34.2 Framework for Instrumental Variables

We consider a binary treatment framework where:

• $D_i \sim Bernoulli(p)$ is a dummy treatment variable.

• $(Y_{0i}, Y_{1i})$ are the potential outcomes under control and treatment.

• The observed outcome is: $$Y_i = Y_{0i} + (Y_{1i} - Y_{0i}) D_i.$$

• We introduce an instrumental variable $Z_i$ satisfying: $$Z_i \perp (Y_{0i}, Y_{1i}, D_{0i}, D_{1i}).$$

  • This means $Z_i$ is independent of potential outcomes and potential treatment status.
  • $Z_i$ must also be correlated with $D_i$ to satisfy the relevance condition.

34.2.1 Constant-Treatment-Effect Model

Under the constant treatment effect assumption (i.e., the treatment effect is the same for all individuals),

$$\begin{aligned} Y_{0i} &= \alpha + \eta_i, \\ Y_{1i} - Y_{0i} &= \rho, \\ Y_i &= Y_{0i} + D_i (Y_{1i} - Y_{0i}) \\ &= \alpha + \eta_i + D_i \rho \\ &= \alpha + \rho D_i + \eta_i. \end{aligned}$$

where:

• $\eta_i$ captures individual-level heterogeneity.
• $\rho$ is the constant treatment effect.

The problem with OLS estimation is that $D_i$ may be correlated with $\eta_i$, leading to endogeneity bias.

34.2.2 Instrumental Variable Solution

A valid instrument $Z_i$ allows us to estimate the causal effect $\rho$ via:

$$\begin{aligned} \rho &= \frac{\text{Cov}(Y_i, Z_i)}{\text{Cov}(D_i, Z_i)} \\ &= \frac{\text{Cov}(Y_i, Z_i) / V(Z_i) }{\text{Cov}(D_i, Z_i) / V(Z_i)} \\ &= \frac{\text{Reduced form estimate}}{\text{First-stage estimate}} \\ &= \frac{E[Y_i |Z_i = 1] - E[Y_i | Z_i = 0]}{E[D_i |Z_i = 1] - E[D_i | Z_i = 0 ]}. \end{aligned}$$

This ratio measures the treatment effect only if $Z_i$ is a valid instrument.

34.2.3 Heterogeneous Treatment Effects and the LATE Framework

In a more general framework where treatment effects vary across individuals,

• Define potential outcomes as: $$Y_i(d,z) = \text{outcome for unit } i \text{ given } D_i = d, Z_i = z.$$

• Define treatment status based on $Z_i$: $$D_i = D_{0i} + Z_i (D_{1i} - D_{0i}).$$

  where:

  • $D_{1i}$ is the treatment status when $Z_i = 1$.
  • $D_{0i}$ is the treatment status when $Z_i = 0$.
  • $D_{1i} - D_{0i}$ is the causal effect of $Z_i$ on $D_i$.

34.2.4 Assumptions for LATE Identification

34.2.4.1 Independence (Instrument Randomization)

The instrument must be as good as randomly assigned:

$$[\{Y_i(d,z); \forall d, z \}, D_{1i}, D_{0i} ] \perp Z_i.$$

This ensures that $Z_i$ is uncorrelated with potential outcomes and potential treatment status.

This assumption let the first-stage equation be the average causal effect of $Z_i$ on $D_i$

$$\begin{aligned} E[D_i |Z_i = 1] - E[D_i | Z_i = 0] &= E[D_{1i} |Z_i = 1] - E[D_{0i} |Z_i = 0] \\ &= E[D_{1i} - D_{0i}] \end{aligned}$$

This assumption also is sufficient for a causal interpretation of the reduced form, where we see the effect of the instrument $Z_i$ on the outcome $Y_i$:

$$E[Y_i |Z_i = 1 ] - E[Y_i|Z_i = 0] = E[Y_i (D_{1i}, Z_i = 1) - Y_i (D_{0i} , Z_i = 0)]$$

34.2.4.2 Exclusion Restriction

This is also known as the existence of the instrument assumption (G. W. Imbens and Angrist 1994). The instrument should only affect $Y_i$ through $D_i$ (i.e., the treatment $D_i$ fully mediates the effect of $Z_i$ on $Y_i$):

$$\begin{aligned} Y_{1i} &= Y_i (1,1) = Y_i (1,0)\\ Y_{0i} &= Y_i (0,1) = Y_i (0,0) \end{aligned}$$

Under this assumption (and assume $Y_{1i, Y_{0i}}$ already satisfy the independence assumption), the observed outcome $Y_i$ can be rewritten as:

$$\begin{aligned} Y_i &= Y_i (0, Z_i) + [Y_i (1 , Z_i) - Y_i (0, Z_i)] D_i \\ &= Y_{0i} + (Y_{1i} - Y_{0i}) D_i. \end{aligned}$$

This assumption let us go from reduced-form causal effects to treatment effects (J. D. Angrist and Imbens 1995).

