9.4 Validity of instruments
5 min read•august 20, 2024
Instrumental variables are crucial in econometrics for addressing issues. They help estimate causal effects when explanatory variables are correlated with error terms. This topic explores the key criteria for valid instruments: and .
Relevance ensures instruments correlate strongly with endogenous regressors, while exogeneity requires they're uncorrelated with error terms. The notes cover , , problems, and trade-offs in instrument selection. Understanding these concepts is vital for proper IV implementation.
Relevance of instruments
- Instruments must be relevant to the endogenous regressors in the model to provide consistent estimates of the causal effect of interest
- The relevance condition requires that the instruments have a strong correlation with the endogenous explanatory variables
Correlation with endogenous regressors
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- Instruments should have a non-zero correlation with the endogenous regressors in the model
- The correlation between the instruments and endogenous regressors can be assessed using the first-stage regression, where the endogenous variables are regressed on the instruments
- A high correlation between the instruments and endogenous regressors indicates that the instruments are relevant and can effectively isolate the exogenous variation in the endogenous variables
- Example: In a study of the effect of education on earnings, a valid instrument could be the distance to the nearest college, as it is likely to be correlated with an individual's level of education
Strength of instruments
- The strength of an instrument refers to the magnitude of its correlation with the endogenous regressors
- Weak instruments, or those with low correlation, can lead to biased and inconsistent estimates in the IV regression
- The strength of instruments can be evaluated using the F-statistic from the first-stage regression
- A high F-statistic (typically greater than 10) suggests that the instruments are strong and relevant
- Example: A study examining the impact of air pollution on health outcomes may use wind direction as an instrument, as it is strongly correlated with the concentration of pollutants in the air
Exogeneity of instruments
- The exogeneity condition requires that the instruments are uncorrelated with the error term in the structural equation
- Instruments should only affect the dependent variable through their influence on the endogenous regressors and not through any other channels
Uncorrelated with error term
- For an instrument to be valid, it must be uncorrelated with the unobserved factors that affect the dependent variable (i.e., the error term)
- If the instrument is correlated with the error term, the IV estimates will be biased and inconsistent
- The assumption of instrument exogeneity cannot be directly tested, as the error term is unobservable
- Researchers must rely on economic theory and intuition to justify the exogeneity of their chosen instruments
Exclusion restriction
- The exclusion restriction states that the instruments should only affect the dependent variable through their impact on the endogenous regressors
- In other words, the instruments should have no direct effect on the dependent variable, other than through the endogenous variables
- Violating the exclusion restriction leads to biased and inconsistent IV estimates
- Example: In a study of the effect of military service on earnings, the draft lottery number may serve as a valid instrument, as it affects earnings only through its impact on the likelihood of military service and not through any other channels
Overidentifying restrictions
- Overidentifying restrictions occur when there are more instruments than endogenous regressors in the model
- Having more instruments than necessary allows for the testing of the validity of the instruments
Surplus of instruments
- When the number of instruments exceeds the number of endogenous regressors, the model is said to be overidentified
- Overidentification provides an opportunity to test the joint validity of the instruments using the Hansen J-statistic or the
- If the overidentifying restrictions are satisfied, the surplus instruments can help improve the efficiency of the IV estimates
Testing validity with restrictions
- The Hansen J-statistic and the Sargan test are used to assess the validity of overidentifying restrictions
- These tests evaluate whether the instruments are uncorrelated with the error term in the structural equation
- A failure to reject the null hypothesis of these tests suggests that the overidentifying restrictions are satisfied and the instruments are valid
- Example: In a study with multiple instruments, such as parental education and siblings' education as instruments for an individual's education, overidentifying restriction tests can be used to assess the validity of these instruments
Weak instruments
- Weak instruments are those that have a low correlation with the endogenous regressors in the model
- The use of weak instruments can lead to biased and inconsistent IV estimates, even in large samples
Bias in IV estimators
- When instruments are weak, the IV estimator can be biased towards the OLS estimator
- The bias of the IV estimator increases as the correlation between the instruments and the endogenous regressors decreases
- In the presence of weak instruments, the IV estimates may be more biased than the OLS estimates, defeating the purpose of using IV methods
