• So far, we have assumed that the assignment of treatment implied the receipt of treatment.
  • Prescribing drug means that the subject takes the drug.
• However, there are many cases in which a subject assigned to treatment does not receive the treatment.
  • Supplied supplements on infant mortality to a village but some of the villagers could not receive the supplements.
• Moreover, there may be cases where subjects not assigned to treatment may receive the treatment.
  • Men who were not drafted to the Vietnam War volunteered.
• Subjects whose treatment receipt status differs from treatment assignment status are called noncompliers.

• This problem arises regardless of whether experimental or observational.
• If the treatment was randomly assigned, the average effect of assignment treatment can be estimated.
• However, if there are noncompliers, this can differ from the average effect of receiving treatment.
• The former is called intention-to-treat (ITT) effect and is distinguished from the latter.
• ITT effects are informative, but may not be robust to the changes in the compliance behaviors.
• In other situations, people’s compliance behavior can be different and hence the ITT may change even if the treatment effect is invariant.

• To describe this situation, we introduce variables representing treatment assignment and treatment receipt status.
• Treatment assignment status: \(Z_i\) takes value \(1\) if assigned and \(0\) otherwise.
• Treatment receipt status: \(W_i^{obs}\) takes value \(1\) if received and \(0\) otherwise.
• The key is that individual treatment receipt status is a function of the individual’s treatment assignment status.
  • A subject may never take the treatment, or always take.
  • A subject may take the treatment only when assigned treatment, or only when not assigned.
• This relation is described by the following potential outcome model: \[ W_i^{obs} = W_i(Z_i) = \begin{cases} W_i(0) & \text{ if } Z_i = 0\\ W_i(1) & \text{ if } Z_i = 1. \end{cases} \]

• We can observe \(W_i(Z_i)\), the treatment receipt status under the observed treatment assignment status.
• However, we never observe \(W_i(1 - Z_i)\), the treatment receipt status under the missing treatment assignment status.

• The potential outcome is now a function of both treatment assignment and receipt status. \[ Y_i^{obs} = Y_i(Z_i, W_i). \]
• We observe: \[ Y_i[Z_i, W_i(Z_i)] = Y_i[1, W_i(1)], Y_i[0, W_i(0)]. \]
• However, we never observe: \[ Y_i[Z_i, W_i(1 - Z_i)] = Y_i[1, W_i(0)], Y_i[0, W_i(1)]. \]

• We still assume that the treatment assignment is random.
• This is true in randomized experiments.
• We have to justify this in observational studies.
• Assumption: Random assignment of \(Z_i\) \[ \mathbb{P}[Z_i = 1 | W_i(0), W_i(1), Y_i(0, 0), Y_i(0, 1), Y_i(1, 0), Y_i(1, 1)] = \mathbb{P}(Z_i = 1). \]

• The subsample sizes by treatment assignment status: \[ N_0 \equiv \sum_{i = 1}^N (1 - Z_i), N_1 \equiv \sum_{i = 1}^N Z_i. \]
• The subsample sizes by treatment receipt status: \[ N_c \equiv \sum_{i = 1}^N (1 - W_i^{obs}), N_t \equiv \sum_{i = 1}^N W_i^{obs}. \]

• The subsample sizes by both treatment assignment and receipt status: \[ N_{0c} \equiv \sum_{i = 1}^N (1 - Z_i) \cdot (1 - W_i^{obs}), N_{0t} \equiv \sum_{i = 1}^N (1 - Z_i) \cdot W_i^{obs}. \] \[ N_{1c} \equiv \sum_{i = 1}^N Z_i \cdot (1 - W_i^{obs}), N_{1t} \equiv \sum_{i = 1}^N Z_i \cdot W_i^{obs}. \]

• The average outcomes by treatment assignment status: \[ \overline{Y}_0^{obs} \equiv \frac{1}{N_0} \sum_{i = 1}^N (1 - Z_i) \cdot Y_i^{obs}, \overline{Y}_1^{obs} \equiv \frac{1}{N_1} \sum_{i = 1}^N Z_i \cdot Y_i^{obs}. \]

