📊Causal Inference Unit 6 – Instrumental variables

What's the Big Idea?

• Instrumental variables (IVs) provide a way to estimate causal effects when there is endogeneity or unmeasured confounding in observational data
• IVs are exogenous variables that affect the outcome only through their effect on the treatment variable
• The IV approach relies on finding a variable that is correlated with the treatment but uncorrelated with the error term in the outcome equation
• IVs allow researchers to isolate the variation in the treatment that is unrelated to the confounding factors and use this variation to estimate the causal effect
• The IV method is particularly useful when randomized controlled trials (RCTs) are not feasible or ethical
• IVs can be used in a variety of settings, including economics, epidemiology, and social sciences
• The validity of the IV approach depends on the strength and validity of the chosen instrument

Why Do We Need This?

• Observational data often suffer from endogeneity, where the treatment variable is correlated with the error term in the outcome equation
  • Endogeneity can arise due to omitted variables, measurement error, or reverse causality
• In the presence of endogeneity, standard regression methods (OLS) yield biased and inconsistent estimates of the causal effect
• Randomized controlled trials (RCTs) are the gold standard for causal inference but are not always possible due to ethical, practical, or financial constraints
• Instrumental variables provide a way to estimate causal effects when RCTs are not feasible and observational data suffer from endogeneity
• IVs can help address important policy questions and inform decision-making in various domains (healthcare, education, labor markets)
• Without IVs, researchers may draw incorrect conclusions about the effectiveness of interventions or policies

Key Concepts to Grasp

• Endogeneity: The correlation between the treatment variable and the error term in the outcome equation
• Exogeneity: The requirement that the instrumental variable is uncorrelated with the error term in the outcome equation
• Relevance: The requirement that the instrumental variable is correlated with the treatment variable
  • Weak instruments (low correlation) can lead to biased estimates and large standard errors
• Exclusion restriction: The assumption that the instrumental variable affects the outcome only through its effect on the treatment variable
  • Violations of the exclusion restriction can invalidate the IV approach
• Two-stage least squares (2SLS): A common estimation method for IV models that involves two regression stages
• Local average treatment effect (LATE): The causal effect estimated by the IV approach, which represents the effect for the subpopulation of "compliers" (those whose treatment status is affected by the instrument)
• Overidentification: The situation where there are more instruments than endogenous variables, allowing for testing the validity of the instruments

How It Works in Practice

• Identify a potential instrumental variable that satisfies the relevance and exogeneity conditions
  • Examples: (distance to college as an IV for education, rainfall as an IV for agricultural output)
• Collect data on the outcome, treatment, instrumental variable, and other relevant covariates
• Estimate the first-stage regression, regressing the treatment variable on the instrumental variable and covariates
  • Check the strength of the instrument using the F-statistic or partial R-squared
• Estimate the second-stage regression, regressing the outcome variable on the predicted values of the treatment from the first stage and covariates
• Interpret the coefficient on the predicted treatment as the local average treatment effect (LATE)
• Assess the validity of the instrument using overidentification tests (if multiple instruments are available) or theoretical arguments
• Report the results, including the LATE estimate, standard errors, and any robustness checks or sensitivity analyses

Common Pitfalls and Challenges

• Weak instruments: Instruments with low correlation with the treatment variable can lead to biased estimates and large standard errors
  • Rule of thumb: First-stage F-statistic should be greater than 10
• Invalid instruments: Instruments that violate the exclusion restriction can lead to biased estimates
  • Example: Using parental education as an IV for child's education, but parental education directly affects child outcomes
• Limited external validity: The LATE estimate applies only to the subpopulation of compliers and may not generalize to the entire population
• Measurement error: Mismeasured variables can lead to biased estimates, and IV methods may amplify the bias
• Finite sample bias: In small samples, IV estimates can be biased towards the OLS estimate
• Nonlinear models: IV methods are more complex and less well-developed for nonlinear models (probit, logit)
• Interpreting the LATE: The LATE may not be the policy-relevant parameter of interest, and interpreting it requires careful consideration of the complier population

Real-World Applications

• Estimating the returns to education using compulsory schooling laws or distance to college as instruments
• Evaluating the effectiveness of job training programs using assignment to training as an instrument
• Assessing the impact of medical treatments using physician prescribing patterns as instruments
• Studying the effect of family size on child outcomes using twin births or sex composition of first two children as instruments
• Analyzing the impact of immigration on labor market outcomes using historical settlement patterns or policy changes as instruments
• Investigating the effect of air pollution on health using wind direction or traffic congestion as instruments
• Estimating the impact of financial aid on college enrollment and completion using discontinuities in aid formulas as instruments

Crunching the Numbers

• First-stage regression: $T_i = \alpha_0 + \alpha_1 Z_i + \alpha_2 X_i + \nu_i$
  • $T_i$: Treatment variable for individual $i$
  • $Z_i$: Instrumental variable for individual $i$
  • $X_i$: Vector of covariates for individual $i$
  • $\nu_i$: Error term in the first-stage equation
• Second-stage regression: $Y_i = \beta_0 + \beta_1 \hat{T}_i + \beta_2 X_i + \epsilon_i$
  • $Y_i$: Outcome variable for individual $i$
  • $\hat{T}_i$: Predicted values of the treatment from the first-stage regression
  • $\epsilon_i$: Error term in the second-stage equation
• The coefficient $\beta_1$ represents the local average treatment effect (LATE)
• Standard errors need to be adjusted for the two-stage estimation process, typically using robust or clustered standard errors
• Overidentification test (Sargan-Hansen test): Regress the residuals from the second-stage regression on the instruments and covariates, and test the joint significance of the instruments
• Weak instrument test (F-test): Conduct an F-test on the excluded instruments in the first-stage regression, with a rule of thumb that the F-statistic should be greater than 10

Going Beyond the Basics

• Heterogeneous treatment effects: Use IV methods to estimate the average treatment effect for different subpopulations or across different levels of the treatment
• Nonlinear models: Develop IV methods for nonlinear models, such as probit or logit, using control function approaches or generalized method of moments (GMM)
• Multiple instruments: Combine multiple instruments to improve efficiency and test overidentifying restrictions
  • Example: Using both compulsory schooling laws and distance to college as instruments for education
• Time-varying treatments and instruments: Extend IV methods to settings with time-varying treatments and instruments, such as panel data or event studies
• Fuzzy regression discontinuity designs: Use IV methods to estimate causal effects in regression discontinuity designs where the treatment assignment is not perfectly determined by the running variable
• Mendelian randomization: Apply IV methods to genetic data, using genetic variants as instruments for modifiable risk factors
• Machine learning and IV: Integrate machine learning techniques (LASSO, random forests) with IV methods to select instruments, control for confounding, or estimate heterogeneous treatment effects