Finite sample properties
- Weak instruments can lead to poor finite sample properties of the IV estimator
- With weak instruments, the IV estimator may have a large standard error and a non-normal sampling distribution, even in large samples
- Confidence intervals based on weak instruments may have incorrect coverage probabilities, leading to invalid inference
Rule of thumb for F-statistic
- A commonly used rule of thumb to assess the strength of instruments is the F-statistic from the first-stage regression
- An F-statistic greater than 10 is often considered a threshold for strong instruments
- However, this rule of thumb should be used with caution, as it may not always be reliable, particularly with multiple endogenous regressors or non-i.i.d. errors
- Example: In a study with a single endogenous regressor, an F-statistic of 5 in the first-stage regression would suggest that the instrument is weak and may lead to biased IV estimates
Instrument exogeneity vs relevance
- When selecting instruments, researchers face a trade-off between the exogeneity and relevance of the instruments
- Instruments that are highly relevant may be more likely to violate the exogeneity condition, while instruments that are strictly exogenous may have weaker relevance
Tradeoffs in instrument selection
- Researchers must carefully consider the balance between instrument exogeneity and relevance when choosing instruments
- Instruments that are more closely related to the endogenous regressors may have a stronger first-stage relationship but may also be more likely to violate the exclusion restriction
- Conversely, instruments that are less related to the endogenous regressors may be more plausibly exogenous but may suffer from weak instrument bias
Consequences of invalid instruments
- Using invalid instruments that violate either the exogeneity or relevance condition can lead to biased and inconsistent IV estimates
- Instruments that are not exogenous (i.e., correlated with the error term) will produce estimates that are biased and inconsistent, even in large samples
- Instruments that are not relevant (i.e., weakly correlated with the endogenous regressors) will lead to estimates that are biased towards the OLS estimator and may have poor finite sample properties
- Example: In a study of the effect of health insurance on health outcomes, using an individual's occupation as an instrument may be problematic, as occupation could be correlated with both insurance status and health outcomes, violating the exogeneity condition
Key Terms to Review (18)
Angrist: Angrist refers to the work and contributions of Joshua Angrist, an influential economist known for his research in econometrics, particularly in the area of instrumental variables and causal inference. His methods help assess the validity of instruments used to estimate causal relationships, providing clarity on how certain variables influence outcomes while mitigating bias from unobserved confounding factors.
Causal interpretation: Causal interpretation refers to the ability to draw conclusions about the cause-and-effect relationships between variables in a study. It’s crucial in econometrics as it helps determine whether a change in one variable leads to a change in another, rather than simply being correlated. Establishing causal interpretation is key when using techniques like instrumental variables, which aim to isolate the causal effect of one variable on another.
Correlation with endogenous regressors: Correlation with endogenous regressors refers to a situation in statistical models where the independent variables are correlated with the error term, leading to biased and inconsistent parameter estimates. This issue arises when an independent variable is affected by the dependent variable or when omitted variables influence both, resulting in a spurious relationship that distorts causal inference.
Endogeneity: Endogeneity refers to a situation in econometric modeling where an explanatory variable is correlated with the error term, which can lead to biased and inconsistent estimates. This correlation may arise due to omitted variables, measurement errors, or simultaneous causality, complicating the interpretation of results and making it difficult to establish causal relationships.
Exogeneity: Exogeneity refers to a condition where an explanatory variable is not correlated with the error term in a regression model. When a variable is exogenous, it suggests that any changes in this variable do not arise from the model's error, making it crucial for establishing causal relationships and ensuring valid inference in econometric analysis.
Hansen J Test: The Hansen J Test is a statistical test used to assess the validity of instruments in econometric models, particularly in the context of instrumental variable estimation. It evaluates whether the instruments are uncorrelated with the errors in the regression model, which is crucial for obtaining consistent estimates. A failure to reject the null hypothesis indicates that the instruments used are appropriate and valid.
Imbens: Imbens refers to the work and contributions of Guido Imbens, particularly in the field of econometrics concerning causal inference and the validity of instruments. His research emphasizes the importance of using appropriate instruments to ensure that estimators are unbiased and consistent, highlighting that the validity of these instruments is crucial for accurate causal interpretations in regression analysis.
