An Instrumental Variable (IV) is a statistical tool used in econometrics to correct for endogeneity, ensuring the reliability and validity of causal inference in regression analysis. The primary function of an IV is to isolate the exogenous variation in an endogenous explanatory variable, thereby providing consistent and unbiased estimators.
Definition§
An Instrumental Variable (IV) is defined as a variable that:
- Is uncorrelated with the error term in the original model.
- Is correlated with the endogenous explanatory variable.
Mathematically, let’s consider a simple linear model:
Significance of Instrumental Variables§
Endogeneity Problem§
Endogeneity arises when an explanatory variable is correlated with the error term, leading to biased and inconsistent parameter estimates. This can occur due to omitted variable bias, measurement error, or simultaneous causality.
Solution through IV§
Instrumental Variables help in overcoming the endogeneity problem by providing a source of variation that is exogenous to the error term. By doing so, IV ensures that the variation in the endogenous explanatory variable is not driven by the factors captured in the error term.
Types of Instrumental Variables§
Valid Instrument§
A valid instrument must satisfy two main conditions: relevance (correlation with the endogenous variable) and exogeneity (no correlation with the error term).
Invalid Instrument§
If an instrument fails either of these conditions, it is considered invalid. Using an invalid instrument can lead to incorrect inferences and biased estimates.
Special Considerations§
Weak Instruments§
A weak instrument is one that has a weak correlation with the endogenous explanatory variable, leading to unreliable estimates. The strength of an instrument is often assessed using the first-stage F-statistic. A rule of thumb is that an F-statistic less than 10 indicates a weak instrument.
Over-Identification§
When more instruments than endogenous variables are available, the model is over-identified. Over-identification allows for testing the validity of instruments through the Sargan-Hansen test, which examines the joint null hypothesis that the instruments are valid.
Examples of Instrumental Variables§
Historical Context§
In the seminal work by Angrist and Krueger (1991), the authors used the quarter of birth as an instrument for educational attainment to study its effect on earnings.
Applicability§
Instrumental Variables are widely used in various fields such as economics, epidemiology, and social sciences to address issues of causality and endogeneity.
Comparisons and Related Terms§
Two-Stage Least Squares (2SLS)§
2SLS is an estimation technique commonly used in IV regression. In the first stage, the endogenous variable is regressed on the instruments, and in the second stage, the predicted values from the first stage are used in the original regression.
Control Function Approach§
Another method for addressing endogeneity involves including a control function derived from the instruments as an additional regressor in the original model.
FAQs§
Q: What makes a good Instrumental Variable?
- A: A good IV is both relevant and exogenous, meaning it is correlated with the endogenous variable and uncorrelated with the error term.
Q: How do I test if my IV is valid?
- A: Validity can be tested using over-identification tests like the Sargan-Hansen test if more instruments than endogenous variables are available.
Q: Can Instrumental Variables be used in non-linear models?
- A: Yes, IV methods have extensions for non-linear models, such as in the IV-Probit model for binary outcomes.
References§
Angrist, J. D., & Krueger, A. B. (1991). Does Compulsory School Attendance Affect Schooling and Earnings? Quarterly Journal of Economics, 106(4), 979-1014.
Wooldridge, J. M. (2010). Econometric Analysis of Cross Section and Panel Data. MIT Press.
Summary§
Instrumental Variables are essential for addressing endogeneity in regression models, ensuring credible causal inference. By leveraging external variability that influences the endogenous explanatory variable but is independent of the error term, IV methods provide robust and unbiased parameter estimates critical for empirical research.
