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st: Bootstrapping prediction standard error


From   rsperling@rcn.com
To   statalist@hsphsun2.harvard.edu
Subject   st: Bootstrapping prediction standard error
Date   Thu, 16 Aug 2007 10:24:08 -0400 (EDT)

I want to bootstrap the standard error of the prediction for the following nonlinear function:

y = (a + b*x^g) * e,

where y and x are scalars, a, b, and g are parameters to be estimated, and e is normally and identically distributed with mean 1 and variance \sigma^2.

I estimate a, b, and g using iteratively re-weighted least squares (IRLS), which, in this case, is equivalent to weighted nonlinear least squares.

Now let's say I run N Bootstrap replications of IRLS to predict N y-values, \hat{y}, given x=x_0, where x_0 is in the original set of data used to estimate a, b, and g. I then calculate the standard error of the N Bootstrap \hat{y}s.

My question is how do I know whether the standard error calculated above is the standard error of the mean response, i.e., \hat{y} or E(y|x=x_0), or the standard error of the prediction error?

To try and make things more concrete, suppose I ask the following two questions:

1) What is the predicted value of y given x = x_0?
2) What is the predicted value of some *future* y, say y_0, given x = x_0?

My understanding is that the first question corresponds to the standard error of the mean response or expected value. And the second question corresponds to the standard error of the prediction error, where the prediction error is defined as e_0 = y_0 - \hat{y}_0.

In other words, the prediction error accounts for "unobserved factors in the error term," e. As such, the standard error of the prediction error should be greater than the standard error of the mean response or expected value.

I hope the question is clear.

Thanks,
Richard

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