34.9 Emerging Research

34.9.1 Heterogeneous Effects in IV Estimation

34.9.1.1 Constant vs. Heterogeneous Treatment Effects

The standard instrumental variables framework assumes that the causal effect of an endogenous regressor $D_i$ on an outcome $Y_i$ is constant across individuals, i.e.:

$$Y_i = \beta_0 + \beta_1 D_i + u_i$$

This implies homogeneous treatment effects, where $\beta_1$ is a structural parameter that applies uniformly to all individuals $i$ in the population. We refer to this as the homogeneous treatment effects model, and it underlies the traditional IV assumptions:

• Linearity with a constant effect $\beta_1$.

• Instrument relevance: $\mathrm{Cov}(Z_i, D_i) \ne 0$.

• Instrument exogeneity: $\mathrm{Cov}(Z_i, u_i) = 0$.

Under these assumptions, the IV estimator $\hat{\beta}_1^{IV}$ consistently estimates the causal effect $\beta_1$.

34.9.1.2 Heterogeneous Treatment Effects and the Problem for IV

In practice, treatment effects often vary across individuals. That is, the effect of $D_i$ on $Y_i$ depends on the individual’s characteristics or other unobserved factors:

$$Y_i = \beta_{1i} D_i + u_i$$

Here, $\beta_{1i}$ represents the individual-specific causal effect, and the population Average Treatment Effect is:

$$ATE = \mathbb{E}[\beta_{1i}]$$

In the presence of treatment effect heterogeneity, the IV estimator $\hat{\beta}_1^{IV}$ does not, in general, estimate the ATE. Instead, it estimates a weighted average of the heterogeneous treatment effects, with weights determined by the instrumental variation in the data.

This distinction is critical:

• OLS estimates a weighted average treatment effect, with weights depending on the variance of $D_i$.

• IV estimates a [Local Average Treatment Effect] (LATE), depending on the instrument $Z_i$.

When there is one endogenous regressor $D_i$ and one instrument $Z_i$, both binary variables, we can interpret the IV estimator as the [Local Average Treatment Effect] under specific assumptions. The setup is:

$$Y_i = \beta_0 + \beta_{1i} D_i + u_i$$

• $D_i \in \{0, 1\}$: The treatment indicator.
• $Z_i \in \{0, 1\}$: The binary instrument.

Assumptions for the LATE Interpretation

1. Instrument Exogeneity

$$Z_i \perp (u_i, v_i)$$ - The instrument is as good as randomly assigned, and is independent of both the structural error term $u_i$ and the unobserved determinants $v_i$ that affect treatment selection.

2. Relevance

$$\mathbb{P}(D_i = 1 | Z_i = 1) \ne \mathbb{P}(D_i = 1 | Z_i = 0)$$ - The instrument must affect the likelihood of receiving treatment $D_i$.

3. Monotonicity (G. W. Imbens and Angrist 1994)

$$D_i(1) \ge D_i(0) \quad \forall i$$

• There are no defiers: no individual who takes the treatment when $Z_i = 0$ but does not take it when $Z_i = 1$.

• Monotonicity is not testable, but must be defended on theoretical grounds.

Under these assumptions, $\hat{\beta}_1^{IV}$ estimates the [Local Average Treatment Effect]:

$$LATE = \mathbb{E}[\beta_{1i} | \text{Compliers}]$$

• Compliers are individuals who receive the treatment when $Z_i = 1$, but not when $Z_i = 0$.
• Local refers to the fact that the estimate pertains to this specific subpopulation of compliers.

Implications:

• The LATE is not the ATE, unless treatment effects are homogeneous, or the complier subpopulation is representative of the entire population.
• Different instruments define different complier groups, leading to different LATEs.

34.9.1.3 Multiple Instruments and Multiple LATEs

When we have multiple instruments $Z_i^{(1)}, Z_i^{(2)}, \dots, Z_i^{(m)}$, each can induce different complier groups:

• Each instrument has its own LATE, corresponding to its own group of compliers.

• If heterogeneous treatment effects exist, these LATEs may differ.

In an overidentified model, where $m > k$, the 2SLS estimator imposes the assumption that all instruments identify the same causal effect $\beta_1$. This leads to the moment conditions:

$$\mathbb{E}[Z_i^{(j)}(Y_i - D_i \beta_1)] = 0 \quad \forall j = 1, \dots, m$$

If instruments identify different LATEs:

• These moment conditions can be inconsistent with one another.

• The Sargan-Hansen J-test may reject, even though each instrument is valid (i.e., exogenous and relevant).

Key Insight: The J-test rejects because the homogeneity assumption is violated—not because instruments are invalid in the exogeneity sense.

