Differentiate between Fixed Effect Model and Random Effect Model.

Fixed Effect Model vs. Random Effect Model

Fixed Effect Model:

1. Nature: In a fixed effect model, the effects of individual entities (e.g., individuals, firms, countries) are assumed to be fixed and specific to each entity. These effects are included as additional parameters in the model.

2. Assumptions:

   • The effects are assumed to be constant across time for each entity.
   • The fixed effects are correlated with the independent variables but are not allowed to be correlated with the error term.

3. Estimation:

   • Fixed effects are estimated separately for each entity, and the model is estimated using only within-entity variation.
   • Fixed effects are usually estimated using dummy variables for each entity, which capture the unobserved heterogeneity.

4. Interpretation:

   • The coefficients of the independent variables in a fixed effect model represent the average effect of those variables across entities, controlling for the entity-specific effects.

Random Effect Model:

1. Nature: In a random effect model, the effects of individual entities are assumed to be random draws from a population of possible effects. These effects are treated as random variables in the model.

2. Assumptions:

   • The random effects are assumed to be uncorrelated with the independent variables and the error term.
   • The random effects are assumed to be independent and identically distributed (i.i.d) across entities.

3. Estimation:

   • Random effects are estimated using techniques such as the method of moments or maximum likelihood estimation.
   • The model is estimated using both within-entity and between-entity variation.

4. Interpretation:

   • The coefficients of the independent variables in a random effect model represent the average effect of those variables across entities, including both the systematic and random components of the effects.

Key Differences:

1. Nature of Effects: Fixed effects are assumed to be specific to each entity and constant over time, while random effects are assumed to be random draws from a population of possible effects.

2. Correlation with Independent Variables: Fixed effects are allowed to be correlated with the independent variables, while random effects are assumed to be uncorrelated with both the independent variables and the error term.

3. Estimation Method: Fixed effects are estimated using within-entity variation only, while random effects are estimated using both within-entity and between-entity variation.

4. Interpretation: Fixed effects capture entity-specific effects that are constant over time, while random effects capture variation in the effects that is not explained by the independent variables.

In conclusion, the choice between a fixed effect model and a random effect model depends on the nature of the data and the research question. Fixed effect models are appropriate when there are entity-specific effects that need to be controlled for, while random effect models are more appropriate when these effects are considered to be random and uncorrelated with the independent variables.