(iv-late-en)=
# Instrumental Variables: Identification and LATE
Uber wants to measure the causal effect of its driver bonus program ($T$) on weekly earnings ($Y$). The problem is immediate: the most motivated and skilled drivers tend to volunteer for the program — not because the bonus pushes them to work more, but because they already work more regardless. An unobservable variable such as driver quality ($U$) determines both program enrollment and earnings, independently of the bonus. Regressing earnings on bonus participation conflates the causal effect with that selection, and no observable control can close that source of confounding.
In this section we introduce the **instrumental variables (IV)** strategy as a response to this problem. The instrument — a third variable that moves the treatment but has no direct effect on the outcome — allows us to recover the causal effect even when the confounder is unobservable. We derive the **Local Average Treatment Effect (LATE)** estimator, show which population it identifies, and explore the implications of different types of response to the instrument.
---
## 1. Objectives
By the end of this section, you will be able to:
- Identify when the back-door criterion fails because the confounder is unobservable.
- State the two conditions a valid instrumental variable must satisfy.
- Derive the LATE estimator as the ratio of the reduced form and first stage.
- Interpret which subpopulation the LATE identifies and why it can differ from the average treatment effect on the treated (TOT).
- Recognize the role of the monotonicity assumption and the consequences of its violation.
---
## 2. The Problem: Unobservable Confounders
The DAG for the Uber problem has three nodes: the treatment $T$ (enrollment in the bonus program), the outcome $Y$ (weekly earnings), and the unobservable confounder $U$ (driver quality).
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The confounder $U$ generates two paths between $T$ and $Y$: the causal path ($T \rightarrow Y$) and the back-door path ($T \leftarrow U \rightarrow Y$). As we saw in the [DAGs section](3_dags.md), to identify the causal effect of $T$ we need to block all back-door paths by conditioning on a set of variables $S$ that satisfies the back-door criterion.
Here the problem is that $U$ is **unobservable**: we cannot include it as a control in the regression. The back-door criterion cannot be satisfied. This is precisely a case of [omitted variable bias](../ch2_regresion_multiple/1_multiple_regression.md): when $U$ is excluded from the regression, the OLS estimate is biased in expectation:
$$\mathbb{E}[\hat{\beta}_{\text{OLS}}] = \beta + \frac{\text{Cov}(T, U)}{\text{Var}(T)} \cdot \delta \neq \beta$$
where $\delta$ is the effect of $U$ on $Y$. The second term is the selection bias: drivers who enroll in the bonus ($T=1$) have, on average, higher quality ($U$ higher), which inflates their observed earnings beyond the causal effect of the bonus. We need a different strategy.
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## 3. The Identification Strategy: The Instrument
An **instrument** $Z$ is a variable that satisfies two conditions.
**First condition — Relevance (first stage):** $Z$ has a non-zero effect on $T$. If $Z$ does not move $T$, it provides no useful information about exogenous variation in the treatment.
**Second condition — Exogeneity and independence:** $Z$ affects $Y$ *only through $T$*, and is independent of the confounder $U$. This implies two restrictions:
- **Exogeneity:** $Z \perp U$ — the instrument is uncorrelated with the confounder.
- **Exclusion restriction:** there is no direct effect $Z \rightarrow Y$ — the only reason $Z$ could change $Y$ is that it first changes $T$.
The full DAG with the instrument represents these conditions directly:
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Note what is **absent** from the DAG: there is no edge between $Z$ and $U$, and no direct edge between $Z$ and $Y$. Those absences are the content of the assumptions. The present edges — $Z \rightarrow T \rightarrow Y$ — mark the only causal channel through which $Z$ can affect $Y$.
> **Exogeneity ($Z \perp U$):** $Z$ is not correlated with unobservable driver quality. When $Z$ comes from a quasi-experimental design — such as algorithmic random assignment — this can be supported with a balance table: the $Z=1$ and $Z=0$ groups are similar on observable pre-treatment characteristics. However, the absence of correlation with *unobservables* is a structural assumption that data alone can never directly refute; it must be argued from the logic of the design.
> **Exclusion restriction ($Z \not\rightarrow Y$ directly):** $Z$ has no direct effect on $Y$. That is, the only way $Z$ could change $Y$ is by first changing $T$. This assumption is entirely untestable with data — it must be argued from economic logic and the structure of the instrument.