• The average treatment receipt by treatment assignment status: \[ \overline{W}_0^{obs} \equiv \frac{1}{N_0} \sum_{i = 1}^N (1 - Z_i) \cdot W_i^{obs}, \overline{W}_1^{obs} \equiv \frac{1}{N_1} \sum_{i = 1}^N Z_i \cdot W_i^{obs}. \]

• The average outcomes by treatment receipt status: \[ \overline{Y}_c^{obs} \equiv \frac{1}{N_c} \sum_{i = 1}^N (1 - W_i^{obs}) \cdot Y_i^{obs}, \overline{Y}_t^{obs} \equiv \frac{1}{N_t} \sum_{i = 1}^N W_i^{obs} \cdot Y_i^{obs}. \]

• The average outcomes by both treatment assignment and receipt status: \[ \overline{Y}_{0c}^{obs} \equiv \frac{1}{N_{0c}} \sum_{i = 1}^N (1 - Z_i) \cdot (1 - W_i^{obs}) \cdot Y_i^{obs}. \] \[ \overline{Y}_{0t}^{obs} \equiv \frac{1}{N_{0t}} \sum_{i = 1}^N (1 - Z_i) \cdot W_i^{obs} \cdot Y_i^{obs}. \] \[ \overline{Y}_{1c}^{obs} \equiv \frac{1}{N_{1c}} \sum_{i = 1}^N Z_i \cdot (1 - W_i^{obs}) \cdot Y_i^{obs}. \] \[ \overline{Y}_{1t}^{obs} \equiv \frac{1}{N_{1t}} \sum_{i = 1}^N Z_i \cdot W_i^{obs} \cdot Y_i^{obs}. \]

• Assume that subject never receives treatment unless assigned treatment: \[ W_i(0) = 0, \forall i. \]
• This holds in most of experiment: subjects will not take new supplements if not supplied.

• Subjects are divided into latent compliance status according to their value of \(W_i(z)\). \[ G_i \equiv \begin{cases} \text{co if } W_i(1) = 1 \\ \text{nc if } W_i(1) = 0. \end{cases} \]

• Underlying compliance status: Table 23.2 of Imbens and Rubin (2015)

• The subsample sizes of compliers and noncompliers \[ N_{co} \equiv \sum_{i = 1}^N 1_{G_i = co}, N_{nc} \equiv \sum_{i = 1}^N 1_{G_i = nc}. \]
• The sample fractions of compliers and noncompliers \[ \pi_{co} \equiv \frac{N_{co}}{N}, \pi_{nt} \equiv \frac{N_{nc}}{N} = 1 - \pi_{co}. \]

• The ITT for the receipt of treatment: \[ ITT_W \equiv \frac{1}{N}\sum_{i = 1}^N[W_i(1) - W_i(0)], \] can be estimated unbiasedly by the average difference in treatment status: \[ \widehat{ITT}_W \equiv \overline{W}_1^{obs} - \overline{W}_0^{obs} = \overline{W}_1^{obs}. \]
• The standard error is also estimated by: \[ \widehat{\mathbb{V}}(\widehat{ITT}_W) \equiv \frac{s_{W, 0}^2}{N_0} + \frac{s_{W, 1}^2}{N_1}. \]

• The ITT for the outcome: \[ ITT_Y \equiv \frac{1}{N}\sum_{i = 1}^N\{Y_i[1, W_i(1)] - Y_i[0, W_i(0)]\}, \] can be estimated unbiasedly by the average difference in treatment status: \[ \widehat{ITT}_Y \equiv \overline{Y}_1^{obs} - \overline{Y}_0^{obs}. \]
• The standard error is also estimated by: \[ \widehat{\mathbb{V}}(\widehat{ITT}_Y) \equiv \frac{s_{Y, 0}^2}{N_0} + \frac{s_{Y, 1}^2}{N_1}. \]

