Risk using Student T

Discussion on Value-at-Risk Models
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FatTail
Posts: 20
Joined: Sun Nov 16, 2008 10:07 am

Re: Risk using Student T

Postby FatTail » Sun Nov 30, 2008 9:28 pm

Attached (in a spreadsheet) is a potential Ad hoc tail estimator for use with t distribution. The idea behind estimator is
- to calculate a standard Hill tail estimator
- calculate a second estimator that is excess over normal of tail but other wise same simple calc as Hill
- blend these estimators via some ratio so that the excess estimator dominates if Hill estimator is small (this means estimator will come out at near zero for a normal distribution)and Hill estimator dominates if Hill estimator is large

It has to use an ad hoc ratio for which i cannot justify but it looks like it might work reasonably.........

coalexander
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Joined: Sun Sep 28, 2008 10:30 pm

Re: Risk using Student T

Postby coalexander » Mon Dec 01, 2008 1:02 pm

Is that the ETL/VaR estimator? What about skewness - or are you using symmetric t?

FatTail
Posts: 20
Joined: Sun Nov 16, 2008 10:07 am

Re: Risk using Student T

Postby FatTail » Mon Dec 01, 2008 1:53 pm

Am not using the ETL/VAR estimator. It turns out that estimator would be very similar to standard Hill Estimator in that it would use exactly same data points as input but use average of ratios over threshold rather than average of ln of ratios over threshold used in Hill.

So the measures are very related and very approximately ETL/VAR= 1/(1-Hill)

I think Hill though is more stable as a basis for an estimator so probably better.

For data analysis intend to just look at one tail at a time. So i think this means this estimator would work for input into either - and as would potetially give two outputs one for each tail maybe suited to non symetric T (although i have never used that)

coalexander
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Joined: Sun Sep 28, 2008 10:30 pm

Re: Risk using Student T

Postby coalexander » Mon Dec 01, 2008 2:11 pm

Want me to find some references to non-symmetric t?

FatTail
Posts: 20
Joined: Sun Nov 16, 2008 10:07 am

Re: Risk using Student T

Postby FatTail » Mon Dec 01, 2008 2:35 pm

No that is ok you are too busy for that -and that is not my focus now.

More interested on any views on the tail estimator in the attached speadsheet- it gets around I think the "Hill Horror Plot" shown in Dowds book P199 and makes output a suitable input suitable for T distibution

coalexander
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Joined: Sun Sep 28, 2008 10:30 pm

Re: Risk using Student T

Postby coalexander » Tue Dec 02, 2008 9:00 am

Hiya

Well, your spreadsheet seems to work well on the data you are giving it. You have a nice 'full' data set with 2000 obs in your experiment. What if you bootstrapped a smaller sample from this, say with only 500 data points - what level of accuracy do you have now for the d.f. estimate?

FatTail
Posts: 20
Joined: Sun Nov 16, 2008 10:07 am

Re: Risk using Student T

Postby FatTail » Tue Dec 02, 2008 9:31 am

Hi tks for your comments/help. I am going to test it as a real estimator rather than as a theoretical estimator soon (a) against randomised T distribution data (b) real data. All these estimators have quite a degree of variance when used in practice so i think you have to apply to lots of different sets of data and take a sort of average outcome.I am quite hopeful that it will provide a good practical solution that will have less cut off problems and less bias when using as input to T. Be good if there was more theoretical justification for it.

Did you say that you had a formula for ETL/VAR ratio for t distribution in the new book. And if so any chance of a preview of the formula on the forum.

Tks

coalexander
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Joined: Sun Sep 28, 2008 10:30 pm

Re: Risk using Student T

Postby coalexander » Tue Dec 02, 2008 9:42 am

Here you are

FatTail
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Joined: Sun Nov 16, 2008 10:07 am

Re: Risk using Student T

Postby FatTail » Tue Dec 02, 2008 8:41 pm

Carol, Thank you very much for that. Of course I will be buying the book which i am looking forward to. Cheers

FatTail
Posts: 20
Joined: Sun Nov 16, 2008 10:07 am

Re: Risk using Student T

Postby FatTail » Sat Jan 17, 2009 10:56 pm

I have been busy and not much time to focus on this recently- but one observation that i find interesting is the quite similar gradient of slope of ETL/VAR as you move from say 90% to 99% cut off for T distribution with different DF - always downwards sloping (ie 90% > 99%) and almost linear and gradient looks very silmilar despite DF (ie gradient similar to normal). Obviously it is not linear across full space because in very far tail it flattens out to be = DF /(DF-1)

Also by observation alpha stable distribution empirically has slope other way for low alpha ( ie 90% less than 99%) i think but must switch as at some point it also converges on normal as alpha goes to 2 - difficult to analyse as we lack analytical CDF i think for alpha stable

GPD can do both upward and doward sloping and maybe this is one way to think of difficult to understand Beta parameter it is this parameter taht controls this slope before you get to very far tail (whereas tail shape/tail index param controls the limiting ratio in very far tail)

A pure Power law (special case of pareto) has a flat gradient for ETL/VAR for diiferent percentiles.

[As we have previously noted on this thread there is vclose relationship between Hill estimator and shape of ETL/VAR]


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