# Panel Data Models: Fixed and Random Effects Linear Models

The linear model with unobserved individual (and unobserved time) effects is

y(i,t) = a(i) + c(t) + b’x(i,t) + e(i,t)

There is one dummy variable coefficient, a(i), for each individual or group. We also allow for ‘two way’ models by allowing for the individual period effect with c(t). The number of groups or individuals allowed in the sample is unlimited. The number of periods is typically small, and is usually not a relevant consideration as regards the capacity of the estimator. In all cases, there is no requirement that the panel be ‘balanced’ (that is, the number of observations may vary by group). LIMDEP allows a large number of different specifications for the linear model of this form.

- Fixed effects (dummy variables) or random effects regression model
- Both one and two way effects models
- Test statistics for the presence of effects (LM test) and fixed vs. random effects (Hausman)
- Nested random effects, nested up to three levels, fit by maximum likelihood
- Two step feasible GLS or maximum likelihood estimation of the random effects model
- Restricted least squares estimates
- Robust forms of the fixed effects estimator covariance matrix (both heteroscedasticity (White) and autocorrelation (Newey-West)
- ‘Panel Corrected Standard Errors’ for the case with large T and large N
- Weighted least squares for groupwise heteroscedasticity
- AR(1) model for autocorrelation in fixed or random effects models
- Hausman and Taylor’s estimator for endogenous right hand side variables
- Instrumental variables estimators for simultaneous equations using first differences or group mean deviations
- Arellano/Bond/Bover’s estimator for dynamic linear models

The linear model with heterogeneous coefficients is

y(i,t) = b(i)’x(i,t) + e(i,t)

This is one form of the random parameters model. LIMDEP supports the Hildreth/Houck/Swamy form of this model using feasible GLS estimators. There is also a full random coefficients treatment. Another form of the panel data version of the linear model allows heterogeneity in both the variance of e(it) across groups as well as variation in the parameters. The linear model is also supported in the classes of full random parameters models and latent class models.

Hausman and Taylor’s estimator for the random effects model overcomes the possible correlation between the independent variables and the random effects. The random effects model is formulated with the possibility that there may be time invariant independent variables.

y(i,t) = b1’x1(i,t) + b2’x2(i,t) + c1’f1(i) + c3’f2(i) + e(i,t) + u(i)

There are four sets of variables in the model, the **x**s which are time varying and the **f**s which are not time varying, and variables subscripted ‘1’ which are uncorrelated with u(i) and the remainder which may be correlated with u(i). A three step procedure which ends with generalized instrumental variables estimation is used for estimation.