Goal of the session

Air pollution and health

An instrument for air pollution

Other ideas of instruments?

• Boat traffic in a port (Moretti and Neidell 2011)
• Thermal inversions (Arceo, Hanna, and Oliva 2016)
• Storms causing traffic jam in airports (Schlenker and Walker 2016)

• Why are they each valid instruments?

Formal intuition

• $y_i = \alpha + \beta x_i + u_i$ with $cov(x, u) \neq 0$

• Problem: $x_i$ is endogenous

• IV solution:

  • Find a variable $z$ that shifts $x$ for reasons unrelated to unobservables in $y$
  • It isolates variation in $x$ that is unrelated to $u$ and recover $\beta$
  • Focuses on the variation that is explained by the instrument

Potential outcomes

• Effect of the instrument on the treatment varies across individuals
• We only observe $z_i$ and $d_i$, not these groups

Identifying assumptions

• Exclusion restriction $\Rightarrow ITT_{\text{never-takers}} = ITT_{\text{always-takers}} = 0$.
• Motonicity $\Rightarrow$ compliers are the only individuals whose treatment status could be altered by the instrument

Validity and exclusion restriction

• Cannot show that the exclusion restriction holds
• Need to convince the audience that it does with logic arguments
• Need to find “clever” instruments
• But some instruments may turn out to be not so great

Exclusion-restriction violations

• Sometimes an instrument affects many (many) other variables $\Rightarrow$ it may affect the outcome through another path (Mellon 2024)

What does the IV estimate?

$$\begin{aligned} \beta_{IV} & := \dfrac{\text{Cov}(y_i, z_i)}{\text{Cov}(d_i, z_i)} = \ ... \\ & = \dfrac{\mathbb{E}[y_i \mid z_i = 1] - \mathbb{E}[y_i \mid z_i = 0]}{\mathbb{E}[d_i \mid z_i = 1] - e[d_i \mid z_i = 0]} = \ ... \\ & = \underbrace{\mathbb{E}\left[y_i(1) - y_i(0) \;\middle|\; d_i(1) = 1,\; d_i(0) = 0 \right]}_{\text{LATE on the compliers}} \end{aligned}$$

LATE - Intuition

• The instrument isolates the local variation in treatment driven by $z$
  
• If treatment effects are heterogeneous, IV does not recover the overall average treatment effect (ATE)
• Instead, it identifies the effect local to those influenced by the instrument

LATE - Example

The LATE differs from the ATE and the ATT

Variation used for identification

Weak Instruments

Consequences

1. Large variance: Confidence intervals can be very wide
2. Finite-sample bias: IV can be biased towards OLS (especially in small samples)
3. Distorted inference: Standard t-tests and F-tests become unreliable

• Testing for relevance: compute the first-stage F-stat (rule of thumb: ok if F-stat > 10)

IV Estimator and 2SLS

• Wald estimator: sample analog of the estimand

$$\widehat{\beta}_W = \dfrac{\widehat{\text{Cov}}(y, z)}{\widehat{\text{Cov}}(d, z)}$$

• Numerically equivalent to Two-Stage Least Squares (2SLS) estimator $\widehat{\beta}_{2SLS}$ obtained through:

  1. Regress $d$ on instrument(s) $z$ and controls and retrieve predicted values

  $$d_i = \pi_0 + \pi_1 z_i + v_i \quad \Rightarrow \quad \widehat{d}_i = \widehat{\mathbb{E}}[d_i \mid z_i]$$

  1. Regress $y$ on predicted treatment $\hat{d}$

  $$y_i = \alpha + \beta \widehat{d}_i + e_i$$

Control function approach

Practical advice

Strengths and weaknesses

• $+$ Compelling identification strategy
• $+$ Can use IVs to address attenuation bias that may result from measurement error in $d$

• $-$ $\beta_{IV}$ has finite sample bias. Stems from randomness in estimates of $d$
• $-$ $\beta_{IV}$ less efficient than OLS, precision further $\searrow$ with weak instruments
• $-$ Weak instruments can make $\beta_{IV}$ way less efficient and even more biased than $\beta_{OLS}$
• $-$ In many settings (e.g., non-linear $d$), 2SLS can be very biased

Exercises

Summary