# Parametric bootstrap in generating returns and hypothesis testing

I am trying to test a hypothesis of a statistic calculated from portfolio returns. To do so I estimate a model on the original returns series and want to obtain 100 bootstrapped series using parametric bootstrap. I am conflicted about two approaches. But first let me say that I obtain the new returns as a mean plus a series of resampled residuals

$r = \mu + \varepsilon$

So the two approaches I am considering:

• Estimate the model using the original returns
• Obtain a new series of returns by using the mean and resampled residuals from the model
• Estimate the model on the new series
• Obtain the new series and so on...

In short I am resampling the new returns each time, so I am obtaining a single new series with each iteration.

• Estimate the model on the original returns
• Obtain 100 new return series by using the mean and resampled residuals from the model (each resampled series is different of course)

I am wondering which of the approaches is a correct method of parametric bootstrap. I am not including any details on the models and test statistics to keep the post simple, as only the method of obtaining the new return series is important here.

edit: I have found a paper that is doing a similar thing to my research, however, it uses a strange bootstrap method, as in some steps it uses the original returns and in some the bootstrapped ones. Equation (1)-(3) mentioned in the fragment are those of a GJR-GARCH model. Is this iterative approach, somehow similar to the one I described as point 1. correct, with taking the original returns in some steps and the new bootstrapped series in other steps?