Lesson M10.L02: Instrumental Variable Estimation
| Field | Detail |
|---|---|
| Module | M10 — Instrumental Variables |
| Level | Graduate (ECMT5001 S1 2027) |
| Duration | 35 min |
| Learning Objective | State and verify the two IV conditions; derive the IV estimator; compute IV estimates from summary statistics; understand the efficiency cost of IV relative to OLS |
| Provenance | Wooldridge, Introductory Econometrics, Ch 15.1–15.2 |
Explanation
The IV Strategy
When \(\text{Cov}(x, u) \neq 0\), OLS is inconsistent. The instrumental variables (IV) strategy identifies a variable \(z\) — the instrument — that is related to \(x\) but not directly related to \(y\) except through \(x\). This allows us to recover the causal effect \(\beta_1\) by using only the "clean" (exogenous) variation in \(x\) that comes from \(z\).
IV Requirements
A valid instrument \(z\) must satisfy two conditions:
Condition 1: Relevance
The instrument must be correlated with the endogenous regressor \(x\). Without this, \(z\) carries no information about \(x\) and the IV estimator is undefined (or very poorly behaved).
Testable: Relevance can be tested with a first-stage \(F\)-test (see M10.L04 on weak instruments).
Condition 2: Exogeneity (Exclusion Restriction)
The instrument affects \(y\) only through \(x\) — not through any other channel. This means \(z\) does not belong in the structural equation as a direct determinant of \(y\).
Not directly testable (in the exactly identified case): This is an assumption that must be justified on theoretical or institutional grounds. It is the central credibility condition for any IV strategy.
Mnemonic: A valid instrument is relevant (talks to \(x\)) and exogenous (doesn't talk to \(u\)).
The IV Estimator
For the simple model \(y = \beta_0 + \beta_1 x + u\) with instrument \(z\), the IV estimator is:
Derivation of consistency:
Under exogeneity (\(\text{Cov}(z, u) = 0\)) and relevance (\(\text{Cov}(z, x) \neq 0\)):
Intuition: The IV estimator divides the "reduced form" (effect of \(z\) on \(y\)) by the "first stage" (effect of \(z\) on \(x\)). If \(z\) only affects \(y\) through \(x\), then the ratio recovers the causal effect of \(x\) on \(y\).
Connection to OLS
Compare IV and OLS:
OLS uses the sample covariance of \(x\) with \(y\) (all variation in \(x\)). IV uses only the variation in \(x\) that is "explained by \(z\)" — discarding the endogenous variation.
Classic Instrument Examples
1. Card (1993) — Proximity to College
- Structural equation: \(\log(\text{wage}) = \beta_0 + \beta_1 \text{educ} + u\)
- Endogeneity: Ability is in \(u\), correlated with education
- Instrument (\(z\)): Whether the individual grew up near a 4-year college
- Relevance: People near colleges attend more schooling (lower cost of access)
- Exogeneity: Living near a college doesn't directly affect wages except through education
2. Angrist & Krueger (1991) — Quarter of Birth
- Instrument (\(z\)): Quarter of birth (Q1, Q2, Q3, Q4)
- Relevance: US compulsory schooling laws mean Q1 babies reach school-leaving age earlier → get less education on average
- Exogeneity: Birth quarter is essentially random with respect to ability/wages (debated)
3. Rainfall as Instrument for Agricultural Income
- Structural equation: \(\text{consumption} = \beta_0 + \beta_1 \text{income} + u\)
- Instrument: Rainfall (affects agricultural output and hence income in farming communities)
- Relevance: Rainfall → crop yields → income
- Exogeneity: Rainfall doesn't directly affect consumption except through income
IV vs OLS: Efficiency Trade-off
| Property | OLS (if exogenous) | IV (endogenous \(x\)) |
|---|---|---|
| Unbiasedness | Unbiased | Biased in finite samples (consistent) |
| Consistency | Consistent | Consistent |
| Efficiency | BLUE | Less efficient (larger SE) |
| SE formula | \(\sigma^2 / \text{Var}(x)\) | Larger (see below) |
Approximate IV variance (large \(n\)):
where \(R^2_{x,z}\) is the R² from regressing \(x\) on \(z\). Since \(R^2_{x,z} \leq 1\):
The stronger the instrument (higher \(R^2_{x,z}\)), the smaller the efficiency loss. A perfect instrument (\(R^2_{x,z} = 1\)) makes IV as efficient as OLS — but such instruments don't exist in practice. The larger SE of IV is the price of addressing endogeneity.