• Using the ITT effect for the outcome by compliance status: \[ ITT_{Y, co} \equiv \frac{1}{N_{co}} \sum_{i = 1: G_i = co}\{Y_i[1, W_i(1)] - Y_i[0, W_i(0)]\}, \] and: \[ ITT_{Y, nc} \equiv \frac{1}{N_{nc}} \sum_{i = 1: G_i = nc}\{Y_i[1, W_i(1)] - Y_i[0, W_i(0)]\}, \] we can decompose \(ITT_Y\) as: \[ ITT_Y = ITT_{Y, co} \cdot ITT_W + ITT_{Y, nc} \cdot (1 - ITT_W). \]

• We cannot identify \(ITT_{Y, co}\) and \(ITT_{Y, nc}\) because we do not know the compliance status of each subject and hence we cannot condition the data on the compliance data.
• \(ITT_{Y, no}\) is not informative, because they never receive the treatment, irrespective of the assignment.
• \(ITT_{Y, co}\) is informative, because this compares the outcome when the treatment is received and not.
• Then, under what kind of additional assumptions, can we identify \(ITT_{Y, co}\)?

• Randomized experiment ensures that the unconfoundedness of \(Z_i\), but not \(W_i^{obs}\)
• However, randomized experiment at least ensures the latent unconfoundedness of \(W_i^{obs}\).
• Lemma: Latent unconfoundedness of \(W_i^{obs}\)
  • The random assignment of \(Z_i\) implies for \(g \in G\): \[ \begin{split} &\mathbb{P}[W_i^{obs} = 1 | Y_i(0, 0), Y_i(0, 1), Y_i(1, 0), Y_i(1, 1), G_i = g]\\ &= \mathbb{P}(W_i^{obs} = 1 | G_i = g). \end{split} \]
• For \(g = nc\), \(W_i^{obs}\) is always \(0\). Therefore, the equation trivially holds
• For \(g = co\), \(W_i^{obs} = Z_i\). Therefore, the random assignment of \(Z_i\) implies the unconfoundedness of \(W_i^{obs}\) for compliers.

• Assumption : Exclusion restriction for noncompliers
  • For all noncompliers: \[ Y_i(0, 0) = Y_i(1, 0). \]
• This rules out that an effect of the assignment on the outcome for noncompliers.
• The random assignment of \(Z_i\) does not justify this assumption.
• This is a substantive assumption to be justified by the contextual knowledge.
• For example, the double blinding makes subjects unaware of the treatment assignment, and hence may justify this assumption if nicely implemented.

• The exclusion restriction for noncompliers is necessary for identifying \(ITT_{Y, co}\).
• The following assumption of exclusion restriction for compliers is not necessary, but only changes the interpretation of \(ITT_{Y, co}\):
• Assumption: Exclusion restriction for compliers:
  • For all compliers, for \(w = 0, 1\): \[ Y_i(0, w) = Y_i(1, w) \equiv Y_i^*(w). \]
• If this assumption holds: \[ ITT_{Y, co} = \frac{1}{N_{co}} \sum_{i = 1: G_i = co}\{Y_i^*(1) - Y_i^*(0)\}, \] which is purely the effect of changing the treatment receipt status.

• The exclusion restriction for noncompliers implies: \[ ITT_{Y, nc} \equiv \frac{1}{N_{nc}} \sum_{i = 1: G_i = nc}\{Y_i[1, W_i(1)] - Y_i[0, W_i(0)]\} = 0. \]
• Therefore: \[ ITT_Y = ITT_{Y, co} \cdot ITT_W + ITT_{Y, nc} \cdot (1 - ITT_W) = ITT_{Y, co} \cdot ITT_W, \] and so: \[ ITT_{Y, co} = \frac{ITT_Y}{ITT_W}. \]
• We call this the local average treatment effect (LATE) or complier average causal effect (CECE), \(\tau_{late}\).

