Content
- Data Loading
- RiskMetrics-IGARCH
- RiskMetrics-IGARCH
- Empirical Quantile
- Quantile Regression
- Extreme Value Approach
- Peaks over the threshold
- Generalized Pareto Distribution
1. Data Loading
da=read.table("d-ibm-0110.txt",header=T)
head(da)
## date return
## 1 20010102 -0.002206
## 2 20010103 0.115696
## 3 20010104 -0.015192
## 4 20010105 0.008719
## 5 20010108 -0.004654
## 6 20010109 -0.010688
ibm=log(da$return+1)
xt = -ibm*100 ##xt is then the loss variable (in percentage)
t.test(xt)
##
## One Sample t-test
##
## data: xt
## t = -0.78176, df = 2514, p-value = 0.4344
## alternative hypothesis: true mean is not equal to 0
## 95 percent confidence interval:
## -0.09296051 0.03996638
## sample estimates:
## mean of x
## -0.02649706
2. RiskMetrics-IGARCH
require(rugarch)
## Loading required package: rugarch
## Loading required package: parallel
##
## Attaching package: 'rugarch'
## The following object is masked from 'package:stats':
##
## sigma
require(fGarch)
## Loading required package: fGarch
## Loading required package: timeDate
## Loading required package: timeSeries
## Loading required package: fBasics
spec3 = ugarchspec(variance.model=list(model="iGARCH", garchOrder=c(1,1)), mean.model=list(armaOrder=c(0,0), include.mean=FALSE),distribution.model="norm", fixed.pars=list(omega=0))
fit = ugarchfit(spec3, data = xt)
summary(fit)
## Length Class Mode
## 1 uGARCHfit S4
fit
##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : iGARCH(1,1)
## Mean Model : ARFIMA(0,0,0)
## Distribution : norm
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## omega 0.000000 NA NA NA
## alpha1 0.057143 0.007172 7.9675 0
## beta1 0.942857 NA NA NA
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## omega 0.000000 NA NA NA
## alpha1 0.057143 0.021965 2.6015 0.009282
## beta1 0.942857 NA NA NA
##
## LogLikelihood : -4501.827
##
## Information Criteria
## ------------------------------------
##
## Akaike 3.5808
## Bayes 3.5831
## Shibata 3.5808
## Hannan-Quinn 3.5816
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.06978 0.7917
## Lag[2*(p+q)+(p+q)-1][2] 2.40452 0.2037
## Lag[4*(p+q)+(p+q)-1][5] 4.59023 0.1891
## d.o.f=0
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.0508 0.8217
## Lag[2*(p+q)+(p+q)-1][5] 2.0768 0.6004
## Lag[4*(p+q)+(p+q)-1][9] 3.5069 0.6732
## d.o.f=2
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[3] 0.06369 0.500 2.000 0.8008
## ARCH Lag[5] 2.89044 1.440 1.667 0.3061
## ARCH Lag[7] 3.50081 2.315 1.543 0.4243
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 0.074
## Individual Statistics:
## alpha1 0.07404
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 0.353 0.47 0.748
## Individual Statistic: 0.35 0.47 0.75
##
## Sign Bias Test
## ------------------------------------
## t-value prob sig
## Sign Bias 0.3907 0.6961
## Negative Sign Bias 0.6239 0.5328
## Positive Sign Bias 0.8428 0.3994
## Joint Effect 3.4079 0.3329
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 45.64 0.0005565
## 2 30 63.01 0.0002572
## 3 40 63.85 0.0072739
## 4 50 87.33 0.0006231
##
##
## Elapsed time : 0.8119991
head(sigma(fit))
## [,1]
## 1970-01-02 07:30:00 1.699656
## 1970-01-03 07:30:00 1.651224
## 1970-01-04 07:30:00 3.069140
## 1970-01-05 07:30:00 3.002544
## 1970-01-06 07:30:00 2.922872
## 1970-01-07 07:30:00 2.840322
tail(sigma(fit))
