[R] OLS standard errors

[Prof Brian Ripley]

Please check your statistical methods lecture notes.  var(e) has divisor 
n-1, and that is not an unbiased estimator of the residual variance when 
'e' are residuals.  From summary.lm (and you are allowed to read the code)

     rdf <- n - p
     if (is.na(z$df.residual) || rdf != z$df.residual)
         warning("residual degrees of freedom in object suggest this is not 
an \"lm\" fit")
     p1 <- 1:p
     r <- z$residuals
     f <- z$fitted.values
     w <- z$weights
     if (is.null(w)) {
         mss <- if (attr(z$terms, "intercept"))
             sum((f - mean(f))^2)
         else sum(f^2)
         rss <- sum(r^2)
     }
     else {
         mss <- if (attr(z$terms, "intercept")) {
             m <- sum(w * f/sum(w))
             sum(w * (f - m)^2)
         }
         else sum(w * f^2)
         rss <- sum(w * r^2)
         r <- sqrt(w) * r
     }
     resvar <- rss/rdf

the correct divisor is n-p.  Since p=3 in your example, that explains a 2% 
difference in variances and hence a 1% difference in SEs.

On Tue, 26 Feb 2008, 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?
>
> Cheers,
> Daniel
>
>
> -------------------------
> cuncta stricte discussurus
>
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>

-- 
Brian D. Ripley,                  ripley at stats.ox.ac.uk
Professor of Applied Statistics,  http://www.stats.ox.ac.uk/~ripley/
University of Oxford,             Tel:  +44 1865 272861 (self)
1 South Parks Road,                     +44 1865 272866 (PA)
Oxford OX1 3TG, UK                Fax:  +44 1865 272595