Re: st: RE: bootstrapping standard errors with several estimated regressors + 1st line:

[Steven Samuels <ssamuels@albany.edu>]

Obstreperous line

The last post with all lines was:

Erasmo, What was the basis for your original thought "that  
bootstrapping would cause statistical significance for all regressors  
to go down"?  I've not seen this in the bootstrap literature. Indeed,  
your example, and that of Maarten, suggest that there is no order  
relation between model-based estimated standard errors and those  
estimated by the bootstrap.

You might be thinking that bootstrapping should cause p-values to  
rise because regressors, as well as responses, are being sampled.  
This is not so. Assume the classical multiple regression model. If  
the X variables are random and  independent of  independent of the  
error terms, then in the usual formula for the V(b), (X'X)^(-1) is  
replaced by its expectation.  (WH Greene, Econometrics, McMillan, 1990).

You might also be thinking that the use of estimated regressors  
should lead to higher higher pvalues, compared to having the "true"  
regressors. This sounds right, although I am not expert in this area,  
but it is irrelevant. Both original and bootstrapped standard errors  
are based on the estimated regressors.

Perhaps you are confusing the estimates of coefficients with  
estimates of standard errors of coefficients. If model assumptions  
are right, then both model-based estimate of standard error and the  
bootstrap estimates of standard error are "good"  estimates of the  
same quantity, the "true" standard error.  However, the model-based  
estimate  benefits from knowing that model is true. For example, in  
OLS, for example, the key assumption is that there is a constant SD.  
The model-based estimate standard error is therefore a function of  
one quantity besides the X'X matrix, namely the residual SD.  The   
bootstrap estimate is valid even if the residual SD is not constant,  
as long as the observations are uncorrelated. The price for this  
greater validity is that, if the model is right, the bootstrap  
estimate of standard error will be more variable then the model-based  
estimate. See Efron & Tibshirani, An Introduction to the Bootstrap,  
Chapman & Hall, 1994.

-Steve


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