• Because we have \(\widehat{ITT}_Y\) and \(\widehat{ITT}_W\), we can estiamte \(\tau_{late}\) by: \[ \widehat{ITT}_{late} \equiv \frac{\widehat{ITT}_Y}{\widehat{ITT}_W}. \]
• The sampling variance is: \[ \begin{split} \mathbb{V}(\widehat{ITT}_{late}) &= \frac{1}{ITT_W^2} \cdot \mathbb{V}(\widehat{ITT}_Y) + \frac{ITT_Y^2}{ITT_W^4} \cdot \mathbb{V}(\widehat{ITT}_W) \\ &- 2 \cdot \frac{ITT_Y}{ITT_W^3} \cdot \mathbb{C}(\widehat{ITT}_Y, \widehat{ITT}_W). \end{split} \]
• We can replace with the sample analogues. However, it can by a poor approximation when \(ITT_W\) is close to 0.

• Take the infinite super-population perspective.
• Define: \[ \tau_{late} = \mathbb{E}[Y_i(1) - Y_i(0)| G_i = co], \] \[ \alpha \equiv \mathbb{E}[Y_i(0)], \] \[ \nu_i \equiv Y_i(1) - Y_i(0) - \tau_{late}, \] \[ \epsilon_i \equiv Y_i(0) - \alpha. \]

• Then, we have a structural equation: \[ Y_i(w) = \alpha + \tau_{late} \cdot w + \epsilon_i + w \cdot \nu_i. \]
• Then, for observations, we have: \[ Y_i^{obs} = Y_i(W_i^{obs}) = \alpha + \tau_{late} \cdot W_i^{obs} + \epsilon_i + W_i^{obs} \cdot \nu_i. \]
• The OLS estimator does not consistently estimate \(\tau_{late}\), because \(\epsilon_i + W_i^{obs} \cdot \nu_i\) is potentially correlated with \(W_i^{obs}\).

• Because of the random assignment of \(Z_i\), we have: \[ \mathbb{E}[\epsilon_i|Z_i] = \mathbb{E}[Y_i(0)|Z_i] - \mathbb{E}[Y_i(0)] = \mathbb{E}[Y_i(0)] - \mathbb{E}[Y_i(0)] = 0. \]
• The one-sided compliance implies: \[ \mathbb{E}[W_i^{obs} \cdot \nu_i| Z_i = 0] = \mathbb{E}[0 \cdot \nu_i| Z_i = 0] = 1. \]

• Moreover, the one-sided noncompliers and the random assignment of \(Z_i\) implies: \[ \begin{split} &\mathbb{E}[W_i^{obs} \cdot \nu_i| Z_i = 1]\\ &= \mathbb{E}[0 \cdot \nu_i | Z_i = 1, W_i = 0] \mathbb{P}(W_i^{obs} = 0| Z_i = 1)\\ &+ \mathbb{E}[1 \cdot \nu_i | Z_i = 1, W_i = 1] \mathbb{P}(W_i^{obs} = 1 | Z_i = 1)\\ &= \mathbb{E}[\nu_i | Z_i = 1, W_i^{obs} = 1, G_i = co] \pi_{co}\\ &= \mathbb{E}[Y_i(1) - Y_i(0) - \tau_{late} | Z_i = 1, G_i = co] \pi_{co}\\ &= \mathbb{E}[Y_i(1) - Y_i(0) - \tau_{late} | G_i = co] \pi_{co}\\ &= 0. \end{split} \]
• Therefore, we have the conditional mean zero property: \[ \mathbb{E}[\epsilon_i + W_i^{obs} \cdot \nu_i| Z_i] = 0. \]

