Instrumental variable estimation

What Is Instrumental Variable Estimation?

Instrumental variable estimation is a statistical technique used in econometrics and other social sciences to estimate causal relationships between variables when ordinary regression analysis methods may yield biased or inconsistent results. It is particularly valuable when dealing with endogeneity, a common problem where an explanatory variable in a model is correlated with the error term. This correlation can arise from issues such as omitted variable bias, simultaneity bias, or measurement error. Instrumental variable estimation helps to isolate the true causal effect of an endogenous variable on an outcome by introducing a third variable, known as an instrumental variable (IV), which satisfies specific conditions.

History and Origin

The concept of instrumental variables has roots in the early 20th century, with significant contributions in econometrics and statistics. While the foundational ideas can be traced back to researchers like Philip G. Wright in the 1920s, the modern understanding and widespread application of instrumental variable estimation gained considerable traction in the latter half of the century. A pivotal moment in the development and clarification of instrumental variable estimation came with the work of economists Joshua Angrist and Guido Imbens. In 2021, they were awarded the Nobel Memorial Prize in Economic Sciences "for their methodological contributions to the analysis of causal relationships."¹⁰ Their research, particularly in the 1990s, clarified how instrumental variables can be used to estimate well-defined causal effects, even in settings where the impact of a "treatment" might vary across individuals, leading to the concept of the local average treatment effect (LATE)).⁹ They showed how this econometric tool could rigorously address causal inference challenges in observational data, making the method more accessible and interpretable for empirical research.⁸

Key Takeaways

• Instrumental variable estimation addresses endogeneity in regression models, providing more reliable estimates of causal effects.
• It requires an instrumental variable that is correlated with the endogenous explanatory variable but uncorrelated with the error term.
• The two-stage least squares (2SLS) method is a common approach to instrumental variable estimation.
• Weak instruments can lead to biased estimates and unreliable hypothesis testing.
• Instrumental variable estimation is widely applied in economics, finance, and social sciences to analyze policy impacts and market behavior.

Formula and Calculation

The most common method for instrumental variable estimation is the two-stage least squares (2SLS)) approach. Consider a structural equation where ( Y ) is the dependent variable, ( X ) is the endogenous explanatory variable, and ( \epsilon ) is the error term:

Here, ( X ) is endogenous, meaning ( \text{Cov}(X, \epsilon) \neq 0 ). To estimate ( \beta_1 ) consistently, an instrumental variable ( Z ) is introduced. The instrument ( Z ) must satisfy two key conditions:

1. Relevance: The instrument ( Z ) must be correlated with the endogenous explanatory variable ( X ). That is, ( \text{Cov}(Z, X) \neq 0 ).
2. Exogeneity (Validity): The instrument ( Z ) must be uncorrelated with the error term ( \epsilon ). That is, ( \text{Cov}(Z, \epsilon) = 0 ). This implies that ( Z ) affects ( Y ) only through its effect on ( X ).

The 2SLS procedure involves two stages:

First Stage: Regress the endogenous variable ( X ) on the instrumental variable ( Z ) and any other truly exogenous variables (if present in the original model). Let ( \hat{X} ) be the predicted values from this regression:

Second Stage: Regress the dependent variable ( Y ) on the predicted values ( \hat{X} ) from the first stage. The coefficient on ( \hat{X} ) in this regression is the instrumental variable estimator for ( \beta_1 ):

The coefficient ( \beta_1 ) estimated from equation (3) is the instrumental variable estimate.

Interpreting the Instrumental Variable Estimation

Interpreting the results from instrumental variable estimation focuses on establishing a causal link between variables, moving beyond mere correlation. The coefficient obtained from an instrumental variable model, such as the ( \beta_1 ) from the 2SLS second stage, represents the estimated causal effect of the endogenous explanatory variable on the dependent variable. This interpretation is valid under the crucial assumptions of instrument relevance and exogeneity.

A key aspect of interpreting instrumental variable estimates, particularly when heterogeneous effects are present, is understanding that the estimate may correspond to the local average treatment effect (LATE)). This means the estimated effect applies to the subpopulation whose "treatment" (the endogenous variable) is actually influenced by the instrument. It is important to assess the statistical significance of the estimated coefficients, similar to traditional regression, to determine if the observed causal effect is likely due to chance.

Hypothetical Example

Consider an economist wanting to determine the causal effect of education (in years of schooling) on an individual's income. A simple regression analysis of income on education might be biased due to omitted variable bias; for instance, an individual's innate ability or family background (unobserved factors) could influence both their education level and their income. This makes education an endogenous variable.

To address this, the economist might use instrumental variable estimation. A hypothetical instrumental variable could be the distance from a person's childhood home to the nearest college.

Assumptions for the Instrument:

• Relevance: Distance to college is assumed to be correlated with years of schooling (i.e., people living closer might be more likely to pursue higher education).
• Exogeneity: Distance to college is assumed to not directly affect an individual's income, except through its influence on their education level. It should also be uncorrelated with other unobserved factors like innate ability that affect income.

