Instrumental Variables Estimation

Instrumental Variables (IV) estimation is a tool economists and social scientists use when they want to deduce a cause-and-effect relationship between two variables—specifically when there's a problem, and hidden factors are messing with results. 

Imagine you're trying to see how pet ownership affects people’s long-term health. The challenge is that factors like income level or personality traits, for example, could influence both an individual’s likelihood to adopt a pet and their long-term health. This creates a problem because these hidden factors make it difficult to know whether it’s actually the pet ownership that’s affecting health, or if baseline health and wealth impact whether someone is more likely to own a pet in the first place. 

That’s where IV estimation comes into play. Instead of directly trying to link pet ownership to health, researchers use an ‘instrument’—a third variable that’s related to pet ownership but doesn’t directly affect health. The key is finding something that influences whether someone adopts a pet, without being influenced by the same hidden factors.

For example, one potential instrument could be the presence of pet adoption campaigns in a specific neighborhood. A greater number of pet adoption advertisements won’t have an impact on the health of the people in the area, but it would likely increase someone’s willingness to adopt. Using this outside variable of adoption campaigns as an instrument helps to isolate the effect of pet adoption on health, stripping away the noise from the hidden factors. 

In more technical terms, the number of pet adoption campaigns is considered an exogenous variable because it’s unrelated to the health outcomes of people in the area (they influence pet adoption rates but don't directly affect health). On the other hand, if we tried to use something like income as a predictor of pet adoption, we might face endogeneity because income could also influence health outcomes, creating a correlation with both adoption and health which leads to biased results. 

To recap, the instrumental variable (the adoption campaigns) helps overcome this bias by isolating the effect of pet adoption on health. Thus, exogeneity ensures the variable only impacts the predictor (adoption rates) and not the outcome (health) and endogeneity occurs when a variable impacts both, confounding the analysis.

Econometrics: The application of statistical and mathematical methods to economic data to analyze relationships, test theories, and make forecasts. It quantifies economic phenomena, providing empirical support for economic theories and helping to evaluate policies.

Endogeneity: When an explanatory variable in a model is correlated with the error term, often due to omitted variables, measurement errors, or reverse causality. This creates biased and inconsistent estimates, leading to inaccurate conclusions.

Exogeneity: When variables that are not correlated with the error term and are determined outside of the model. Exogenous variables provide unbiased estimates, making them reliable for causal inference in statistical models.

Instrument: A variable correlated with the endogenous explanatory variable but uncorrelated with the error term.

Overidentification: Occurs when the number of instruments exceeds the number of endogenous variables in an instrumental variables (IV) regression model. This allows for testing the validity of the instruments using overidentification tests (e.g., the Sargan-Hansen test), which assess whether the instruments are exogenous (uncorrelated with the error term) and appropriate for the model.

Underidentification: The opposite of overidentification, underidentification occurs when there are fewer instruments than endogenous variables.

Exclusion Restriction: The assumption that the instrument affects the dependent variable only through the endogenous explanatory variable.

Weak Instruments: Instruments that are weakly correlated with the endogenous variable, leading to biased estimates.

Causal Inference: Concluding causal relationships based on the use of instruments to address endogeneity.

Natural Experiments: Real-world situations that provide exogenous variation used as instruments.

Simultaneity Bias: Bias that arises when causality between two variables runs in both directions, which IV estimation can help address.

Instrumental variables estimation first emerged as a crucial technique in econometrics to address the problem of endogeneity (where explanatory variables are correlated with the error term), leading to biased and inconsistent parameter estimates. The concept was pioneered by Philip G. Wright in 1928, who applied it to agricultural economics, particularly in his work on supply and demand models. Wright recognized that traditional methods failed when variables influencing the outcome were also correlated with unobserved factors, leading to erroneous conclusions.¹

Throughout the mid-20th century, the method gained traction through the contributions of economists like Lawrence Klein and Herman Wold, who expanded its theoretical foundations and practical applications. IV estimation became especially important in scenarios where randomized controlled trials were impractical or impossible, like in the field of economics or other social sciences. For example, when estimating the effect of education on earnings, the issue of endogeneity arises because unobserved factors like a student’s inherent ability might influence both education and earnings. An instrumental variable—like geographic proximity to a college—could be used to isolate the exogenous variation in education, allowing for a more accurate estimation of its causal effect.

