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I'm using the dataset daxreturns, with the description of the data like:

"This data set contains transformed standardized residuals of daily log returns of 15 major German stocks represented in the index DAX observed from January 2005 to August 2009. Each time series is filtered using a GARCH(1,1) model with Student $t$ innovations.".

So I take a look at the data and it seems like the minimum value is positive. It seems to me that, the standardized residuals obtained from a GARCH(1,1) model with Student-$t$ innovations are always positive? Can someone help me to explain this idea?

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    $\begingroup$ The standardized residuals obtained from a GARCH(1,1) model with Student-$t$ innovations should not always be positive. The word transformed might be key. Is there any explanation of what that could be? $\endgroup$ Apr 30, 2020 at 14:35
  • $\begingroup$ rdocumentation.org/packages/VineCopula/versions/2.3.0/topics/… can you take a look at this? $\endgroup$
    – tmtran99
    Apr 30, 2020 at 14:52
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    $\begingroup$ What do you think about my answer? If it is clear, you may upvote it and accept it by clicking on the tick mark to the left. Otherwise you may ask for further clarification. This is how Cross Validated works. $\endgroup$ May 5, 2020 at 12:32
  • $\begingroup$ Ping... (See my comment above). $\endgroup$ May 23, 2021 at 9:43

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The standardized residuals obtained from a GARCH(1,1) model with Student-$t$ innovations should not always be positive. The qualifier transformed might be key here. I think these might be standardized returns that have been transformed using probability integral transform (PIT) so as to lie within $[0,1]$. I think so because the data is used for fitting a copula, and it is customary to fit a copula on data that lies in a $[0,1]^k$ hypercube. See slide 13 of Brechman "Statistical inference of vine copulas using the R-package VineCopula" (2013).

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