The Estimation Process (2SLS):

1. First Stage: The economist would first regress years of schooling (endogenous variable) on the distance to college (instrumental variable) and any other control variables (e.g., age). This stage predicts the "exogenous" variation in education due to college proximity.
2. Second Stage: Next, the economist would regress income on the predicted years of schooling obtained from the first stage. The coefficient on the predicted years of schooling would then provide an instrumental variable estimate of the causal effect of education on income, mitigating the bias from omitted abilities or family background.

This process provides a more robust estimate of the true return to education, a crucial input for financial modeling and policy analysis.

Practical Applications

Instrumental variable estimation is a powerful tool with diverse applications across financial and economic research, particularly where establishing causality is critical.

• Monetary Policy and Inflation: Central banks and economists use instrumental variables to understand the causal impact of monetary policy decisions on inflation and other macroeconomic indicators. For example, researchers at the Federal Reserve Bank of San Francisco have explored the effects of the COVID-19 pandemic on inflation using instrumental variables approaches to disentangle various contributing factors.⁷ This helps in refining economic forecasting and policy adjustments.
• Corporate Finance: In corporate finance, instrumental variable estimation can be employed to assess the causal effect of factors like leverage, investment decisions, or corporate governance structures on firm performance, where these variables might be endogenous due to simultaneity bias or reverse causality. The International Monetary Fund (IMF), for instance, has utilized instrumental variables in analyzing the relationship between markups, investment, and firm-level dynamics.⁶
• Fiscal Policy Evaluation: Policymakers and researchers use instrumental variables to evaluate the causal impact of fiscal policies, such as tax changes or government spending, on economic outcomes like GDP growth, employment, or fiscal balances. An IMF working paper used an instrumental variable strategy to estimate the causal effect of fiscal rules on fiscal balances, exploiting the geographical diffusion of such rules as an instrument.⁵
• Labor Economics: Beyond the hypothetical example of education and income, instrumental variable estimation is frequently used in labor economics to study the causal effects of minimum wages on employment, job training programs on earnings, or immigration on native wages. These applications aim to isolate specific policy effects from confounding factors.

Limitations and Criticisms

Despite its utility, instrumental variable estimation is not without limitations and criticisms. A primary concern is the validity of the chosen instrumental variable. If an instrument does not truly satisfy the exogeneity condition—meaning it is correlated with the error term—the instrumental variable estimates will be biased and inconsistent, potentially even more so than ordinary least squares (OLS) estimates.

Another significant challenge is the issue of "weak instruments." Wea⁴k instruments are those that have a very low correlation with the endogenous explanatory variable. When instruments are weak, instrumental variable estimators can suffer from large biases, and their sampling distributions can be highly non-normal, leading to unreliable hypothesis testing and incorrect inferences. Res³earchers James H. Stock and Motohiro Yogo have conducted extensive work on identifying and testing for weak instruments, highlighting the need for robust methods to address this problem.

Fu²rthermore, even when instruments are valid and strong, the instrumental variable estimate typically represents a local average treatment effect (LATE)), rather than a global average treatment effect for the entire population. Thi¹s means the estimated causal impact applies only to the subpopulation whose behavior is influenced by the instrument. While valuable for specific contexts, it can limit the generalizability of the findings. The difficulty in finding truly valid and strong instruments often makes instrumental variable estimation challenging in practice.

Instrumental Variable Estimation vs. Ordinary Least Squares (OLS)

Instrumental variable estimation and ordinary least squares (OLS)) are both fundamental techniques in regression analysis, but they serve different purposes and are appropriate under different conditions, particularly concerning endogeneity.

The confusion between the two often arises when researchers initially apply OLS but suspect endogeneity. In such cases, instrumental variable estimation becomes a necessary, albeit more complex, alternative to obtain reliable causal estimates.

FAQs

What is an instrumental variable?

An instrumental variable (IV) is a third variable used in regression analysis to estimate the causal effect of an explanatory variable on an outcome when the explanatory variable is endogenous. For an IV to be valid, it must be correlated with the endogenous explanatory variable (relevance) but uncorrelated with the error term in the outcome equation (exogeneity).

When should I use instrumental variable estimation?

You should use instrumental variable estimation when you suspect that one or more of your explanatory variables are endogenous, meaning they are correlated with the error term in your model. Common reasons for endogeneity include omitted variable bias, simultaneity bias (where variables mutually influence each other), or measurement error in the explanatory variable.

What are the challenges of instrumental variable estimation?

The main challenges are finding a truly valid and strong instrument. A valid instrument must be exogenous variable (uncorrelated with the error term), which is often difficult to confirm empirically. A strong instrument is highly correlated with the endogenous variable; if the correlation is weak, it leads to "weak instruments" which can produce biased results and unreliable statistical significance.

Can instrumental variable estimation be used for economic forecasting?

While instrumental variable estimation is primarily used to establish causal inference and obtain unbiased estimates of parameters, these robust causal estimates can then be incorporated into larger financial modeling and forecasting frameworks. By understanding the true causal relationships, forecasters can build more accurate predictive models.