🎳Intro to Econometrics Unit 9 – Instrumental Variables & Two-Stage LS

Key Concepts

• Instrumental variables (IV) address endogeneity issues in regression models when explanatory variables are correlated with the error term
• Endogeneity arises from omitted variables, measurement error, or simultaneous causality leading to biased and inconsistent OLS estimates
• Valid instruments are correlated with the endogenous explanatory variable but uncorrelated with the error term
  • Instruments should be relevant (strong correlation with the endogenous variable) and exogenous (no direct effect on the dependent variable)
• Two-stage least squares (2SLS) is a common method for implementing IV estimation
  • First stage regresses the endogenous variable on the instrument(s) and other exogenous variables
  • Second stage uses the predicted values from the first stage in place of the endogenous variable
• IV and 2SLS aim to obtain consistent estimates of the causal effect of the explanatory variable on the dependent variable
• Weak instruments (low correlation with the endogenous variable) can lead to biased IV estimates and large standard errors
• Overidentification occurs when there are more instruments than endogenous variables allowing for testing the validity of the instruments

Problem of Endogeneity

• Endogeneity violates the assumption of zero conditional mean of the error term $E[u|X] = 0$ required for unbiased and consistent OLS estimates
• Omitted variable bias occurs when a relevant variable is excluded from the model and is correlated with both the dependent and explanatory variables
  • Example: estimating the effect of education on earnings without controlling for ability
• Measurement error in the explanatory variable leads to attenuation bias (downward bias) in the OLS estimates
  • Example: using self-reported income instead of actual income
• Simultaneous causality or reverse causality arises when the dependent variable also affects the explanatory variable
  • Example: estimating the effect of police on crime while crime levels influence police allocation
• Endogeneity causes the explanatory variable to be correlated with the error term leading to biased and inconsistent estimates of the causal effect
• IV methods aim to isolate the exogenous variation in the endogenous explanatory variable to obtain consistent estimates

Instrumental Variables (IV) Explained

• Instrumental variables (IV) are used to address endogeneity by finding a source of exogenous variation in the endogenous explanatory variable
• An instrument $Z$ is a variable that is correlated with the endogenous explanatory variable $X$ but uncorrelated with the error term $u$
  • $Cov(Z, X) \neq 0$ (relevance condition)
  • $Cov(Z, u) = 0$ (exogeneity condition)
• The instrument affects the dependent variable $Y$ only through its effect on the endogenous explanatory variable $X$
  • Example: using distance to college as an instrument for education when estimating the effect of education on earnings
• IV estimation isolates the exogenous variation in $X$ that is uncorrelated with the error term to obtain a consistent estimate of the causal effect
• The IV estimator is given by $\beta_{IV} = \frac{Cov(Z, Y)}{Cov(Z, X)}$ which is consistent under the relevance and exogeneity conditions
• Multiple instruments can be used for a single endogenous variable to improve efficiency and allow for overidentification tests
• The reduced form equation regresses the dependent variable directly on the instrument(s) and other exogenous variables

Criteria for Valid Instruments

• Relevance: the instrument must be correlated with the endogenous explanatory variable
  • Weak instruments (low correlation) can lead to biased IV estimates and large standard errors
  • The first-stage F-statistic tests the strength of the instrument(s) with a rule of thumb of F > 10 indicating a strong instrument
• Exogeneity: the instrument must be uncorrelated with the error term in the structural equation
  • The instrument should not have a direct effect on the dependent variable other than through the endogenous explanatory variable
  • Overidentifying restrictions tests (e.g., Sargan-Hansen test) can be used to assess the validity of multiple instruments
• Exclusion restriction: the instrument should not be correlated with any omitted variables that affect the dependent variable
  • This condition is not directly testable and relies on theoretical justification
• Monotonicity: the effect of the instrument on the endogenous variable should be monotonic (always positive or always negative) for all individuals
  • This assumption is required for the interpretation of the local average treatment effect (LATE) in the presence of heterogeneous treatment effects
• External validity: the IV estimate may not be generalizable to the entire population if the effect of the endogenous variable varies across individuals
  • The IV estimate represents the LATE for the subpopulation affected by the instrument (compliers)

