6.2 Two-stage least squares (2SLS)

Two-stage least squares (2SLS) is a powerful tool for estimating causal effects when dealing with endogeneity. It uses instrumental variables to isolate exogenous variation in the explanatory variable, allowing researchers to overcome biases from omitted variables or reverse causality.

The 2SLS method involves two stages: first, regressing the endogenous variable on instruments, then using predicted values to estimate the causal effect. This approach provides consistent estimates when OLS fails, but requires finding valid instruments that satisfy relevance and exclusion conditions.

Overview of 2SLS

• Two-stage least squares (2SLS) is an instrumental variable estimation method used to estimate causal effects in the presence of endogeneity
• 2SLS consists of two stages: the first stage regresses the endogenous variable on the instrumental variables, and the second stage uses the predicted values from the first stage to estimate the causal effect
• 2SLS is widely used in causal inference when randomized experiments are not feasible or ethical, and it relies on finding suitable instrumental variables that satisfy certain conditions

Motivation for 2SLS

Endogeneity problem

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• Endogeneity occurs when an explanatory variable is correlated with the error term in a regression model, violating the assumption of exogeneity
• Endogeneity can arise due to omitted variables, measurement error, or simultaneous causality, leading to biased and inconsistent estimates of causal effects
• Examples of endogeneity include the relationship between education and earnings (ability bias) or the impact of police on crime rates (reverse causality)

Limitations of OLS

• Ordinary least squares (OLS) estimation assumes that the explanatory variables are exogenous and uncorrelated with the error term
• In the presence of endogeneity, OLS estimates are biased and inconsistent, leading to incorrect conclusions about causal relationships
• OLS cannot disentangle the causal effect of an endogenous variable from the confounding factors that affect both the explanatory variable and the outcome

Instrumental variables (IVs)

Definition of IVs

• An instrumental variable (IV) is a variable that is correlated with the endogenous explanatory variable but uncorrelated with the error term in the regression model
• IVs provide exogenous variation in the endogenous variable, allowing for the estimation of causal effects
• Examples of IVs include distance to college as an instrument for education, or rainfall as an instrument for agricultural output

Relevance condition

• The relevance condition requires that the IV is sufficiently correlated with the endogenous explanatory variable
• A strong correlation between the IV and the endogenous variable is necessary for the IV to be a valid instrument and to avoid the weak instruments problem
• The first-stage F-statistic is commonly used to assess the strength of the IV, with a rule of thumb suggesting an F-statistic greater than 10 for a strong instrument

Exclusion restriction

• The exclusion restriction assumes that the IV affects the outcome variable only through its effect on the endogenous explanatory variable
• This means that the IV should have no direct effect on the outcome, and any effect should be mediated entirely through the endogenous variable
• Violations of the exclusion restriction can lead to biased estimates and invalidate the IV approach

First stage of 2SLS

Regressing endogenous variable on IVs

• In the first stage of 2SLS, the endogenous explanatory variable is regressed on the IV(s) and any other exogenous control variables
• This regression estimates the relationship between the IV(s) and the endogenous variable, isolating the exogenous variation in the endogenous variable
• The predicted values from the first-stage regression are used in the second stage to estimate the causal effect

Assessing IV strength

• The strength of the IV is assessed using the first-stage F-statistic, which tests the joint significance of the IV(s) in predicting the endogenous variable
• A high F-statistic (greater than 10) indicates a strong IV, while a low F-statistic suggests a weak instrument problem
• Weak instruments can lead to biased and imprecise estimates in the second stage, and they may perform poorly in small samples

Weak instruments problem

• The weak instruments problem arises when the IV is only weakly correlated with the endogenous variable, leading to biased and imprecise estimates in the second stage
• Weak instruments can cause the 2SLS estimator to be biased towards the OLS estimator, and the bias may not disappear even in large samples
• Solutions to the weak instruments problem include using multiple IVs, estimating the reduced form directly, or employing alternative estimation methods (limited information maximum likelihood)

Second stage of 2SLS

Using first-stage predictions

• In the second stage of 2SLS, the predicted values of the endogenous variable from the first stage are used as a regressor in the main regression model
• By using the predicted values, which are based on the exogenous variation in the IV, the second stage isolates the causal effect of the endogenous variable on the outcome
• The second-stage regression includes the predicted endogenous variable and any other exogenous control variables

Estimating causal effect

• The coefficient on the predicted endogenous variable in the second stage represents the causal effect of the endogenous variable on the outcome
• This causal effect is interpreted as the local average treatment effect (LATE) for the subpopulation affected by the IV (compliers)
• Standard errors in the second stage need to be adjusted to account for the two-stage estimation process, typically using robust or clustered standard errors

Properties of 2SLS estimator

Consistency

• The 2SLS estimator is consistent, meaning that it converges in probability to the true causal effect as the sample size increases, under the assumptions of IV validity
• Consistency requires that the IV satisfies the relevance condition and the exclusion restriction, and that the sample size is large enough for the asymptotic properties to hold
• Consistency ensures that the 2SLS estimator provides an unbiased estimate of the causal effect in large samples

