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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