A diverse group constructing a data-driven bridge over economic disparity.

Decoding the Data: How Functional Regression Can Help Us Understand Immigration's Impact

Soraya Malik in Business & Economy July 2025 • 4 min read.

"New statistical methods offer a clearer picture of how immigration affects native wages, skill by skill."

In an era defined by unprecedented data availability, our capacity to analyze complex phenomena is ever-expanding. Functional data analysis (FDA) has emerged as a powerful statistical tool, allowing researchers to study intricate relationships between function-valued variables—curves, density functions, and images. One area ripe for this type of sophisticated exploration is the economic impact of immigration, where traditional models often fall short in capturing the nuances of skill-based wage effects.

Traditional linear regression, while useful, struggles when explanatory variables are endogenous—influenced by factors within the model itself. This is especially true when studying immigration, where factors like omitted variables or measurement errors can skew results, obscuring the true impact on native-born workers' earnings. Recently, statisticians have begun to tackle this issue within functional regression, but practical applications have been limited due to the complexities of estimator distributions and statistical inference.

A groundbreaking paper introduces new Functional Principal Component Analysis (FPCA)-based instrumental variable estimators. This technique not only addresses the endogeneity problem but also provides detailed asymptotic properties, allowing for more robust conclusions. Applying their methodology to the study of immigration, these researchers offer fresh insights into how immigration affects native wages across different skill levels.

What is Functional Instrumental Variable Regression (FIVR)?

A diverse group constructing a data-driven bridge over economic disparity.

Functional regression steps beyond standard regression by treating variables as functions. Imagine tracking a worker's wage not as a single number, but as a curve representing their earning potential across different skill levels. Functional regression allows us to model this type of data directly, uncovering relationships that traditional methods might miss.

Endogeneity occurs when the explanatory variables are correlated with the error term in a regression model, leading to biased estimates. In the context of immigration, this might arise because immigrants tend to cluster in specific industries or regions, which could independently affect wages, regardless of the immigrants' direct impact.

Omitted Variable Bias: Failing to account for relevant factors (e.g., local economic conditions, technological changes) that influence both immigration patterns and native wages.
Measurement Error: Inaccuracies in measuring immigrants' skills or education levels.
Reverse Causality: The possibility that wage levels attract immigration, rather than immigration driving wage changes.

FIVR combines the strengths of functional regression with instrumental variable techniques. Instrumental variables are additional factors that are correlated with the endogenous explanatory variable (immigration) but not with the error term (other factors that directly affect native wages). By using these instruments, researchers can isolate the exogenous variation in immigration, providing a less biased estimate of its impact on wages. FPCA reduces the dimensionality of the functional data, making the estimation process more manageable and stable. This ensures the model is both accurate and practical.

Why This Matters: Implications and Future Directions

The development of consistent FPCA-based instrumental variable estimators represents a significant step forward in economic analysis. By addressing the challenges of endogeneity and providing detailed methods for statistical inference, these new tools allow for a more accurate and nuanced understanding of complex economic phenomena. These methods offer valuable insights for policymakers seeking to understand and manage the economic effects of immigration, potentially leading to more effective and equitable labor market policies.

About this Article -

This article was crafted using a human-AI hybrid and collaborative approach. AI assisted our team with initial drafting, research insights, identifying key questions, and image generation. Our human editors guided topic selection, defined the angle, structured the content, ensured factual accuracy and relevance, refined the tone, and conducted thorough editing to deliver helpful, high-quality information.See our About page for more information.

This article is based on research published under:

DOI-LINK: 10.1017/s0266466624000252,

Title: Functional Instrumental Variable Regression With An Application To Estimating The Impact Of Immigration On Native Wages

Subject: econ.em stat.me

Authors: Dakyung Seong, Won-Ki Seo

Published: 25-10-2021

Everything You Need To Know

What is Functional Instrumental Variable Regression (FIVR) and how does it improve economic analysis?

Functional Instrumental Variable Regression (FIVR) is a sophisticated statistical technique that combines functional regression with instrumental variable methods. Functional regression treats variables as functions, such as a curve representing a worker's wage potential across different skill levels. Instrumental variables are factors correlated with the endogenous explanatory variable (like immigration) but not with the error term in the regression. This helps isolate the exogenous variation, reducing bias. FIVR addresses the endogeneity problem, where explanatory variables are influenced by factors within the model, providing a more nuanced understanding of complex economic phenomena, particularly the impact of immigration on native wages.

How does endogeneity affect the study of immigration's impact on wages, and what are some sources of this bias?

Endogeneity introduces bias in the study of immigration's wage effects when explanatory variables are correlated with the error term in a regression model. In the context of immigration, this correlation can occur due to several factors. Omitted variable bias arises when relevant factors, like local economic conditions or technological changes, influence both immigration patterns and native wages, leading to inaccurate estimates. Measurement errors in assessing immigrants' skills or education levels also contribute to this bias. Additionally, reverse causality may occur if wage levels attract immigration, rather than immigration directly influencing wage changes. These biases can distort the true impact on native-born workers' earnings.

What role does Functional Principal Component Analysis (FPCA) play in FIVR, and why is it important?

Functional Principal Component Analysis (FPCA) plays a crucial role in Functional Instrumental Variable Regression (FIVR) by reducing the dimensionality of functional data. This simplification makes the estimation process more manageable and stable. FPCA transforms complex functional data into a more accessible form, allowing researchers to derive robust conclusions and ensuring the model is both accurate and practical. The use of FPCA in FIVR enables the analysis of function-valued variables, like wage curves, facilitating a deeper understanding of the relationships between immigration and wages across different skill levels.

What are the limitations of traditional linear regression when analyzing the economic impact of immigration?

Traditional linear regression struggles to capture the nuances of skill-based wage effects in immigration studies because it often oversimplifies complex relationships. The primary limitation is its inability to handle endogeneity effectively. Standard models are not designed to account for factors like omitted variables, measurement errors, or reverse causality, which can skew results and obscure the true impact of immigration on native-born workers' earnings. This method treats variables as single numbers, which can not reflect on the detailed wage changes across different skill levels, therefore providing a less detailed view compared to functional regression methods.

What are the implications of the new Functional Principal Component Analysis (FPCA)-based instrumental variable estimators for policymakers?

The development of FPCA-based instrumental variable estimators offers significant implications for policymakers. These estimators provide a more accurate and nuanced understanding of the economic effects of immigration by addressing the challenges of endogeneity and enabling detailed statistical inference. This allows policymakers to gain valuable insights into the impact of immigration on native wages across different skill levels. With this new understanding, policymakers can design more effective and equitable labor market policies. Such policies can help manage the economic effects of immigration, potentially leading to fairer outcomes for native-born workers and contributing to a more stable and prosperous economy.