Random effects model
random effects model, random effects model vs fixed effects modelIn statistics, a random effects model, also called a variance components model, is a statistical model where the model parameters are random variables It is a kind of hierarchical linear model, which assumes that the data being analysed are drawn from a hierarchy of different populations whose differences relate to that hierarchy In econometrics, random effects models are used in the analysis of hierarchical or panel data when one assumes no fixed effects it allows for individual effects The random effects model is a special case of the fixed effects model
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
Contents
 1 Qualitative description
 2 Simple example
 3 Variance components
 4 Unbiasedness
 5 See also
 6 Notes
 7 Further reading
 8 External links
Qualitative description
Random effect models assist in controlling for unobserved heterogeneity when the heterogeneity is constant over time and correlated with independent variables 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[citation needed]
Two common assumptions are made about the individual specific effect: the random effects assumption and the fixed effects assumption The random effects assumption 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[citation needed]
Simple example
Suppose m large elementary schools are chosen randomly from among thousands in a large country Suppose also that n pupils of the same age are chosen randomly at each selected school Their scores on a standard aptitude test are ascertained Let Yij be the score of the jth pupil at the ith school A simple way to model the relationships of these quantities is
Y i j = μ + U i + W i j , =\mu +U_+W_,\,}where μ is the average test score for the entire population In this model Ui is the schoolspecific random effect: it measures the difference between the average score at school i and the average score in the entire country The term Wij is the individualspecific random effect, ie, it's the deviation of the jth pupil’s score from the average for the ith school
The model can be augmented by including additional explanatory variables, which would capture differences in scores among different groups For example:
Y i j = μ + β 1 S e x i j + β 2 R a c e i j + β 3 P a r e n t s E d u c i j + U i + W i j , =\mu +\beta _\mathrm _+\beta _\mathrm _+\beta _\mathrm _+U_+W_,\,}where Sexij is the dummy variable for boys/girls, Raceij is the dummy variable for white/black pupils, and ParentsEducij records, say, the average education level of a child’s parents This is a mixed model, not a purely random effects model, as it introduces fixedeffects terms for Sex, Race, and Parents' Education
Variance components
The variance of Yij is the sum of the variances τ2 and σ2 of Ui and Wij respectively
Let
Y ¯ i ∙ = 1 n ∑ j = 1 n Y i j }_=}\sum _^Y_}be the average, not of all scores at the ith school, but of those at the ith school that are included in the random sample Let
Y ¯ ∙ ∙ = 1 m n ∑ i = 1 m ∑ j = 1 n Y i j }_=}\sum _^\sum _^Y_}be the grand average
Let
S S W = ∑ i = 1 m ∑ j = 1 n Y i j − Y ¯ i ∙ 2 ^\sum _^Y_}_^\,} S S B = n ∑ i = 1 m Y ¯ i ∙ − Y ¯ ∙ ∙ 2 ^}_}_^\,}be respectively the sum of squares due to differences within groups and the sum of squares due to difference between groups Then it can be shown[citation needed] that
1 m n − 1 E S S W = σ 2 }ESSW=\sigma ^}and
1 m − 1 n E S S B = σ 2 n + τ 2 }ESSB=}}+\tau ^}These "expected mean squares" can be used as the basis for estimation of the "variance components" σ2 and τ2
Unbiasedness
In general, random effects are efficient, and should be used over fixed effects if the assumptions underlying them are believed to be satisfied For random effects to work in the school example it is necessary that the schoolspecific effects be uncorrelated to the other covariates of the model This can be tested by running fixed effects, then random effects, and doing a Hausman specification test If the test rejects, then random effects is biased and fixed effects is the correct estimation procedure
See also
 Bühlmann model
 Hierarchical linear modeling
 Fixed effects
 MINQUE
 Covariance estimation
 Conditional variance
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:101002/sim3478
Further reading
 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
 Hsiao, Cheng 2003 Analysis of Panel Data Second ed New York: Cambridge University Press pp 73–92 ISBN 0521522714
 Wooldridge, Jeffrey M 2013 "Random Effects Estimation" Introductory Econometrics: A Modern Approach Fifth international ed Mason, OH: SouthWestern pp 474–478 ISBN 9781111534394
External links
 Fixed and random effects models
 How to Conduct a MetaAnalysis: Fixed and Random Effect Models
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