LATE with Multiple Instruments
The multiple-instruments extension is easy to see. This is essentially the same as a result we discussed in the grouped-data context. Consider a pair of dummy instruments, Zu and Z2i. Without loss of generality, assume these dummies are mutually exclusive (if not, then we can work with a mutually exclusive set of three dummies, Zii(1—Z2i),Z2i(1—Zii), and ZiiZ2i). The two dummies can be used to construct Wald estimators. Again, without loss of generality assume monotonicity is satisfied for each with a positive first stage (if not, we can recode the dummies so this is true). Both therefore estimate a version of E[Y1i—Yoi|D1i >Doi], though the population with D1i >Doi differs for Z1i and Z2i.
Instead of Wald estimators, we can use Z1i and Z2i together in a 2SLS procedure. Since these two dummies and a constant exhaust the information in the instrument set, this 2SLS procedure is the same as grouped-data estimation using conditional means defined given Z1i and Z2i (whether or not the instruments are correlated). As in Angrist (1991), the resulting grouped-data estimator is a linear combination of the underlying Wald estimators. In other words, it is a linear combination of the instrument-specific LATEs using the instruments one at a time (in fact, it is the efficient linear combination in a traditional homoskedastic linear constant-effects model).
This argument is not quite complete since we haven’t shown that the linear combination of LATEs produced by 2SLS is also a weighted average (i. e., the weights are non-negative and sum to one). The relevant weighting formulas appear in Imbens and Angrist (1994) and Angrist and Imbens (1995). The formulas are a little messy, so here we lay out a simple version based on the two-instrument example. The example shows that 2SLS using Z1i and Z2i together is a weighted average of IV estimates using Z1i and Z2i one at a time. Let
C0V(yi, Zji) ; . _ 1 2 Cov(Di; Zji)’ ;
denote the two IV estimands using Zu and Z2j.
The (population) first stage fitted values for 2SLS are Dj = ^j^Z^ + ^i2Z2i – By virtue of the IV
interpretation of 2SLS, the 2SLS estimand is
^ = _________ ^iiCov(Dj, Zij)___________
^iiCov(Dj, Zij) + ^2iCov(Dj, Z2j)
is a number between zero and one that depends on the relative strength of each instrument in the first stage. Thus, we have shown that 2SLS is a weighted average of causal effects for instrument-specific compliant subpopulations. Suppose, for example, that Zij denotes twins births and Z2j indicates same-sex sibships in families with two or more children, both instruments for family size as in Angrist and Evans (1998). A multiple second birth increases the likelihood of having a third child by about.6 while a same-sex sibling pair increases the likelihood of a third birth by about.07. When these two instruments are used together, the resulting 2SLS estimates are a weighted average of the Wald estimates produced by using the instruments one at a time.