[R] OLS standard errors

[Daniel Malter]

Hi,

the standard errors of the coefficients in two regressions that I computed
by hand and using lm() differ by about 1%. Can somebody help me to identify
the source of this difference? The coefficient estimates are the same, but
the standard errors differ. 

####Simulate data
	
	happiness=0
	income=0
	gender=(rep(c(0,1,1,0),25))
		for(i in 1:100){
  			happiness[i]=1000+i+rnorm(1,0,40)
 			income[i]=2*i+rnorm(1,0,40)
  			}

####Run lm()

	reg=lm(happiness~income+factor(gender))
	summary(reg)

####Compute coefficient estimates "by hand"

	x=cbind(income,gender)
	y=happiness

	z=as.matrix(cbind(rep(1,100),x))
	beta=solve(t(z)%*%z)%*%t(z)%*%y

####Compare estimates

	cbind(reg$coef,beta)  ##fine so far, they both look the same
	
	reg$coef[1]-beta[1]
	reg$coef[2]-beta[2]
	reg$coef[3]-beta[3]	##differences are too small to cause a 1%
difference

####Check predicted values

	estimates=c(beta[2],beta[3])

	predicted=estimates%*%t(x)
	predicted=as.vector(t(as.double(predicted+beta[1])))
	
	cbind(reg$fitted,predicted)		##inspect fitted values
	as.vector(reg$fitted-predicted)	##differences are marginal

#### Compute errors

	e=NA
	e2=NA
	for(i in 1:length(happiness)){
  		e[i]=y[i]-predicted[i]   ##for "hand-computed" regression
  		e2[i]=y[i]-reg$fitted[i] ##for lm() regression
  	}

#### Compute standard error of the coefficients

  sqrt(abs(var(e)*solve(t(z)%*%z)))	##for "hand-computed" regression
  sqrt(abs(var(e2)*solve(t(z)%*%z)))	##for lm() regression using e2 from
above

	##Both are the same

####Compare to lm() standard errors of the coefficients again

	summary(reg)


The diagonal elements of the variance/covariance matrices should be the
standard errors of the coefficients. Both are identical when computed by
hand. However, they differ from the standard errors reported in
summary(reg). The difference of 1% seems nonmarginal. Should I have
multiplied var(e)*solve(t(z)%*%z) by n and divided by n-1? But even if I do
this, I still observe a difference. Can anybody help me out what the source
of this difference is?

Cheers,
Daniel


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cuncta stricte discussurus