• We can rewrite the structural equation as: \[ \begin{split} Y_i^{obs} &= \alpha + \tau_{late} \cdot \mathbb{E}[W_i^{obs}|Z_i] - \tau_{late} \cdot \mathbb{E}[W_i^{obs}|Z_i] + \tau_{late} \cdot W_i^{obs}\\ &+ \epsilon_i + W_i^{obs} \cdot \nu_i\\ &= \alpha + \tau_{late} \cdot \mathbb{E}[W_i^{obs}|Z_i] + \eta_i \\ &= \alpha + \tau_{late} \cdot \{\pi_0 + \pi_1 \cdot Z_i \} + \eta_i,\\ &= \alpha + \tau_{late} \cdot \{\pi_0 + \pi_1 \cdot Z_i \} + \eta_i, \end{split} \] where: \[ \pi_0 \equiv \mathbb{E}[W_i^{obs}|Z_i = 0] = 0, \] \[ \pi_1 \equiv \mathbb{E}[W_i^{obs}|Z_i = 1] - \mathbb{E}[W_i^{obs}|Z_i = 0] = \mathbb{E}[W_i^{obs}|Z_i = 1] = \pi_{co}. \] under the one-sided noncompliers.

• Therefore, we have the reduced-form equation: \[ Y_i^{obs} = \alpha + \gamma \cdot Z_i + \eta_i, \gamma \equiv \pi_{co} \cdot \tau_{late}. \]
• We know: \[ \hat{\gamma}_{ols} = \overline{Y}_1^{obs} - \overline{Y}_0^{obs} = \widehat{ITT}_Y. \]
• Therefore, the instrumental variable estimator for \(\tau_{late}\), which is \(\hat{\gamma}_{ols}/\hat{\pi}_{co}\), is numerically the same with \(\widehat{ITT}_{Y, co}\).

• In experiments, we may be able to ensure that subjects who are not assigned treatmet do not receives treatment.
• However, especially in observational studies, subjects may voluntarily take treatment, even if assigned treatment.
• In such a situation, we need to generalize the compliance status.

• We consider 4 compliance status: Allways takers, never takers, compliers, and defiers. \[ G_i = \begin{cases} nt &\text{ if } W_i(0) = 0, W_i(1) = 0,\\ co &\text{ if } W_i(0) = 0, W_i(1) = 1,\\ df &\text{ if } W_i(0) = 1, W_i(1) = 0,\\ at &\text{ if } W_i(0) = 1, W_i(1) = 1. \end{cases} \]

• The are two underlying compliance status for each pair of \(Z_i\) and \(W_i^{obs}\): Table 24.3 of Imbens and Rubin (2015)

• The ITT analysis is unchanged with the two-sided compliers

• With the two-sided noncompliers, \(ITT_Y\) is a mixture of ITT effects of 4 compliance status. \[ \begin{split} ITT_Y &= ITT_{Y, nt} \cdot \pi_{nt} + ITT_{Y, co} \cdot \pi_{co} \\ &+ ITT_{Y, df} \cdot \pi_{df} + ITT_{Y, at} \cdot \pi_{at}. \end{split} \]

• Assumption: Exclusion restriction for nevertakers
  • For all nevertakers: \[ Y_i(0, 0) = Y_i(1, 0). \]
• Assumption: Exclusion restriction for alwaystakers
  • For all alwaystakers: \[ Y_i(0, 1) = Y_i(1, 1). \]

\[ \begin{split} ITT_{Y, nt} & = \frac{1}{N_{nt}} \sum_{i: G_i = nt}\{Y_i[1, W_i(1)] - Y_i[0, W_i(0)] \}\\ &=\frac{1}{N_{nt}} \sum_{i: G_i = nt}\{Y_i(1, 0) - Y_i(0, 0) \}\\ &=0. \end{split} \]

\[ \begin{split} ITT_{Y, at} & = \frac{1}{N_{at}} \sum_{i: G_i = at}\{Y_i[1, W_i(1)] - Y_i[0, W_i(0)] \}\\ &=\frac{1}{N_{at}} \sum_{i: G_i = at}\{Y_i(1, 1) - Y_i(0, 1) \}\\ &=0. \end{split} \] - Therefore, \(ITT_Y\) is a mixture of compliers and defiers.

