Em xin gửi lời cảm ơn chân thành đến thầy Nguyễn Tuấn Duy đã hướng dẫn, hỗ trợ cho em trong suốt quá trình học tập và thực hiện bài tiểu luận này.
Vì kiến thức bản thân còn nhiều hạn chế nên không thể tránh khỏi những sai sót nên em rất mong nhận được những góp ý từ thầy cũng như các bạn để có thể hoàn thiện hơn kiến thức của mình.Một lần nữa, em xin chân thành cảm ơn!
Chuẩn bị dữ liệu
library(nortest)
library(kSamples)## Loading required package: SuppDists##
## Attaching package: 'kSamples'## The following object is masked from 'package:nortest':
##
## ad.testlibrary(VineCopula)
library(FinTS)## Loading required package: zoo##
## Attaching package: 'zoo'## The following objects are masked from 'package:base':
##
## as.Date, as.Date.numericlibrary(goftest)##
## Attaching package: 'goftest'## The following object is masked from 'package:kSamples':
##
## ad.test## The following objects are masked from 'package:nortest':
##
## ad.test, cvm.testlibrary(readxl)
library(rugarch)## Loading required package: parallel##
## Attaching package: 'rugarch'## The following object is masked from 'package:stats':
##
## sigmalibrary(fGarch)## NOTE: Packages 'fBasics', 'timeDate', and 'timeSeries' are no longer
## attached to the search() path when 'fGarch' is attached.
##
## If needed attach them yourself in your R script by e.g.,
## require("timeSeries")sp500 <- read_excel("d:/thayduy/sp500.xlsx")
vni <- read_excel("d:/thayduy/vni.xlsx")
vni.ts <- ts(vni[, 2])
sp500.ts <- ts(sp500[, 2])Cơ sở lý thuyết mô hình Garch: Mô hình GARCH (Generalized Autoregressive Conditional Heteroskedasticity) là một mô hình thống kê được phát triển bởi Robert F. Engle vào những năm 1980 để mô hình hóa sự biến đổi không đều của dữ liệu tài chính. Mô hình này được sử dụng rộng rãi trong lĩnh vực tài chính để phân tích và dự đoán rủi ro.
Mô hình GARCH kết hợp các thành phần autoregressive (AR) và moving average (MA) để mô hình hóa sự phụ thuộc của phương sai có điều kiện vào các giá trị đã qua xử lý cũng như các giá trị độ biến động trước đó. Một mô hình GARCH(p, q) có thể được biểu diễn như sau:
σ²(t) = ω + α₁ε²(t-1) + … + αₚε²(t-p) + β₁σ²(t-1) + … + β_qσ²(t-q)
Trong đó:
σ²(t) là phương sai có điều kiện của biến ngẫu nhiên tại thời điểm t. ε(t) là biến ngẫu nhiên có kỳ vọng bằng 0 và phương sai đẳng nhất. ω, α₁, …, αₚ, β₁, …, β_q là các tham số của mô hình. p là số lags cho thành phần autoregressive (AR) và q là số lags cho thành phần moving average (MA). Thường thì các tham số α và β trong mô hình GARCH có giá trị từ 0 đến 1 để đảm bảo tính chất dương và tích cực của phương sai. Trong trường hợp p = 0 và q = 0, mô hình GARCH trở thành mô hình ARCH.
Mô hình GARCH được ước lượng thông qua các phương pháp tối đa hóa hàm hợp lý hoặc phương pháp bình phương tối thiểu. Các giá trị ước lượng của các tham số được xác định để tối thiểu hóa sai số giữa dữ liệu quan sát và dữ liệu được dự đoán từ mô hình GARCH.
Mô hình GARCH được sử dụng rộng rãi trong các ứng dụng tài chính như dự báo giá trị tài sản, quản lý rủi ro tài chính, và đánh giá tác động của biến động thị trường lên các quyết định đầu tư và giao dịch.
sp500.garch11n.spec <- ugarchspec(variance.model = list(model = "gjrGARCH", garchOrder = c(1, 1)),
mean.model = list(armaOrder = c(2, 2)),
distribution.model = "norm")
sp500.garch11n.fit <- ugarchfit(spec = sp500.garch11n.spec, data = sp500.ts)
sp500.garch11n.fit##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : gjrGARCH(1,1)
## Mean Model : ARFIMA(2,0,2)
## Distribution : norm
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## mu 4520.631354 6.465239 699.2211 0.000000
## ar1 0.021776 0.000151 144.2794 0.000000
## ar2 0.979536 0.000012 79890.4707 0.000000
## ma1 0.935717 0.000006 154944.8689 0.000000
## ma2 -0.047631 0.000164 -290.8071 0.000000
## omega 4.147176 0.638125 6.4990 0.000000
## alpha1 0.003724 0.002228 1.6716 0.094597
## beta1 0.886552 0.006688 132.5546 0.000000
## gamma1 0.217449 0.015771 13.7882 0.000000
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## mu 4520.631354 6.000575 753.3664 0.00000
## ar1 0.021776 0.000145 150.4695 0.00000
## ar2 0.979536 0.000015 65776.2258 0.00000
## ma1 0.935717 0.000007 139710.3740 0.00000
## ma2 -0.047631 0.000127 -375.3213 0.00000
## omega 4.147176 0.817447 5.0733 0.00000
## alpha1 0.003724 0.002537 1.4679 0.14214
## beta1 0.886552 0.009959 89.0202 0.00000
## gamma1 0.217449 0.020233 10.7474 0.00000
##
## LogLikelihood : -15965.26
##
## Information Criteria
## ------------------------------------
##
## Akaike 8.7291
## Bayes 8.7444
## Shibata 8.7291
## Hannan-Quinn 8.7345
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.002854 0.9574
## Lag[2*(p+q)+(p+q)-1][11] 5.825179 0.6023
## Lag[4*(p+q)+(p+q)-1][19] 9.597926 0.5363
## d.o.f=4
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 6.521 0.01066
## Lag[2*(p+q)+(p+q)-1][5] 7.557 0.03792
## Lag[4*(p+q)+(p+q)-1][9] 9.004 0.08112
## d.o.f=2
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[3] 0.8113 0.500 2.000 0.3677
## ARCH Lag[5] 2.0502 1.440 1.667 0.4603
## ARCH Lag[7] 2.7434 2.315 1.543 0.5633
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 3.7051
