# [R-sig-ME] Efficiency of random and fixed effects estimators

Daniel Malter daniel at umd.edu
Tue Aug 23 06:06:21 CEST 2011

```Hi all,

I am statistically confused tonight. When the assumptions to a random effects estimator are warranted, the random effects estimator should be the more efficient estimator than the fixed effects estimator because it uses fewer degrees of freedom (estimating just the variance parameter of the normal rather than using one df for each included fixed effect). Consequently, the standard errors of the coefficients should be smaller in random than in fixed effects estimation, I thought, provided that RE assumptions are warranted. However, I do not find this to be the case in this simulated example.

For the sake of the example, assume you measure subjects' happiness before exposing them to a happy or sad movie, and then you measure their happiness again after watching the movie. Here, "id" marks the subject, "obs" marks the pre- and post-treatment observations of a subject, "d" is the treatment indicator (whether the subject watched the happy or sad movie), "base.happy" is the ~N(0,1)-distributed individual effect a(i), happy is the measured happiness for each subject pre- and post-treatment, respectively, and the error term u(i,t) is also distributed ~N(0,1).

id<-rep(c(1:100),each=2)
obs<-rep(c(0:1),100)
d<-rep(sample(c(-1,1),100,replace=T),each=2)
base.happy<-rep(rnorm(100),each=2)
happy<-base.happy+1.5*d*obs+rnorm(200)

data<-data.frame(id,obs,d,happy)

# Now run the random and fixed effects models

require(lme4)
require(plm)

reg.re1<-lmer(happy~factor(obs)*factor(d)+(1|id))
summary(reg.re1)

reg.re2<-plm(happy~factor(obs)*factor(d),index=c('id','obs'),model="random",data=data)
summary(reg.re2)

reg.fe1<-lm(happy~factor(id)+factor(obs)*factor(d))
summary(reg.fe1)

reg.fe2<-plm(happy~factor(obs)*factor(d),index=c('id','obs'),model="within",data=data)
summary(reg.fe2)

I am confused why FE and RE models are virtually equally efficient in this case. Can somebody lift my confusion?

Thanks much,
Daniel

```