Fixed effects models are a powerful tool for controlling unobserved confounding in panel data. They estimate causal effects by leveraging within-group variation over time, assuming time-invariant confounders are correlated with explanatory variables.

These models offer advantages like reducing omitted variable bias, but have limitations such as inability to estimate time-invariant effects. Understanding when to use fixed effects versus alternatives like random effects is crucial for accurate causal inference in panel data analysis.

Definition of fixed effects models

• Fixed effects models are a type of regression model used to control for unobserved confounding in panel data or repeated measures data
• Assume that the unobserved confounders are time-invariant and correlated with the explanatory variables
• Estimate the effect of the time-varying explanatory variables while controlling for the fixed effects

Assumptions in fixed effects models

• The unobserved confounders are time-invariant within each individual or group
• The explanatory variables are strictly exogenous conditional on the fixed effects
• The idiosyncratic errors are uncorrelated with the explanatory variables and the fixed effects
• The idiosyncratic errors are homoscedastic and serially uncorrelated

Advantages of fixed effects models

Control for unobserved confounders

• Fixed effects models can control for unobserved confounders that are constant over time within each individual or group
• This helps to reduce omitted variable bias and obtain more consistent estimates of the causal effect
• Examples of unobserved confounders include individual ability, motivation, or family background

Consistent estimates with panel data

• Fixed effects models can provide consistent estimates of the causal effect even if the unobserved confounders are correlated with the explanatory variables
• This is because the fixed effects absorb the time-invariant confounders and the within-group variation is used to estimate the effect
• Panel data with multiple observations per individual or group over time is required to estimate fixed effects models

Limitations of fixed effects models

Cannot estimate time-invariant effects

• Fixed effects models cannot estimate the effect of time-invariant variables because they are absorbed by the fixed effects
• This includes variables such as gender, race, or region that do not change over time within each individual or group
• If the effect of time-invariant variables is of interest, random effects models or other approaches may be more appropriate

Inefficient with low within-group variation

• Fixed effects models rely on the within-group variation in the explanatory variables to estimate the effect
• If there is little variation in the explanatory variables within each individual or group over time, the fixed effects estimates may be imprecise and inefficient
• This can lead to large standard errors and low statistical power to detect significant effects

Fixed effects vs random effects models

• Fixed effects models assume that the unobserved confounders are correlated with the explanatory variables, while random effects models assume they are uncorrelated
• Fixed effects models control for all time-invariant confounders, while random effects models only control for the included time-invariant variables
• Fixed effects models are consistent under weaker assumptions, but random effects models are more efficient if the assumptions hold
• The choice between fixed and random effects models depends on the nature of the unobserved confounders and the research question

Estimating fixed effects models

Within transformation

• The within transformation subtracts the individual or group means from each variable to remove the fixed effects
• The transformed variables are then used in a standard OLS regression to estimate the fixed effects model
• This approach is equivalent to including a dummy variable for each individual or group in the regression

First differences

• First differencing subtracts the lagged values from each variable to remove the fixed effects
• The differenced variables are then used in a standard OLS regression to estimate the fixed effects model
• This approach is equivalent to the within transformation if there are only two time periods per individual or group

Least squares dummy variable approach

• The least squares dummy variable (LSDV) approach includes a dummy variable for each individual or group in the regression
• This directly estimates the fixed effects as the coefficients on the dummy variables
• The LSDV approach is computationally equivalent to the within transformation, but may be less efficient with many individuals or groups

Interpreting fixed effects estimates

• The fixed effects estimate represents the average change in the outcome variable for a one-unit change in the explanatory variable, holding the fixed effects constant
• The interpretation is within-group: it compares the outcome for the same individual or group at different times with different values of the explanatory variable
• The fixed effects estimate does not generalize to the population level or to other individuals or groups not included in the sample

Extensions of fixed effects models

Fixed effects vector decomposition

• Fixed effects vector decomposition (FEVD) is a method to estimate the effects of time-invariant variables in a fixed effects model
• FEVD decomposes the fixed effects into a part explained by the time-invariant variables and an unexplained part
• The unexplained part is then used as an additional regressor in a standard fixed effects model to estimate the effects of both time-varying and time-invariant variables

Interacted fixed effects

• Interacted fixed effects models include interactions between the individual or group fixed effects and time fixed effects
• This allows the unobserved confounders to have different effects at different times, relaxing the assumption of time-invariant effects
• Interacted fixed effects models can control for unobserved confounders that vary over time in a limited way, such as individual-specific time trends

Applications of fixed effects models

Panel data examples

• Fixed effects models are commonly used with panel data on individuals, households, firms, or countries over time
• Examples include estimating the effect of education on earnings, the impact of minimum wage laws on employment, or the relationship between democracy and economic growth
• Panel data allows researchers to control for unobserved individual-specific confounders and estimate causal effects

Difference-in-differences designs

• Difference-in-differences (DiD) designs are a special case of fixed effects models used to estimate the effect of a policy or treatment
• DiD compares the change in outcomes before and after the policy for a treatment group to the change in outcomes for a control group
• The DiD estimate is equivalent to a fixed effects estimate with a binary treatment variable interacted with time fixed effects

Specification tests for fixed effects

Hausman test

• The Hausman test compares the fixed effects and random effects estimates to test for correlation between the unobserved confounders and the explanatory variables
• If the test rejects the null hypothesis of no correlation, fixed effects models are preferred over random effects models
• The Hausman test helps to choose between fixed and random effects models based on the assumptions about the unobserved confounders

Testing for serial correlation

• Serial correlation in the idiosyncratic errors can lead to biased standard errors and invalid inference in fixed effects models
• Tests for serial correlation, such as the Breusch-Godfrey test or the Arellano-Bond test, can be used to detect serial correlation in fixed effects models
• If serial correlation is present, robust standard errors or alternative estimators (such as first differences or generalized least squares) may be needed

Best practices for fixed effects models

Choosing between fixed and random effects

• The choice between fixed and random effects models should be based on the assumptions about the unobserved confounders and the research question
• If the unobserved confounders are likely to be correlated with the explanatory variables, fixed effects models are preferred
• If the unobserved confounders are likely to be uncorrelated with the explanatory variables and the goal is to estimate the effects of time-invariant variables, random effects models may be more appropriate

Reporting fixed effects results

• When reporting fixed effects results, it is important to clearly state the assumptions and limitations of the model
• The fixed effects estimates should be interpreted as within-group effects, not population-level effects
• The number of individuals or groups and the number of time periods should be reported to assess the precision and reliability of the estimates
• Specification tests and robustness checks should be reported to support the validity of the fixed effects model