Fixed Effect Model and Random Effect Model in Panel Data Analysis

What is Fixed Effect Model and Random Effect Model?

Fixed Effect Model (FEM) and Random Effect Model (REM) are two common approaches to analyzing panel data, which involves data collected over time for the same entities (e.g., individuals, firms, countries).

Fixed Effect Model (FEM)

The Fixed Effect Model assumes that individual-specific effects (the unobserved variables that vary across entities but are constant over time) are correlated with the independent variables. This model accounts for these individual-specific effects by including a separate intercept for each entity.

When to Use Fixed Effect Model

• When you believe that individual-specific effects are correlated with the independent variables.
• When you want to control for unobserved heterogeneity (variables that are not included in the model but vary across entities).

Equation of Random Effect: 

$Y_{it} = \alpha_i + \beta X_{it} + \epsilon_{it}$

Random Effect Model (REM)

The Random Effect Model assumes that the individual-specific effects are uncorrelated with the independent variables. It treats these effects as random and part of the error term.

When to Use Random Effect Model:

• When you believe that individual-specific effects are uncorrelated with the independent variables.
• When you have a large number of entities and the time dimension is short, making it reasonable to treat individual effects as random.

Equation of Random Effect:

$Y_{it} = \alpha + \beta X_{it} + u_i + \epsilon_{it}$

Here, 

ui​ is the random effect for entity i, , assumed to be uncorrelated with $X_{it}$

​

Choosing Between FEM and REM

Hausman Test:

A statistical test used to decide between FEM and REM. The null hypothesis is that the preferred model is REM. If the test is significant, it suggests that REM is not appropriate because individual-specific effects are correlated with the independent variables, thus FEM should be used.

In practice:

• FEM is preferred when individual-specific effects are likely correlated with the independent variables.
• REM is preferred for efficiency if individual-specific effects are uncorrelated with the independent variables, as it generally provides more efficient (lower variance) estimates.

The Hausman test will provide a test statistic and a p-value.

If the p-value is small (typically less than 0.05), reject the null hypothesis, which suggests that the Random Effect Model is not appropriate and that the Fixed Effect Model should be used.

If the p-value is large, you do not reject the null hypothesis, and the Random Effect Model is preferred.

Stata Code

* Load data use: your_data.dta, clear 

 * Set the panel data structure: xtset entity time 

 * Estimate Fixed Effect Model: xtreg Y X1 X2, fe 

 * Estimate Random Effect Model: xtreg Y X1 X2, re 

 * Perform Hausman Test: hausman fe re