Instrumental variables regression

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What Is Instrumental Variables Regression?

Instrumental variables (IV) regression is a statistical technique used in econometrics to estimate the causal effect of one variable on another when a simple regression analysis would yield biased results due to endogeneity. It falls under the broader financial category of quantitative methods. This method addresses situations where an independent variable is correlated with the error term, often due to omitted variables, measurement error, or simultaneity. By employing an "instrumental variable," IV regression aims to isolate the exogenous variation in the problematic independent variable, thereby providing a more accurate estimate of the true causal relationships.

History and Origin

The concept of instrumental variables regression emerged in the early 20th century, primarily to tackle challenges in estimating economic relationships. American economist Philip G. Wright is credited with first proposing the use of instrumental variables estimation in 1928, as a solution to the identification problem in econometrics¹¹. His work, detailed in an appendix to his 1928 book, The Tariff on Animal and Vegetable Oils, demonstrated how to estimate supply and demand elasticities when observed data on price and quantity alone were insufficient¹⁰. Wright referred to these additional factors, which affected one curve (supply or demand) without affecting the other, as "external factors". His foundational insights, though initially overlooked for decades, laid the groundwork for modern econometric methods, particularly in addressing issues of simultaneous equations in econometric models⁹.

Key Takeaways

• Instrumental variables (IV) regression is an econometric method used to estimate causal effects when independent variables are endogenous.
• It addresses issues like omitted variable bias, measurement error, and simultaneity by using an instrumental variable.
• An effective instrumental variable must be correlated with the endogenous independent variable but uncorrelated with the error term.
• IV regression provides more reliable estimates of causal effects than ordinary least squares (OLS) when endogeneity is present.
• It is widely applied in economics, finance, and other social sciences for policy evaluation and empirical analysis.

Formula and Calculation

Instrumental variables regression typically involves a two-stage process when implemented using two-stage least squares (2SLS). Consider a linear model where (Y) is the dependent variable, (X) is the endogenous independent variable, and (Z) is the instrumental variable. Let (\epsilon) be the error term.

The structural equation is:
[ Y = \beta_0 + \beta_1 X + \epsilon ]

Here, (X) is endogenous, meaning (\text{Cov}(X, \epsilon) \neq 0). We need an instrument (Z) that satisfies two conditions:

1. Relevance: (\text{Cov}(Z, X) \neq 0) (The instrument is correlated with the endogenous variable).
2. Exogeneity: (\text{Cov}(Z, \epsilon) = 0) (The instrument is uncorrelated with the error term).

The two stages of 2SLS are:

Stage 1: Regress the endogenous variable on the instrumental variable and any exogenous covariates.
[ X = \gamma_0 + \gamma_1 Z + \nu ]
From this, we obtain the predicted values of (X), denoted as (\hat{X}). This step purges the endogenous component of (X), leaving only the variation explained by the instrument (Z).

Stage 2: Regress the dependent variable on the predicted values of the endogenous variable.
[ Y = \delta_0 + \delta_1 \hat{X} + \mu ]
The coefficient (\delta_1) from this second stage is the IV estimator of (\beta_1). This process leverages the correlation between the instrument and the endogenous variable to achieve consistent estimation.

Interpreting the Instrumental Variables Regression

Interpreting the results of instrumental variables regression requires careful consideration of the causal effect being estimated. Unlike ordinary least squares (OLS), which only measures correlation, IV regression aims to uncover the true causal impact of an endogenous variable on an outcome variable. The coefficient obtained from IV regression represents the change in the dependent variable for a one-unit change in the endogenous variable that is caused by the variation in the instrumental variable. This provides a more robust estimate, free from the bias that would arise from ignored confounding variables or simultaneous relationships.

Hypothetical Example

Imagine a researcher wants to determine the causal effect of education on income. A simple OLS regression might show a positive correlation, but it could be biased because unobserved factors like individual ability or motivation (which also affect income) are correlated with education.

Here, education is the endogenous variable, and income is the dependent variable.

To use instrumental variables regression, the researcher needs an instrumental variable that is correlated with education but not directly with innate ability or motivation. A hypothetical instrument could be proximity to a college campus at the age of 18.

• Relevance: Proximity to a college campus is likely to be correlated with the level of education attained, as it reduces costs and increases access.
• Exogeneity: Proximity to a college campus is arguably uncorrelated with an individual's innate ability or motivation, after controlling for other observable factors.

Stage 1: The researcher would first regress "years of education" (endogenous variable) on "proximity to college campus" (instrumental variable) and other exogenous variables like parental income or high school GPA. This stage generates predicted years of education.