## Individual Statistics:
## mu 0.00486
## ar1 0.45221
## ar2 0.45648
## ma1 0.08847
## ma2 0.09187
## omega 0.07301
## alpha1 0.52322
## beta1 0.42949
## gamma1 0.41921
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 2.1 2.32 2.82
## Individual Statistic: 0.35 0.47 0.75
##
## Sign Bias Test
## ------------------------------------
## t-value prob sig
## Sign Bias 0.9815 0.3264
## Negative Sign Bias 1.3917 0.1641
## Positive Sign Bias 0.2208 0.8253
## Joint Effect 2.0807 0.5558
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 96.77 2.054e-12
## 2 30 105.03 1.508e-10
## 3 40 107.76 2.321e-08
## 4 50 129.48 3.437e-09
##
##
## Elapsed time : 2.999571# Xây dựng mô hình GARCH(1,1) với phân phốichuẩn cho Vni Index
vni.garch11n.spec <- ugarchspec(variance.model = list(model = "gjrGARCH", garchOrder = c(1, 1)),
mean.model = list(armaOrder = c(2, 2)),
distribution.model = "norm")
vni.garch11n.fit <- ugarchfit(spec = vni.garch11n.spec, data = vni.ts)
vni.garch11n.fit##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : gjrGARCH(1,1)
## Mean Model : ARFIMA(2,0,2)
## Distribution : norm
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## mu 1218.81786 4.018580 303.2957 0
## ar1 0.12549 0.000500 251.0812 0
## ar2 0.87576 0.000541 1619.3475 0
## ma1 0.98418 0.016982 57.9557 0
## ma2 0.10122 0.016728 6.0512 0
## omega 1.05202 0.200794 5.2393 0
## alpha1 0.06261 0.011805 5.3035 0
## beta1 0.85373 0.011874 71.8974 0
## gamma1 0.16531 0.022356 7.3945 0
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## mu 1218.81786 5.847442 208.4361 0.000000
## ar1 0.12549 0.000307 408.4271 0.000000
## ar2 0.87576 0.000521 1681.6757 0.000000
## ma1 0.98418 0.016212 60.7073 0.000000
## ma2 0.10122 0.015002 6.7476 0.000000
## omega 1.05202 0.254376 4.1357 0.000035
## alpha1 0.06261 0.017195 3.6413 0.000271
## beta1 0.85373 0.015228 56.0629 0.000000
## gamma1 0.16531 0.032149 5.1421 0.000000
##
## LogLikelihood : -12587.34
##
## Information Criteria
## ------------------------------------
##
## Akaike 6.8832
## Bayes 6.8985
## Shibata 6.8832
## Hannan-Quinn 6.8887
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 3.182 0.074476
## Lag[2*(p+q)+(p+q)-1][11] 8.232 0.000397
## Lag[4*(p+q)+(p+q)-1][19] 14.666 0.036535
## d.o.f=4
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.0001076 0.9917
## Lag[2*(p+q)+(p+q)-1][5] 2.4899503 0.5080
## Lag[4*(p+q)+(p+q)-1][9] 5.1409573 0.4090
## d.o.f=2
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[3] 0.2272 0.500 2.000 0.6336
## ARCH Lag[5] 1.0371 1.440 1.667 0.7220
## ARCH Lag[7] 2.4705 2.315 1.543 0.6185
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 3.9993
## Individual Statistics:
## mu 0.0008057
## ar1 0.4150025
## ar2 0.4440455
## ma1 0.6219894
## ma2 0.2453643
## omega 0.2385878
## alpha1 0.5298608
## beta1 0.7848942
## gamma1 1.2214374
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 2.1 2.32 2.82
## Individual Statistic: 0.35 0.47 0.75
##
## Sign Bias Test
## ------------------------------------
## t-value prob sig
## Sign Bias 2.115 0.03446 **
## Negative Sign Bias 2.577 0.01001 **
## Positive Sign Bias 1.260 0.20765
## Joint Effect 8.339 0.03950 **
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 78.86 2.917e-09
## 2 30 102.52 3.844e-10
## 3 40 116.61 1.153e-09
## 4 50 125.25 1.331e-08
##
##
## Elapsed time : 2.025054sp500.garch11t.spec <- ugarchspec(variance.model = list(model = "gjrGARCH", garchOrder = c(1, 1)),
mean.model = list(armaOrder = c(2, 2)),
distribution.model = "std")
sp500.garch11t.fit <- ugarchfit(spec = sp500.garch11t.spec, data = sp500.ts)
sp500.garch11t.fit##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : gjrGARCH(1,1)
## Mean Model : ARFIMA(2,0,2)
## Distribution : std
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## mu 4509.084120 14.735835 3.0599e+02 0.00000
## ar1 1.737506 0.000002 9.9921e+05 0.00000
## ar2 -0.737343 0.000005 -1.4902e+05 0.00000
## ma1 -0.773426 0.016201 -4.7740e+01 0.00000
## ma2 0.009184 0.016244 5.6541e-01 0.57180
## omega 3.909650 0.688356 5.6797e+00 0.00000
## alpha1 0.000000 0.000009 5.9000e-04 0.99953
## beta1 0.886838 0.007950 1.1156e+02 0.00000
## gamma1 0.224325 0.017706 1.2670e+01 0.00000
## shape 8.839162 0.665206 1.3288e+01 0.00000
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## mu 4509.084120 6.032684 7.4744e+02 0.000000
## ar1 1.737506 0.000002 9.5777e+05 0.000000
## ar2 -0.737343 0.000005 -1.4273e+05 0.000000
## ma1 -0.773426 0.015222 -5.0811e+01 0.000000
## ma2 0.009184 0.015388 5.9686e-01 0.550598
## omega 3.909650 0.914551 4.2749e+00 0.000019
## alpha1 0.000000 0.000022 2.4600e-04 0.999804
## beta1 0.886838 0.010834 8.1859e+01 0.000000
## gamma1 0.224325 0.019700 1.1387e+01 0.000000
## shape 8.839162 0.682263 1.2956e+01 0.000000
##
## LogLikelihood : -15929.5
##
## Information Criteria
## ------------------------------------
##
## Akaike 8.7101
## Bayes 8.7271
## Shibata 8.7101
## Hannan-Quinn 8.7161
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.1658 0.6839
## Lag[2*(p+q)+(p+q)-1][11] 3.3172 1.0000
## Lag[4*(p+q)+(p+q)-1][19] 6.9775 0.9103
## d.o.f=4
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 6.043 0.01396
