IV in Randomized Trials

The language of the LATE framework is based on an analogy between IV and randomized trials. But some instruments really come from randomized trials. If the instrument is a randomly assigned offer of treatment, then LATE is the effect of treatment on those who comply with the offer but are not treated otherwise. An especially important case is when the instrument is generated by a randomized trial with one-sided non­compliance. In many randomized trials, participation is voluntary among those randomly assigned to receive treatment. On the other hand, no one in the control group has access to the experimental intervention. Since the group that receives (i. e., complies with) the assigned treatment is a self-selected subset of those offered treatment, a comparison between those actually treated and the control group is misleading. The selection bias in this case is almost always positive: those who take their medicine in a randomized trial tend to be healthier; those who take advantage of randomly assigned economic interventions like training programs tend to earn more anyway.

IV using the randomly assigned treatment intended as an instrumental variable for treatment received solves this sort of compliance problem. Moreover, LATE is the effect of treatment on the treated in this case. Suppose the instrument, Zj, is a dummy variable indicating random assignment to a treatment group, while Dj is a dummy indicating whether treatment was actually received. In practice, because of non-compliance, Dj is not equal to Zj. An example is the randomized evaluation of the JTPA training program, where only 60 percent of those assigned to be trained received training, while roughly 2 percent of those assigned to the control group received training anyway (Bloom, et al., 1997). Non-compliance in the JTPA arose from lack of interest among participants and the failure of program operators to encourage participation. Since the compliance problem in this case is largely confined to the treatment group, LATE using random assignment, zi, as an instrument for treatment received, Di, is the effect of treatment on the treated.

This use of IV to solve the compliance problems is illustrated in Table 4.4.1, which presents results from the JTPA experiment. The outcome variable of primary interest in the JTPA experiment is total earnings in the 30-month period after random assignment. Columns 1-2 of the table show the difference in earnings between those who were trained and those who were not (the estimates in column 2 are from a regression model that adjusts for a number of individual characteristics measured at the beginning of the experiment. The contrast reported in columns 1-2 is on the order of $4,000 for men and $2,200 for women, in both cases a large treatment effect that amounts to about 20 percent of average earnings. But these estimates are misleading because they compare individuals according to Di, the actual treatment received. Since individuals assigned to the treatment group were free to decline (and 40% did so), this comparison throws away the random assignment unless the decision to accept treatment is itself independent of potential outcomes. This seems unlikely.

Comparisons by Training Status

Comparisons by Assignment Status

Instrumental Variable Estimates

Without

Covariates

(1)

With

Covariates

(2)

Without

Covariates

(3)

With

Covariates

(4)

Without

Covariates

(5)

With

Covariates

(6)

A. Men

3,970

3,754

1,117

970

1,825

1,593

(555)

(536)

(569)

(546)

(928)

(895)

B. Women

2,133

2,215

1,243

1,139

1,942

1,780

(345)

(334)

(359)

(341)

(560)

(532)

Table 4.4.1: Results from the JTPA experiment: OLS and IV estimates of training impacts

Notes: The table reports OLS, reduced-form, and IV estimates of the effect of subsidized training on earnings in the JTPA experiment. Columns (1) and (2) show differences in earnings by training status; columns (3) and (4) show differences by random-assignment status. Columns (5) and (6) report the result of using random-assignment status as an instrument for training. The covariates used in columns (2), (5) and (6) are High school or GED, Black, Hispanic, Married, Worked less than 13 weeks in past year, AFDC (for women), plus indicators for the service strategy recommended, age group and second follow-up survey. Robust standard errors are shown in parenthesis.

to

 

Columns 3 and 4 of Table 4.4.1. compare individuals according to whether they were offered treatment. In other words, this comparison is based on randomly assigned Zj. In the language of clinical trials, the contrast in columns 3-4 is known as the intention-to-treat (ITT) effect. The intention-to-treat effects in the table are on the order $1,200 (somewhat less with covariates). Since Zj was randomly assigned, the ITT effect have a causal interpretation: they tell us the causal effect of the offer of treatment, building in the fact that many of those offered will decline. For this reason, the ITT effect is too small relative to the average causal effect on those who were in fact treated. Columns 5 and 6 put the pieces together and give us the most interesting effect: intention-to-treat divided by the difference in compliance rates between treatment and control groups as originally assigned (about.6). These figures, roughly $1,800, estimate the effect of treatment on the treated.

