Supplement to “testing identifying assumptions in fuzzy regression discontinuity designs”

We provide a detailed analytical discussion of how our testing approach differs from and complements the existing approach in terms of assessing the local continuity (LC) assumption. Let X ∈ X ⊂ Rdx be observable covariates. Assuming that all the probability densities in the following equations are well defined, we can write fYd(r),T|r−r0 ||R,X(y, t|r, x) = fR|Yd(r),T|r−r0 |,X(r|y, t, x) fX|R(x|r) fR(r) fYd(r),T|r−r0 |,X(y, t, x), (B.1) where fYd(r),T|r−r0 ||R,X(y, t|r, x) denotes the conditional density of (Yd(r), T|r−r0|) given R, X. On the right-hand side of the equation, the continuity of fR|Yd(r),T|r−r0 |,X(r|y, t, x) in r near r0 is essentially Lee (2008)’s stronger local continuity (SLC) assumption (with different notation), which was originally introduced in the sharp RD framework and later discussed in Dong (2018) in the context of the FRD setting. Since the SLC assumption is not directly testable, the existing literature has derived Date: Saturday 20th March, 2021. a. School of Social Sciences, Waseda University, yarai@waseda.jp. b. Institute of Economics, Academia Sinica; Department of Finance, National Central University; Department of Economics, National Chengchi University, ychsu@econ.sinica.edu.tw. c. Department of Economics, University College London, t.kitagawa@ucl.ac.uk. d. Department of Economics, University of Toronto, ismael.mourifie@utoronto.ca. e. Department of Economics, University of Toronto, yuanyuan.wan@utoronto.ca. Acknowledgement: Financial support from the ESRC through the ESRC Centre for Microdata Methods and Practice (CeMMAP) (grant number RES-589-28-0001), the ERC through the ERC starting grant (grant number EPP-715940), the Japan Society for the Promotion of Science through the Grants-in-Aid for Scientific Research No. 15H03334, Ministry of Science and Technology of Taiwan (MOST107-2410-H-001-034-MY3), Academia Sinica Taiwan through the Career Development Award, and Waseda University Grant for Special Research Projects is gratefully acknowledged.