## [,1]
## 1976-11-15 07:30:00 0.8289833
## 1976-11-16 07:30:00 0.8050096
## 1976-11-17 07:30:00 0.7868684
## 1976-11-18 07:30:00 0.7664697
## 1976-11-19 07:30:00 0.7559542
## 1976-11-20 07:30:00 0.7344453
coef(fit)
## omega alpha1 beta1
## 0.00000000 0.05714296 0.94285704
vt = coef(fit)[3]*sigma(fit)[2515]^2+coef(fit)[2]*xt[2515]^2 ## volatility prediction; xt[2515]^2 because mean equation=0
sqrt(vt)*qnorm(.95) #VaR 0.95
## [,1]
## 1976-11-20 07:30:00 1.17328
sqrt(vt)*qnorm(.99) #VaR 0.99
## [,1]
## 1976-11-20 07:30:00 1.659392
sqrt(vt)*qnorm(.999) #VaR 0.999
## [,1]
## 1976-11-20 07:30:00 2.204273
3. IGARCH
xt1 = -ibm #loss variable: negative log returns
(m1 = garchFit(~garch(1,1),data=xt1,trace=F))
##
## Title:
## GARCH Modelling
##
## Call:
## garchFit(formula = ~garch(1, 1), data = xt1, trace = F)
##
## Mean and Variance Equation:
## data ~ garch(1, 1)
## <environment: 0x00000000236edd80>
## [data = xt1]
##
## Conditional Distribution:
## norm
##
## Coefficient(s):
## mu omega alpha1 beta1
## -6.0097e-04 4.3781e-06 1.0113e-01 8.8412e-01
##
## Std. Errors:
## based on Hessian
##
## Error Analysis:
## Estimate Std. Error t value Pr(>|t|)
## mu -6.010e-04 2.393e-04 -2.511 0.012044 *
## omega 4.378e-06 1.160e-06 3.774 0.000161 ***
## alpha1 1.011e-01 1.851e-02 5.463 4.67e-08 ***
## beta1 8.841e-01 1.991e-02 44.413 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Log Likelihood:
## 7114.066 normalized: 2.828654
##
## Description:
## Mon Apr 15 17:33:02 2019 by user: Sirius
predict(m1,3)
## meanForecast meanError standardDeviation
## 1 -0.0006009667 0.007824302 0.007824302
## 2 -0.0006009667 0.008043298 0.008043298
## 3 -0.0006009667 0.008253382 0.008253382
-.000601+qnorm(.95)*.0078243 #VaR 0.95
## [1] 0.01226883
-.000601+qnorm(.99)*.0078243 #VaR 0.99
## [1] 0.01760104
-.000601+qnorm(.999)*.0078243 #VaR 0.999
## [1] 0.0235779
m2=garchFit(~garch(1,1),data=xt1,trace=F,cond.dist="std")
m2
##
## Title:
## GARCH Modelling
##
## Call:
## garchFit(formula = ~garch(1, 1), data = xt1, cond.dist = "std",
## trace = F)
##
## Mean and Variance Equation:
## data ~ garch(1, 1)
## <environment: 0x00000000240e8238>
## [data = xt1]
##
## Conditional Distribution:
## std
##
## Coefficient(s):
## mu omega alpha1 beta1 shape
## -4.1127e-04 1.9223e-06 6.4480e-02 9.2863e-01 5.7513e+00
##
## Std. Errors:
## based on Hessian
##
## Error Analysis:
## Estimate Std. Error t value Pr(>|t|)
## mu -4.113e-04 2.254e-04 -1.824 0.06811 .
## omega 1.922e-06 7.417e-07 2.592 0.00954 **
## alpha1 6.448e-02 1.323e-02 4.874 1.09e-06 ***
## beta1 9.286e-01 1.407e-02 65.993 < 2e-16 ***
## shape 5.751e+00 6.080e-01 9.459 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Log Likelihood:
## 7218.69 normalized: 2.870254
##
## Description:
## Mon Apr 15 17:33:03 2019 by user: Sirius
predict(m2,3)
## meanForecast meanError standardDeviation
## 1 -0.0004112737 0.008100874 0.008100874
## 2 -0.0004112737 0.008191121 0.008191121
## 3 -0.0004112737 0.008279774 0.008279774
-.0004113+qstd(.95,nu=5.751)*.0081009
## [1] 0.01240096
-.0004113+qstd(.99,nu=5.751)*.0081009
## [1] 0.02045082
-.0004113+qstd(.999,nu=5.751)*.0081009
## [1] 0.03456563
4. Empirical Quantile
quantile(xt,c(.95,.99))
## 95% 99%
## 2.653783 5.013151
- Empirical expected shortfall or CVaR. 1st method
