Testing rank similarity in the local average treatment effects model

Abstract

This paper develops a test for the rank similarity condition of the nonseparable instrumental variable quantile regression model using the local average treatment effect model. When the instrument takes more than two values or multiple binary instruments are available, there exist multiple complier groups for which the marginal distributions of potential outcomes are identified. A testable implication is obtained by comparing the distributions of ranks across complier groups. We propose a test procedure in a semiparametric quantile regression specification. We establish the weak convergence of the test statistic and the validity of the bootstrap critical value. We illustrate the test with an empirical example of the effects of fertility on women’s labor supply.

Keywords:

• Bootstrap
• heterogeneous treatment effect model
• identification
• rank similarity
• treatment effects

JEL classification:

• C12
• C14
• C31
• C35
• C36

Acknowledgments

The authors are grateful to Alfonso Flores-Lagunes, David Jacho-Chavez, Hiroaki Kaido, Christoph Rothe, Bernard Salanie, and participants in seminars at Duke University, Southern Economic Association Meeting, Triangle Econometrics Conference, Asian Meeting of the Econometric Society, Summer Meeting of Western Economic Association International, New York Camp Econometrics, North American Summer Meeting of the Econometric Society, and Summer Meeting of Western Economic Association International for helpful comments and discussion. All remaining errors are our own.

Notes

1 Dong and Shen (2018) derive some testable implication from a binary instrument without additional assumptions. It is possible because they consider unconditional rank similarity, which is neither sufficient nor necessary for conditional rank similarity. To see this, consider the following example presented in Vytlacil (2015): Suppose that the potential outcome is determined by a linear regression model, $Y_d={\alpha }_d+X{\beta }_d+{\epsilon }_d$ for $d\in \{0,1\},$ and ${\epsilon }_1=\sigma {\epsilon }_0.$ In this example, the rank similarity of Chernozhukov and Hansen (2005) holds. However, the null hypothesis in Dong and Shen (2018) is $U_0|X\sim U_1|X,$ which holds if and only if ${\beta }_1=\sigma {\beta }_0.$ Thus, their null hypothesis is neither necessary nor sufficient for our null hypothesis.

2 Kitagawa (2015) and Mourifié and Wan (2017) require that the ordering of the instruments is known ex ante. If the ordering is unknown, Frandsen et al. (2019) can be more appropriate.

3 For more discussion on refutability and nonverifiability, refer to Breusch (1986).

4 This result corresponds to Lemma 3.2 in Abadie et al. (2002).

5 Abadie et al. (2002) derive the limiting distribution of ${\widehat{\theta }}_c(\tau )-{\theta }_{c,0}(\tau )$ at a fixed quantile level under the assumptions. Kim and Park (2018) show that $\sqrt{n}({\widehat{\theta }}_c(·)-{\theta }_{c,0}(·))$ weakly converges to a Gaussian process under the assumptions.