[R] Regressions with fixed-effect in R

[Daniel Malter]

if the plm function only puts out one r-squared, it should be the within
r-squared, but I could be wrong. Stata, for example, gives you a within, a
between, and an overall r-squared. Here is what they do.

set.seed(1)
x=rnorm(100)
fe=rep(rnorm(10),each=10)
id=rep(1:10,each=10)
ti=rep(1:10,10)
e=rnorm(100)
y=x+fe+e

data=data.frame(y,x,id,ti)

library(plm)
reg=plm(y~x,model="within",index=c("id","ti"),data=data)
summary(reg)

cat("R-squared: ", 1-83.908/178.5)

#Let's compute the squared residuals of this regression
SSR=sum(residuals(reg)^2)

#let's compute the total squares of the ys
SS0=sum((y-mean(y))^2)
SS0 #Note that this is not the TSS given by plm

#Now, let's demean y and x for each individual separately
y.dem=y-tapply(y,id,mean)[id]
x.dem=x-tapply(x,id,mean)[id]

#and regress them
#note that we do not estimate the intercept because we have demeaned the
data
reg.fe=lm(y.dem~-1+x.dem)
summary(reg.fe)
#The coefficient is correct, i.e., the same as in plm
#Note that the standard error is wrong, however. We would need to account
for
#that we are losing degrees of freedom by taking out the fixed effects.


#now let's look at the sum of squares after demeaning y
SSR.y.dem=sum((y.dem-mean(y.dem))^2)
SSR.y.dem #Note, this is the Total sum of squares given by plm

#Now, we know that the total sum of squares
#not accounting for fixed effects is

# TSS=SS0=331.7986

#However, we know that after taking out the fixed effects (demeaning y)
# the total sum of squares is
# SSR.y.dem=178.5050

#The within R-squared is then the variance explained by x AFTER having taken
out the fixed effects
# So the R-squared computable from the plm output is in fact the within
R-squared
cat("Within R-squared: ", 1-SSR/SSR.y.dem)

#which is identical to the r-squared in our hand-computed FE regression
summary(reg.fe)$r.squared


#The two other R-squareds Stata would give you are:

#The overall r-squared
#which is the r-squared of a pooled OLS of y on x WITHOUT accounting for the
fixed effects

#Pooled OLS
reg1=lm(y~x)
summary(reg1)

#This is what Stata shows as overall R-squared
summary(reg1)$r.squared

#The second R-squared Stata shows is the between R-squared
# which is the R-squared of regressing the mean of the individual y(i)
# on the mean(s) of the individual X(i)

#Get the means of y and x for each individual
y.means=tapply(y,id,mean)[id]
x.means=tapply(x,id,mean)[id]

#Regress them on each other
reg2=lm(y.means~x.means)

#This is what Stata shows as between R-squared
summary(reg2)$r.squared


So you see that the R-squared computable from the plm output is indeed the
within R-squared.

For comparison, look at the Stata output:

Fixed-effects (within) regression               Number of obs      =      
100
Group variable: id                              Number of groups   =       
10

R-sq:  within  = 0.5299                         Obs per group: min =       
10
       between = 0.1744                                        avg =     
10.0
       overall = 0.3367                                        max =       
10

                                                F(1,89)            =   
100.34
corr(u_i, Xb)  = 0.0547                         Prob > F           =   
0.0000

------------------------------------------------------------------------------
           y |      Coef.   Std. Err.      t    P>|t|     [95% Conf.
Interval]
-------------+----------------------------------------------------------------
           x |   1.111187   .1109319    10.02   0.000     .8907682   
1.331607
       _cons |   .3155321   .0978457     3.22   0.002     .1211147   
.5099495
-------------+----------------------------------------------------------------
     sigma_u |  1.2318621
     sigma_e |  .97097297
         rho |  .61679513   (fraction of variance due to u_i)
------------------------------------------------------------------------------
F test that all u_i=0:     F(9, 89) =    16.05               Prob > F =
0.0000

HTH,
Daniel



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