## Lag[2*(p+q)+(p+q)-1][5] 7.063 0.05007
## Lag[4*(p+q)+(p+q)-1][9] 8.501 0.10261
## d.o.f=2
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[3] 0.757 0.500 2.000 0.3843
## ARCH Lag[5] 2.018 1.440 1.667 0.4672
## ARCH Lag[7] 2.649 2.315 1.543 0.5822
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 6.0365
## Individual Statistics:
## mu 0.00640
## ar1 0.57471
## ar2 0.57428
## ma1 0.12334
## ma2 0.04307
## omega 0.08502
## alpha1 1.13609
## beta1 0.60841
## gamma1 0.84416
## shape 0.26746
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 2.29 2.54 3.05
## Individual Statistic: 0.35 0.47 0.75
##
## Sign Bias Test
## ------------------------------------
## t-value prob sig
## Sign Bias 0.6034 0.5463
## Negative Sign Bias 1.3199 0.1870
## Positive Sign Bias 0.1487 0.8818
## Joint Effect 1.7725 0.6209
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 53.54 3.877e-05
## 2 30 64.54 1.628e-04
## 3 40 72.28 9.430e-04
## 4 50 91.86 2.024e-04
##
##
## Elapsed time : 4.729817vni.garch11t.spec <- ugarchspec(variance.model = list(model = "gjrGARCH", garchOrder = c(1, 1)),
mean.model = list(armaOrder = c(2, 2)),
distribution.model = "std")
vni.garch11t.fit <- ugarchfit(spec = vni.garch11t.spec, data = vni.ts)
vni.garch11t.fit##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : gjrGARCH(1,1)
## Mean Model : ARFIMA(2,0,2)
## Distribution : std
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## mu 1219.321520 1.789390 681.4173 0e+00
## ar1 1.984264 0.000059 33812.3159 0e+00
## ar2 -0.984253 0.000027 -37132.7919 0e+00
## ma1 -0.878670 0.000071 -12357.2706 0e+00
## ma2 -0.097394 0.000319 -305.6148 0e+00
## omega 1.056696 0.235911 4.4792 7e-06
## alpha1 0.056433 0.003214 17.5565 0e+00
## beta1 0.862466 0.013455 64.1006 0e+00
## gamma1 0.160201 0.017604 9.1003 0e+00
## shape 6.844339 0.760206 9.0033 0e+00
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## mu 1219.321520 4.274749 285.23818 0.000000
## ar1 1.984264 0.000282 7026.71468 0.000000
## ar2 -0.984253 0.001179 -834.98721 0.000000
## ma1 -0.878670 0.000208 -4220.28123 0.000000
## ma2 -0.097394 0.002018 -48.26073 0.000000
## omega 1.056696 0.808014 1.30777 0.190952
## alpha1 0.056433 0.313161 0.18021 0.856991
## beta1 0.862466 0.082852 10.40976 0.000000
## gamma1 0.160201 0.425245 0.37673 0.706376
## shape 6.844339 1.711906 3.99808 0.000064
##
## LogLikelihood : -12526.66
##
## Information Criteria
## ------------------------------------
##
## Akaike 6.8506
## Bayes 6.8676
## Shibata 6.8506
## Hannan-Quinn 6.8567
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 2.689 0.101029
## Lag[2*(p+q)+(p+q)-1][11] 7.707 0.004239
## Lag[4*(p+q)+(p+q)-1][19] 14.311 0.046869
## d.o.f=4
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.04464 0.8327
## Lag[2*(p+q)+(p+q)-1][5] 2.45620 0.5152
## Lag[4*(p+q)+(p+q)-1][9] 5.23275 0.3960
## d.o.f=2
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[3] 0.5892 0.500 2.000 0.4427
## ARCH Lag[5] 1.2254 1.440 1.667 0.6674
## ARCH Lag[7] 2.5761 2.315 1.543 0.5969
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 6.6162
## Individual Statistics:
## mu 0.002072
## ar1 0.437587
## ar2 0.437624
## ma1 0.080884
## ma2 0.122955
## omega 0.261107
## alpha1 0.238379
## beta1 0.441295
## gamma1 0.733527
## shape 0.800464
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 2.29 2.54 3.05
## Individual Statistic: 0.35 0.47 0.75
##
## Sign Bias Test
## ------------------------------------
## t-value prob sig
## Sign Bias 2.0677 0.038738 **
## Negative Sign Bias 2.8702 0.004125 ***
## Positive Sign Bias 0.9206 0.357307
## Joint Effect 9.1554 0.027295 **
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 39.46 0.003845
## 2 30 49.52 0.010157
## 3 40 56.42 0.035136
## 4 50 68.33 0.035292
##
##
## Elapsed time : 3.140952sp500.garch11st.spec <- ugarchspec(variance.model = list(model = "sGARCH", garchOrder = c(1, 1)),
mean.model = list(armaOrder = c(2, 2)),
distribution.model = "sstd")
sp500.garch11st.fit <- ugarchfit(spec = sp500.garch11st.spec, data = sp500.ts)
sp500.garch11st.fit##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : sGARCH(1,1)
## Mean Model : ARFIMA(2,0,2)
## Distribution : sstd
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## mu 4504.667828 14.868568 302.9658 0.000000
## ar1 1.801695 0.000183 9824.7438 0.000000
## ar2 -0.801580 0.000097 -8276.5352 0.000000
## ma1 -0.851305 0.017353 -49.0570 0.000000
## ma2 0.017647 0.016806 1.0500 0.293704
## omega 6.164229 1.350358 4.5649 0.000005
## alpha1 0.166313 0.016009 10.3890 0.000000
## beta1 0.832687 0.014166 58.7789 0.000000
## skew 0.954689 0.025497 37.4430 0.000000
## shape 6.574201 0.725328 9.0638 0.000000
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## mu 4504.667828 18.305859 246.07793 0.000000
## ar1 1.801695 0.002932 614.57806 0.000000
## ar2 -0.801580 0.001299 -617.19825 0.000000
## ma1 -0.851305 0.063024 -13.50761 0.000000
## ma2 0.017647 0.022140 0.79707 0.425412
## omega 6.164229 2.026640 3.04160 0.002353
## alpha1 0.166313 0.039495 4.21097 0.000025
## beta1 0.832687 0.027519 30.25852 0.000000
## skew 0.954689 0.266620 3.58072 0.000343
## shape 6.574201 1.116755 5.88688 0.000000
##
## LogLikelihood : -16005.11
##
## Information Criteria
## ------------------------------------
##
## Akaike 8.7514
## Bayes 8.7684
## Shibata 8.7514
## Hannan-Quinn 8.7575
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.08554 0.7699