In the late 20th century, the technique became widely adopted, driven by the increasing availability of large observational datasets and the rise of natural experiments. Researchers recognized that IV estimation could help establish causality in complex models where direct experimentation was not feasible. Today, IV estimation is a standard tool in econometrics and is extensively used across various fields, including economics, epidemiology, and political science, to address issues of endogeneity and to draw more reliable inferences from observational data.²

Philip G. Wright: Wright introduced the concept of instrumental variables in his 1928 book The Tariff on Animal and Vegetable Oils, where he used IV methods to address simultaneity in supply and demand models. His pioneering work laid the theoretical foundation for IV estimation, helping econometricians deal with endogeneity issues.

Joshua Angrist: A prominent figure in causal inference and MIT professor of economics, Angrist is known for applying IV methods to important economic questions, particularly through natural experiments. His influential 1991 paper with Alan Krueger, Does Compulsory School Attendance Affect Schooling and Earnings?, used the quarter of the year when children were born (which we’ll discuss more later) as an instrument to study the effect of education on earnings. Angrist also co-authored the empirical research textbook Mostly Harmless Econometrics. 

Guido Imbens: Imbens is a professor at Stanford University and has contributed extensively to the academic understanding of causal inference using IV. Imbens' main contributions are in the context of natural experiments and his work on local average treatment effects (LATE) has been highly influential. In his 1994 paper with Angrist, Identification and Estimation of Local Average Treatment Effects, they provided a framework for estimating causal effects when instruments vary treatment status. Imbens is also co-author of Causal Inference for Statistics, Social, and Biomedical Sciences. 

Herman Wold: Swedish economist and statistician who contributed to the theoretical foundation of instrumental variable (IV) estimation through his work on structural modeling and systems of equations. His development of recursive models and confluence analysis provided key insights into dealing with simultaneity and endogeneity, concepts closely related to IV estimation. Although not directly focused on IV methods, Wold's contributions to structural equation modeling, including the Wold decomposition in time series analysis, laid important groundwork for the field. 

IV estimation has had major consequences in the field of econometrics and beyond; its ability to provide more accurate and consistent estimates in situations where endogeneity would otherwise bias results has been a huge help to economists. Having access to more accurate estimates has allowed researchers to draw more reliable conclusions about causal relationships in economic models, particularly in observational studies where randomized controlled trials aren’t always feasible.

The method has also helped broaden the scope of econometric analysis by enabling the study of complex systems when there’s the concern of simultaneous causality or omitted variable bias. In labor economics, for example, it’s been crucial for estimating the returns to education, where traditional methods might overestimate effects due to factors like ability bias. But by using instruments or measurements like proximity to schools, researchers can isolate the exogenous variation in education and therefore have been able to create more credible estimates.¹

Lastly, the IV estimation method has greatly influenced policy-making. It’s helped by providing empirical evidence that is less prone to bias. Governments and organizations have relied more and more on IV-based studies to inform decisions, particularly in areas like healthcare, education, and public finance. The rise of natural experiments, where real-world events provide exogenous variation, has further cemented IV estimation as a powerful tool in both academic research and practical policy analysis. This methodological advancement has led to more evidence-based policy interventions, improving both their effectiveness and ability to target specific populations directly.¹

One of the most significant controversies surrounding the estimation method is related to the question of the validity of instruments. For an instrument to be valid, it must satisfy two key conditions: relevance and exogeneity. 

Relevance condition

The instrument used is correlated with the endogenous explanatory variable. For example, if you’re trying to find out whether years of schooling relates to lifetime earnings (an example we’ll explore more later) you might be concerned that education is endogenous because it could be correlated with other unobserved factors. In this case, you might decide to use an instrumental variable like "distance to the nearest college" as your instrument. 

