Overview
The Random Effects (RE) model is the last method for panel data analysis discussed in this series of topics. Unlike the Fixed Effects (FE) model, which focuses on withingroup variations, the RE model treats the unobserved entityspecific effects as random and uncorrelated with the explanatory variables. After delving into the RE model first, we address probably the most critical choice to make when working with panel data: deciding between an FE or RE model.
This is an overview of the content:
 The RE model
 Error term structure
 Estimation in R
 Twoway RE model
 Choice between a FE and RE model
The RE model
Let’s continue with the model where we estimate the relationship of market and stock value on the gross investment of firms, using Grunfeld
data. This is the regression equation:
\(invest_{it} = \beta_0 + \beta_1 value_{it} + \beta_2 capital_{it} + \alpha_i + \epsilon_{it}\)
where,
 $invest_{it}$ is the gross investment of firm
i
in yeart
 $value_{it}$ is the market value of assets of firm
i
in yeart
 $capital_{it}$ is the stock value of plant and equipment of firm
i
in yeart
 $\alpha_i$ is the fixed effect for firm
i
 $\epsilon_{it}$ is the error term, which includes all other unobserved factors that affect investment but are not accounted for by the independent variables or the fixed effects.
The fixed effects $\alpha_i$ represent the timeinvariant unobserved heterogeneity that differs across firms. The RE model assumes that $\alpha_i$ is uncorrelated with the explanatory variables, allowing for the inclusion of timeinvariant variables such as a person’s gender or education level,
Error term in the RE model
The error term (capturing everything unobserved in the model) consists of two components:
 The individualspecific error component: $\alpha_i$.
This captures the unobserved heterogeneity varying across individuals but constant over time. It is assumed to be uncorrelated with the explanatory variables.
 The timevarying error component: $\epsilon_{it}$.
This component accounts for the withinfirm variation in gross investment over time. It captures the fluctuations and changes that occur within each firm over different periods. While these timevarying effects can be correlated within the same firm, they are assumed to be uncorrelated across different firms. Note that this correlation of the error term across time is allowed in a FE model as well.
Estimation in R
To estimate the RE model in R, use the plm()
function and specify the model type as "random"
.
# Load packages & data
library(plm)
library(AER)
data(Grunfeld)
# Model estimation
model_random < plm(invest ~ value + capital,
data = Grunfeld,
index = c("firm", "year"),
model = "random")
summary(model_random)
The estimated coefficients capture the average effect of the independent variable (X) on the dependent variable (Y) while accounting for both withinentity and betweenentity effects. This means that the coefficients represent the average effect of X on Y when X changes within each entity (e.g., firm) over time and when X varies between different entities.
Twoway Random Effects model
Extend the RE model to a twoway
model by including timespecific effects. These timespecific effects are also unobservable and assumed to be uncorrelated with the independent variables, just like the entityspecific effects.
Include effect = "twoways"
within the plm()
function in R:
# Estimate twoway RE model
model_random_twoway < plm(invest ~ value + capital,
data = Grunfeld,
index = c("firm", "year"),
model = "random",
effect = "twoway")
Choice between Fixed or Random Effects
When deciding between an FE and RE model for panel data analysis, considering the structure of your data is the most important thing to do.
FE model preference
If there is a correlation between unobserved effects and the independent variables, the FE model is preferred as it controls for timeinvariant heterogeneity. This is particularly valuable when dealing with observational data where inherent differences among entities (e.g., universities or companies) could affect the outcome.
RE model preference
Opt for the RE model if you are not worried about invariant unobserved effects in the error term. This is often the case in experimental settings where you have control over treatment assignments.
One important reason to use the RE model over the FE model is when you have a specific interest in the coefficients of timeinvariant variables. Unlike the FE model, the RE model allows for leaving these timeinvariant variables in and estimating their impact on the outcome variable while still accounting for unobserved entityspecific differences.
Practical example: University performance analysis
Imagine you want to understand if the availability of research grants at universities affects student performance. You have panel data from universities per year, with variables describing their students' performance and research grants. You believe that universities have unobserved universityspecific effects, such as its reputation, that could influence both the research grant and student performance but are not included as variables in the data.
Use the FE model
 …if you believe unobserved effects are correlated with grants availability and student performance. You want to control for universityspecific factors and isolate grants impact within each university over time.
 For example, you suspect that universities with stronger reputations are more likely to secure research grants.
Use the RE model

…if you believe that unobserved universityspecific effects are random and not directly tied to grant availability.

RE allows you to add multilevel fixed effects. For example, universitylevel FE (capturing differences across universities) and studentlevel FE (capturing differences across students within universities) can be added simultaneously.

For example, grant decisions are made by an external committee and largely independent of universityspecific characteristics. Moreover, you want to control for both universitylevel and studentlevel variations.
FE Model  RE Model  

When to Use 
Unobserved university specific effects are correlated with research grants and student performance. 
Unobserved effects are random and not directly tied to grant availability. Allows multilevel fixed effects (on university and studentlevel) in the same model. 
Example  You believe universities with stronger reputations are more likely to secure research grants. 
You believe grant decisions are made by external committees and are independent of university specific characteristics. 
Hausman test
To determine the appropriate model, a Hausman test can be conducted to test the endogeneity of the entityspecific effects.
 The null hypothesis states no correlation between the independent variables and the entityspecific effects $\alpha_i$. If $H_{0}$ is true, the RE model is preferred.
 The alternative hypothesis states a correlation between the independent variables and the entityspecific effects($\alpha_i$). If $H_{0}$ is rejected, the FE model is preferred.
The Hausman test can be performed in R with the phtest()
function from the package plm
. Specify the FE and RE model as arguments in this function. Note that the models included as arguments should be estimated with plm
. Therefore, the Within model is also estimated with plm()
first (instead of with feols()
from the fixest
package like in the Fixed Effects model article).
# Estimate Twoway FE (Within) model
model_within_twoway < plm(invest ~ value + capital,
data = Grunfeld,
index = c("firm", "year"),
model = "within",
effect = "twoway")
# Perform Hausman test
phtest(model_within_twoway,
model_random_twoway)
The pvalue is 0.0013, which is lower than 0.05. Thus the $H_{0}$ is rejected and an FE model is preferred according to the Hausman test.
Recommended reading
A recommended paper for delving deeper into Random Effects Models is Wooldridge (2019), which introduces strategies for allowing unobserved heterogeneity to be correlated with observed covariates in unbalanced panels.
The Random Effects (RE) model is a method for panel data analysis that treats unobserved entityspecific effects as random and uncorrelated with the explanatory variables.
One distinct advantage of the RE model is its flexibility in allowing the inclusion of timeinvariant variables, a feature not available in the FE model. Additionally, the RE model allows you to include multilevel fixed effects.
The key difference between the RE and FE model is:
 In a FE model, the unobserved effects are assumed to be correlated with the independent variables
 In a RE model, the unobserved effects are assumed to be uncorrelated with the independent variables