## Lag[2*(p+q)+(p+q)-1][11] 4.43616 0.9976
## Lag[4*(p+q)+(p+q)-1][19] 8.00025 0.7943
## d.o.f=4
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 11.01 0.000907
## Lag[2*(p+q)+(p+q)-1][5] 11.49 0.003785
## Lag[4*(p+q)+(p+q)-1][9] 14.54 0.004620
## d.o.f=2
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[3] 0.00454 0.500 2.000 0.9463
## ARCH Lag[5] 1.37907 1.440 1.667 0.6247
## ARCH Lag[7] 4.38398 2.315 1.543 0.2946
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 4.9619
## Individual Statistics:
## mu 0.001412
## ar1 0.444725
## ar2 0.444392
## ma1 0.106257
## ma2 0.079719
## omega 0.290701
## alpha1 2.560262
## beta1 1.898357
## skew 0.104883
## shape 1.252211
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 2.29 2.54 3.05
## Individual Statistic: 0.35 0.47 0.75
##
## Sign Bias Test
## ------------------------------------
## t-value prob sig
## Sign Bias 2.3205 0.020370 **
## Negative Sign Bias 0.4806 0.630832
## Positive Sign Bias 2.2623 0.023739 **
## Joint Effect 19.8309 0.000184 ***
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 54.14 3.142e-05
## 2 30 68.66 4.620e-05
## 3 40 70.03 1.665e-03
## 4 50 84.13 1.333e-03
##
##
## Elapsed time : 8.141798vni.garch11st.spec <- ugarchspec(variance.model = list(model = "sGARCH", garchOrder = c(1, 1)),
mean.model = list(armaOrder = c(2, 2)),
distribution.model = "sstd")
vni.garch11st.fit <- ugarchfit(spec = vni.garch11st.spec, data = vni.ts)
vni.garch11st.fit##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : sGARCH(1,1)
## Mean Model : ARFIMA(2,0,2)
## Distribution : sstd
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## mu 1219.383003 4.517547 269.9215 0.0e+00
## ar1 1.904015 0.000200 9527.3446 0.0e+00
## ar2 -0.903995 0.000114 -7947.0212 0.0e+00
## ma1 -0.798995 0.017466 -45.7468 0.0e+00
## ma2 -0.068383 0.017202 -3.9753 7.0e-05
## omega 0.954583 0.226404 4.2163 2.5e-05
## alpha1 0.122886 0.014240 8.6295 0.0e+00
## beta1 0.873309 0.012927 67.5545 0.0e+00
## skew 1.117406 0.030538 36.5903 0.0e+00
## shape 6.092223 0.634124 9.6073 0.0e+00
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## mu 1219.383003 1.976433 616.9616 0.000000
## ar1 1.904015 0.002546 747.7583 0.000000
## ar2 -0.903995 0.001228 -735.9860 0.000000
## ma1 -0.798995 0.057811 -13.8208 0.000000
## ma2 -0.068383 0.049071 -1.3935 0.163456
## omega 0.954583 0.374139 2.5514 0.010729
## alpha1 0.122886 0.017866 6.8781 0.000000
## beta1 0.873309 0.019324 45.1939 0.000000
## skew 1.117406 0.243558 4.5878 0.000004
## shape 6.092223 0.854340 7.1309 0.000000
##
## LogLikelihood : -12540.08
##
## Information Criteria
## ------------------------------------
##
## Akaike 6.8580
## Bayes 6.8749
## Shibata 6.8580
## Hannan-Quinn 6.8640
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 3.026 8.193e-02
## Lag[2*(p+q)+(p+q)-1][11] 11.454 2.252e-13
## Lag[4*(p+q)+(p+q)-1][19] 18.653 1.384e-03
## d.o.f=4
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.4601 0.49756
## Lag[2*(p+q)+(p+q)-1][5] 5.4380 0.12160
## Lag[4*(p+q)+(p+q)-1][9] 8.6764 0.09461
## d.o.f=2
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[3] 0.3733 0.500 2.000 0.5412
## ARCH Lag[5] 1.0711 1.440 1.667 0.7119
## ARCH Lag[7] 2.4472 2.315 1.543 0.6233
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 5.7612
## Individual Statistics:
## mu 0.0003554
## ar1 0.2544755
## ar2 0.2551012
## ma1 0.1495863
## ma2 0.0674933
## omega 0.3777895
## alpha1 0.3363812
## beta1 0.4972430
## skew 0.2949211
## shape 0.8519623
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 2.29 2.54 3.05
## Individual Statistic: 0.35 0.47 0.75
##
## Sign Bias Test
## ------------------------------------
## t-value prob sig
## Sign Bias 2.961 3.091e-03 ***
## Negative Sign Bias 5.033 5.072e-07 ***
## Positive Sign Bias 2.117 3.429e-02 **
## Joint Effect 30.289 1.200e-06 ***
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 39.15 0.004227
## 2 30 56.97 0.001450
## 3 40 56.68 0.033372
## 4 50 62.05 0.099824
##
##
## Elapsed time : 2.658036sp500.garch11g.spec <- ugarchspec(variance.model = list(model = "gjrGARCH", garchOrder = c(1, 1)),
mean.model = list(armaOrder = c(2, 2)),
distribution.model = "ged")
sp500.garch11g.fit <- ugarchfit(spec = sp500.garch11g.spec, data = sp500.ts)
sp500.garch11g.fit##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : gjrGARCH(1,1)
## Mean Model : ARFIMA(2,0,2)
## Distribution : ged
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## mu 4512.251099 17.929761 251.662653 0.000000
## ar1 0.543276 0.000107 5093.113106 0.000000
## ar2 0.457616 0.000164 2786.922737 0.000000
## ma1 0.422435 0.016086 26.261223 0.000000
## ma2 -0.017651 0.016967 -1.040311 0.298195
## omega 3.900803 0.809438 4.819150 0.000001
## alpha1 0.000000 0.001899 0.000017 0.999987
## beta1 0.886213 0.008817 100.507575 0.000000
## gamma1 0.225573 0.018163 12.419281 0.000000
## shape 1.435296 0.048326 29.700479 0.000000
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## mu 4512.251099 3.418087 1.3201e+03 0.00000
