Fixed effects model
Part of a series on Statistics 
Regression analysis 

Models 

Estimation 
Background 
In econometrics and statistics, a fixed effects model is a statistical model that represents the observed quantities in terms of explanatory variables that are treated as if the quantities were nonrandom. This is in contrast to random effects models and mixed models in which either all or some of the explanatory variables are treated as if they arise from random causes. Contrast this to the biostatistics definitions,^{[1]}^{[2]}^{[3]}^{[4]} as biostatisticians use "fixed" and "random" effects to respectively refer to the populationaverage and subjectspecific effects (and where the latter are generally assumed to be unknown, latent variables). Often the same structure of model, which is usually a linear regression model, can be treated as any of the three types depending on the analyst's viewpoint, although there may be a natural choice in any given situation.
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. If we assume fixed effects, we impose time independent effects for each entity that are possibly correlated with the regressors.
Contents
 1 Qualitative description
 2 Formal description
 3 Equality of Fixed Effects (FE) and First Differences (FD) estimators when T=2
 4 Hausman–Taylor method
 5 Testing fixed effects (FE) vs. random effects (RE)
 6 Steps in Fixed Effects Model for sample data
 7 See also
 8 Notes
 9 References
 10 External links
Qualitative description
Such models assist in controlling for unobserved heterogeneity when this heterogeneity is constant over time. This constant can be removed from the data through differencing, for example by taking a first difference which will remove any time invariant components of the model.
There are two common assumptions made about the individual specific effect, the random effects assumption and the fixed effects assumption. The random effects assumption (made in a random effects model) is that the individual specific effects are uncorrelated with the independent variables. The fixed effect assumption is that the individual specific effect is correlated with the independent variables. If the random effects assumption holds, the random effects model is more efficient than the fixed effects model. However, if this assumption does not hold, the random effects model is not consistent. The DurbinWuHausman test is often used to discriminate between the fixed and the random effects model.
Formal description
Consider the linear unobserved effects model for observations and time periods:
 for and
where is the dependent variable observed for individual at time is the timevariant regressor matrix, is the unobserved timeinvariant individual effect and is the error term. Unlike , cannot be observed by the econometrician. Common examples for timeinvariant effects are innate ability for individuals or historical and institutional factors for countries.
Unlike the Random effects (RE) model where the unobserved is independent of for all , the FE model allows to be correlated with the regressor matrix . Strict exogeneity, however, is still required.
Since is not observable, it cannot be directly controlled for. The FE model eliminates by demeaning the variables using the within transformation:
where and . Since is constant, and hence the effect is eliminated. The FE estimator is then obtained by an OLS regression of on .
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.^{[5]} 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.^{[6]} 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.^{[7]} This approach is the most computationally and memory efficient, but it requires proficient programming skills and access to the model programming code; although, it can be programmed even in SAS.^{[8]}^{[9]} Finally, each of the above alternatives can be improved if the seriesspecific 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.^{[10]}
Equality of Fixed Effects (FE) and First Differences (FD) estimators when T=2
For the special two period case (), the FE estimator and the FD estimator are numerically equivalent. This is because the FE estimator effectively "doubles the data set" used in the FD estimator. To see this, establish that the fixed effects estimator is:
Since each can be rewritten as , we'll rewrite the line as:
Hausman–Taylor method
Need to have more than one timevariant regressor () and timeinvariant regressor () and at least one and one that are uncorrelated with .
Partition the and variables such that where and are uncorrelated with . Need .
Estimating via OLS on using and as instruments yields a consistent estimate.
Testing fixed effects (FE) vs. random effects (RE)
We can test whether a model is appropriate using a Hausman test.
 :
 :
If is true, both and are consistent, but only is efficient. If is true, is consistent and is not.
 where
The Hausman test is a specification test so a large test statistic might be indication that there might be Errors in Variables (EIV) or our model is misspecified. If the FE assumption is true, we should find that .
A simple heuristic is that if there could be EIV.
Steps in Fixed Effects Model for sample data
 Calculate group and grand means
 Calculate k=number of groups, n=number of observations per group, N=total number of observations (k x n)
 Calculate SStotal (or total variance) as: (Each score  Grand mean)^2 then summed
 Calculate SStreat (or treatment effect) as: (Each group mean Grand mean)^2 then summed x n
 Calculate SSerror (or error effect) as (Each score  Its group mean)^2 then summed
 Calculate dftotal: N1, dftreat: k1 and dferror k(n1)
 Calculate Mean Square MStreat: SStreat/dftreat, then MSerror: SSerror/dferror
 Calculate obtained f value: MStreat/MSerror
 Use Ftable or probability function, to look up critical f value with a certain significance level
 Conclude as to whether treatment effect significantly affects the variable of interest
See also
Notes
 ↑ Diggle, Peter J.; Heagerty, Patrick; Liang, KungYee; Zeger, Scott L. (2002). Analysis of Longitudinal Data (2nd ed.). Oxford University Press. pp. 169–171. ISBN 0198524846.
 ↑ Fitzmaurice, Garrett M.; Laird, Nan M.; Ware, James H. (2004). Applied Longitudinal Analysis. Hoboken: John Wiley & Sons. pp. 326–328. ISBN 0471214876.
 ↑ Laird, Nan M.; Ware, James H. (1982). "RandomEffects Models for Longitudinal Data". Biometrics. 38 (4): 963–974. JSTOR 2529876.
 ↑ Gardiner, Joseph C.; Luo, Zhehui; Roman, Lee Anne (2009). "Fixed effects, random effects and GEE: What are the differences?". Statistics in Medicine. 28: 221–239. doi:10.1002/sim.3478.
 ↑ Garcia, Oscar. (1983). "A stochastic differential equation model for the height growth of forest stands". Biometrics: 1059–1072.
 ↑ Tait, David; Cieszewski, Chris J.; Bella, Imre E. (1986). "The stand dynamics of lodgepole pine". Can. J. For. Res. 18: 1255–1260.
 ↑ Strub, Mike; Cieszewski, Chris J. (2006). "Base–age invariance properties of two techniques for estimating the parameters of site index models". Forest Science. 52 (2): 182–186.
 ↑ Strub, Mike; Cieszewski, Chris J. (2003). "Fitting global site index parameters when plot or tree site index is treated as a local nuisance parameter In: Burkhart HA, editor. Proceedings of the Symposium on Statistics and Information Technology in Forestry; 2002 September 8–12; Blacksburg, Virginia: Virginia Polytechnic Institute and State University": 97–107.
 ↑ Cieszewski, Chris J.; Harrison, Mike; Martin, Stacey W. (2000). "Practical methods for estimating nonbiased parameters in selfreferencing growth and yield models" (PDF). PMRC Technical Report. 2000 (7): 12.
 ↑ Schnute, Jon; McKinnell, Skip (1984). "A biologically meaningful approach to response surface analysis". Can. J. Fish. Aquat. 41: 936–953.
References
 Christensen, Ronald (2002). Plane Answers to Complex Questions: The Theory of Linear Models (Third ed.). New York: Springer. ISBN 0387953612.
 Gujarati, Damodar N.; Porter, Dawn C. (2009). "Panel Data Regression Models". Basic Econometrics (Fifth international ed.). Boston: McGrawHill. pp. 591–616. ISBN 9780071276252.
 Wooldridge, Jeffrey M. (2013). "Fixed Effects Estimation". Introductory Econometrics: A Modern Approach (Fifth international ed.). Mason, OH: SouthWestern. pp. 466–474. ISBN 9781111534394.