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Fixed Effect

category_specifier : "Causal Inference"

Reference Docs: Difference in Difference | Omitted Variable Bias

Motivation

Do I think unobserved, stable traits are driving both my treatment and outcome?

If yes, and you have panel data - fixed effects is your friend.

Definition

In panel data analysis, a Fixed Effects model gives each entity its own intercept term to capture time-invariant characteristics. It includes both unit fixed effects for entities and time fixed effects for period-specific factors.

Fixed effects regression analyzes within-entity variation: how changes in an entity over time relate to changes in predictors. Each entity acts as its own control, accounting for permanent characteristics like a store's location or size.

The model equation for two-way fixed effects is:

\(Y_{it} = \alpha_i + \lambda_t + \beta X_{it} + \epsilon_{it}\)

where \(α_i\) is the entity fixed effect and λt is the time fixed effect.

Unlike random effects models, fixed effects don't assume entity differences are random. This approach controls for "all unobserved heterogeneity that is constant over time" without measuring it directly.

Why It Matters / Motivation

Fixed effects help address omitted variable bias in panel data by controlling for unobserved characteristics that are constant over time without directly observing them.

Fixed effects are crucial in difference-in-differences ( Difference in Difference analysis, allowing comparison of treated and control groups while accounting for baseline differences and time trends.

Key benefits:

  • They eliminate bias from unobserved, time-invariant factors, making our coefficient estimates closer to causal.
  • They allow us to use within-unit variation, focusing on changes rather than levels.
  • They handle panel data questions naturally by analyzing changes within an entity.

Examples

  • Sales → Ad spending, but each store has its own baseline popularity → include store fixed effects.
  • Test scores → Class size, but some students are naturally high-performing → use student fixed effects.
  • Policy change → Employment, but each state has a fixed culture of regulation → include state fixed effects.

How Fixed Effects Are Implemented and Interpreted

Fixed effects models use dummy variables for each entity and time period (minus one reference).

In interpretation, coefficients show the effect of a variable on Y while controlling for time-invariant entity differences. For example, in a store fixed effects model, a price coefficient shows how changes in price affect sales within the same store over time.

Key considerations include degrees of freedom (N-1 dummies for N entities) and within R-squared vs overall R-squared. While fixed effects control for time-invariant factors, they don't guarantee causality - time-varying confounders may still exist.

Who Uses Fixed Effects

  • Economists/Academics: Used extensively in empirical research, especially for wage gaps and health outcomes
  • Policy Analysts: Evaluate program effects across regions over time
  • Data Scientists: Analyze user behavior, A/B tests, and customer patterns
  • Financial Analysts: Study firm performance and market trends
  • Educational Researchers: Control for school-level differences in student outcomes

Fixed effects methods may appear under different names in industry, such as "categorical embeddings" in machine learning or "entity encoders" in deep learning.