34.2.4.3 Monotonicity (No Defiers)

We assume that $Z_i$ affects $D_i$ in a monotonic way:

$$D_{1i} \geq D_{0i}, \quad \forall i.$$

• This assumption lets us assume that there is a first stage, in which we examine the proportion of the population that $D_i$ is driven by $Z_i$. It implies that $Z_i$ only moves individuals toward treatment, but never away. This rules out “defiers” (i.e., individuals who would have taken the treatment when not assigned but refuse when assigned).
• This assumption is used to solve to problem of the shifts between participation status back to non-participation status.

  • Alternatively, one can solve the same problem by assuming constant (homogeneous) treatment effect (G. W. Imbens and Angrist 1994), but this is rather restrictive.

  • A third solution is the assumption that there exists a value of the instrument, where the probability of participation conditional on that value is 0 J. Angrist and Imbens (1991).

Under monotonicity,

$$\begin{aligned} E[D_{1i} - D_{0i} ] = P[D_{1i} > D_{0i}]. \end{aligned}$$

34.2.5 Local Average Treatment Effect Theorem

Given Independence, Exclusion, and Monotonicity, we obtain the LATE result (J. D. Angrist and Pischke 2009, 4.4.1):

$$\begin{aligned} \frac{E[Y_i | Z_i = 1] - E[Y_i | Z_i = 0]}{E[D_i |Z_i = 1] - E[D_i |Z_i = 0]} = E[Y_{1i} - Y_{0i} | D_{1i} > D_{0i}]. \end{aligned}$$

This states that the IV estimator recovers the causal effect only for compliers—units whose treatment status changes due to $Z_i$.

IV only identifies treatment effects for switchers (compliers):

Switcher Type	Compliance Type	Definition
Switchers	Compliers	$D_{1i} > D_{0i}$ (take treatment if $Z_i = 1$, not if $Z_i = 0$)
Non-switchers	Always-Takers	$D_{1i} = D_{0i} = 1$ (always take treatment)
Non-switchers	Never-Takers	$D_{1i} = D_{0i} = 0$ (never take treatment)

• IV estimates nothing for always-takers and never-takers since their treatment status is unaffected by $Z_i$ (Similar to the fixed-effects models).

34.2.6 IV in Randomized Trials (Noncompliance)

• In randomized trials, if compliance is imperfect (i.e., compliance is voluntary), where individuals in the treatment group will not always take the treatment (e.g., selection bias), intention-to-treat (ITT) estimates are valid but contaminated by noncompliance.
• IV estimation using random assignment ($Z_i$) as an instrument for actual treatment received ($D_i$) recovers the LATE.

$$\begin{aligned} \frac{E[Y_i |Z_i = 1] - E[Y_i |Z_i = 0]}{E[D_i |Z_i = 1]} = \frac{\text{Intent-to-Treat Effect}}{\text{Compliance Rate}} = E[Y_{1i} - Y_{0i} |D_i = 1]. \end{aligned}$$

Under full compliance, LATE = Treatment Effect on the Treated (TOT).

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References

Angrist, Joshua D, and Guido W Imbens. 1995. “Two-Stage Least Squares Estimation of Average Causal Effects in Models with Variable Treatment Intensity.” Journal of the American Statistical Association 90 (430): 431–42.

Angrist, Joshua D, and Jörn-Steffen Pischke. 2009. Mostly Harmless Econometrics: An Empiricist’s Companion. Princeton university press.

Angrist, Joshua, and Guido Imbens. 1991. “Sources of Identifying Information in Evaluation Models.” National Bureau of Economic Research; National Bureau of Economic Research Cambridge, Mass., USA.

Imbens, Guido W, and Joshua D Angrist. 1994. “Identification and Estimation of Local Average Treatment Effects.” Econometrica 62 (2): 467–75.