• Moreover, we assume there are no defiers.
• This is a substantive assumption that needs to be justified from the contextual knowledge.
• For example, suppose that:
  • \(Z_i\): draft to the Vietnam war.
  • \(W_i^{obs}\): go to the Vietnam war.
  • Compliers: goes to the Vietnam war if drafted, while does not if not drafted.
  • Defiers: volunteers to the Vietnam war if not drafted, while does not go if drafted.
• Formally:
• Asumption: Monotonicity
  • For all subjects \(W_i(1) \ge W_i(0)\).

• Under the exclusion restriction for nevertakes, alwaystakers, and the monotonicity, we have: \[ ITT_Y = ITT_{Y, co} \cdot \pi_{co}. \]
• Moreover: \[ \begin{split} ITT_W &= \mathbb{E}[W_i(1) - W_i(0)| G_i = nt] \cdot \pi_{nt}\\ &+ \mathbb{E}[W_i(1) - W_i(0)| G_i = at] \cdot \pi_{at}\\ &+ \mathbb{E}[W_i(1) - W_i(0)| G_i = co] \cdot \pi_{co}\\ &+ \mathbb{E}[W_i(1) - W_i(0)| G_i = df] \cdot \pi_{df}\\ &=\pi_{co}. \end{split} \]

• Therefore, \(ITT_{Y, co}\) is identified as: \[ ITT_{Y, co} = \frac{ITT_Y}{ITT_W}. \]

• The estimation and inference is, therefore, the same as the one-sided noncompliers.
• The connection to the instrumental variable estimation is the same as well.

• The treatment assignment \(Z\) is often interpreted as the instrumental variable in econometrics.
• In econometrics, we often use multi-valued or even continuous variable instead of a binary variable as \(Z\).
• How do we generalize the LATE to the multi-valued instrumental variables?

• Consider the following framework: \[ Y(1) = \mu_1(X) + U_1, \] \[ Y(0) = \mu_0(X) + U_0, \] \[ D = 1\{\mu_D(Z, X) \ge V\}. \]
• We observe \(Y^{obs} = Y(1) \cdot D + Y(0) \cdot (1 - D)\).

• Note that the additive separability of the potential outcome is without loss of generality, because if: \[ Y(1) = \tilde{\mu}_1(X, \tilde{U}_1), \] we can redefine: \[ \mu_1(X) \equiv \mathbb{E}[\tilde{\mu}_1(X, \tilde{U}_1)|X], \] and: \[ U_1 \equiv Y(1) - \mathbb{E}[\tilde{\mu}_1(X, \tilde{U}_1)|X]. \]

1. Instrument independence: \((U_1, U_0, V) \perp \!\!\! \perp Z | X\).
2. Exclusion restriction: \(\mu_D(Z, X)\) has a non-degenerate distribution given \(X\).
3. Scalar \(V\) is continuously distributed.
4. \(\mathbb{E}[|Y(1)|]\) and \(\mathbb{E}[|Y(0)|]\) are finite.
5. \(0 < \mathbb{P}[D = 1| X] < 1\).

• Vytlacil (2002) showed that these assumptions are equivalent to the identifying assumptions for LATE.
• In particular, the threshold crossing selection equation with scalar \(V\) can be interpreted as the monotonicity assumption.

• Under the first assumption, the propensity score can be written as: \[ \begin{split} e(z, x) &\equiv \mathbb{P}(D = 1 | z, x) \\ &= \mathbb{P}[V \le \mu_D(z, x) | z, x]\\ &= \mathbb{P}\{F_{V|X}(V|x) \le F_{V|X}[\mu_D(z, x)|x]|z, x\}\\ &= \mathbb{P}\{U_D \le F_{V|X}[\mu_D(z, x)|x]|z, x\}, U_D|X \sim U[0, 1]\\ &= F_{V|X}[\mu_D(z, x)|x]. \end{split} \]
• Then, under the third assumption, we have: \[ D =1\{e(Z, X) \ge U_D\}, U_D|X \sim U[0, 1]. \]

• In this framework, define the marginal treatment effect (MTE) as follows: \[ \tau_{mte}(x, u_D) \equiv \mathbb{E}[Y(1) - Y(0)| X = x, U_D = u_D], \] as a function of \((x, u_D)\).
• We can define the other estimands such as the average treatment effect and local average treatment effect as a function of the marginal treatment effects.

