Extreme Value Theory (EVT): Hill estimator

[ac_1]

Hi Tom,

Here's my loop for risk algo Historical Simulation (HS), calculate VaR and ES:

Code: Select all

*===============================
do i = 1, %allocend() - WIN
*
   clear dlp1
   clear y
   set dlp1 1+i WIN+i = dlp

   order(index=ix,nodecreasing) dlp1 1+i WIN+i
   set y 1+i WIN+i = dlp1(ix)

   comp op = fix(WIN*alpha)+i; * alpha smallest
   comp VaR = - y(op) * SCALE

   sstats(mean) 1+i fix(WIN*alpha)+i y>>mu
   comp ES = - mu * SCALE

   comp VaR_1(WIN+1+i) = VaR
   comp ES_1(WIN+1+i) = ES


   disp 'obs' WIN+i+1 'HS 1% VaR=' VaR 'HS 1% ES=' ES
*
end do i

I ought to get a similar picture of multi-step ahead forecasts (sqrt-of-time) for Extreme Value Theory (EVT): Hill estimator, even though HS and EVT are completely different methods.

How would I implement EVT in RATS?

Amarjit

[TomDoan]

Tsay's book has a whole chapter (Chapter 7) on that. There are four examples in the textbook examples.

[ac_1]

Thanks! Based on the HS loop (fixed rolling window) and using @HillGEV, it would be something like the following, however what are the formulae for VaR and ES as positive numbers?

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comp WIN = 1000
comp SCALE = 100.0

comp CT = 100; * threshold for tail observations

*===============================
do i = 1, %allocend() - WIN
*
   clear dlp1
   clear y
   clear ysort

   set dlp1 1+i WIN+i = dlp
   set y 1+i WIN+i = -dlp1

   order(index=ix,nodecreasing) y 1+i WIN+i
   set ysort 1+i WIN+i = y(ix)

   @HillGEV(span=CT,tail=left) ysort 1+i WIN+i
   sstat 1+i i+CT %%hill>>sumlog
   comp iota = -(CT/sumlog)

   if iota <= 1.0 {
      comp VaR_EVT = %NA
      comp ES_EVT  = %NA
   }

   comp threshold = ysort(i+CT+1) * SCALE

   comp VaR_EVT = ?
   comp ES_EVT  = ?

   comp VaR_1(WIN+1+i) = VaR_EVT
   comp ES_1(WIN+1+i)  = ES_EVT

   disp 'obs' WIN+i+1 'EVT iota=' iota 'VaR=' VaR_EVT 'ES=' ES_EVT
*
end do i

Although, @HillGEV is designed for GEV asymptotics, not classical Pareto Hill estimator:

1/ihat = (1/k) summation from j=1 to k [log( X(j)/X(k+1))].

I'd like the classical estimator to complement various risk algos.

So, maybe:

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set logratios 1+i WE+i = log( ysort / ysort(i+CT+1) )
sstat 1+i i+CT logratios>>sumlog
comp iota = CT / sumlog

comp VaR_EVT = ?
comp ES_EVT  = ?

?

[TomDoan]

Although, @HillGEV is designed for GEV asymptotics, not classical Pareto Hill estimator:

1/ihat = (1/k) summation from j=1 to k [log( X(j)/X(k+1))].

I'd like the classical estimator to complement various risk algos.

That's what the procedure computes.

[ac_1]

Although, @HillGEV is designed for GEV asymptotics, not classical Pareto Hill estimator:

1/ihat = (1/k) summation from j=1 to k [log( X(j)/X(k+1))].

I'd like the classical estimator to complement various risk algos.

That's what the procedure computes.

Alright if I use

Code: Select all

   @HillGEV(span=CT,tail=left) ysort 1+i WIN+i
   sstat 1+i i+CT %%hill>>sumlog
   comp iota = -(CT/sumlog)

   comp threshold = ysort(i+CT+1) * SCALE

then what are the VaR_EVT and ES_EVT formulae?

Code: Select all

  comp VaR_EVT = TSay (2010), p.355, eqn (7.28), if so, how do I extract alphahat, betahat, Xihat from the procedure? Or is there another VaR_EVT formulae?
  comp ES_EVT  = ?

[TomDoan]

The tail index doesn't provide enough information to compute the VaR and the ES. Those need complete distributions. tsay3p355 does that. The thing to note is that the results are sensitive to the number of extreme values that are included, which is the point of the earlier search.

[ac_1]

The tail index doesn't provide enough information to compute the VaR and the ES. Those need complete distributions. tsay3p355 does that. The thing to note is that the results are sensitive to the number of extreme values that are included, which is the point of the earlier search.

Are you saying I cannot do a rolling analysis: fixed window T=1000, n=100 (the tail), p=0.01, in each loop, EVT method, and calculate VaR and ES using @HILLGEV, but rather using

nonlin alpha beta k
dec series r

compute alpha=.8,beta=-2.0,k=-.2; * guess values

frml gevmin = %loggevdensity(-r,k,-beta,alpha)
maximize(METHOD=BFGS) gevmin 1 gend

compute var01=%invgev((1-.01)**n,k,-beta,alpha)

?

I am not certain how to proceed, given my aim: backtest and forecast multi-step ahead VaR and ES using EVT to complement other various risk algos.

[TomDoan]

Is there a reason you can't put that in a loop?

If you want the ES for a GEV, you will have to find someone who has figured that out.