This can be interpreted as the average treatment effect on individuals with

# This can be interpreted as the average treatment

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whereuD=P(Di= 1|Zi=z) andu0D=P(Di= 1|Zi=z0). Thus, in the limit, asu0DuD– i.e. as theinstrument shifts the probability of participation by an infinitessimal quantity – LATE approaches MTE.It turns out that the MTE function can be estimated given sufficient variation in the instrument. Whatis necessary is to first estimate the functionP(Zi).Then it is straightforward to show thatMTE=∂pE[Yi|Xi=x, P(Zi) =p]. I demonstrate this below:E[Yi|Xi=x, P(Zi) =p]=E DiY1i+ (1-Di)Y0i|Xi=x, P(Zi) =p=E Y0i|Xi=x, P(Zi) =p+E Di(Y1i-Y0i)|Xi=x, P(Zi) =p=E Y0i|Xi=x+E Y1i-Y0i|Xi=x, P(Zi) =p, Di= 1p=E Y0i|Xi=x+E Y1i-Y0i|Xi=x, UDp p=E Y0i|Xi=x+pˆ0E Y1i-Y0i|Xi=x, UD=uDduDThus, by Leibniz’s rule:∂pE[Yi|Xi=x, P(Zi) =p]P(Zi)=p=E Y1i-Y0i|Xi=x, UD=p=ΔMT E(x, p)So what is required in order to identify the distribution of treatment effect heterogeneity captured by theMTE function is to have a powerful instrument vectorZithat can trace out the support of ΔMT E(x, p). Withsuch an instrument one can estimateE[Yi|Xi=x, P(Zi) =p] either parametrically or nonparametricallyand then compute derivatives.Such exercises are referred to as “Local IV” after the paper by Heckmanand Vytlacil (1999). Carneiro and Lee (2009) provide a feasible approach to estimation and inference in theLocal IV model and a brief application.Local IV has recently taken off with applications by Doyle (2007, 2008) and Maestas, Mullen, and Strand(2013) exploiting the “judges” design which involves use of group dummies as instruments.See also therecent extension and application by Brinch, Mogstad, and Wiswall (2012).This is an area where morecareful theoretical and practical research would be welcome as we still don’t know much about the finitesample performance of the LIV estimators that are being used in the literature.49Weak InstrumentsIn an influential paper, Bound, Jaeger, and Baker (1995) demonstrated that weak instruments (instrumentsexhibiting little relationship with the treatment variable) could yield highly misleading inferences whenconventional asymptotic tests are used.Today it is well known that weak instruments tend to performpoorly in practice. Conventional asymptotics provide a poor guide to finite sample inference and the pointestimates of IV are biased towards those of OLS. Many an econometrician in the last decade has gotten ajob trying to deal with these problems.For surveys of the weak IV literature see Stock, Wright, and Yogo (2002) and Andrews and Stock (2005).The standard benchmark for a “weak” instrument is a first stage F-test less than 10. This number comes
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