sxt = sort(xt) # Sorting into increasing order
0.95*2515
## [1] 2389.25
(es=sum(sxt[2390:2515])/(2515-2389))# ES 0.95
## [1] 3.994857
0.99*2515
## [1] 2489.85
(es=sum(sxt[2490:2515])/(2515-2489))# ES 0.99
## [1] 6.074108
- Empirical expected shortfall or CVaR. 2nd method
idx=c(1:2515)[xt >= 2.653783]
mean(xt[idx])
## [1] 3.994857
idx1=c(1:2515)[xt >= 5.013151]
mean(xt[idx1])
## [1] 6.074108
- Empirical median shortfall
median(xt[idx])
## [1] 3.527853
median(xt[idx1])
## [1] 5.781306
5. Quantile Regression
require(quantreg)
## Loading required package: quantreg
## Loading required package: SparseM
##
## Attaching package: 'SparseM'
## The following object is masked from 'package:base':
##
## backsolve
da = read.table("d-ibm-rq.txt",header=T)
head(da)
## nibm vol vix
## 1 -0.109478400 0.01700121 29.99
## 2 0.015308580 0.01614694 26.60
## 3 -0.008681209 0.03786369 26.97
## 4 0.004664864 0.03602087 28.67
## 5 0.010745530 0.03403132 29.84
## 6 -0.009408600 0.03211091 27.99
nibm=da$nibm
vol=da$vol
vix=da$vix/100
length(nibm)
## [1] 2514
length(vol)
## [1] 2514
length(vix)
## [1] 2514
fit1 <- rq(nibm ~ vol + vix, tau = 0.95)
summary(fit1)
##
## Call: rq(formula = nibm ~ vol + vix, tau = 0.95)
##
## tau: [1] 0.95
##
## Coefficients:
## Value Std. Error t value Pr(>|t|)
## (Intercept) -0.00104 0.00257 -0.40317 0.68686
## vol 1.17724 0.22268 5.28661 0.00000
## vix 0.02809 0.01615 1.73977 0.08202
names(fit1)
## [1] "coefficients" "x" "y" "residuals"
## [5] "dual" "fitted.values" "formula" "terms"
## [9] "xlevels" "call" "tau" "rho"
## [13] "method" "model"
fit1$coefficients
## (Intercept) vol vix
## -0.001036872 1.177238592 0.028091939
coef(fit1)
## (Intercept) vol vix
## -0.001036872 1.177238592 0.028091939
plot(nibm,type='l',main='Negative daily log returns of IBM stock with 95th quantiles(red line)')
lines(1:2514,fit1$fitted.values,col='red')
tail(volatility(m1))#volatility of GARCH(1,1)
## [1] 0.008923548 0.008653557 0.008516047 0.008299314 0.008230775 0.008018202
vfit=-.00104+1.17724*volatility(m1)[2515]+0.02809*17.75/100 #17.75 is 2010-12-31 VIX
vfit
## [1] 0.01338532
fit2 <- rq(nibm ~ vol + vix, tau = 0.99)
summary(fit2)
##
## Call: rq(formula = nibm ~ vol + vix, tau = 0.99)
##
## tau: [1] 0.99
##
## Coefficients:
## Value Std. Error t value Pr(>|t|)
## (Intercept) 0.01182 0.00831 1.42190 0.15518
## vol 1.03129 0.73125 1.41032 0.15857
## vix 0.04409 0.05335 0.82641 0.40865
6. Extreme Value Approach
require(evir)
## Loading required package: evir
## Warning: package 'evir' was built under R version 3.4.4
par(mfcol=c(2,1))
(m2=gev(xt,block=21))
## $n.all
## [1] 2515
##
## $n
## [1] 120
##
## $data
## [1] 4.0335654 4.6038703 6.9818569 4.6158081 2.9868660 1.8749682
## [7] 5.1245927 3.6876663 3.3919828 2.3115108 2.0342513 2.4236342
## [13] 4.8790114 4.7451211 10.6674712 7.3298915 3.1040820 6.3058018
## [19] 4.9832257 4.1026182 7.0740111 5.3694069 3.2279420 3.3725360
## [25] 5.6782014 2.8647448 3.1710505 1.7130900 4.1022015 2.0702832
## [31] 4.0106625 1.2247698 2.1467793 3.8022791 1.2162666 1.2266932
## [37] 2.0414975 1.9473382 2.0080266 1.8147677 1.4965425 2.0119037
## [43] 1.8123238 1.7842230 1.6585787 1.1930891 1.6205604 1.2909976
## [49] 1.9149181 1.5308581 1.0340277 8.6620544 2.2660828 2.3848119
## [55] 1.1244989 1.7883969 1.4590932 1.2093837 0.6296783 2.6523663
## [61] 