asymptotic bias. Choosing two bandwidths $h_{1}$ and $h_{2}$ is a problem. McCrary (2008) stated that the choice of $h_{1}$ does not matter much whereas the choice of $h_{2}$ does, which is questionable though; (McCrary 2008, p. 705) suggested $h_{1}=2SD(S)N^{-1/2}$ (then $h_{2}$ may be chosen by visual inspection).