The Random Effects Model is also sometimes called the Error Components Model or the Two-error structure approach. The Fixed Effects Model has some problems such as the use of dummy variables in the LSDV approach or the inability to include time-invariant independent variables. Hence, the Random Effects Model provides a different approach to account for cross-sectional and time-specific effects in panel data. The random effects model includes the cross-sectional and time-specific effects as random error components in the model equation.
The difference between the Fixed Effects Model and the Random Effects arises from one crucial assumption. Under Random Effects, the cross-sectional and time-specific effects are assumed to be independent of the explanatory variables. Hence, the random error components are not correlated with the independent variables. On the other hand, the cross-sectional and time-specific effects are correlated with the independent variables in the Fixed Effects ‘Within’ Model.
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Specification of the Random Effects Model
Random Effects Model with Cross-sectional and Time-specific Effects
We can express the Random Effects model with both cross-sectional and time-specific effects as:
In the equation, the intercept “α” can be considered the mean intercept of all the cross-sections and time periods. The cross-sectional effects are given by “μi” where each individual “i” has a different value that is added to the mean intercept “α“. Similarly, time-specific effects are represented by “vt” and each time period “t” has a different value that is added to the mean intercept “α“.
Both the cross-sectional and time-specific effects given by “μi” and “vt” are random error components in the above equations. That is, these effects capture the deviations of the individuals (cross-sectional units) and time periods from the mean intercept value of “α“.
This equation, however, cannot be estimated straightforwardly. It is just a simplified expression to illustrate the model. The Random Effects Model has to be estimated using Generalized Least Squares (GLS) or Maximum Likelihood Approach. Here, we will discuss the basics of the GLS estimation of the Random Effects Model.
the Random Effects Model: Estimation
Let us consider a basic model with only cross-sectional effects:
In this model, we have included only the cross-sectional or individual-specific effects using “μi“. This “μi” is assumed to be independently and identically distributed with mean “mμ” and variance “σμ2“. The error “εit” is the usual residual or idiosyncratic error term that is normally distributed with mean 0 and variance “σε2“. The variances “σμ2” and “σε2” of both the errors are crucial for the estimation of the Random Effects Model. These variances are used to apply the GLS transformation in the model.
GLS Transformation in the Random Effects Model
In Generalized Least Squares (GLS), we transform the variables in the model using some weights. For example, the Weighted Least Squares (WLS) Model is a special case of GLS where we transform using weights based on the nature of heteroscedasticity.
In the case of the Random Effects Model, the weights for the GLS transformation of variables are defined as:
Any variable xit in the Random Effects Model is transformed using “θ” as shown above. In turn, this “θ” is a function of the variances of the idiosyncratic error and the random error component. However, these weights must be estimated, they are not known to us beforehand. That is, we must estimate the “σμ2” and “σε2” first, only then we can estimate the “θ” and transform the variables.
These variances (“σμ2” and “σε2“) and “θ” are estimated using the Fixed Effects ‘Within’ Model and the ‘Between’ Effects Model. Hence, the estimation of the Random Effects Model is a combination of the ‘Within’ and ‘Between’ Effects models. The calculations of the variances and the model can get complicated, so we will not discuss those here.
Considering everything, the estimation of the Random Effects Model using GLS is equivalent to estimating the following equation:
Software programs help us easily estimate this equation and its coefficients. They automatically take care of estimating the variances “σμ2” and “σε2“, and “θ” first and then estimate the Random Effects Model. The reported results often contain the values of “θ“, “σμ2” and “σε2” along with the coefficients.
Fixed Effects vs Random Effects Model
Critical Assumption: Fixed and Random Effects
The Random Effects (cross-sectional and time-specific) are assumed to be independent or uncorrelated to the independent variables in the model. If this assumption is violated and the effects are correlated to the independent variables, the Random Effects estimates are biased and inconsistent.
As a result, we should not apply the Random Effects model if the cross-sectional or time effects are correlated with any of the independent variables. In such cases, we can use the Fixed Effects model where these effects are correlated to the independent variables.
Including the Time-invariant variables
As discussed in the Fixed Effects Model, we cannot include time-invariant independent variables in the Fixed Effects model because the fixed effects capture all the heterogeneity of the units. However, this is not the case in the Random Effects Model.
We can include time-invariant variables in the Random Effects model. This gives it a huge advantage over the Fixed Effects model. The assumption stated above makes the random error components uncorrelated with the independent variables. Therefore, they do not interfere with the estimation of the coefficients of the time-invariant variables as they do in the Fixed Effects Model.
Choosing between Fixed and Random Effects
Finally, we need to know whether we should estimate the Fixed Effects or the Random Effects Model. We must decide which model is more appropriate based on the dataset and research objectives. This decision mostly comes down to 2 considerations.
First, we must decide the importance of time-invariant variables in our research. Random Effects model can be preferred we must include certain time-invariant variables in our study. This is because the Fixed Effects Model cannot accommodate time-invariant variables. Therefore, we may have to use the Random Effects Model or Correlated Random Effects model.
Second, we must ensure whether the critical assumption of the Random Effects model is satisfied or not. If the cross-sectional or time effects are correlated with the independent variables, then we cannot use the Random Effects Model. We have to use the Fixed Effects mode in such a case. Here, we can use the Hausman Test to choose between the Fixed and Random Effects Models.
Further, we can also check whether the Random Effects are significant in the model or not. The Lagrange Multiplier Test for Random Effects can be used for this purpose.