Stage 2: Next, the researcher would regress "income" (dependent variable) on the "predicted years of education" from Stage 1. The coefficient on predicted years of education would then represent a less biased estimate of the causal effect of education on income. This two-stage process helps to isolate the variation in education that is purely driven by the instrument, allowing for a more accurate assessment of its impact on income.

Practical Applications

Instrumental variables regression is widely used in various fields of finance and economics to estimate causal effects where direct methods are insufficient due to endogeneity. In labor economics, for instance, IV regression has been employed to estimate the impact of welfare reform on labor markets and wages, by using policy-related instruments to account for the endogenous nature of welfare caseloads⁸. Similarly, researchers use IV to analyze the relationship between inflation and unemployment, particularly when considering how regional labor market tightness influences price pressures⁶, ⁷.

Another significant application is in assessing the effects of central bank policy interventions on financial markets. For example, studies have utilized instrumental variables to understand how changes in the Federal Reserve's balance sheet, such as those arising from quantitative easing, impact equity market valuations⁴, ⁵. This method helps to disentangle the direct effects of monetary policy from other confounding factors influencing market dynamics.

Limitations and Criticisms

While instrumental variables regression offers a powerful solution to endogeneity, it is not without limitations. A primary challenge lies in finding truly valid instrumental variables. An instrument must satisfy both the relevance and exogeneity conditions: it must be sufficiently correlated with the endogenous regressor (relevance) and uncorrelated with the error term (exogeneity). If the instrument is only weakly correlated with the endogenous variable, it can lead to "weak instruments" problems, resulting in large bias and poor finite-sample properties of the IV estimator², ³. This means that even a small violation of the exogeneity assumption can lead to significant bias in the IV estimates when instruments are weak.

Another criticism centers on the exogeneity assumption itself, which is often difficult to prove empirically and relies on strong theoretical justifications. If the instrumental variable affects the dependent variable through channels other than the endogenous regressor, the IV estimate will be biased. Furthermore, IV regression can sometimes produce estimates with larger standard errors compared to OLS, particularly when the instruments are weak, making statistical inference less precise. Researchers at the Federal Reserve Board highlight the crucial role of the identification condition and the challenges of detecting lack of identification in generalized method of moments models, of which linear instrumental variables models are a special case¹.

Instrumental Variables Regression vs. Ordinary Least Squares

The core distinction between instrumental variables (IV) regression and ordinary least squares (OLS) lies in how they handle endogeneity. OLS assumes that the independent variables are exogenous, meaning they are uncorrelated with the error term. When this assumption is violated, typically due to omitted variables, measurement error, or simultaneity, OLS estimates become biased and inconsistent, failing to capture the true causal relationships.

In contrast, instrumental variables regression is specifically designed to address endogeneity. It introduces an instrumental variable that serves to isolate the exogenous variation in the problematic independent variable. By using this instrument, IV regression purges the endogenous variables of their correlation with the error term, thereby producing consistent and unbiased estimates of the causal effect. While OLS is simpler to implement and provides efficient estimates when its assumptions hold, IV regression is necessary when endogeneity is present to ensure valid statistical inference.

FAQs

When should I use instrumental variables regression?

You should use instrumental variables regression when you suspect that one or more of your independent variables are endogenous, meaning they are correlated with the error term in your regression model. This often happens due to unobserved factors influencing both the independent and dependent variables, measurement errors, or when variables simultaneously determine each other.

What makes a good instrumental variable?

A good instrumental variable must meet two critical conditions:

1. Relevance: It must be sufficiently correlated with the endogenous independent variable you are trying to instrument.
2. Exogeneity: It must be uncorrelated with the error term of the main regression equation. This means it should only affect the dependent variable through its effect on the endogenous independent variable, not directly or through other omitted factors.

Can instrumental variables regression be used with multiple endogenous variables?

Yes, instrumental variables regression can be extended to handle multiple endogenous variables. Techniques like two-stage least squares (2SLS) can accommodate multiple endogenous regressors, requiring at least as many valid instruments as there are endogenous variables to achieve identification.

What is the "weak instruments" problem?

The "weak instruments" problem occurs when the instrumental variable is only weakly correlated with the endogenous independent variable. In such cases, the IV estimates can be severely biased, even if the instrument is perfectly exogenous. This issue can lead to unreliable statistical inference and inflated standard errors.

How does instrumental variables regression relate to supply and demand analysis?

Instrumental variables regression is particularly useful in supply and demand analysis. When estimating supply or demand curves, price and quantity are simultaneously determined, making them endogenous. An instrumental variable, such as a factor that shifts only the supply curve (like a weather shock affecting crop yield) or only the demand curve (like a change in consumer tastes), can be used to identify and estimate the true slope of the other curve.