Panel Data Analysis — Fixed and Random Effects

Panel data combines cross-sectional and time-series dimensions — the same units are observed over multiple time periods. This structure allows controlling for unobserved heterogeneity.

Panel Data Structure

| Entity | Time 1 | Time 2 | Time 3 |

|--------|--------|--------|--------|

| Unit 1 | | | |

| Unit 2 | | | |

| Unit 3 | | | |

Notation: — outcome for unit at time

The Pooled OLS Problem

Fixed Effects (FE) Model

Random Effects (RE) Model

FE vs RE Comparison

| Feature | Fixed Effects | Random Effects |

|---------|--------------|----------------|

| Assumption | correlated with | uncorrelated with |

| Time-invariant variables | Cannot estimate | Can estimate |

| Efficiency | Less efficient | More efficient |

| Consistency | Consistent even if correlated | Consistent only if uncorrelated |

| Estimation | Demeaning / LSDV | GLS |

Hausman Test

The Hausman test compares FE and RE to determine which is appropriate.

| Decision | Interpretation |

|---------|---------------|

| Reject | Use Fixed Effects (correlation exists) |

| Fail to reject | Use Random Effects (more efficient) |

First Differences Alternative

Differencing eliminates the entity-specific effect, just like demeaning.

Time Fixed Effects

Controls for factors that change over time but are constant across entities (e.g., economic shocks, policy changes).

Python Implementation

import numpy as np

import pandas as pd

import statsmodels.api as sm

from linearmodels.panel import PanelOLS, RandomEffects

from linearmodels.panel import compare

import matplotlib.pyplot as plt



np.random.seed(42)



# Simulate panel data

n_entities = 100

n_periods = 10

n = n_entities * n_periods



entity_id = np.repeat(np.arange(n_entities), n_periods)

time_id = np.tile(np.arange(n_periods), n_entities)



# Entity effects

alpha = np.random.randn(n_entities) * 2

alpha_panel = alpha[entity_id]



# Covariates

X = np.random.randn(n)

Y = 5 + alpha_panel + 0.8 * X + np.random.randn(n) * 1.5



df = pd.DataFrame({

    'Y': Y, 'X': X,

    'entity': entity_id,

    'time': time_id

}).set_index(['entity', 'time'])



# Fixed Effects

fe_model = PanelOLS.from_formula('Y ~ 1 + X', data=df, entity_effects=True)

fe_result = fe_model.fit()

print("Fixed Effects:")

print(fe_result.summary.tables[1])



# Random Effects

re_model = RandomEffects.from_formula('Y ~ 1 + X', data=df)

re_result = re_model.fit()

print("\nRandom Effects:")

print(re_result.summary.tables[1])



# Compare

print("\nModel Comparison:")

print(compare({'FE': fe_result, 'RE': re_result}))

Worked Example

Key Takeaways

Related Topics

• See Multilevel Modeling for hierarchical structures

• See Difference-in-Differences for policy evaluation

• See Instrumental Variables for endogeneity issues