The relevance condition would be satisfied if “distance to the nearest college” does indeed influence how much education a person receives. In this case, you could check the correlation between the distance to the nearest college and the years of education. If there is a strong negative correlation, the instrument is relevant.

Exogeneity condition

The variable used is not correlated with the error term in the structural equation. Unfortunately, this condition can be untestable. In the same education example, to satisfy the exogeneity condition, “distance to the nearest college” must also not be correlated with the error term in the earnings equation. If “distance to college” is related to unobserved factors that also affect earnings (e.g., regions closer to colleges might have better overall economic opportunities), then the exogeneity condition would be violated. In this case, the “distance to nearest college” instrument would almost certainly be correlated with the error term, making it invalid and leading to biased IV estimates.

Critics argue that finding instruments that meet these conditions is inherently difficult, and researchers often have to rely on weak or invalid instruments. This leads to sometimes biased and inconsistent estimates. 

Local Average Treatment Effect (LATE)

Additionally, the concept of Local Average Treatment Effect (LATE) helps us in understanding the scope of causal inference. LATE refers to the causal effect estimated only for the subset of the population that’s affected by the instrument, known as the "compliers." In our example, LATE would measure the effect of education on income only for people who go to college because they happen to live closer to one (the "compliers").  It doesn’t tell you the effect of education on everyone, just on those whose decision to attend college was influenced by the distance.

In other words, LATE focuses on the subgroup of people whose behavior changes because of the instrument, rather than the entire population. This underscores the importance of carefully interpreting IV results, as they can provide insights into specific causal relationships that might not extend universally.

Weak instruments 

Another major point of contention is the strength of instruments, which relates to how well the instruments explain the variation in the endogenous variable. Weak instruments (which are only weakly correlated with the endogenous variable), can lead to severe biases in IV estimates. This has led to a growing concern about the reliability of IV estimates in practice, particularly in cases where instruments are not sufficiently strong. If and when economists use weak instruments, it can lead to problems like large standard errors, misleading confidence intervals, or even false IV estimates.

One common method used to avoid this is to check the F-statistic from the first stage of the two-stage least squares (2SLS) regression; an F-statistic below 10 typically signals a weak instrument. Using weak instruments risks producing estimates that are as biased as (or even worse than), ordinary least squares (OLS) estimates. Therefore, ensuring instrument strength is vital for accurate IV estimation.

Finally, there’s a broader debate about the implications of relying on IV estimation for causal inference. While IV methods are often promoted as a solution to endogeneity, some scholars argue that the assumptions required for IV estimation are often unrealistic or too stringent in practice. This has sparked a discussion about whether IV estimation can be truly considered a "gold standard" for causal inference or whether its limitations should prompt a more cautious interpretation of results derived from this method. Additionally, the focus on finding the ‘perfect’ instruments can sometimes distract researchers from addressing other important aspects of model specification and identification, potentially leading to an over-reliance on IV methods without sufficient scrutiny of their underlying assumptions.³

Schooling and Lifetime Earnings

In their 1991 paper, "Does Compulsory School Attendance Affect Schooling and Earnings?"⁴ two scientists named Joshua Angrist and Alan Krueger investigated the impact of education on earnings. Their study is one example of the many ways IV estimation can be used to address endogeneity issues when we’re investigating causal relationships.

Angrist and Krueger sought to estimate the return on education investment (an increasingly relevant topic as university tuition rates continue to skyrocket). Specifically, they looked at how additional years of schooling affect earnings. However, they faced the classic endogeneity problem: individuals' decisions regarding how much education to pursue are likely influenced by a milieu of additional factors like innate ability, family background, social/cultural pressure, and intrinsic motivation, which are all also correlated with earnings. Without being properly addressed, this endogeneity issue could lead to biased and inconsistent estimates, on what they were truly trying to estimate. To overcome this, Angrist and Krueger used the individuals’ birth month as an instrumental variable. Their idea was that individuals born in different parts of the year are subject to different school regulations (when they can/can’t enroll), which in turn affects the minimum amount of schooling they receive.