2.1040814 1.4236865 1.5996261 1.8530637 2.7150255 2.1617994
## [67] 1.6564436 1.1022525 2.1935841 0.8557511 2.0563997 0.9094227
## [73] 3.3537135 3.0913950 1.4987757 2.4177945 3.2943735 2.0471111
## [79] 3.4377182 1.7755704 1.6698650 3.2461210 5.6767196 3.6600708
## [85] 2.7332143 2.5186535 1.4527008 2.3683247 2.7861553 2.8494129
## [91] 1.7898222 2.8629954 6.0170475 6.1018324 5.9323344 2.7901656
## [97] 3.5234509 5.1061742 4.8006072 1.9594729 2.0101692 3.0112876
## [103] 1.2639544 1.4527008 1.4558462 5.0808149 1.2491697 1.2677015
## [109] 2.9428811 0.8985247 0.7097125 2.2935008 4.0058744 3.0624170
## [115] 2.5279860 2.0077205 0.9940241 3.4183660 1.5317720 0.7572600
##
## $block
## [1] 21
##
## $par.ests
## xi sigma mu
## 0.251353 1.028910 1.965850
##
## $par.ses
## xi sigma mu
## 0.08847742 0.09013351 0.10932034
##
## $varcov
## [,1] [,2] [,3]
## [1,] 0.007828254 -0.001080741 -0.003453668
## [2,] -0.001080741 0.008124049 0.006145413
## [3,] -0.003453668 0.006145413 0.011950936
##
## $converged
## [1] 0
##
## $nllh.final
## [1] 210.1579
##
## attr(,"class")
## [1] "gev"
#### Model checking ####
#plot(m2)
aa=1.0-(-21*log(1.0-0.05))^{-0.251353} #aa=1.0-(-n*log(1.0-prob))^{-xi}
1.965850 - (1.028910/0.251353)*aa #mu - (sigma/xi)*aa. VaR 0.95 with evt
## [1] 1.890084
aaa=1.0-(-21*log(1.0-0.01))^{-0.251353} #aaa=1.0-(-n*log(1.0-prob))^{-xi}
1.965850 - (1.028910/0.251353)*aaa #mu - (sigma/xi)*aaa. VaR 0.99 with evt
## [1] 3.924483
10^(0.251353)*3.924483 #VaR 0.99 for 10 trading days
## [1] 7.000603
7. Peaks over the threshold
require(evir)
(m3=pot(xt,1.0)) ## Threshold is set to 1%.
## $n
## [1] 2515
##
## $period
## [1] 1 2515
##
## $data
## [1] 1.530858 1.074553 1.139063 2.445357 4.033565 3.370467 2.432958
## [8] 1.857648 1.005943 1.535935 3.090776 3.644098 4.603870 2.607296
## [15] 2.657090 3.600134 1.009276 6.981857 3.902361 3.548315 5.883429
## [22] 4.754874 5.251088 1.593021 4.615808 2.010883 1.649022 1.270436
## [29] 2.495377 2.659349 1.484058 1.533295 2.986866 1.516440 2.171099
## [36] 2.299129 1.874968 1.535631 1.841349 1.205335 5.124593 1.685526
## [43] 2.670955 3.994738 1.273981 1.233982 1.555739 1.823627 1.126825
## [50] 1.102960 2.145860 1.803567 3.687666 1.129657 2.369656 1.449251
## [57] 3.298301 2.745650 3.154124 3.391983 1.434135 1.390320 1.606638
## [64] 2.311511 2.034251 1.847970 1.811407 1.271145 1.043324 2.423634
## [71] 1.591091 1.241778 1.905745 1.509638 1.896368 1.227503 4.826945
## [78] 3.337376 2.381023 1.039383 4.879011 1.117319 2.584000 4.745121
## [85] 3.310084 2.994800 1.176797 2.466674 1.224061 1.868651 1.111252
## [92] 1.950703 1.102253 1.874357 3.625017 10.667471 5.567234 1.625642
## [99] 1.209485 2.333516 2.511578 7.329891 3.104082 1.153123 1.457673
## [106] 1.191166 1.492990 1.234995 2.212803 2.951738 1.111049 1.722042
## [113] 1.934287 1.720415 1.118937 1.567826 3.470118 2.442692 4.033878
## [120] 1.590786 6.305802 3.038911 2.312636 1.314806 2.842828 4.983226
## [127] 2.139525 4.346925 1.955190 3.101606 2.823796 1.490046 1.958351
## [134] 1.226390 1.855407 2.467289 1.631639 4.102618 2.124713 1.869262
## [141] 3.190561 3.114191 7.074011 1.367305 5.882156 1.583674 2.713073
## [148] 3.455314 2.549834 5.833593 3.532256 5.369407 1.413037 3.408640
## [155] 3.227942 1.737609 1.042819 1.028269 1.331323 2.423122 1.799901
## [162] 3.372536 