In many applications including econometrics[1] and biostatistics[2][3][4][5][6] a fixed effects model refers to a regression model in which the group means are fixed (non-random) as opposed to a random effects model in which the group means are a random sample from a population.
[7][6] Generally, data can be grouped according to several observed factors.
In panel data where longitudinal observations exist for the same subject, fixed effects represent the subject-specific means.
In panel data analysis the term fixed effects estimator (also known as the within estimator) is used to refer to an estimator for the coefficients in the regression model including those fixed effects (one time-invariant intercept for each subject).
Such models assist in controlling for omitted variable bias due to unobserved heterogeneity when this heterogeneity is constant over time.
This heterogeneity can be removed from the data through differencing, for example by subtracting the group-level average over time, or by taking a first difference which will remove any time invariant components of the model.
However, if this assumption does not hold, the random effects estimator is not consistent.
The Durbin–Wu–Hausman test is often used to discriminate between the fixed and the random effects models.
Unlike the random effects model where the unobserved
Strict exogeneity with respect to the idiosyncratic error term
At least three alternatives to the within transformation exist with variations.
One is to add a dummy variable for each individual
This is numerically, but not computationally, equivalent to the fixed effect model and only works if the sum of the number of series and the number of global parameters is smaller than the number of observations.
[10] The dummy variable approach is particularly demanding with respect to computer memory usage and it is not recommended for problems larger than the available RAM, and the applied program compilation, can accommodate.
Second alternative is to use consecutive reiterations approach to local and global estimations.
[11] This approach is very suitable for low memory systems on which it is much more computationally efficient than the dummy variable approach.
The third approach is a nested estimation whereby the local estimation for individual series is programmed in as a part of the model definition.
[13][14] Finally, each of the above alternatives can be improved if the series-specific estimation is linear (within a nonlinear model), in which case the direct linear solution for individual series can be programmed in as part of the nonlinear model definition.
, the first difference and fixed effects estimators are numerically equivalent.
follows a random walk, however, the first difference estimator is more efficient.
Gary Chamberlain's method, a generalization of the within estimator, replaces
value, rather than the sum of squared residuals, should be minimized.
[18] This can be directly achieved from substitution rules: then the values and standard deviations for
can be determined via classical ordinary least squares analysis and variance-covariance matrix.
Random effects estimators may be inconsistent sometimes in the long time series limit, if the random effects are misspecified (i.e. the model chosen for the random effects is incorrect).
However, the fixed effects model may still be consistent in some situations.
For example, if the time series being modeled is not stationary, random effects models assuming stationarity may not be consistent in the long-series limit.
Then, as the series becomes longer, the model revises estimates for the mean of earlier periods upwards, giving increasingly biased predictions of coefficients.
However, a model with fixed time effects does not pool information across time, and as a result earlier estimates will not be affected.
In situations like these where the fixed effects model is known to be consistent, the Durbin-Wu-Hausman test can be used to test whether the random effects model chosen is consistent.