How do we know the that ITT-divided-by-compliance is the effect of treatment on the treated? We can recognize ITT as the reduced-form effect of the randomly assigned offer of treatment, our instrument in this case. The compliance rate is the first stage associated with this instrument, and the Wald estimand, as always, is the reduced-form divided by the first-stage. In general this equals LATE, but because we have (almost) no always-takers, the treated population consists (almost) entirely of compliers. The IV estimates in column 5 and 6 of Table 4.4.1 are therefore consistent estimates of the effect of treatment on the treated.

This conclusion is important enough that it warrants an alternative derivation. To the best of our knowledge the first person to point out that the IV formula can be used to estimate the effect of treatment on the treated in a randomized trial with one-sided non-compliance was Howard Bloom (1984). Here is Bloom’s result with a simple direct proof.

Подпись:THE BLOOM RESULT. Suppose the assumptions of the LATE theorem hold, and E [d;|z; =

Подпись: E[Yii - Yoj|Dj = 1].E[Yj|Zj = 1] – E[Yj|Zj =0]
E [d j I z j = 1]

Proof. E[Yj|Zj = 1] = E[Yjo + (yij Yoj)dj|Zj = 1], while E[Yj|Zj = 0] = E[Yjo|Zj = 0] because E[Dj|Zj = 0] = 0. Therefore

E[Yj|Zj = 1] – E[Yj|Zj = 0] = E[(Yij – Yoj)Dj|Zj = 1]

by independence. But

E[(Yij – Yoj)Dj |Zj = 1] = E[Yij – Yoj|Dj = 1, Zj = 1]P[Dj = 1|Zj = 1]

while E[Dj|Zj = 0] = 0 means Dj = 1 implies Zj = 1. Hence, E[Yij—Yoj|Dj = 1,Zj = 1] = E[yij—Yoj|Dj = 1] ■

In addition to telling us how to analyze randomized trials with non-compliance, the LATE framework

opens the door to cleverly-designed randomized experiments in settings where it’s impossible or unethical to compel treatment compliance. A famous example from the field of Criminology is the Minneapolis Domestic Violence Experiment (MDVE). The MDVE was a pioneering effort to determine the best police response to domestic violence (Sherman and Berk, 1984). In general, police use a number of strategies when on a domestic violence call. These include referral to counseling, separation orders, and arrest. A vigorous debate swirls around the question of whether a hard-line response – arrest and at least temporary incarceration – is productive, especially in view of the fact that domestic assault charges are frequently dropped.

As a result of this debate, the city of Minneapolis authorized a randomized trial where the police response to a domestic disturbance was determined in part by random assignment. The research design used randomly shuffled color-coded charge sheets telling the responding officers to arrest some perpetrators while referring others to counseling or separating the parties. In practice, however, the police were free to overrule the random assignment. For example, an especially dangerous or drunk offender was arrested no matter what. As a result, the actual response often deviated from the randomly assigned response, though the two are highly correlated.

Most published analyses of the MDVE data recognize this compliance problem and focus on ITT effects,

i. e., an analysis using the original random assignment and not the treatment actually delivered. But the MDVE data can also be used to get the average causal effect on compliers, in this case those who were arrested because they were randomly assigned to be but would not have been arrested otherwise. The MDVE is analyzed in this spirit in Angrist (2006). Because everyone in the MDVE who was assigned to be arrested was in fact arrested, there are no never-takers. This is an interesting twist and the flip-side of the Bloom scenario: here, we have Du = 1 for everybody. Consequently, LATE is the effect of treatment on the non-treated, i. e.,

E[Yu – Yoi|Dii > Doi] = E[ Yii – Yoi |Dj = 0],

where Dj indicates arrest. The IV estimates using MDVE data show that arrest reduces repeat offenses sharply, in this case, among the subpopulation that was not arrested.[62]

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