**The instrument in the Uber example:** Suppose Uber randomly sends a digital push notification offering the bonus ($Z = 1$ if the driver received the notification, $Z = 0$ otherwise). The probability of receiving the notification is independent of driver quality — it is assigned by a platform algorithm that does not observe $U$ (**exogeneity**). The notification itself contains no information about driving strategies or hourly rates — its only possible effect on earnings is to induce enrollment in the bonus program (**exclusion restriction**, plausible as long as the notification does not reveal the value of the bonus or provide productivity tips).
> **Note:** The First Condition (relevance) is testable from the data — we can estimate the first stage and evaluate whether $Z$ has a significant effect on $T$. The Second Condition (exogeneity and exclusion restriction), by contrast, consists of theoretical assumptions that cannot be directly verified from the data. Our confidence in them rests entirely on our understanding of the institutional setting and the logic of the design.
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## 4. The LATE Estimator: Three Observable Quantities
The instrument $Z$ allows us to recover the causal effect of $T$ on $Y$ even though $U$ is never observed. The key is that, under the two instrument assumptions, three quantities can be estimated without bias from the data.
### 4.1 First stage
The **first stage** measures how much $Z$ moves the treatment $T$:
$$\hat{\alpha} = \mathbb{E}[T \mid Z=1] - \mathbb{E}[T \mid Z=0]$$
This estimator is unbiased because $Z \perp U$: the groups $Z=1$ and $Z=0$ have the same distribution of the confounder $U$ in expectation, so the difference in $T$ across groups reflects only the causal effect of $Z$ on $T$.
### 4.2 Reduced form
The **reduced form** measures how much $Z$ moves the outcome $Y$:
$$\hat{\gamma} = \mathbb{E}[Y \mid Z=1] - \mathbb{E}[Y \mid Z=0]$$
This estimator is also unbiased. By $Z \perp U$, the $Z=1$ and $Z=0$ groups are comparable in $U$. And by the exclusion restriction, the only channel through which $Z$ affects $Y$ is the causal path $Z \rightarrow T \rightarrow Y$. Consequently, $\hat{\gamma}$ captures exactly the total effect of $Z$ on $Y$ operating through the treatment.
### 4.3 LATE as a ratio
The **Local Average Treatment Effect (LATE)** is:
$$\hat{\beta}_{\text{LATE}} = \frac{\hat{\gamma}}{\hat{\alpha}} = \frac{\mathbb{E}[Y \mid Z=1] - \mathbb{E}[Y \mid Z=0]}{\mathbb{E}[T \mid Z=1] - \mathbb{E}[T \mid Z=0]}$$
**Why does this ratio work?** The reduced form captures the effect of $Z$ on $Y$ through the channel $Z \rightarrow T \rightarrow Y$. That effect is the product of two components: the fraction of units that change their treatment in response to $Z$ (the first stage $\alpha$) multiplied by the treatment effect on those units ($\beta$). Formally:
$$\hat{\gamma} \xrightarrow{p} \alpha \cdot \beta$$
Dividing by $\hat{\alpha}$ cancels the first stage and recovers $\beta$ for the subpopulation that responds to the instrument:
$$\hat{\beta}_{\text{LATE}} \xrightarrow{p} \frac{\alpha \cdot \beta}{\alpha} = \beta$$
**Formal result:**
$$\beta_{\text{LATE}} = \frac{\mathbb{E}[Y \mid Z=1] - \mathbb{E}[Y \mid Z=0]}{\mathbb{E}[T \mid Z=1] - \mathbb{E}[T \mid Z=0]}$$
Under exogeneity ($Z \perp U$), the exclusion restriction (no direct $Z \rightarrow Y$), relevance ($\alpha \neq 0$), and monotonicity (see Section 5), the LATE is a consistent estimator of the average causal effect for the *complier* subpopulation — units whose treatment is determined by the instrument. The formal derivation appears in the [appendix](#appendix-iv-late).
**In the Uber example:** if drivers who receive the notification are 30 percentage points more likely to enroll in the bonus (first stage = 0.30), and their weekly earnings are $45 higher on average (reduced form = 45), then the LATE is:
$$\hat{\beta}_{\text{LATE}} = \frac{45}{0.30} = \$150 \text{ per driver who enrolls because of the notification}$$
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## 5. Consolidation via Simulation: Scenario A
With the identification logic and the formal derivation of the LATE in hand, the first simulation lets you observe the three steps of the Wald estimator in action, with data generated exactly according to the DAG we just analyzed.
Before exploring the dashboard, predict:
- What happens to OLS bias when you increase the confounding intensity ($\delta$)?