1.571687 1.618020 1.480201 1.592310 1.462848 1.445294
## [169] 2.127062 1.123892 1.773840 5.678201 1.048477 2.574457 2.547064
## [176] 1.404010 1.156664 1.753891 2.134724 2.864745 3.171051 2.303119
## [183] 3.038911 1.376835 1.713090 1.073845 1.561427 1.206246 1.432715
## [190] 1.017358 2.895827 4.102201 2.828940 2.284804 1.439714 2.070283
## [197] 1.372476 1.108623 1.109027 1.699154 4.010663 1.467921 1.216267
## [204] 1.032209 1.590278 1.224770 1.098006 1.782696 2.047009 2.146779
## [211] 1.260005 3.802279 1.216267 1.136231 1.226693 1.986501 2.041497
## [218] 1.057168 1.447729 1.336390 1.397013 1.947338 1.567319 2.008027
## [225] 1.599016 1.333552 1.814768 1.413037 1.482028 1.026854 1.496543
## [232] 1.023621 1.551473 2.011904 1.322507 1.234995 1.415978 1.136939
## [239] 1.812324 1.541419 1.784223 1.658579 1.193089 1.079607 1.328181
## [246] 1.620560 1.290998 1.080011 1.914918 1.530858 1.023722 1.034028
## [253] 1.611109 1.385656 1.105791 8.662054 1.538169 4.706226 1.490655
## [260] 2.071100 2.266083 2.030884 2.037415 1.047871 2.384812 1.873949
## [267] 1.124499 1.788397 1.465182 1.348654 1.126724 1.459093 1.209384
## [274] 1.195214 1.087189 1.384642 1.168096 2.652366 1.446512 2.104081
## [281] 1.563357 1.100332 1.423686 1.491264 1.599626 1.151504 1.755825
## [288] 1.853064 1.089210 1.460615 2.715026 1.427744 2.161799 1.303053
## [295] 1.656444 1.654207 1.102253 2.193584 2.056400 1.186208 3.353714
## [302] 1.079910 3.091395 1.091536 1.495933 1.005640 1.498776 1.356864
## [309] 2.417795 1.055248 1.555942 3.294373 2.047111 1.338316 3.437718
## [316] 1.190458 2.011598 1.394174 1.526492 1.277526 1.775570 1.157068
## [323] 1.669865 3.246121 2.219551 1.519994 2.150049 1.043930 1.504562
## [330] 4.577486 5.676720 1.753688 2.483075 1.167084 2.019250 2.181728
## [337] 2.218426 2.252990 1.179327 1.772517 1.824137 3.205326 3.660071
## [344] 1.073643 2.489533 2.277540 1.074452 2.130842 2.260866 1.059897
## [351] 2.733214 1.214040 2.134826 1.098714 1.053126 1.204728 2.518653
## [358] 1.249271 1.196125 1.452701 2.368325 1.135725 1.765595 1.045951
## [365] 1.041101 1.254030 1.612227 2.786155 2.159040 1.278741 1.840534
## [372] 2.808368 1.274183 2.849413 1.137242 1.789822 1.052520 1.730589
## [379] 1.095984 1.410704 1.642719 1.670780 2.314274 2.765181 2.862995
## [386] 1.936938 3.228872 4.026590 2.246342 4.242119 6.017047 5.018023
## [393] 1.248967 2.764050 5.065560 5.479310 1.726621 1.414456 5.840378
## [400] 4.025445 6.101832 2.740203 2.980478 3.774860 4.887307 2.821431
## [407] 1.356458 3.693166 4.717023 3.612367 5.268795 5.729019 5.932334
## [414] 4.086366 2.590465 2.790166 2.166807 1.848887 1.709835 1.626964
## [421] 2.885946 2.551578 3.523451 1.487712 2.466366 3.735509 1.302242
## [428] 3.436476 2.859908 5.106174 3.291686 1.447831 2.271709 1.927457
## [435] 2.752845 1.038676 4.800607 2.805797 1.735676 1.108016 1.330005
## [442] 1.959473 1.077989 1.629505 1.190357 1.393464 1.189142 2.010169
## [449] 1.431294 1.093760 1.302242 1.341255 3.011288 1.446715 1.232059
## [456] 1.263954 1.452701 1.018267 1.158788 1.455846 1.439917 5.080815
## [463] 1.600439 1.917365 1.856425 1.249170 1.114487 1.023015 1.267701
## [470] 1.215356 1.052520 2.942881 2.750687 2.063443 1.105084 2.139525
## [477] 1.939589 1.471778 1.125409 1.148571 2.293501 1.479592 4.005874
## [484] 1.041404 