Worked Example
Problem: Estimate the return to schooling in a wage equation.
Instrument: Proximity to college (binary: \(z_i = 1\) if grew up within 25 miles of a 4-year college).
Summary statistics (hypothetical, from \(n = 3{,}010\) observations):
| Statistic | Value |
|---|---|
| \(\overline{\log(\text{wage})}\) | 6.26 |
| \(\overline{\text{educ}}\) | 13.17 |
| \(\bar{z}\) | 0.68 |
| \(\widehat{\text{Cov}}(z, \log(\text{wage}))\) | 0.0514 |
| \(\widehat{\text{Cov}}(z, \text{educ})\) | 0.3851 |
| \(\widehat{\text{Cov}}(\text{educ}, \log(\text{wage}))\) | 0.3022 |
| \(\widehat{\text{Var}}(\text{educ})\) | 7.513 |
Step 1: OLS estimate
OLS says one extra year of education raises wages by approximately 4.0%.
Step 2: IV estimate
IV says one extra year of education raises wages by approximately 13.4%.
Step 3: Interpret the gap
\(\hat{\beta}_1^{\text{IV}} = 0.134 > \hat{\beta}_1^{\text{OLS}} = 0.040\)
The higher IV estimate is consistent with: (a) ability bias pulling OLS down (if high-ability people go to school and earn more, OLS should be upward biased — but Card found upward IV, suggesting ability bias is not dominant here); OR (b) IV estimates a Local Average Treatment Effect (LATE) — the return for "compliers" (those induced into more education by proximity to college) may differ from the average return. This is the LATE interpretation, studied in M10 extension topics.
Step 4: Compare relevance
\(\widehat{\text{Cov}}(z, \text{educ}) = 0.385 > 0\) — people near colleges get significantly more education. Relevance is supported.
Exogeneity — we must argue that proximity to college does not affect wages through any channel other than education (e.g., urban/rural location, peer effects). Card controls for additional covariates to bolster this assumption.
Common Misconception
"Any variable that is correlated with \(x\) makes a valid instrument."
Only Condition 1 (relevance) is satisfied. Relevance alone is insufficient — the instrument must also satisfy exogeneity: \(\text{Cov}(z, u) = 0\). If \(z\) has a direct effect on \(y\) or is correlated with omitted variables in \(u\), the IV estimator is also inconsistent:
Even a small violation of exogeneity can cause substantial bias when the instrument is weak (small \(\text{Cov}(z,x)\)). This is why instrument validity is the most contested part of any IV analysis — finding a relevant instrument is easy; finding an exogenous one is hard.
Practice Prompts
-
Card (1993) uses proximity to college as an instrument for education. A critic argues that "people who grow up near colleges live in urban areas, and urban areas have higher wages for reasons unrelated to education." How does this critique threaten the exogeneity condition? How might Card's paper address this?
-
Using the data in the worked example, verify the IV formula by computing: \((a)\) the reduced-form coefficient \(\hat{\pi}_{\text{RF}} = \widehat{\text{Cov}}(z,y)/\widehat{\text{Var}}(z)\) and \((b)\) the first-stage coefficient \(\hat{\pi}_{\text{FS}} = \widehat{\text{Cov}}(z,x)/\widehat{\text{Var}}(z)\). Confirm that \(\hat{\pi}_{\text{RF}}/\hat{\pi}_{\text{FS}} = \hat{\beta}_1^{\text{IV}}\) (Wald estimator). [Assume \(\widehat{\text{Var}}(z) = 0.218\).]
-
In the wage–education example, OLS gives \(\hat{\beta}_1 = 0.040\) and IV gives \(\hat{\beta}_1 = 0.134\). Under what two scenarios could IV be higher than OLS? (Hint: think about the direction of ability bias and the LATE interpretation.)
Further Resources
- Wooldridge, Introductory Econometrics (7th ed.), Ch 15.1–15.2
- Card, D. (1993), "Using Geographic Variation in College Proximity to Estimate the Return to Schooling," NBER Working Paper 4483
- Angrist, J.D. & Pischke, J.-S. (2009), Mostly Harmless Econometrics, Ch 4
- Imbens, G. & Angrist, J.D. (1994), "Identification and Estimation of Local Average Treatment Effects," Econometrica