Weighted regression I have a response variable, y.hat, that is an estimate of animal abundance.  I know the standard error of y.hat.  I'm skeptical of a recommendation to use the uncertainty in y.hat as a weight when I regress or calibrate y.hat to another variable.  There are a few parts to consider.  First, the standard error of y.hat tends to increase with y.hat.  So large estimates of abundance will have less weight than lower estimates, which seemingly causes the fit to be biased low.  Second, the independent variable is positively correlated with y.hat, so this means that there is more uncertainty on the right-hand side of the plot. This results in heteroscedasticity, which is when I think WLS is appropriate.  What I think we have here is a potential trade-off between bias (due to weights covarying with y) vs. accommodating heteroscedasticity.
If the uncertainty was randomly assigned to each data pair, I still don't see why we'd want to use the uncertainty as weights.  Here's a little R code to simulate a data when uncertainty is random (default) vs. a function of y.hat (commented out).  Rather than regressing, this code calibrates x to y.hat using a mean of ratios.  The result is that using weights results in a biased estimate of the true ratio (2) when uncertainty is correlated to y.hat, and an unbiased but relatively imprecise estimate when uncertainty is not correlated to y.hat.
Am I right that using uncertainty in the estimate of y as a weight is inappropriate in this context?  
N    <- 6
reps <- 5000
out1 <- matrix(NA, reps, 2)
for (i in 1:reps){
  x         <- runif(N, 10,  30)
  y.hat     <- rnorm(N, 2*x, 10)
  #se       <- -0.1 + 0.3*y.hat
  se        <- rnorm(N, 7, 4)
  w         <- 1/se^2
  out1[i,1] <- mean(y/x)
  out1[i,2] <- sum(y/x*w) / sum(w)
}
hist(out1[,1], 50)
hist(out1[,],  50)

 A: If you are going to use weighted least squares (WLS) to address heteroscedasticity, you need weightings that reflect the overall pattern of variation in the conditional variance of the dependent variable, in your case $Var[y.hat | x]$.  You have identified one source of variance, the standard error in estimates of $y.hat$, and you have noted that this gives rise to heteroscedasticity since the standard error increases with $y.hat$.  However, this may not be the only source of heteroscedasticity.
Suppose you could measure $y.hat$ with complete accuracy, and you calculated a regression of $y.hat$ on $x$.  Very likely, you would still find that there was still some variability of $y.hat$ about the line of best fit.  Quite possibly, you would find that the variability of $y.hat$ tends to increase with $y.hat$.  If this is the case, weightings that only reflect the standard error of the estimates of $y.hat$ will not be appropriate.
You also raise the issue of weightings causing bias. This could occur if your model of heteroscedasticity has variability increasing with $y.hat$ and the weightings are based on the observed values of $y.hat$. To see this, consider two observations with the same $x$ value, one with $y.hat$ above trend and one with $y.hat$ below trend.  Then, using the inverse of $y.hat$ as the weight, the below trend observation will have greater weight, tending to result in downward bias.  This type of bias can be avoided by a 2-stage procedure, first using ordinary least squares (OLS), then applying WLS using as weightings the OLS-fitted values of $y.hat$ for each value of $x$.  This will result in both the above-trend and below-trend observations receiving the same weighting, so avoiding bias.