2.116644 1.038676 1.486697 3.062417 1.295455 2.079268
## [491] 2.527986 1.016650 1.536342 1.185499 1.092041 1.249170 2.007720
## [498] 1.072026 1.285831 3.418366 1.168905 1.531772 1.172750 1.005842
## attr(,"times")
## [1] 3 6 10 14 21 23 27 28 30 33 34 35 37 39
## [15] 40 42 46 47 48 50 52 54 60 63 64 68 70 71
## [29] 77 84 85 90 91 101 102 103 115 118 125 127 129 130
## [43] 131 137 141 144 150 152 157 159 161 166 168 171 172 173
## [57] 175 178 179 182 183 191 198 205 217 225 226 229 232 237
## [71] 248 252 255 256 257 259 261 262 263 266 267 271 274 280
## [85] 281 283 286 293 297 302 304 305 309 310 313 314 317 321
## [99] 324 328 333 334 337 341 344 345 348 349 352 353 356 357
## [113] 358 359 360 364 365 366 367 369 373 377 378 379 383 387
## [127] 388 391 394 395 397 408 411 412 413 414 416 417 419 423
## [141] 424 428 429 430 432 434 435 436 438 440 443 448 452 454
## [155] 464 465 471 472 477 481 482 485 488 491 492 497 498 499
## [169] 505 510 511 512 514 516 520 523 529 536 538 546 556 558
## [183] 561 565 568 575 579 585 592 594 595 605 607 608 613 619
## [197] 621 623 624 631 636 638 643 645 649 663 671 674 682 684
## [211] 688 700 716 725 753 758 760 769 770 776 787 797 799 800
## [225] 802 806 825 827 834 835 841 846 879 881 885 890 892 899
## [239] 902 905 906 934 949 973 982 987 995 1006 1016 1031 1039 1066
## [253] 1072 1074 1075 1076 1078 1079 1085 1090 1093 1107 1110 1111 1124 1125
## [267] 1153 1157 1161 1173 1184 1186 1201 1207 1213 1241 1243 1244 1261 1269
## [281] 1279 1285 1292 1312 1323 1325 1330 1337 1350 1361 1365 1375 1385 1388
## [295] 1389 1402 1428 1438 1484 1513 1519 1534 1545 1546 1548 1552 1555 1575
## [309] 1580 1605 1606 1614 1626 1649 1652 1655 1659 1664 1666 1672 1679 1682
## [323] 1697 1707 1709 1712 1718 1720 1722 1723 1724 1727 1730 1732 1734 1738
## [337] 1745 1748 1749 1756 1758 1759 1761 1762 1763 1766 1768 1772 1775 1781
## [351] 1782 1784 1789 1791 1798 1799 1803 1812 1817 1821 1828 1839 1841 1854
## [365] 1855 1856 1863 1867 1870 1874 1877 1881 1883 1889 1901 1902 1906 1912
## [379] 1913 1917 1918 1922 1926 1927 1929 1932 1936 1938 1941 1946 1948 1949
## [393] 1950 1951 1952 1953 1954 1955 1958 1962 1963 1965 1966 1973 1974 1976
## [407] 1977 1978 1980 1981 1983 1984 1990 1993 1996 1998 2003 2005 2006 2015
## [421] 2017 2020 2023 2025 2030 2038 2041 2042 2044 2046 2051 2052 2054 2055
## [435] 2056 2063 2070 2077 2080 2088 2089 2098 2099 2102 2103 2107 2108 2111
## [449] 2115 2123 2129 2135 2137 2139 2142 2150 2168 2177 2179 2197 2200 2211
## [463] 2214 2216 2221 2240 2252 2254 2263 2265 2269 2275 2277 2281 2282 2286
## [477] 2337 2342 2345 2347 2349 2350 2359 2363 2369 2381 2383 2386 2387 2398
## [491] 2400 2401 2416 2417 2423 2425 2427 2429 2434 2464 2482 2489 2491 2493
##
## $span
## [1] 2514
##
## $threshold
## [1] 1
##
## $p.less.thresh
## [1] 0.7996024
##
## $n.exceed
## [1] 504
##
## $run
## [1] NA
##
## $par.ests
## xi sigma mu beta
## 0.1073161 0.8913773 -0.5633651 1.0591516
##
## $par.ses
## xi sigma mu
## 0.05468298 0.13084522 0.16921433
##
## $varcov
## [,1] [,2] [,3]
## [1,] 0.002990229 -0.006738203 0.00795378
## [2,] -0.006738203 0.017120472 -0.02111842
## [3,] 0.007953780 -0.021118424 0.02863349
##
## $intensity
## [1] 0.2004773
##
## $nllh.final
## [1] 1900.967
##
## $converged
## [1] 0
##
## attr(,"class")
## [1] "potd"
## Model checking
#plot(m3)