- Does the LATE move with $\delta$? Why shouldn't it?
- What happens when $\delta = 0$? Is the instrument necessary in that case?
- When you change $\beta$, do the three quantities (first stage, reduced form, LATE) respond as the theory predicts?
The **means table** on the right panel shows exactly the Wald computation: the first row computes $\hat{\alpha}$ as the difference in means of $T$ across $Z=1$ and $Z=0$ groups; the second row computes $\hat{\gamma}$ as the difference in means of $Y$. The three tiles below synthesize the result: first stage, reduced form, and LATE = $\hat{\gamma}/\hat{\alpha}$. The bar chart on the right compares OLS and LATE against the true value of $\beta$: as you increase $\delta$, the red bar (OLS) moves away from the true value while the blue bar (LATE) remains centered on it.
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## 6. Who Does the LATE Identify? Compliance Types
The LATE does not identify the average effect for the entire population, nor even for the treated. It identifies the effect for a specific subpopulation: the **compliers** — units whose treatment status changes in response to the instrument.
To understand this, it is useful to classify all individuals according to how they would respond to the instrument under both possible values of $Z$.
### 6.1 The compliance taxonomy
In the Uber context, where $Z$ is receiving the bonus notification and $T$ is enrolling in the program. We will imagine four types of individuals — following the taxonomy of {cite:t}`angrist2015mastering`:
| | **Not enrolled if Z = 0** | **Enrolled if Z = 0** |
|---|---|---|
| **Enrolled if Z = 1** | **Camila** — Complier | **Alvaro** — Always-taker |
| **Not enrolled if Z = 1** | **Normando** — Never-taker | **Diego** — Defier |
**Camila — Compliers ($\pi_C$):** Enroll if and only if they receive the notification. These are exactly the drivers whose enrollment decision is moved by $Z$. They are the subpopulation identified by the LATE.
**Alvaro — Always-takers ($\pi_A$):** Enroll regardless of whether they receive the notification — they are drivers so motivated that they would participate in any case. Their treatment does not respond to $Z$, so they contribute nothing to the first stage.
**Normando — Never-takers ($\pi_N$):** Do not enroll regardless of whether they receive the notification. They also do not respond to the instrument or contribute to the first stage.
**Diego — Defiers ($\pi_D$):** Do the opposite of what $Z$ prescribes: they enroll when they do *not* receive the notification and do not enroll when they *do* receive it. Their existence violates the monotonicity assumption discussed below.
### 6.2 LATE vs. Treatment Effect on the Treated (TOT)
The **average treatment effect on the treated (TOT)** is the average causal effect for all who actually participate in the bonus. Assuming each type of individual has its own treatment effect, it follows:
$$\text{TOT} = \mathbb{E}[Y(1) - Y(0) \mid T = 1] = \frac{\pi_C \cdot \beta_C + \pi_A \cdot \beta_A}{\pi_C + \pi_A}$$
The treated are compliers ($\pi_C$) and always-takers ($\pi_A$). Because the LATE estimates an effect on the basis of response to the instrument, it only averages over compliers:
$$\text{LATE} = \mathbb{E}[Y(1) - Y(0) \mid \text{complier}] = \beta_C$$
**When do they diverge?** When always-takers have a different treatment effect from compliers ($\beta_A \neq \beta_C$). For instance, if always-takers voluntarily enroll because they benefit more ($\beta_A > \beta_C$), then TOT > LATE: the instrument identifies the effect for the drivers "at the margin" — those the nudge moves — not for those who would have enrolled anyway and who may be the ones who benefit most.
**When do they coincide?** When there are no always-takers ($\pi_A = 0$) or when all types have the same treatment effect ($\beta_A = \beta_C$). In the case of a perfectly complied randomized experiment, everyone is a complier and LATE = TOT = ATE.
### 6.3 The Monotonicity Assumption
For the LATE to have a causal interpretation, we need an additional assumption that rules out defiers:
**Monotonicity:** $T_i(Z=1) \geq T_i(Z=0)$ for all $i$.
In words: receiving the instrument can never *reduce* the probability of taking the treatment. In the Uber example: no driver is discouraged from enrolling in the bonus program by receiving the notification.
