Introduction to Fixed Effects: A guide with R examples

Fixed effects play a central role in statistical analysis and in panel data analysis in particular. Fixed effects models are designed to control for unobserved heterogeneity by eliminating the effects of unobserved variables that remain constant over time. This makes them a powerful tool in analyzing data collected across multiple time points and/or units. In this blog post, I will introduce you to the concept of fixed effects and show you how to apply them in R.

What are fixed effects?

Fixed effects models focus on the variation within a unit (e.g. an individual or a company) over time. They control for all time-invariant factors that could influence the outcome by exploiting differences within each unit. This is done by introducing a separate intercept for each unit that includes all unobserved, time-invariant characteristics of that unit.

When should fixed effects be used?

Fixed effects models are particularly useful if you want to:

1. To control unobserved heterogeneity that exists between units (e.g. different individuals or countries) but remains constant over time within a unit.
2. Analyze panel data in which the same unit is observed over several time periods.
3. Ensure that estimates are not biased by unobserved variables that are correlated with the explanatory variables.

Example of a fixed effects model in R

Suppose we have a data set that tracks the annual salaries of employees over several years. We want to find out how the number of hours worked and work experience affect salary, while controlling for time-invariant characteristics such as gender or ethnicity.

Simulate data set

First, we create a simulated data set:

set.seed(123)
n <- 100  # Anzahl der Mitarbeiter
T <- 5    # Anzahl der Jahre

# Simulierter Datensatz
df <- data.frame(
  MitarbeiterID = rep(1:n, each = T),
  Jahr = rep(2000:2004, times = n),
  Gehalt = rnorm(n * T, mean = 50000, sd = 10000),
  Arbeitsstunden = rnorm(n * T, mean = 40, sd = 5),
  Erfahrung = rep(rnorm(n, mean = 10, sd = 5), each = T),
  Geschlecht = rep(sample(c("Männlich", "Weiblich"), n, replace = TRUE), each = T)
)

Estimate fixed effects model

To estimate a fixed effects model, we use the package plmwhich was specially developed for analyzing panel data:

install.packages("plm")
library(plm)

# Fixed Effects Modell
fe_model <- plm(Gehalt ~ Arbeitsstunden + Erfahrung, 
                data = df, 
                index = c("MitarbeiterID", "Jahr"), 
                model = "within")

summary(fe_model)

In this model, we control for all time-invariant employee characteristics by using only the within-employee variation over time.

Interpretation of the results

The coefficients of the fixed effects model indicate the average change in salary for a unit change in working hours or professional experience within an employee over time. As time-invariant characteristics such as gender and ethnicity are eliminated by the fixed effects, the model focuses exclusively on the effects of variables that change within employees over time.

Advantages and limitations of fixed effects

Advantages:

• Control for unobserved heterogeneity: Fixed effects eliminate the bias due to time-invariant unobserved variables.
• Simple interpretation: The results are easy to interpret as they focus on the variation within the units.

Limits:

• No estimation of time-invariant variables: Fixed effects models cannot estimate coefficients for time-invariant variables, as these are eliminated by the transformation.
• Lower efficiency with little variation within the units: If the explanatory variables hardly change within the units, the model can become inefficient.

Conclusion

Fixed effects models are an indispensable tool in panel data analysis because they make it possible to control for unobserved heterogeneity and thus avoid bias in the estimates. They are particularly useful in situations where time-invariant characteristics of the units, such as gender or ethnicity, might play a role but are not explicitly measured. With the plm package in R, you can easily estimate and interpret fixed effects models. This makes it a powerful tool in analyzing data collected over time.