Scenario-Based Problem: Instrumental Variables Identification and Endogeneity
Research question
Estimate the causal effect of years of schooling (S) on log annual earnings (Y) in a cross-sectional sample of N = 5,000 individuals born 1980–1985. The baseline structural model is:
Y_i = β S_i + X_i′γ + u_i,
where X includes age, gender, race, parental education, and region fixed effects. Schooling S_i may be endogenous due to unobserved ability and measurement error, implying Cov(S_i, u_i) ≠ 0.
Candidate instruments
- Z1: Indicator equal to 1 if there was a 4-year college within 10 km of the individual’s residence at age 17 (following the proximity strategy in Card 1995).
- Z2: Count of 2- and 4-year colleges within 25 km at age 17.
Both Z1 and Z2 are measured historically (at age 17), prior to observed schooling and earnings. Researchers argue that proximity affects schooling decisions through lower costs of attendance but has no direct effect on adult earnings, conditional on X and region fixed effects.
Selected estimation results (controlling for X and region fixed effects)
- OLS (Y on S, X): β̂_OLS = 0.074; SE = 0.006.
- First stage with Z1 only: S on Z1 and X
- π̂1 = 0.350; SE = 0.080; t-stat = 4.375; first-stage F on Z1 = 19.14.
- Reduced form with Z1 only: Y on Z1 and X
- 2SLS with Z1 and X:
- β̂_IV (implied by Wald ratio) = 0.035 / 0.350 = 0.100.
- First stage with Z1 and Z2 jointly: S on Z1, Z2, and X
- Joint F for instruments = 23.4; partial R^2 of instruments = 0.020.
- Overidentification (2SLS with Z1 and Z2): Hansen J = 1.80, df = 1, p = 0.18.
- Endogeneity (Hausman test comparing OLS vs. 2SLS with Z1): χ^2(1) = 4.60, p = 0.032.
Assumptions to evaluate
- Relevance: Cov(Zk, S | X) ≠ 0 for k ∈ {1, 2}.
- Exclusion: Cov(Zk, u | X) = 0.
- Monotonicity (for LATE interpretation): For any two values z′ > z of the binary proximity instrument Z1, S_i(z′) ≥ S_i(z) for all i (no “defiers”) (Imbens and Angrist 1994).
Tasks
- Identification logic (short answer)
- State precisely why OLS is potentially inconsistent in this setting. Frame your answer in terms of omitted variables and measurement error. Then state the moment condition that justifies IV estimation with Z1 (and with Z1, Z2 jointly).
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Instrument selection (multiple choice; select all that apply and justify)
Which statements are correct?
A. Z1 is relevant if the first-stage F-statistic exceeds conventional weak-IV thresholds.
B. Z1 is valid if its coefficient is statistically significant in the first stage.
C. Z1 is valid if, conditional on X and region fixed effects, it affects Y only through S.
D. Z2 is redundant if Z1 is already strong in the first stage; adding Z2 cannot improve identification.
E. If both Z1 and Z2 are valid, 2SLS identifies β consistently and the J test should not reject asymptotically.
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First-stage diagnostics (calculation + interpretation)
- Using the first-stage with Z1 only, compute and interpret the first-stage F-statistic. Based on the Staiger–Stock rule-of-thumb (F ≳ 10), assess whether weak instrument concerns are likely to be severe. Briefly note limitations of the rule-of-thumb and when Stock–Yogo critical values would be preferable.
- Wald/2SLS estimation (calculation)
- Using the reduced form and first stage with Z1 only, compute the Wald estimator for β. Report the point estimate and interpret it as an approximate percentage effect on earnings per additional year of schooling.
- Comparing OLS and IV (short answer)
- Explain why β̂_IV > β̂_OLS in these results. Discuss at least two mechanisms consistent with this pattern (e.g., classical measurement error in S, or negative selection of marginal students whose schooling decisions are more sensitive to proximity).
- Endogeneity test (interpretation)
- Interpret the reported Hausman test (χ^2(1) = 4.60, p = 0.032). What does this imply about the consistency of OLS and the necessity of IV in this application? State the null and alternative hypotheses clearly.
- Overidentification (interpretation; short answer)
- Interpret the Hansen J statistic (1.80, df = 1, p = 0.18) obtained when using Z1 and Z2 jointly. What does this test assess? Why does a failure to reject not prove instrument validity? Under what circumstances can the J test have low power?
- Exclusion restriction analysis (applied reasoning)
- Critically evaluate the exclusion restriction for Z1 and Z2 in this setting. Discuss at least two plausible threats (e.g., local labor market opportunities, urban amenities correlated with earnings). Propose empirical strategies to probe these threats (e.g., richer geographic controls, pre-trend falsification with outcomes measured before schooling completion, covariate balance tests, or using historical college openings).
- LATE interpretation (conceptual)
- Define the population for which β̂_IV identifies a Local Average Treatment Effect when Z1 is used. Characterize “compliers” in this context and discuss external validity: to whom does this estimate generalize, and to whom might it not?
- Weak instruments—what if? (short answer)
- Suppose in an alternative subsample the first-stage F falls to 5. Explain the consequences for bias and inference in 2SLS. Name at least two methods that provide more reliable inference under weak instruments (e.g., LIML, Fuller, Anderson–Rubin test, conditional likelihood ratio). Cite the relevant literature.
- Alternative instrument critique (applied reasoning)
- A colleague proposes using the local unemployment rate at age 17 as an instrument for schooling. Analyze whether it likely satisfies relevance and exclusion. If exclusion is doubtful, suggest design modifications or alternative strategies.
Answer format
- Provide numerical answers where requested (with brief interpretation).
- For conceptual questions, use concise, evidence-based reasoning tied to the identification assumptions and reported diagnostics.
- When referencing diagnostics, explicitly connect conclusions to the reported statistics and their sampling uncertainty.
References
- Angrist, J. D., and A. B. Krueger. 1991. Does Compulsory School Attendance Affect Schooling and Earnings? Quarterly Journal of Economics 106(4): 979–1014.
- Angrist, J. D., and J.-S. Pischke. 2009. Mostly Harmless Econometrics: An Empiricist’s Companion. Princeton University Press.
- Card, D. 1995. Using Geographic Variation in College Proximity to Estimate the Return to Schooling. In Aspects of Labour Market Behaviour: Essays in Honour of John Vanderkamp, edited by L. N. Christofides et al., 201–222. University of Toronto Press.
- Hansen, L. P. 1982. Large Sample Properties of Generalized Method of Moments Estimators. Econometrica 50(4): 1029–1054.
- Hausman, J. A. 1978. Specification Tests in Econometrics. Econometrica 46(6): 1251–1271.
- Imbens, G. W., and J. D. Angrist. 1994. Identification and Estimation of Local Average Treatment Effects. Econometrica 62(2): 467–475.
- Sargan, J. D. 1958. The Estimation of Economic Relationships Using Instrumental Variables. Econometrica 26(3): 393–415.
- Staiger, D., and J. H. Stock. 1997. Instrumental Variables Regression with Weak Instruments. Econometrica 65(3): 557–586.
- Stock, J. H., and M. Yogo. 2005. Testing for Weak Instruments in Linear IV Regression. In Identification and Inference for Econometric Models: Essays in Honor of Thomas Rothenberg, edited by D. W. K. Andrews and J. H. Stock, 80–108. Cambridge University Press.