Asymptotic normality

• The 2SLS estimator is asymptotically normally distributed, which allows for the construction of confidence intervals and hypothesis testing
• Asymptotic normality holds under the assumptions of IV validity and the usual regularity conditions for the central limit theorem
• The asymptotic variance of the 2SLS estimator is larger than that of the OLS estimator, reflecting the uncertainty introduced by the two-stage estimation process

Efficiency vs OLS

• The 2SLS estimator is less efficient than the OLS estimator when the explanatory variables are exogenous, meaning that it has a larger variance
• The efficiency loss of 2SLS compared to OLS depends on the strength of the IV and the degree of endogeneity in the model
• However, when endogeneity is present, the OLS estimator is biased and inconsistent, making the 2SLS estimator a preferred choice despite its lower efficiency

Assumptions for 2SLS validity

Linear model specification

• 2SLS assumes that the relationship between the endogenous variable, the IV, and the outcome is linear in parameters
• Misspecification of the functional form can lead to biased estimates and invalidate the 2SLS approach
• Non-linear relationships can be accommodated by transforming variables or using non-linear 2SLS estimators

Homoskedasticity

• 2SLS assumes that the error term in the second-stage regression has constant variance (homoskedasticity)
• Violations of homoskedasticity (heteroskedasticity) can lead to inefficient estimates and incorrect standard errors
• Heteroskedasticity-robust standard errors can be used to account for non-constant variance in the error term

No multicollinearity

• 2SLS assumes that there is no perfect multicollinearity among the explanatory variables, including the IV and the endogenous variable
• Perfect multicollinearity can lead to non-identification of the model and inability to estimate the causal effect
• Near multicollinearity (high correlation among explanatory variables) can lead to imprecise estimates and large standard errors

Testing 2SLS assumptions

Overidentification tests

• Overidentification tests are used when there are more IVs than endogenous variables (overidentified model) to test the validity of the exclusion restriction
• The Sargan-Hansen test is commonly used, which tests the joint null hypothesis that all IVs are valid (uncorrelated with the error term)
• Rejecting the null hypothesis suggests that at least one of the IVs violates the exclusion restriction, and the model may need to be reconsidered

Hausman test for endogeneity

• The Hausman test is used to compare the OLS and 2SLS estimates and test for the presence of endogeneity in the model
• The test is based on the difference between the OLS and 2SLS estimates, which should be small if the explanatory variables are exogenous
• Rejecting the null hypothesis of the Hausman test indicates the presence of endogeneity, and the 2SLS estimator is preferred over OLS

Interpreting 2SLS results

Local average treatment effect (LATE)

• The 2SLS estimator identifies the local average treatment effect (LATE), which is the causal effect for the subpopulation affected by the IV (compliers)
• Compliers are individuals whose treatment status (endogenous variable) is influenced by the IV, while always-takers and never-takers are not affected by the IV
• The LATE may differ from the average treatment effect (ATE) for the entire population, depending on the heterogeneity of treatment effects

Generalizability of findings

• The generalizability of 2SLS findings depends on the similarity between the compliers and the target population of interest
• If the compliers are a non-representative subgroup, the LATE may not be informative about the causal effect for the entire population
• Assessing the external validity of 2SLS results requires careful consideration of the characteristics of the compliers and the context of the study

Limitations of 2SLS

Finding suitable instruments

• The main challenge in implementing 2SLS is finding suitable instruments that satisfy the relevance condition and the exclusion restriction
• Instruments that are weakly correlated with the endogenous variable or violate the exclusion restriction can lead to biased and inconsistent estimates
• Identifying credible IVs often requires deep institutional knowledge and theoretical justification, and the validity of IVs cannot be tested directly

Sensitivity to assumptions

• 2SLS estimates are sensitive to violations of the assumptions, such as non-linearity, heteroskedasticity, or invalid instruments
• Small violations of the exclusion restriction can lead to substantial bias in the 2SLS estimator, especially when the instruments are weak
• Sensitivity analysis and robustness checks are important to assess the stability of 2SLS results under different assumptions and specifications

Extensions of 2SLS

Multiple endogenous variables

• 2SLS can be extended to handle multiple endogenous variables by using multiple instruments and estimating the first stage for each endogenous variable separately
• The second stage includes the predicted values of all endogenous variables and estimates their causal effects simultaneously
• Identification in the presence of multiple endogenous variables requires at least as many instruments as endogenous variables (order condition) and sufficient variation in the instruments (rank condition)

Nonlinear models

• 2SLS can be adapted to estimate causal effects in nonlinear models, such as probit, logit, or Poisson regression
• The two-stage residual inclusion (2SRI) method is commonly used, where the residuals from the first stage are included as an additional regressor in the second stage
• Nonlinear 2SLS estimators have different properties than linear 2SLS and require careful interpretation of the marginal effects

Panel data settings

• 2SLS can be applied to panel data settings, where multiple observations are available for each individual or unit over time
• Panel data 2SLS estimators can control for unobserved time-invariant heterogeneity by including individual fixed effects or using first-differencing
• Instruments in panel data settings can be time-varying or time-invariant, and the relevance and exclusion conditions need to be satisfied in the presence of fixed effects or first-differencing