## ar1 0.543276 0.000051 1.0600e+04 0.00000
## ar2 0.457616 0.000076 6.0485e+03 0.00000
## ma1 0.422435 0.014773 2.8595e+01 0.00000
## ma2 -0.017651 0.017640 -1.0006e+00 0.31700
## omega 3.900803 0.980849 3.9770e+00 0.00007
## alpha1 0.000000 0.002029 1.6000e-05 0.99999
## beta1 0.886213 0.011836 7.4871e+01 0.00000
## gamma1 0.225573 0.019445 1.1601e+01 0.00000
## shape 1.435296 0.052061 2.7569e+01 0.00000
##
## LogLikelihood : -15918.89
##
## Information Criteria
## ------------------------------------
##
## Akaike 8.7043
## Bayes 8.7213
## Shibata 8.7043
## Hannan-Quinn 8.7103
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.2449 0.6207
## Lag[2*(p+q)+(p+q)-1][11] 6.3586 0.2687
## Lag[4*(p+q)+(p+q)-1][19] 10.4916 0.3889
## d.o.f=4
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 6.396 0.01144
## Lag[2*(p+q)+(p+q)-1][5] 7.510 0.03893
## Lag[4*(p+q)+(p+q)-1][9] 8.913 0.08468
## d.o.f=2
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[3] 0.9679 0.500 2.000 0.3252
## ARCH Lag[5] 2.1829 1.440 1.667 0.4322
## ARCH Lag[7] 2.8114 2.315 1.543 0.5499
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 4.0045
## Individual Statistics:
## mu 0.008951
## ar1 0.405633
## ar2 0.406207
## ma1 0.151596
## ma2 0.027755
## omega 0.076584
## alpha1 0.903002
## beta1 0.564906
## gamma1 0.742678
## shape 0.265892
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 2.29 2.54 3.05
## Individual Statistic: 0.35 0.47 0.75
##
## Sign Bias Test
## ------------------------------------
## t-value prob sig
## Sign Bias 0.5616 0.5744
## Negative Sign Bias 1.2802 0.2005
## Positive Sign Bias 0.1264 0.8994
## Joint Effect 1.6660 0.6445
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 35.07 0.013694
## 2 30 45.38 0.027034
## 3 40 60.70 0.014579
## 4 50 79.23 0.004023
##
##
## Elapsed time : 7.253886vni.garch11g.spec <- ugarchspec(variance.model = list(model = "gjrGARCH", garchOrder = c(1, 1)),
mean.model = list(armaOrder = c(2, 2)),
distribution.model = "ged")
vni.garch11g.fit <- ugarchfit(spec = vni.garch11g.spec, data = vni.ts)
vni.garch11g.fit##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : gjrGARCH(1,1)
## Mean Model : ARFIMA(2,0,2)
## Distribution : ged
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## mu 1223.166910 4.272875 286.2632 0e+00
## ar1 1.896661 0.000078 24437.8028 0e+00
## ar2 -0.896589 0.000035 -25449.5471 0e+00
## ma1 -0.790893 0.016135 -49.0182 0e+00
## ma2 -0.071190 0.015651 -4.5486 5e-06
## omega 1.026726 0.226976 4.5235 6e-06
## alpha1 0.057609 0.002442 23.5905 0e+00
## beta1 0.859044 0.013476 63.7468 0e+00
## gamma1 0.164692 0.018048 9.1250 0e+00
## shape 1.398839 0.045859 30.5029 0e+00
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## mu 1223.166910 4.739653 258.07100 0.000000
## ar1 1.896661 0.000086 21947.93339 0.000000
## ar2 -0.896589 0.000317 -2826.69869 0.000000
## ma1 -0.790893 0.018982 -41.66489 0.000000
## ma2 -0.071190 0.017268 -4.12268 0.000037
## omega 1.026726 0.265229 3.87109 0.000108
## alpha1 0.057609 0.066113 0.87136 0.383558
## beta1 0.859044 0.021328 40.27692 0.000000
## gamma1 0.164692 0.089409 1.84200 0.065476
## shape 1.398839 0.048510 28.83619 0.000000
##
## LogLikelihood : -12525.93
##
## Information Criteria
## ------------------------------------
##
## Akaike 6.8502
## Bayes 6.8672
## Shibata 6.8502
## Hannan-Quinn 6.8563
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 2.613 0.1060078
## Lag[2*(p+q)+(p+q)-1][11] 8.468 0.0001217
## Lag[4*(p+q)+(p+q)-1][19] 15.386 0.0215209
## d.o.f=4
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.02935 0.8640
## Lag[2*(p+q)+(p+q)-1][5] 2.53775 0.4979
## Lag[4*(p+q)+(p+q)-1][9] 5.32873 0.3827
## d.o.f=2
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[3] 0.3758 0.500 2.000 0.5399
## ARCH Lag[5] 1.0328 1.440 1.667 0.7232
## ARCH Lag[7] 2.3828 2.315 1.543 0.6367
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 5.5278
## Individual Statistics:
## mu 0.0009833
## ar1 0.4619020
## ar2 0.4632270
## ma1 0.4475272
## ma2 0.0548165
## omega 0.2186256
## alpha1 0.2706208
## beta1 0.4749190
## gamma1 0.7263245
## shape 1.4150164
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 2.29 2.54 3.05
## Individual Statistic: 0.35 0.47 0.75
##
## Sign Bias Test
## ------------------------------------
## t-value prob sig
## Sign Bias 1.867 0.06202 *
## Negative Sign Bias 2.550 0.01080 **
## Positive Sign Bias 0.961 0.33660
## Joint Effect 7.464 0.05850 *
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 42.55 0.001495
## 2 30 51.75 0.005806
## 3 40 57.57 0.027921
## 4 50 67.79 0.038889
##
##
## Elapsed time : 3.979952sp500.garch11sg.spec <- ugarchspec(variance.model = list(model = "gjrGARCH", garchOrder = c(1, 1)),
mean.model = list(armaOrder = c(2, 2)),
distribution.model = "sged")
sp500.garch11sg.fit <- ugarchfit(spec = sp500.garch11sg.spec, data = sp500.ts)
sp500.garch11sg.fit##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : gjrGARCH(1,1)