34.9.1.4 Illustration: Multiple Instruments, Different LATEs

Consider the following example:

• $Z_i^{(1)}$ identifies a LATE of 1.0.

• $Z_i^{(2)}$ identifies a LATE of 2.0.

• If both instruments are included in an overidentified IV model, the 2SLS estimator tries to reconcile these LATEs as if they were identifying the same $\beta_1$, leading to:

  • An average of these LATEs.

  • A possible rejection of the overidentification restrictions via the J-test.

This scenario is common in:

• Labor economics (e.g., different instruments for education identify different populations).

• Marketing and pricing experiments (e.g., different price instruments impact different customer segments).

34.9.1.5 Practical Implications for Empirical Research

1. Be Clear About Whose Effect You’re Estimating

• Different instruments often imply different complier groups.
• Understanding who the compliers are is essential for policy implications.

2. Interpret the J-Test Carefully

• A rejection may indicate treatment effect heterogeneity, not necessarily instrument invalidity.
• Supplement the J-test with:
  • Subgroup analysis.
  • Sensitivity analysis.
  • Local Instrumental Variable or Marginal Treatment Effects frameworks.

3. Use Structural Models When Needed

• If you need an ATE, consider parametric or semi-parametric structural models that explicitly model heterogeneity.

4. Don’t Assume LATE = ATE

• Be cautious in generalizing LATE estimates beyond the complier subpopulation.

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34.9.2 Zero-Valued Outcomes

For outcomes that take zero values, log transformations can introduce interpretation issues. Specifically, the coefficient on a log-transformed outcome does not directly represent a percentage change (J. Chen and Roth 2023). We have to distinguish the treatment effect on the intensive (outcome: 10 to 11) vs. extensive margins (outcome: 0 to 1), and we can’t readily interpret the treatment coefficient of log-transformed outcome regression as percentage change. In such cases, researchers use alternative methods:

34.9.2.1 Proportional LATE Estimation

When dealing with zero-valued outcomes, direct log transformations can lead to interpretation issues. To obtain an interpretable percentage change in the outcome due to treatment among compliers, we estimate the proportional Local Average Treatment Effect (LATE), denoted as $\theta_{ATE\%}$.

Steps to Estimate Proportional LATE:

1. Estimate LATE using 2SLS:

   We first estimate the treatment effect using a standard Two-Stage Least Squares regression: $$Y_i = \beta D_i + X_i + \epsilon_i,$$ where:

   • $D_i$ is the endogenous treatment variable.
   • $X_i$ includes any exogenous controls.
   • $\beta$ represents the LATE in levels for the mean of the control group’s compliers.

2. Estimate the control complier mean ($\beta_{cc}$):

   Using the same 2SLS setup, we estimate the control mean for compliers by transforming the outcome variable (Abadie, Angrist, and Imbens 2002): $$Y_i^{CC} = -(D_i - 1) Y_i.$$ The estimated coefficient from this regression, $\beta_{cc}$, captures the mean outcome for compliers in the control group.

3. Compute the proportional LATE:

   The estimated proportional LATE is given by: $$\theta_{ATE\%} = \frac{\hat{\beta}}{\hat{\beta}_{cc}},$$ which provides a direct percentage change interpretation for the outcome among compliers induced by the instrument.

4. Obtain standard errors via non-parametric bootstrap:

   Since $\theta_{ATE\%}$ is a ratio of estimated coefficients, standard errors are best obtained using non-parametric bootstrap methods.

5. Special case: Binary instrument

   If the instrument is binary, $\theta_{ATE\%}$ for the intensive margin of compliers can be directly estimated using Poisson IV regression (ivpoisson in Stata).

34.9.2.2 Bounds on Intensive-Margin Effects

Lee (2009) proposed a bounding approach for intensive-margin effects, assuming that compliers always have positive outcomes regardless of treatment (i.e., intensive-margin effect). These bounds help estimate treatment effects without relying on log transformations. However, this requires a monotonicity assumption for compliers where they should still have positive outcome regardless of treatment status.

References

Abadie, Alberto, Joshua Angrist, and Guido Imbens. 2002. “Instrumental Variables Estimates of the Effect of Subsidized Training on the Quantiles of Trainee Earnings.” Econometrica 70 (1): 91–117.

Chen, Jiafeng, and Jonathan Roth. 2023. “Logs with Zeros? Some Problems and Solutions.” The Quarterly Journal of Economics, qjad054.

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

Lee, David S. 2009. “Training, Wages, and Sample Selection: Estimating Sharp Bounds on Treatment Effects.” The Review of Economic Studies, 1071–1102.