• For example, we can write LATE as: \[ \begin{split} \tau_{late}(x) &= \mathbb{E}[Y(1) - Y(0)| X = x, G = co]\\ &= \mathbb{E}[Y(1) - Y(0)| X = x, D(z) = 1, D(z') = 0]\\ &= \mathbb{E}[Y(1) - Y(0)| X = x, U_D \le e(z, x), U_D \ge e(z', x)]\\ &=\frac{1}{e(z, x) - e(z', x)} \int_{e(z', x)}^{e(z, x)} \tau_{mte}(x, u_D) du_D. \end{split} \]

• We define the local IV (LIV) estimand as: \[ \tau_{liv}(x, e) \equiv \frac{\partial}{\partial e}\mathbb{E}[Y|X = x, e(Z, x) = e]. \]
• We can nonparametrically estimate LIV \(\tau_{liv}(x, e)\) at every \(e\) in the support of the conditional distribution of \(e(Z, X)\) given \(X = x\).

\[ \begin{split} \mathbb{E}[Y|e(Z) = e] &= \mathbb{E}[Y(0)| e(z) = e]\\ &+ \mathbb{E}\{D\cdot[Y(1) - Y(0)]| e(z) = e\} \\ &= \mathbb{E}[Y(0)| e(z) = e]\\ &+ \mathbb{E}[Y(1) - Y(0)| D = 1, e(z) = e] \cdot \mathbb{P}[D = 1|e(z) = e]\\ &= \mathbb{E}[Y(0)| e(z) = e] + \mathbb{E}[Y(1) - Y(0)| D = 1, e(z) = e] \cdot e\\ &= \mathbb{E}[Y(0)] + \mathbb{E}[Y(1) - Y(0)| D = 1, e(z) = e] \cdot e\\ &= \mathbb{E}[Y(0)] + (\mu_1 - \mu_0) \cdot e\\ &+ \mathbb{E}[U_1 - U_0| D = 1, e(z) = e] \cdot e\\ &= \mathbb{E}[Y(0)] + (\mu_1 - \mu_0) \cdot e + \mathbb{E}[U_1 - U_0| U_D \le e] \cdot e\\ &= \mathbb{E}[Y(0)] + (\mu_1 - \mu_0) \cdot e\\ &+ \int_0^e \mathbb{E}[U_1 - U_0| U_d = u_D] d u_D. \end{split} \]

• By taking the derivatives with respect to \(e\), we have: \[ \begin{split} \tau_{liv}(e) &= \frac{\partial}{\partial e} \mathbb{E}[Y|e(Z) = e] \\ &= \mu_1 - \mu_0 + \mathbb{E}[U_1 - U_0| U_d = e] \\ &= \tau_{mte}(e). \end{split} \]

set.seed(1)
N <- 1000
e <- 0.5
tau <- c(1, 2)

df_latent <-
  dplyr::bind_rows(
    tibble::tibble(
      g = "nc",
      y_0 = rnorm(N),
      y_1 = tau[1] + rnorm(N)
    ),
    tibble::tibble(
      g = "co",
      y_0 = rnorm(N),
      y_1 = tau[2] + rnorm(N)
    )   
  )
df_latent

## # A tibble: 2,000 x 3
##    g        y_0    y_1
##    <chr>  <dbl>  <dbl>
##  1 nc    -0.626  2.13 
##  2 nc     0.184  2.11 
##  3 nc    -0.836  0.129
##  4 nc     1.60   1.21 
##  5 nc     0.330  1.07 
##  6 nc    -0.820 -0.663
##  7 nc     0.487  1.81 
##  8 nc     0.738 -0.912
##  9 nc     0.576 -0.247
## 10 nc    -0.305  2.00 
## # ... with 1,990 more rows

df_latent %>%
  dplyr::filter(g == "co") %>%
  dplyr::summarise(mean(y_1) - mean(y_0))