## Mean Model : ARFIMA(2,0,2)
## Distribution : sged
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## mu 4511.517157 18.602161 2.4253e+02 0.000000
## ar1 0.845287 0.000003 3.2388e+05 0.000000
## ar2 0.155479 0.000087 1.7855e+03 0.000000
## ma1 0.124380 0.016170 7.6919e+00 0.000000
## ma2 -0.013719 0.016357 -8.3870e-01 0.401640
## omega 4.035782 0.864130 4.6703e+00 0.000003
## alpha1 0.000000 0.001682 7.0000e-06 0.999994
## beta1 0.881813 0.009081 9.7109e+01 0.000000
## gamma1 0.240396 0.020389 1.1791e+01 0.000000
## skew 0.951365 0.017393 5.4700e+01 0.000000
## shape 1.410620 0.048076 2.9342e+01 0.000000
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## mu 4511.517157 3.090829 1.4596e+03 0.00000
## ar1 0.845287 0.000003 3.0406e+05 0.00000
## ar2 0.155479 0.000077 2.0146e+03 0.00000
## ma1 0.124380 0.014840 8.3812e+00 0.00000
## ma2 -0.013719 0.017264 -7.9462e-01 0.42683
## omega 4.035782 0.982165 4.1091e+00 0.00004
## alpha1 0.000000 0.001691 7.0000e-06 0.99999
## beta1 0.881813 0.012149 7.2584e+01 0.00000
## gamma1 0.240396 0.022252 1.0804e+01 0.00000
## skew 0.951365 0.018145 5.2430e+01 0.00000
## shape 1.410620 0.053102 2.6565e+01 0.00000
##
## LogLikelihood : -15915.4
##
## Information Criteria
## ------------------------------------
##
## Akaike 8.7030
## Bayes 8.7216
## Shibata 8.7029
## Hannan-Quinn 8.7096
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.4449 0.5048
## Lag[2*(p+q)+(p+q)-1][11] 6.1940 0.3625
## Lag[4*(p+q)+(p+q)-1][19] 10.1764 0.4392
## d.o.f=4
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 6.854 0.008846
## Lag[2*(p+q)+(p+q)-1][5] 7.808 0.032881
## Lag[4*(p+q)+(p+q)-1][9] 9.085 0.078081
## d.o.f=2
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[3] 0.7243 0.500 2.000 0.3947
## ARCH Lag[5] 1.8278 1.440 1.667 0.5105
## ARCH Lag[7] 2.3670 2.315 1.543 0.6400
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 4.0535
## Individual Statistics:
## mu 0.01598
## ar1 0.51898
## ar2 0.51949
## ma1 0.18997
## ma2 0.03984
## omega 0.12720
## alpha1 0.65555
## beta1 0.38954
## gamma1 0.37942
## skew 0.29984
## shape 0.26201
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 2.49 2.75 3.27
## Individual Statistic: 0.35 0.47 0.75
##
## Sign Bias Test
## ------------------------------------
## t-value prob sig
## Sign Bias 0.59476 0.5520
## Negative Sign Bias 1.53416 0.1251
## Positive Sign Bias 0.01542 0.9877
## Joint Effect 2.37381 0.4985
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 25.91 0.13264
## 2 30 32.11 0.31486
## 3 40 50.40 0.10434
## 4 50 66.37 0.04977
##
##
## Elapsed time : 17.96521vni.garch11sg.spec <- ugarchspec(variance.model = list(model = "gjrGARCH", garchOrder = c(1, 1)),
mean.model = list(armaOrder = c(2, 2)),
distribution.model = "sged")
vni.garch11sg.fit <- ugarchfit(spec = vni.garch11sg.spec, data = vni.ts)
vni.garch11sg.fit##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : gjrGARCH(1,1)
## Mean Model : ARFIMA(2,0,2)
## Distribution : sged
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## mu 1223.571848 4.276084 286.1431 0.000000
## ar1 1.897315 0.000064 29710.4476 0.000000
## ar2 -0.897265 0.000024 -37828.9152 0.000000
## ma1 -0.795370 0.019493 -40.8024 0.000000
## ma2 -0.070855 0.019711 -3.5947 0.000325
## omega 0.945420 0.214470 4.4082 0.000010
## alpha1 0.055543 0.002115 26.2645 0.000000
## beta1 0.869494 0.012645 68.7606 0.000000
## gamma1 0.141940 0.015185 9.3476 0.000000
## skew 1.088952 0.020504 53.1088 0.000000
## shape 1.399488 0.045642 30.6623 0.000000
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## mu 1223.571848 4.494494 272.23794 0.000000
## ar1 1.897315 0.000239 7948.04798 0.000000
## ar2 -0.897265 0.000403 -2226.38494 0.000000
## ma1 -0.795370 0.026341 -30.19527 0.000000
## ma2 -0.070855 0.027961 -2.53411 0.011273
## omega 0.945420 0.263719 3.58495 0.000337
## alpha1 0.055543 0.076681 0.72433 0.468861
## beta1 0.869494 0.022913 37.94804 0.000000
## gamma1 0.141940 0.104487 1.35845 0.174320
## skew 1.088952 0.035915 30.32053 0.000000
## shape 1.399488 0.049508 28.26795 0.000000
##
## LogLikelihood : -12516.68
##
## Information Criteria
## ------------------------------------
##
## Akaike 6.8457
## Bayes 6.8644
## Shibata 6.8457
## Hannan-Quinn 6.8524
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 2.257 0.133017
## Lag[2*(p+q)+(p+q)-1][11] 8.071 0.000855
## Lag[4*(p+q)+(p+q)-1][19] 15.050 0.027648
## d.o.f=4
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.3421 0.5586
## Lag[2*(p+q)+(p+q)-1][5] 3.9787 0.2567
## Lag[4*(p+q)+(p+q)-1][9] 6.7616 0.2200
## d.o.f=2
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[3] 0.5985 0.500 2.000 0.4392
## ARCH Lag[5] 1.1237 1.440 1.667 0.6966
## ARCH Lag[7] 2.4344 2.315 1.543 0.6260
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 6.827
## Individual Statistics:
## mu 0.0009516
## ar1 0.3957184
## ar2 0.3980590
## ma1 0.3026770
## ma2 0.0326900
## omega 0.2333046
## alpha1 0.3841668
## beta1 0.6698102
## gamma1 1.0337438
## skew 0.4493683
## shape 1.6905764
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 2.49 2.75 3.27