**Why does it matter?** Without monotonicity, the first-stage denominator mixes compliers and defiers:
$$\mathbb{E}[T \mid Z=1] - \mathbb{E}[T \mid Z=0] = \pi_C - \pi_D$$
And the IV estimator converges to:
$$\hat{\beta}_{\text{IV}} \xrightarrow{p} \frac{\pi_C \cdot \beta_C}{\pi_C - \pi_D}$$
When $\pi_D > 0$, this ratio does not equal $\beta_C$ (the LATE for compliers) nor any positively-weighted average of treatment effects across types of individuals. The estimator loses its causal interpretation.
Monotonicity is not directly testable — individual compliance types are counterfactual. It must be argued from the design logic and economic context.
---
## 7. Consolidation via Simulation: Scenario B
With the compliance taxonomy clear, the second scenario of the simulation lets you explore how the LATE, TOT, and IV estimator change when you vary the population composition.
Before using the controls, predict:
- What happens to the LATE when you increase $\beta_A$ above $\beta_C$ (making always-takers' effect greater than compliers')?
- If $\beta_A = \beta_C$, does the fraction of always-takers matter?
- What happens to the IV estimator when you introduce defiers ($\pi_D > 0$)? In which direction does it move?
- If $\pi_D$ approaches $\pi_C$, what happens to the first-stage denominator?
The right panel shows three bars: TOT (purple), true LATE (blue), and the IV estimate (green/red). When $\pi_D = 0$, the IV estimate coincides with the LATE. As you increase $\pi_D$, the green bar turns red and diverges from both targets — and the monotonicity violation warning banner appears. The dynamic text below the chart explains what is happening with the estimate under each configuration.
Monotonicity is the assumption that gives the IV denominator its interpretation: the first stage measures the fraction of compliers, not an arbitrary mixture of types. When that interpretation fails, the instrument no longer identifies a causally meaningful effect.
---
(appendix-iv-late)=
## Appendix: Consistency of the LATE Estimator
We show that, under the four instrument assumptions, $\hat{\beta}_{\text{LATE}} = \hat{\gamma}/\hat{\alpha}$ converges in probability to $\beta_C$, the average causal effect for compliers.
**Assumptions:**
1. **Relevance:** $\alpha = \mathbb{E}[T \mid Z=1] - \mathbb{E}[T \mid Z=0] \neq 0$.
2. **Exogeneity:** $Z \perp U$ (the instrument is independent of the confounder).
3. **Exclusion restriction:** $Y_i(z, t) = Y_i(t)$ for all $z$ — the potential outcome does not depend directly on $Z$, only on $T$.
4. **Monotonicity:** $T_i(Z=1) \geq T_i(Z=0)$ for all $i$ (no defiers).
**Step 1 — Decompose the first stage:**
Under monotonicity, $T_i(Z=1) - T_i(Z=0) \in \{0, 1\}$: the difference is 1 for compliers and 0 for always-takers and never-takers (defiers are ruled out by assumption). Therefore:
$$\alpha = \mathbb{E}[T \mid Z=1] - \mathbb{E}[T \mid Z=0] = \mathbb{E}[T_i(1) - T_i(0)] = \pi_C$$
The first stage identifies without bias the fraction of compliers.
**Step 2 — Decompose the reduced form:**
By the exclusion restriction and exogeneity:
$$\mathbb{E}[Y \mid Z=1] - \mathbb{E}[Y \mid Z=0] = \mathbb{E}[Y_i(T_i(1)) - Y_i(T_i(0))]$$
For always-takers: $T_i(1) = T_i(0) = 1$, so $Y_i(T_i(1)) - Y_i(T_i(0)) = 0$.
For never-takers: $T_i(1) = T_i(0) = 0$, same, difference = 0.
Only compliers contribute ($T_i(1) = 1, T_i(0) = 0$):
$$\gamma = \mathbb{E}[Y_i(1) - Y_i(0) \mid \text{complier}] \cdot \pi_C = \beta_C \cdot \pi_C$$
**Step 3 — The ratio:**
$$\beta_{\text{LATE}} = \frac{\gamma}{\alpha} = \frac{\beta_C \cdot \pi_C}{\pi_C} = \beta_C \qquad \blacksquare$$
The estimators $\hat{\alpha}$ and $\hat{\gamma}$ are sample mean differences that converge to their population values by the Law of Large Numbers. By the delta method, their ratio converges to $\beta_C$.
## References
The interactive dashboard in this section also draws on {cite:t}`angrist2009mostly`, {cite:t}`imbensangrist1994identification`, and {cite:t}`staigerstock1997instrumental`, in addition to {cite:t}`angrist2015mastering`.
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