Regression Discontinuity Designs With Sample Selection

Abstract

This article extends the standard regression discontinuity (RD) design to allow for sample selection or missing outcomes. We deal with both treatment endogeneity and sample selection. Identification in this article does not require any exclusion restrictions in the selection equation, nor does it require specifying any selection mechanism. The results can therefore be applied broadly, regardless of how sample selection is incurred. Identification instead relies on smoothness conditions. Smoothness conditions are empirically plausible, have readily testable implications, and are typically assumed even in the standard RD design. We first provide identification of the “extensive margin” and “intensive margin” effects. Then based on these identification results and principle stratification, sharp bounds are constructed for the treatment effects among the group of individuals that may be of particular policy interest, that is, those always participating compliers. These results are applied to evaluate the impacts of academic probation on college completion and final GPAs. Our analysis reveals striking gender differences at the extensive versus the intensive margin in response to this negative signal on performance.

KEY WORDS:

• Extensive margin
• Fuzzy design
• Gender differences
• Intensive margin
• Missing outcomes
• Performance standard
• Regression discontinuity
• Sample selection

ACKNOWLEDGMENT

The author would like to thank anonymous referees for valuable comments.

Notes

1 Identification of the standard RD design has been discussed in Hahn, Todd, and van der Klaauw (2001), Lee (2008), and Dong (2016). Inference was discussed by Porter (2003), Imbens and Kalyanaraman (2012), Calonico, Cattaneo, and Titiunik (2014), Cattaneo, Frandsen, and Titiunik (2015), Otsu, Xu, and Matsushita (2015), and Feir, Lemieux, and Marmer (2016). See Cattaneo, Titiuni, and Vazquez-Bare (2016) for a comparison of different inference approaches for the standard RD design.

2 In particular, Staub (2014) discussed bounds under two alternative assumptions. The first assumption assumes that treatment effects are nonnegative for everyone. The second assumption assume that treatment effects are nonnegative for switchers and have the same sign for always participants, and further that one knows that $\text{ATE}>0$ or $\text{ATE}<0$.

3 Y*_t ≡ Y*(t, S_t) for t = 0, 1.

4 Assume T = h(R, V) for unobservables V, which can be a vector. Without loss of generality, one can write T = h₁(R, V)Z + h₀(R, V)(1 − Z). The function h_z(R, V) for z = 0, 1 describes the treatment assignment below or above the cutoff. Define then T_z(r) ≡ h_z(r, V) for z = 0, 1.

5 Alternatively, one could assume that $F_{Y_t^{*},S_t,\Theta |R}(y,s|r)$ for any Θ ∈ {A, N, C} is continuous at r₀.

6 That is, smoothness conditions need to hold for the selected sample in order for Equation (3(3) $$\begin{array}{ccc}& & E\left[g\left(Y_1^{*}\right)|S_1=1,R=r_0,C\right]-E\left[g\left(Y_0^{*}\right)|S_0=1,R=r_0,C\right]\hfill \\ & & =\frac{{lim}_{r\downarrow r_0}E\left[g\left(Y\right)|R=r,S=1\right]-{lim}_{r\uparrow r_0}E\left[g\left(Y\right)|R=r,S=1\right]}{{lim}_{r\downarrow r_0}E\left[T|R=r,S=1\right]-{lim}_{r\uparrow r_0}E\left[T|R=r,S=1\right]}.\hfill \end{array}$$(3) ) to identify a causal effect for the selected sample.

7 In practice, the fuzzy RD estimator along with its robust bias-corrected inference can be conveniently implemented using the Stata command rdrobust.ado (https://sites.google.com/site/rdpackages/rdrobust).

8 Zhang and Rubin (2003) and Imai (2008) discussed similar bounds in the context of randomized experiments with perfect compliance. See also Lee (2009), Blanco, Flores, and Flores-Lagunes (2013), and Chen and Flores (2014) for construction of bounds in evaluating the effects of Job Corps.

9 In practice, these quantiles can be conveniently estimated by using the RD quantile treatment effect estimator proposed by Frandsen, Frölich, and Melly (2012), after replacing T with ST and (1 − T) with S(1 − T) to deal with sample selection.

10 Fletcher and Tokmouline (2010) also used the THEOP data, but all the data used in this article are obtained and processed independently.

11 In practice, when a student is considered as scholastically deficient, he or she may only be given an academic warning. However, a quick survey administered to the relevant academic deans suggests that students are generally placed on probation in this case.

12 The close-to-cutoff sample is used to produce sample summary statistics and figures only.

13 All our figures are conveniently generated using the Stata command, rdplot.ado. Details can be found in Calonico, Cattaneo, and Titiunik, (2015).

14 Students whose first semester GPAs are exactly 2.0 are not included in our sample, considering possible rounding at this value. We assume that observations away from 2.0 are correctly measured.

15 For notational convenience, in all the tables, I drop C and R = r₀ in the conditioning set. Nevertheless, all estimates are among the compliers at the probation threshold.