## Individual Statistic: 0.35 0.47 0.75
##
## Sign Bias Test
## ------------------------------------
## t-value prob sig
## Sign Bias 2.0975 0.036015 **
## Negative Sign Bias 3.1903 0.001433 ***
## Positive Sign Bias 0.8166 0.414206
## Joint Effect 10.8651 0.012478 **
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 24.43 0.1803
## 2 30 38.54 0.1108
## 3 40 43.91 0.2712
## 4 50 45.00 0.6360
##
##
## Elapsed time : 11.51499sp500.garch12n.spec <- ugarchspec(variance.model = list(model = "gjrGARCH", garchOrder = c(1, 2)),
mean.model = list(armaOrder = c(2, 2)),
distribution.model = "norm")
sp500.garch12n.fit <- ugarchfit(spec = sp500.garch12n.spec, data = sp500.ts)
sp500.garch12n.fit##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : gjrGARCH(1,2)
## Mean Model : ARFIMA(2,0,2)
## Distribution : norm
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## mu 4520.545429 6.445692 701.3281 0.00000
## ar1 0.021625 0.000152 142.6807 0.00000
## ar2 0.979687 0.000013 77205.1053 0.00000
## ma1 0.935990 0.000006 154299.3976 0.00000
## ma2 -0.047484 0.000168 -281.9232 0.00000
## omega 4.153237 0.820517 5.0617 0.00000
## alpha1 0.003500 0.002382 1.4696 0.14168
## beta1 0.886643 0.151924 5.8361 0.00000
## beta2 0.000000 0.138542 0.0000 1.00000
## gamma1 0.217714 0.029373 7.4121 0.00000
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## mu 4520.545429 6.014451 751.6139 0.000000
## ar1 0.021625 0.000147 147.3485 0.000000
## ar2 0.979687 0.000016 61534.2770 0.000000
## ma1 0.935990 0.000007 139382.4615 0.000000
## ma2 -0.047484 0.000131 -362.1476 0.000000
## omega 4.153237 0.899601 4.6168 0.000004
## alpha1 0.003500 0.002696 1.2979 0.194321
## beta1 0.886643 0.146061 6.0704 0.000000
## beta2 0.000000 0.136546 0.0000 1.000000
## gamma1 0.217714 0.024231 8.9850 0.000000
##
## LogLikelihood : -15965.26
##
## Information Criteria
## ------------------------------------
##
## Akaike 8.7297
## Bayes 8.7466
## Shibata 8.7296
## Hannan-Quinn 8.7357
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.00221 0.9625
## Lag[2*(p+q)+(p+q)-1][11] 5.82583 0.6018
## Lag[4*(p+q)+(p+q)-1][19] 9.60143 0.5357
## d.o.f=4
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 6.474 0.01095
## Lag[2*(p+q)+(p+q)-1][8] 8.623 0.06988
## Lag[4*(p+q)+(p+q)-1][14] 11.061 0.13420
## d.o.f=3
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[4] 0.9732 0.500 2.000 0.3239
## ARCH Lag[6] 2.2458 1.461 1.711 0.4390
## ARCH Lag[8] 3.0183 2.368 1.583 0.5400
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 9.9444
## Individual Statistics:
## mu 0.005099
## ar1 0.454196
## ar2 0.458442
## ma1 0.087862
## ma2 0.091217
## omega 0.074982
## alpha1 0.528589
## beta1 0.430564
## beta2 0.431613
## gamma1 0.418935
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 2.29 2.54 3.05
## Individual Statistic: 0.35 0.47 0.75
##
## Sign Bias Test
## ------------------------------------
## t-value prob sig
## Sign Bias 0.9509 0.3417
## Negative Sign Bias 1.3754 0.1691
## Positive Sign Bias 0.2193 0.8265
## Joint Effect 2.0181 0.5687
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 96.83 1.999e-12
## 2 30 105.54 1.246e-10
## 3 40 107.74 2.338e-08
## 4 50 128.39 4.886e-09
##
##
## Elapsed time : 2.189599vni.garch12n.spec <- ugarchspec(variance.model = list(model = "gjrGARCH", garchOrder = c(1, 2)),
mean.model = list(armaOrder = c(2, 2)),
distribution.model = "norm")
vni.garch12n.fit <- ugarchfit(spec = vni.garch12n.spec, data = vni.ts)
vni.garch12n.fit##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : gjrGARCH(1,2)
## Mean Model : ARFIMA(2,0,2)
## Distribution : norm
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## mu 1.2189e+03 4.017527 303.38621 0.000000
## ar1 1.2324e-01 0.000502 245.70704 0.000000
## ar2 8.7801e-01 0.000544 1615.36162 0.000000
## ma1 9.8660e-01 0.017062 57.82427 0.000000
## ma2 1.0149e-01 0.016779 6.04835 0.000000
## omega 1.0628e+00 0.226468 4.69286 0.000003
## alpha1 6.3244e-02 0.013044 4.84847 0.000001
## beta1 8.3873e-01 0.139198 6.02548 0.000000
## beta2 1.3377e-02 0.124193 0.10771 0.914224
## gamma1 1.6729e-01 0.028852 5.79807 0.000000
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## mu 1.2189e+03 5.942770 2.0510e+02 0.000000
## ar1 1.2324e-01 0.000311 3.9623e+02 0.000000
## ar2 8.7801e-01 0.000529 1.6592e+03 0.000000
## ma1 9.8660e-01 0.016397 6.0169e+01 0.000000
## ma2 1.0149e-01 0.015157 6.6956e+00 0.000000
## omega 1.0628e+00 0.274679 3.8692e+00 0.000109
## alpha1 6.3244e-02 0.017550 3.6037e+00 0.000314
## beta1 8.3873e-01 0.151059 5.5524e+00 0.000000
## beta2 1.3377e-02 0.136051 9.8324e-02 0.921675
## gamma1 1.6729e-01 0.039193 4.2683e+00 0.000020
##
## LogLikelihood : -12587.33
##
## Information Criteria
## ------------------------------------
##
## Akaike 6.8838
## Bayes 6.9007
## Shibata 6.8838
## Hannan-Quinn 6.8898
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 3.207 0.0733323
## Lag[2*(p+q)+(p+q)-1][11] 8.271 0.0003279
## Lag[4*(p+q)+(p+q)-1][19] 14.706 0.0355068
## d.o.f=4
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.001 0.9748
## Lag[2*(p+q)+(p+q)-1][8] 4.406 0.4400
## Lag[4*(p+q)+(p+q)-1][14] 8.951 0.2856
## d.o.f=3
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[4] 0.8807 0.500 2.000 0.3480