riskmeasures(m3,c(.95,.99,.999))
## p quantile sfall
## [1,] 0.950 2.585581 3.962676
## [2,] 0.990 4.745218 6.381938
## [3,] 0.999 8.561586 10.657100
8. Generalized Pareto distribution
(m4=gpd(xt,1.0)) ### Threshold 1%
## $n
## [1] 2515
##
## $data
## [1] 1.530858 1.074553 1.139063 2.445357 4.033565 3.370467 2.432958
## [8] 1.857648 1.005943 1.535935 3.090776 3.644098 4.603870 2.607296
## [15] 2.657090 3.600134 1.009276 6.981857 3.902361 3.548315 5.883429
## [22] 4.754874 5.251088 1.593021 4.615808 2.010883 1.649022 1.270436
## [29] 2.495377 2.659349 1.484058 1.533295 2.986866 1.516440 2.171099
## [36] 2.299129 1.874968 1.535631 1.841349 1.205335 5.124593 1.685526
## [43] 2.670955 3.994738 1.273981 1.233982 1.555739 1.823627 1.126825
## [50] 1.102960 2.145860 1.803567 3.687666 1.129657 2.369656 1.449251
## [57] 3.298301 2.745650 3.154124 3.391983 1.434135 1.390320 1.606638
## [64] 2.311511 2.034251 1.847970 1.811407 1.271145 1.043324 2.423634
## [71] 1.591091 1.241778 1.905745 1.509638 1.896368 1.227503 4.826945
## [78] 3.337376 2.381023 1.039383 4.879011 1.117319 2.584000 4.745121
## [85] 3.310084 2.994800 1.176797 2.466674 1.224061 1.868651 1.111252
## [92] 1.950703 1.102253 1.874357 3.625017 10.667471 5.567234 1.625642
## [99] 1.209485 2.333516 2.511578 7.329891 3.104082 1.153123 1.457673
## [106] 1.191166 1.492990 1.234995 2.212803 2.951738 1.111049 1.722042
## [113] 1.934287 1.720415 1.118937 1.567826 3.470118 2.442692 4.033878
## [120] 1.590786 6.305802 3.038911 2.312636 1.314806 2.842828 4.983226
## [127] 2.139525 4.346925 1.955190 3.101606 2.823796 1.490046 1.958351
## [134] 1.226390 1.855407 2.467289 1.631639 4.102618 2.124713 1.869262
## [141] 3.190561 3.114191 7.074011 1.367305 5.882156 1.583674 2.713073
## [148] 3.455314 2.549834 5.833593 3.532256 5.369407 1.413037 3.408640
## [155] 3.227942 1.737609 1.042819 1.028269 1.331323 2.423122 1.799901
## [162] 3.372536 1.571687 1.618020 1.480201 1.592310 1.462848 1.445294
## [169] 2.127062 1.123892 1.773840 5.678201 1.048477 2.574457 2.547064
## [176] 1.404010 1.156664 1.753891 2.134724 2.864745 3.171051 2.303119
## [183] 3.038911 1.376835 1.713090 1.073845 1.561427 1.206246 1.432715
## [190] 1.017358 2.895827 4.102201 2.828940 2.284804 1.439714 2.070283
## [197] 1.372476 1.108623 1.109027 1.699154 4.010663 1.467921 1.216267
## [204] 1.032209 1.590278 1.224770 1.098006 1.782696 2.047009 2.146779
## [211] 1.260005 3.802279 1.216267 1.136231 1.226693 1.986501 2.041497
## [218] 1.057168 1.447729 1.336390 1.397013 1.947338 1.567319 2.008027
## [225] 1.599016 1.333552 1.814768 1.413037 1.482028 1.026854 1.496543
## [232] 1.023621 1.551473 2.011904 1.322507 1.234995 1.415978 1.136939
## [239] 1.812324 1.541419 1.784223 1.658579 1.193089 1.079607 1.328181
## [246] 1.620560 1.290998 1.080011 1.914918 1.530858 1.023722 1.034028
## [253] 1.611109 1.385656 1.105791 8.662054 1.538169 4.706226 1.490655
## [260] 2.071100 2.266083 2.030884 2.037415 1.047871 2.384812 1.873949
## [267] 1.124499 1.788397 1.465182 1.348654 1.126724 1.459093 1.209384
## [274] 1.195214 1.087189 1.384642 1.168096 2.652366 1.446512 2.104081
## [281] 1.563357 1.100332 1.423686 1.491264 1.599626 1.151504 1.755825
## [288] 1.853064 1.089210 1.460615 2.715026 1.427744 2.161799 