## # A tibble: 1 x 1
##   `mean(y_1) - mean(y_0)`
##                     <dbl>
## 1                    2.00

df_latent <-
  df_latent %>%
  dplyr::mutate(z = (runif(length(g)) < e))
df_latent

## # A tibble: 2,000 x 4
##    g        y_0    y_1 z    
##    <chr>  <dbl>  <dbl> <lgl>
##  1 nc    -0.626  2.13  TRUE 
##  2 nc     0.184  2.11  TRUE 
##  3 nc    -0.836  0.129 FALSE
##  4 nc     1.60   1.21  TRUE 
##  5 nc     0.330  1.07  FALSE
##  6 nc    -0.820 -0.663 TRUE 
##  7 nc     0.487  1.81  TRUE 
##  8 nc     0.738 -0.912 TRUE 
##  9 nc     0.576 -0.247 TRUE 
## 10 nc    -0.305  2.00  FALSE
## # ... with 1,990 more rows

df_latent <-
  df_latent %>%
  dplyr::mutate(
    w = ifelse(
      g == "nc",
      FALSE,
      z
    )
  )
df_latent

## # A tibble: 2,000 x 5
##    g        y_0    y_1 z     w    
##    <chr>  <dbl>  <dbl> <lgl> <lgl>
##  1 nc    -0.626  2.13  TRUE  FALSE
##  2 nc     0.184  2.11  TRUE  FALSE
##  3 nc    -0.836  0.129 FALSE FALSE
##  4 nc     1.60   1.21  TRUE  FALSE
##  5 nc     0.330  1.07  FALSE FALSE
##  6 nc    -0.820 -0.663 TRUE  FALSE
##  7 nc     0.487  1.81  TRUE  FALSE
##  8 nc     0.738 -0.912 TRUE  FALSE
##  9 nc     0.576 -0.247 TRUE  FALSE
## 10 nc    -0.305  2.00  FALSE FALSE
## # ... with 1,990 more rows

df_observed <-
  df_latent %>%
  dplyr::mutate(y = y_0 * (1 - w) + y_1 * w) %>%
  dplyr::select(y, z, w)
head(df_observed)

## # A tibble: 6 x 3
##        y z     w    
##    <dbl> <lgl> <lgl>
## 1 -0.626 TRUE  FALSE
## 2  0.184 TRUE  FALSE
## 3 -0.836 FALSE FALSE
## 4  1.60  TRUE  FALSE
## 5  0.330 FALSE FALSE
## 6 -0.820 TRUE  FALSE

itt_w <-
  df_observed %>%
  dplyr::filter(z == 1) %>%
  dplyr::summarise(w = sum(w) / length(w)) %>%
  dplyr::pull(w)
itt_w

## [1] 0.5024777

df_observed %>%
  lm(
    data = .,
    formula = w ~ z
  ) %>%
  modelsummary(fmt = 6)

itt_y <-
  df_observed %>%
  dplyr::group_by(z) %>%
  dplyr::summarise(y = mean(y)) %>%
  dplyr::ungroup() %>%
  dplyr::summarise(y = sum(y * z) - sum(y * (1 - z))) %>%
  dplyr::pull(y)
itt_y

## [1] 1.014518

df_observed %>%
  lm(
    data = .,
    formula = y ~ z
  ) %>%
  modelsummary(fmt = 6)

tau_late <-
  itt_y / itt_w
tau_late

## [1] 2.019031

df_observed %>%
  estimatr::iv_robust(
    data = .,
    formula = y ~ w | z
  ) %>%
  modelsummary(fmt = 6)

• Chapter 23-24, Guido W. Imbens and Donald B. Rubin, 2015, Causal Inference for Statistics, Social, and Biomedical Sciences, Cambridge University Press.
• Section 9, Athey, Susan, and Guido Imbens. 2016. “The Econometrics of Randomized Experiments.” arXiv [stat.ME]. arXiv. http://arxiv.org/abs/1607.00698.