# Re: [R] OLS standard errors

From: Peter Dalgaard <p.dalgaard_at_biostat.ku.dk>
Date: Tue, 26 Feb 2008 08:57:54 +0100

Daniel Malter wrote:
> 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?
>
>
The degrees of freedom in a regression analysis is n minus the number of parameters, three in your case. I.e. the variance var(e) does not know about this and divides by n-1 where it should have been n-3, so.....

> Cheers,
> Daniel
>
>
> -------------------------
> cuncta stricte discussurus
>
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> and provide commented, minimal, self-contained, reproducible code.
>

--
O__  ---- Peter Dalgaard             ุster Farimagsgade 5, Entr.B
c/ /'_ --- Dept. of Biostatistics     PO Box 2099, 1014 Cph. K
(*) \(*) -- University of Copenhagen   Denmark          Ph:  (+45) 35327918
~~~~~~~~~~ - (p.dalgaard_at_biostat.ku.dk)                  FAX: (+45) 35327907

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