Using when the participants were born is a valid instrument under the assumption that it affects educational attainment (relevance condition) but otherwise it’s uncorrelated with the error term in the earnings equation (exogeneity condition). In this context, the individuals who were born earlier in the year were more likely to start school at a younger age and therefore they were likely required to attend school for a longer period due to compulsory schooling laws. Angrist and Krueger used this otherwise ‘random’ regulation around birth date and schooling regulation to isolate the causal effect of education on earnings from other potentially confounding factors.

Their findings revealed that the returns to education were higher than what they estimated using traditional methods of estimations where they didn’t account for the influence of other variables like innate ability. Thus, their study became a cornerstone in labor economics, demonstrating the power of IV methods in addressing and accounting for outside variables, as well as the importance of doing so.

Vietnam Era Draft and Lifetime Earnings

In another well-known case study from Joshua Angrist we can clearly see IV estimation at play. In this 1990 paper titled "Lifetime Earnings and the Vietnam Era Draft Lottery: Evidence from Social Security Administrative Records,” he looked at the impact of military service during the Vietnam War on veterans' future earnings. A central challenge in estimating this effect is the selection bias that arises because individuals who serve in the military are usually systematically different from those who do not. For example, those who volunteer for military service may come from different cultural or geographic upbringings (they may come from a military family that has a history of honoring those who serve, or they may come from a city where the military is frowned upon). Many people who serve are also motivated by financial need, and choose to go into the armed forces as a way to pay their tuition or pay off student loans. Obviously, this background or need could also influence their earnings later in life. This selection bias would lead to biased estimates if it’s not properly addressed.

Thus, to overcome this issue, Angrist used the Vietnam draft lottery as an instrumental variable. Since the draft lottery randomly assigned numbers to birthdates, with lower numbers indicating a higher likelihood of being drafted, the random assignment created a natural experiment where men with low lottery numbers were more likely to serve in the military, regardless of their personal characteristics or socioeconomic status. The lottery number thus served as an instrument because it was correlated with military service (relevance condition) but, due to its random nature, wasn’t correlated with other determinants of earnings (exogeneity condition).

Using the draft lottery as an instrument, the researchers were able to estimate the causal effect of military service on earnings: they found that Vietnam veterans earned significantly less over their lifetimes compared to their non-veteran counterparts, which then sparked and continues to perpetuate a conversation about the economic costs of military service. 

Correlation vs. Causation: It’s important to better understand how to differentiate and disentangle correlation and causation. Instrumental Variables Estimation is a way to support that process, but it’s necessary to first understand why they can often be confused in the first place. This article outlines how and why correlation is so often confused with causation, and the potential pitfalls when we conflate the two. 

Illusory correlation: Have you ever wondered why we assume some things are related but then later find out they aren’t? Illusory correlation bias is when we see an association between two variables (events, actions, ideas, etc.) when they aren’t actually associated. Read this explanation of the illusory correlation bias to better understand the effects of this bias in daily life and how we can overcome it.

1. Wikipedia contributors. (n.d.). Instrumental variables estimation. In Wikipedia, The Free Encyclopedia. Retrieved August 25, 2024, from https://en.wikipedia.org/wiki/Instrumental_variables_estimation
2. Stock, J. H. (n.d.). The history of IV regression. Harvard University. https://scholar.harvard.edu/stock/content/history-iv-regression
3. FasterCapital. (n.d.). The theory behind instrumental variables. FasterCapital. Retrieved August 25, 2024, from https://fastercapital.com/topics/the-theory-behind-instrumental-variables.html
4. Angrist, J. D., & Krueger, A. B. (1991). Does Compulsory School Attendance Affect Schooling and Earnings? The Quarterly Journal of Economics, 106(4), 979–1014. https://doi.org/10.2307/2937954 
5. Angrist, J. D. (1990). Lifetime Earnings and the Vietnam Era Draft Lottery: Evidence from Social Security Administrative Records. The American Economic Review, 80(3), 313–336. http://www.jstor.org/stable/2006669