Two-Stage Least Squares (2SLS) Method

• Two-stage least squares (2SLS) is a common method for implementing IV estimation when the endogenous explanatory variable is continuous
• The first stage regresses the endogenous explanatory variable $X$ on the instrument(s) $Z$ and other exogenous variables $W$:
  • $X = \delta_0 + \delta_1 Z + \delta_2 W + v$
  • This stage isolates the exogenous variation in $X$ that is uncorrelated with the error term in the structural equation
• The second stage regresses the dependent variable $Y$ on the predicted values of $X$ from the first stage ($\hat{X}$) and other exogenous variables $W$:
  • $Y = \beta_0 + \beta_1 \hat{X} + \beta_2 W + u$
  • The predicted values $\hat{X}$ are uncorrelated with the error term $u$ by construction
• The 2SLS estimator is consistent and asymptotically normal under the relevance and exogeneity conditions
• Standard errors in the second stage need to be adjusted to account for the two-step estimation process
  • This can be done using the Huber-White sandwich estimator or bootstrapping
• 2SLS can be extended to multiple endogenous variables and multiple instruments (e.g., three-stage least squares)

Implementing IV and 2SLS

• Identify the endogenous explanatory variable(s) and potential instruments based on theoretical considerations and institutional knowledge
• Check the relevance condition by regressing the endogenous variable on the instrument(s) and testing for significance (first-stage F-statistic)
  • If the instruments are weak, consider finding stronger instruments or using alternative methods (e.g., limited information maximum likelihood)
• Assess the exogeneity condition using overidentification tests if there are more instruments than endogenous variables
  • Sargan-Hansen J-test for overidentifying restrictions
  • Failure to reject the null hypothesis supports the validity of the instruments
• Estimate the first-stage regression and obtain the predicted values of the endogenous variable
• Estimate the second-stage regression using the predicted values from the first stage in place of the endogenous variable
• Interpret the IV estimates as the local average treatment effect (LATE) for the subpopulation affected by the instrument (compliers)
  • The LATE may differ from the average treatment effect (ATE) if the effect of the endogenous variable varies across individuals
• Report the first-stage F-statistic, overidentification test results, and adjusted standard errors in addition to the IV estimates
• Conduct robustness checks using alternative instruments, subsamples, or estimation methods to assess the sensitivity of the results

Limitations and Challenges

• Finding valid instruments that satisfy the relevance and exogeneity conditions can be difficult in practice
  • Instruments that are theoretically justified may be weakly correlated with the endogenous variable leading to biased estimates
  • Instruments that are strongly correlated with the endogenous variable may have direct effects on the dependent variable violating the exclusion restriction
• Weak instruments can lead to biased IV estimates, large standard errors, and incorrect inference
  • The bias of the IV estimator is inversely proportional to the strength of the instrument (first-stage F-statistic)
  • Weak instrument robust inference methods (e.g., Anderson-Rubin test) can be used to construct confidence intervals
• The LATE interpretation of the IV estimate may not be generalizable to the entire population if the effect of the endogenous variable varies across individuals
  • The IV estimate represents the effect for the subpopulation affected by the instrument (compliers) which may differ from the average treatment effect (ATE)
• IV estimation can be less efficient than OLS when the instruments are weak or the sample size is small
  • The standard errors of the IV estimator are larger than those of the OLS estimator in the absence of endogeneity
• Measurement error in the instrument can lead to biased IV estimates and incorrect inference
  • The bias is proportional to the degree of measurement error and inversely proportional to the strength of the instrument
• IV estimation relies on strong assumptions (relevance, exogeneity, exclusion restriction, monotonicity) that are not directly testable and may be violated in practice
  • Sensitivity analysis using alternative instruments and estimation methods can help assess the robustness of the results

Real-World Applications

• Estimating the returns to education using compulsory schooling laws or distance to college as instruments for educational attainment
  • Addresses the endogeneity of education due to omitted variables (e.g., ability) or measurement error
• Evaluating the effect of military service on earnings using the Vietnam War draft lottery as an instrument for veteran status
  • Exploits the random assignment of draft eligibility based on birth dates to identify the causal effect of military service
• Assessing the impact of air pollution on health outcomes using wind direction or traffic congestion as instruments for pollutant concentrations
  • Addresses the endogeneity of pollution due to omitted variables (e.g., industrial activity) or measurement error
• Estimating the effect of immigration on native wages using historical settlement patterns or policy changes as instruments for immigrant inflows
  • Deals with the endogeneity of immigration due to self-selection or reverse causality (e.g., immigrants may be attracted to areas with higher wages)
• Analyzing the impact of financial development on economic growth using legal origins or geographic characteristics as instruments for financial institutions
  • Addresses the endogeneity of finance due to omitted variables (e.g., institutional quality) or reverse causality (e.g., growth may lead to financial development)
• Evaluating the effect of health insurance on healthcare utilization using Medicaid eligibility rules or employer-provided coverage as instruments for insurance status
  • Deals with the endogeneity of insurance due to self-selection or omitted variables (e.g., health status)