7.2 Fixed effects and random effects models

Fixed and random effects models are crucial tools in panel data analysis. They help researchers account for unobserved heterogeneity across units, offering different approaches to estimating the impact of explanatory variables on outcomes.

Choosing between fixed and random effects depends on data characteristics and research goals. Fixed effects control for time-invariant factors, while random effects balance efficiency and consistency. Understanding these models is key for accurate impact evaluation in econometrics.

Fixed vs Random Effects Models

Model Assumptions and Characteristics

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• Fixed effects models assume individual-specific effects correlate with explanatory variables, while random effects models assume no correlation
• Fixed effects control for time-invariant unobserved heterogeneity by allowing each cross-sectional unit to have its own intercept
• Random effects treat individual-specific effects as random variables drawn from a probability distribution
• Fixed effects models provide more robust but less efficient estimates compared to random effects models
• Fixed effects focus on within-unit variation, while random effects capture both within and between-unit variation

Model Selection Considerations

• Nature of the data and research question determine choice between fixed and random effects
• Fixed effects preferred for analyzing within-unit variation (changes over time within entities)
• Random effects suitable for examining both within and between-unit variation (differences across entities)
• Hausman test formally assesses whether fixed or random effects model is more appropriate
• Consider efficiency-consistency trade-off (random effects more efficient, fixed effects more consistent)

Model Selection for Research Questions

Data Characteristics Assessment

• Evaluate potential correlation between individual-specific effects and explanatory variables
• Assess importance of time-invariant variables (fixed effects cannot estimate their coefficients)
• Examine relative significance of within-unit vs between-unit variation for research objectives
• Consider sample size and number of time periods in panel data set (affects estimation precision)
• Analyze potential for omitted variable bias and need to control for unobserved heterogeneity

Diagnostic Approaches

• Utilize Hausman test to guide model selection process (compares fixed and random effects estimates)
• Employ Breusch-Pagan Lagrange Multiplier test to check for presence of random effects
• Conduct F-test to compare pooled OLS model with fixed effects model
• Consider Honda test as one-sided alternative to Breusch-Pagan for testing random effects
• Interpret diagnostic test results in context of research question and data structure

Estimating and Interpreting Effects

Estimation Techniques

• Use within estimator or least squares dummy variable (LSDV) approach for fixed effects models
• Apply generalized least squares (GLS) estimator for random effects models
• Employ feasible GLS when variance components are unknown in random effects models
• Calculate different R-squared values for panel data models (within, between, overall)
• Conduct hypothesis tests on individual coefficients and overall model significance

Coefficient Interpretation

• Interpret fixed effects coefficients as average within-unit effect of time-varying variables
• Understand random effects coefficients as weighted average of within and between-unit effects
• Assess magnitude and economic significance of estimated effects, not just statistical significance
• Consider practical implications of coefficient estimates for policy or decision-making
• Compare coefficient estimates across different model specifications for robustness

Testing for Individual or Time Effects

Individual Effects Tests

• F-test compares pooled OLS model with fixed effects model to assess joint significance of individual-specific effects
• Breusch-Pagan Lagrange Multiplier (LM) test checks for presence of random effects against pooled OLS model
• Honda test provides one-sided version of Breusch-Pagan test for random effects
• Hausman test compares fixed effects and random effects models to detect correlation between individual effects and explanatory variables

Time Effects Considerations

• Include time dummies in model to capture time-specific effects
• Test joint significance of time dummies to determine their relevance
• Evaluate need for two-way error component models accounting for both individual and time effects
• Analyze patterns in time effects to identify potential structural breaks or trends in the data
• Consider interaction between time effects and key explanatory variables for dynamic relationships