Instrument strength: Instrument strength refers to the ability of an instrumental variable to significantly affect the endogenous explanatory variable in a regression model. A strong instrument is critical for reliable estimation, as it ensures that the variation in the endogenous variable is sufficiently captured, leading to more accurate causal inferences in the context of econometric analysis.
Instrumental Variable Assumption: The instrumental variable assumption is a key concept in econometrics that asserts that an instrument used in regression analysis must be correlated with the endogenous explanatory variable and must be uncorrelated with the error term of the regression model. This assumption is crucial because it helps to establish a causal relationship between variables by isolating the variation in the endogenous variable that is not affected by omitted variable bias or measurement error. If this assumption holds, it allows for consistent estimation of the causal effect of one variable on another.
Limited Information Maximum Likelihood (LIML): Limited Information Maximum Likelihood (LIML) is an estimation technique used in econometrics that focuses on estimating the parameters of a specific equation within a larger system of equations while treating the other equations as fixed. This method is particularly useful in situations where there are concerns about the validity and strength of instruments, which can affect the reliability of estimates. By using LIML, researchers can obtain more consistent estimates even when instruments are weak or invalid.
Omitted variable bias: Omitted variable bias occurs when a model leaves out one or more relevant variables that influence both the dependent variable and one or more independent variables. This leads to biased and inconsistent estimates, making it difficult to draw accurate conclusions about the relationships being studied. Understanding this bias is crucial when interpreting results, ensuring proper variable selection, and assessing model specifications.
Orthogonality Condition: The orthogonality condition refers to the requirement that an instrument in an econometric model must be uncorrelated with the error term of the structural equation it is intended to explain. This condition ensures that the instrument can provide valid estimates by isolating exogenous variation, which is crucial for obtaining unbiased and consistent parameter estimates in models with endogenous variables.
Overidentification: Overidentification occurs when there are more instruments available than the number of endogenous variables in a model. This situation allows for the possibility of testing the validity of the instruments, which can lead to better model estimates. Additionally, overidentification can be critical for assessing joint hypotheses about parameters, ensuring that the model does not suffer from weak instruments that could bias the results.
Relevance: Relevance refers to the necessity of an instrument to be strongly associated with the endogenous variable in an econometric model, which is essential for the proper identification of causal relationships. It emphasizes the importance of selecting instruments that can effectively explain the variation in the endogenous variables while not directly affecting the dependent variable, ensuring that the estimates produced are valid and reliable.
Sargan Test: The Sargan test is a statistical test used to assess the validity of instrumental variables in econometric models, particularly in the presence of endogeneity. It evaluates whether the instruments used in an estimation are correlated with the error term of the regression model, which would violate the assumptions of valid instruments. A failure of the Sargan test suggests that the instruments may not be valid, potentially leading to biased and inconsistent estimates.
Treatment effect: The treatment effect refers to the impact of a specific intervention or treatment on an outcome variable, helping to measure how much a certain factor influences the result being studied. Understanding this effect is crucial for assessing causal relationships, especially in observational studies and experiments, as it allows researchers to isolate the influence of a treatment from other confounding variables. This concept is key when using methods like instrumental variables and fixed effects models, as they help identify the true effect of the treatment in the presence of potential biases.
Two-stage least squares (2sls): Two-stage least squares (2SLS) is an estimation technique used to provide consistent estimates of parameters in a regression model when there is endogeneity or correlation between the independent variables and the error term. This method employs instrumental variables to remove bias by first predicting the values of the endogenous variables using instruments and then substituting those predicted values back into the original equation for final estimation. Its effectiveness hinges on the validity of the instruments used, addressing issues related to weak instruments and allowing for diagnostic tests like the Hausman test.
Weak instrument: A weak instrument refers to an instrumental variable that does not have a strong correlation with the endogenous explanatory variable it is intended to replace in a regression model. This concept is crucial because using weak instruments can lead to biased and inconsistent parameter estimates, particularly in two-stage least squares (2SLS) estimation. The validity of instruments hinges on their strength, as weak instruments may fail to adequately control for unobserved confounding factors.