1.303053
## [295] 1.656444 1.654207 1.102253 2.193584 2.056400 1.186208 3.353714
## [302] 1.079910 3.091395 1.091536 1.495933 1.005640 1.498776 1.356864
## [309] 2.417795 1.055248 1.555942 3.294373 2.047111 1.338316 3.437718
## [316] 1.190458 2.011598 1.394174 1.526492 1.277526 1.775570 1.157068
## [323] 1.669865 3.246121 2.219551 1.519994 2.150049 1.043930 1.504562
## [330] 4.577486 5.676720 1.753688 2.483075 1.167084 2.019250 2.181728
## [337] 2.218426 2.252990 1.179327 1.772517 1.824137 3.205326 3.660071
## [344] 1.073643 2.489533 2.277540 1.074452 2.130842 2.260866 1.059897
## [351] 2.733214 1.214040 2.134826 1.098714 1.053126 1.204728 2.518653
## [358] 1.249271 1.196125 1.452701 2.368325 1.135725 1.765595 1.045951
## [365] 1.041101 1.254030 1.612227 2.786155 2.159040 1.278741 1.840534
## [372] 2.808368 1.274183 2.849413 1.137242 1.789822 1.052520 1.730589
## [379] 1.095984 1.410704 1.642719 1.670780 2.314274 2.765181 2.862995
## [386] 1.936938 3.228872 4.026590 2.246342 4.242119 6.017047 5.018023
## [393] 1.248967 2.764050 5.065560 5.479310 1.726621 1.414456 5.840378
## [400] 4.025445 6.101832 2.740203 2.980478 3.774860 4.887307 2.821431
## [407] 1.356458 3.693166 4.717023 3.612367 5.268795 5.729019 5.932334
## [414] 4.086366 2.590465 2.790166 2.166807 1.848887 1.709835 1.626964
## [421] 2.885946 2.551578 3.523451 1.487712 2.466366 3.735509 1.302242
## [428] 3.436476 2.859908 5.106174 3.291686 1.447831 2.271709 1.927457
## [435] 2.752845 1.038676 4.800607 2.805797 1.735676 1.108016 1.330005
## [442] 1.959473 1.077989 1.629505 1.190357 1.393464 1.189142 2.010169
## [449] 1.431294 1.093760 1.302242 1.341255 3.011288 1.446715 1.232059
## [456] 1.263954 1.452701 1.018267 1.158788 1.455846 1.439917 5.080815
## [463] 1.600439 1.917365 1.856425 1.249170 1.114487 1.023015 1.267701
## [470] 1.215356 1.052520 2.942881 2.750687 2.063443 1.105084 2.139525
## [477] 1.939589 1.471778 1.125409 1.148571 2.293501 1.479592 4.005874
## [484] 1.041404 2.116644 1.038676 1.486697 3.062417 1.295455 2.079268
## [491] 2.527986 1.016650 1.536342 1.185499 1.092041 1.249170 2.007720
## [498] 1.072026 1.285831 3.418366 1.168905 1.531772 1.172750 1.005842
##
## $threshold
## [1] 1
##
## $p.less.thresh
## [1] 0.7996024
##
## $n.exceed
## [1] 504
##
## $method
## [1] "ml"
##
## $par.ests
## xi beta
## 0.1072777 1.0591719
##
## $par.ses
## xi beta
## 0.05465781 0.07450330
##
## $varcov
## [,1] [,2]
## [1,] 0.002987476 -0.002914008
## [2,] -0.002914008 0.005550741
##
## $information
## [1] "observed"
##
## $converged
## [1] 0
##
## $nllh.final
## [1] 587.0113
##
## attr(,"class")
## [1] "gpd"
names(m4)
## [1] "n" "data" "threshold" "p.less.thresh"
## [5] "n.exceed" "method" "par.ests" "par.ses"
## [9] "varcov" "information" "converged" "nllh.final"
par(mfcol=c(2,2))
#plot(m4)
riskmeasures(m4,c(.95,.99,.999))
## p quantile sfall
## [1,] 0.950 2.585569 3.962557
## [2,] 0.990 4.745063 6.381556
## [3,] 0.999 8.560890 10.655928
m5=gpd(xt,1.2) ### Threshold 1.2%
riskmeasures(m5,c(.95,.99,.999))
## p quantile sfall
## [1,] 0.950 2.611388 3.960077
## [2,] 0.990 4.745530 6.266801
## [3,] 0.999 8.282208 10.089481
m6=gpd(xt,0.8) ### Threshold 0.8%
riskmeasures(m6,c(.95,.99,.999))
## p quantile sfall
## [1,] 0.950 2.586713 3.961958
## [2,] 0.990 4.744087 6.375946
## [3,] 0.999 8.548816 10.633237