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I have seen a few sources that describe the wild bootstrap as so:

For each bootstrap instance, compute a new $y$ based on:

$y^*_i = \hat{y}_i+\hat{\epsilon}_iv_i$

Where $y^*_i$ is the $i^{th}$ bootstrap instance, $\hat{y}_i$ is the original data, $\hat{\epsilon}_i$ is the residual and $v_i$ is a bootstrap variable with mean zero variance one.

One example would be the wikipedia page on bootstrapping for example here.

However, I've seen other sources saying that the wild bootstrap instead uses the standardised residuals like so:

For each bootstrap instance, compute a new $y$ based on:

$y^*_i = \hat{y}_i+\frac{\hat{\epsilon}_i}{\sigma}v_i$

Where $y^*_i$ is the $i^{th}$ bootstrap instance, $\hat{y}_i$ is the original data, $\hat{\epsilon}_i$ is the residual, $v_i$ is a bootstrap variable with mean zero variance one and $\sigma$ is the estimated standard deviation of the $y_i$ variables.

An example of this would be the Wu 1986 paper here.

Which of these is correct? I don't understand the logic behind using the standardised residuals here.

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The article you have linked for the formula where the residual is divided by sigma is not the original paper from Wu (1986), but instead the link leads to a paper from Prášková (2002) called "Bootstrap in nonstationary autoregression", so I don't know where you found the information about the residual being "standardized". Further, this formula does not really depict a standardized residual, since the standardized residual is the original residual divided by the estimated standard error of the residuals not the estimated standard deviation of y (see here).

In Wu's original article (p. 1282) he describes the ith unstandardized residual being transformed (divided by the square root of 1 minus the ith leverage value)and multiplied by a bootstrap variable t* (or v*). Variations of this version of the wild bootstrap can be found in e.g. MacKinnon (2011), or Cribari-Neto (2004). In other iterations of the wild bootstrap, f.i. the one implemented in SPSS, the residual is not transformed prior to multiplying it with the bootstrap variable t*, but it is again the unstandardized residual that is used.

I personally would not think it correct to use standardized residuals, since the goal of each bootstrap iteration is to compute a vector of bootstrapped y-values with which the unstandardized regression coefficients can be computed. This would require the y-values (and thus the residuals) to remain in their original metric.

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  • $\begingroup$ The argument for standardising them is that they are then more nearly exchangeable; it might well be better to rescale them to the original variance afterwards, for the reason you give. $\endgroup$ Commented Sep 13, 2023 at 7:12
  • $\begingroup$ Okay, thanks! Do you by any chance have any literature that discusses this, since I have not come upon using standardised residuals in the context of the wild bootstrap? $\endgroup$ Commented Sep 13, 2023 at 9:07
  • $\begingroup$ I'm going off memories from last century, so no, sorry. $\endgroup$ Commented Sep 13, 2023 at 21:31

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