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Regression Discontinuity with Errors in the Running Variable: Effect on Truthful Margin

Myoung-Jae Lee

Abstract

In regression discontinuity (RD) with a running variable S crossing a known cutoff c, an unexpectedly small break magnitude is due to S being a mis-measured version of the genuine running variable G. Has all been lost, and is RD useless when G≠S? This paper proves three things. First, when P(G=S)=0, nonparametric RD identification fails. Second, when P(G=S)>0, although the usual RD effect on the margin E(·|G=c) is not nonparametrically identified, the “effect on the truthful margin” E(·|G=S=c) is. Third, under a no-selection-problem assumption, the effect on the truthful margin becomes the effect on the margin; the no-selection-problem assumption is unnecessary, as long as the effect on the truthful margin is taken as a parameter of interest.

JEL Classification: C14; C21

Corresponding author: Myoung-Jae Lee, Department of Economics, Korea University, Seoul 136-701, South Korea, Tel.: +82-2-3290-2229, Fax: +82-2-926-3601, E-mail:

Acknowledgments

This research has been supported by the National Research Foundation of Korea Grant funded by the Korean Government (NRF-2014S1A5B1014360). This research has been also partially funded by a Korea University Grant.

Appendix

Example of (2.2)

Suppose, for an error term ε,

S=G+V and D=1[cG]1[0G+ε]

where ε|(G=g, V=v) is symmetric around 0 with Fε|G=g,V=v (e) continuous in (g, v, e). This type of D can occur if 1[c≤G] is a legal eligibility for a program and 1[0≤G+ε] is the individual participation decision (Battistin and Rettore 2008, and Battistin et al. 2009). Then we have

J(g,s)=1[cg]P(0g+ε|G=g,S=s)=1[cg]P(εg|G=g,V=sg)=1[cg]Fε|G=g,V=sg(g).

This is continuous in sg, and discontinuous at g=c for any s with Fε|G=c,V=sc (c)>0; if the support of ε|(G=g, V=v) is (–∞, ∞) ∀(g, v), then J(g, s) is discontinuous at g=cs.

Proof of Theorem 1

Observe

P(Ss|G=g)=P(g+Vs|G=g)=FV(sg)  as VG,fS|G=g(s)=P(Ss|G=g)s=fV(sg),  fS,G(s,g)=fV(sg)fG(g),fS(s)=fS,G(s,g)g=fV(sg)fG(g)g.

With |fV (v)|≤μ for some μ, it holds that fV(sg)fG(g)gμ. Using this dominance, the continuity of fV (v) in v gives the continuity of fS(s)=fV(sg)fG(g)g due to the dominated convergence.

For SRD with D=1[c≤G], it holds that

E(D|S=s)=1[cg]fG|S=s(g)g=1[cg]fS,G(s,g)fS(s)g=cfV(sg)fG(g)gfS(s).

This is continuous in s because fV and fS are continuous. For FRD, replace 1[c≤g] with J(g, s) to obtain

E(D|S=s)=J(g,s)fV(sg)fG(g)gfS(s)

which is also continuous in s, again invoking the dominated convergence (J(g, s) is bounded as D is bounded).

Proof of Theorem 2

First, for Theorem 2(i), observe that

P(Ss|G=g)=P{Qg+(1Q)Ws|G=g}=P(Ws|Q=0,G=g)P(Q=0|G=g)+1[gs]P(Q=1|G=g)=FW|Q=0,G=g(s)P(Q=0|G=g)+1[gs]P(Q=1|G=g);P(Ss)=FW|Q=0,G=g(s)P(Q=0|G=g)fG(g)g+sP(Q=1|G=g)fG(g)gfS(s)=fW|Q=0,G=g(s)P(Q=0|G=g)fG(g)g+P(Q=1|G=s)fG(s);

since fW|Q=0,G=g (w) is uniformly bounded over (g, w), we can invoke the dominated convergence: the continuity of fW|Q=0,G=g (w) in w a.e. PG and the continuity of P(Q=1|G=g) and fG (g) render the continuity of fS (s). This proves (i).

Second, for Theorem 2(ii), observe that

P(Gg|S=s)=P(Gg|S=s,Q=0)P(Q=0|S=s)+P(Gg|S=s,Q=1)P(Q=1|S=s)=P(Gg|W=s,Q=0)P(Q=0|S=s)+P(Gg|G=s,Q=1)P(Q=1|S=s)=FG|W=s,Q=0(g)P(Q=0|S=s)+1[sg]P(Q=1|S=s).

For SRD D=1[cG], setting g=c and reversing the inequality Gg to cG in the last display gives

(A.1)E(D|S=s)=P(cG|S=s)={1FG|W=s,Q=0(c)}P(Q=0|S=s)+1[cs]P(Q=1|S=s).

(A.1) is in fact a special case of FRD, for which

(A.2)E(D|S=s)=E{J(G,S)|S=s}=E{J(G,S)|S=s,Q=0}P(Q=0|S=s)+E{J(G,S)|S=s,Q=1}P(Q=1|S=s)=E{J(G,S)|S=s,Q=0}P(Q=0|S=s)+J(s,s)P(Q=1|S=s).

From this, we can obtain E(D|S=c+)–E(D|S=c), which is, however, hard to interpret. Instead, invoke the continuity of E{J(G, S)|S=s, Q=0} and P(Q=1|S=s) to see the first term of (A.2) drop out of E(D|S=c+)–E(D|S=c) to lead to (3.2), and then (3.3) under SRD. To see when the continuity of E{J(G, S)|S=s, Q=0} in s holds, rewrite (A.2) as

(A.3)J(g,s)fG|W=s,Q=0(g)gP(Q=0|S=s)+J(s,s)P(Q=1|S=s).

If fG|W=s,Q=0(g) is continuous in s for a.e. PG and |fG|W=s,Q=0(g)|≤μ(g) for any s with μ(g)g< for some function μ(g) as assumed in Theorem 2(ii), then E{J(G,S)|S=s,Q=0}=J(g,s)fG|W=s,Q=0(g)g is continuous in s.

Third, for Theorem 2(iii), working analogously to what was done for E(D|S=s), we have

(A.4)E(Y|S=s)=L(g,s)fG|W=s,Q=0(g)gP(Q=0|S=s)+L(s,s)P(Q=1|S=s).

This, along with L(g, s) satisfying the same (dis-) continuity condition in (2.2) for J(g, s), proves Theorem 2(iii) because

AR={L(c+,c+)L(c,c)}P(Q=1|S=c){J(c+,c+)J(c,c)}P(Q=1|S=c)=L(c+,c+)L(c,c)J(c+,c+)J(c,c)

where the right and left limits for G and S are along the same sequence, as the limits are taken on J(s, s) and L(s, s).

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Published Online: 2015-11-4
Published in Print: 2017-1-1

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