Fixed-Effects Model

Summary

The fixed-effects (FE) model (equivalently: within-person mean centering) controls for unobserved time-invariant confounders by focusing only on within-person deviations from individual means. It targets contemporaneous effects of X on Y. Key causal assumptions: no lagged dynamics, no time-varying confounders, and homogeneous slopes across persons. Violations of any of these bias the FE estimate.

Overview

The FE approach demeans each person’s observations, removing all between-person variation. Equivalently, person-level dummy variables are included. The estimator then captures the average within-person association between deviations of X and Y from their person-specific means.

It is widely used in economics (panel data), epidemiology, and increasingly in psychology via experience-sampling and diary study designs.

Causal DAG (Box 1)

Definition: Fixed-Effects Causal Graph

The standard FE model assumes the DAG (Hamaker & Muthén 2020, adapted in Box 1):

• $U$ (unobserved time-invariant confounder) → each $X_t$ and each $Y_t$
• $X_t\to Y_t$ (contemporaneous effect — the estimand)
• No cross-lagged paths: $X_{t-1}\nrightarrow Y_t$, $Y_{t-1}\nrightarrow X_t$
• No autoregressive paths among Y: $Y_{t-1}\nrightarrow Y_t$
• $X$ treated as exogenous (no arrows from $Y$ to $X$)

By demeaning, the FE estimator controls for $U$ without measuring it. The estimated coefficient on $X_t$ reflects the causal effect of X on Y within persons, under the stated assumptions.

What FE Controls and Does Not Control

Source of Variation	FE Controls?	Reason
Time-invariant confounders ($U$)	Yes	Demeaning removes all person-level variance
Time-varying confounders	No	Vary within-person; survive demeaning
Lagged effects ($X_{t-1}\to Y_t$)	No	Assumed absent in the FE model
Heterogeneous slopes (${\beta }_i$)	No	FE estimates one average slope for all
Reciprocal dynamics ($Y\to X$)	No	$X$ treated as exogenous

Assumptions for Causal Identification

FE Causal Assumptions (Box 1)

For the FE estimate to identify a causal contemporaneous effect of $X$ on $Y$:

1. No lagged dynamics: $X_{t-1}$ does not affect $Y_t$ (no cross-lagged paths from X to Y); $Y_{t-1}$ does not affect $Y_t$ beyond what is already captured (no autoregressive paths among Y that would create endogeneity)
2. Strict exogeneity / no time-varying confounders: All confounders affecting both $X_t$ and $Y_t$ have constant effects over the study duration (so they are removed by demeaning)
3. Homogeneous slopes: The within-person effect of $X$ on $Y$ is the same for all individuals (no heterogeneous ${\beta }_i$)

Limitations

1. Lagged dynamics: If talkativeness today causally affects well-being tomorrow (not just today), the FE contemporaneous estimate misses this causal pathway entirely

2. Time-varying confounders: Social events, stress, fatigue — anything that changes within a person and affects both $X$ and $Y$ — confounds the within-person estimate

3. Heterogeneous slopes: Different people may have different $X\to Y$ effects. FE estimates a population average that may not represent anyone’s true effect. If slope heterogeneity correlates with X levels, estimates are further biased (Rüttenauer & Ludwig 2020)

4. No reciprocal dynamics: The FE model treats X as exogenous — it cannot model the feedback loop $Y\to X$ that is often theoretically important in psychology

5. Consistency: The causal effect only makes sense if there is a well-defined intervention on X (see Estimands in Longitudinal Research, Box 4 on psychological interventions)

Connections

• The canonical within-persons approach; see Within-Between Persons Causal Inference for when it helps
• The DAG formalises assumptions in Directed Acyclic Graphs notation
• Compare to Cross-Lagged and Dynamic Panel Models — for lagged and reciprocal effects
• In economics: FE is standard for panel data with unobserved heterogeneity; see Regression and the CEF for the CEF interpretation

See Also

• Cross-Lagged and Dynamic Panel Models — for lagged reciprocal effects; addresses some FE limitations
• Within-Between Persons Causal Inference — when FE is and is not sufficient for causal claims
• Estimands in Longitudinal Research — how to define the right target before choosing FE
• Directed Acyclic Graphs — the formal causal graph underlying FE