## ARCH Lag[6] 1.2326 1.461 1.711 0.6821
## ARCH Lag[8] 5.2827 2.368 1.583 0.2203
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 4.1898
## Individual Statistics:
## mu 0.0008017
## ar1 0.4151388
## ar2 0.4443553
## ma1 0.6197154
## ma2 0.2456410
## omega 0.2377222
## alpha1 0.5258808
## beta1 0.7800819
## beta2 0.7926746
## gamma1 1.2142585
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 2.29 2.54 3.05
## Individual Statistic: 0.35 0.47 0.75
##
## Sign Bias Test
## ------------------------------------
## t-value prob sig
## Sign Bias 2.144 0.03214 **
## Negative Sign Bias 2.549 0.01084 **
## Positive Sign Bias 1.283 0.19966
## Joint Effect 8.286 0.04046 **
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 78.63 3.194e-09
## 2 30 104.84 1.623e-10
## 3 40 119.39 4.403e-10
## 4 50 126.89 7.904e-09
##
##
## Elapsed time : 1.722155sp500.garch12t.spec <- ugarchspec(variance.model = list(model = "gjrGARCH", garchOrder = c(1, 2)),
mean.model = list(armaOrder = c(2, 2)),
distribution.model = "std")
sp500.garch12t.fit <- ugarchfit(spec = sp500.garch12t.spec, data = sp500.ts)
sp500.garch12t.fit##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : gjrGARCH(1,2)
## Mean Model : ARFIMA(2,0,2)
## Distribution : std
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## mu 4510.508157 16.792526 2.6860e+02 0.000000
## ar1 1.592614 0.000002 7.9666e+05 0.000000
## ar2 -0.592356 0.000007 -8.7346e+04 0.000000
## ma1 -0.628161 0.015992 -3.9278e+01 0.000000
## ma2 0.000773 0.016173 4.7780e-02 0.961891
## omega 4.155380 0.933324 4.4522e+00 0.000008
## alpha1 0.000000 0.000061 7.2000e-05 0.999942
## beta1 0.884634 0.165148 5.3566e+00 0.000000
## beta2 0.000000 0.149934 1.0000e-06 0.999999
## gamma1 0.228732 0.032984 6.9347e+00 0.000000
## shape 8.839840 0.973406 9.0814e+00 0.000000
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## mu 4510.508157 3.878199 1.1630e+03 0.000000
## ar1 1.592614 0.000002 7.7207e+05 0.000000
## ar2 -0.592356 0.000007 -8.4224e+04 0.000000
## ma1 -0.628161 0.015020 -4.1820e+01 0.000000
## ma2 0.000773 0.016047 4.8155e-02 0.961593
## omega 4.155380 0.967266 4.2960e+00 0.000017
## alpha1 0.000000 0.000088 5.0000e-05 0.999960
## beta1 0.884634 0.160483 5.5123e+00 0.000000
## beta2 0.000000 0.148037 1.0000e-06 0.999999
## gamma1 0.228732 0.026533 8.6208e+00 0.000000
## shape 8.839840 1.045624 8.4541e+00 0.000000
##
## LogLikelihood : -15929.7
##
## Information Criteria
## ------------------------------------
##
## Akaike 8.7108
## Bayes 8.7294
## Shibata 8.7107
## Hannan-Quinn 8.7174
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.1671 0.6827
## Lag[2*(p+q)+(p+q)-1][11] 3.7675 1.0000
## Lag[4*(p+q)+(p+q)-1][19] 7.4982 0.8579
## d.o.f=4
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 6.171 0.01299
## Lag[2*(p+q)+(p+q)-1][8] 8.263 0.08335
## Lag[4*(p+q)+(p+q)-1][14] 11.056 0.13443
## d.o.f=3
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[4] 1.034 0.500 2.000 0.3093
## ARCH Lag[6] 2.242 1.461 1.711 0.4397
## ARCH Lag[8] 3.108 2.368 1.583 0.5234
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 11.5132
## Individual Statistics:
## mu 0.02608
## ar1 0.57675
## ar2 0.57624
## ma1 0.11191
## ma2 0.01959
## omega 0.11449
## alpha1 1.17477
## beta1 0.63749
## beta2 0.61199
## gamma1 0.84912
## shape 0.25724
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 2.49 2.75 3.27
## Individual Statistic: 0.35 0.47 0.75
##
## Sign Bias Test
## ------------------------------------
## t-value prob sig
## Sign Bias 0.5553 0.5787
## Negative Sign Bias 1.2914 0.1966
## Positive Sign Bias 0.1601 0.8728
## Joint Effect 1.7142 0.6338
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 53.93 3.379e-05
## 2 30 62.62 2.882e-04
## 3 40 72.20 9.643e-04
## 4 50 94.62 9.946e-05
##
##
## Elapsed time : 6.643258vni.garch12t.spec <- ugarchspec(variance.model = list(model = "gjrGARCH", garchOrder = c(1, 2)),
mean.model = list(armaOrder = c(2, 2)),
distribution.model = "std")
vni.garch12t.fit <- ugarchfit(spec = vni.garch12t.spec, data = vni.ts)
vni.garch12t.fit##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : gjrGARCH(1,2)
## Mean Model : ARFIMA(2,0,2)
## Distribution : std
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## mu 1219.337840 1.787411 6.8218e+02 0.000000
## ar1 1.984317 0.000060 3.3339e+04 0.000000
## ar2 -0.984306 0.000027 -3.6385e+04 0.000000
## ma1 -0.878794 0.000051 -1.7103e+04 0.000000
## ma2 -0.097346 0.000173 -5.6221e+02 0.000000
## omega 1.056524 0.270053 3.9123e+00 0.000091
## alpha1 0.056439 0.009606 5.8757e+00 0.000000
## beta1 0.862428 0.171172 5.0384e+00 0.000000
## beta2 0.000001 0.152316 6.0000e-06 0.999995
## gamma1 0.160263 0.024967 6.4190e+00 0.000000
## shape 6.846286 0.761196 8.9941e+00 0.000000
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## mu 1219.337840 4.379518 2.7842e+02 0.000000
## ar1 1.984317 0.000232 8.5658e+03 0.000000
## ar2 -0.984306 0.001153 -8.5360e+02 0.000000
## ma1 -0.878794 0.000168 -5.2395e+03 0.000000
## ma2 -0.097346 0.002956 -3.2933e+01 0.000000
## omega 1.056524 1.477066 7.1529e-01 0.474433
## alpha1 0.056439 0.267054 2.1134e-01 0.832622
## beta1 0.862428 0.821853 1.0494e+00 0.294008
## beta2 0.000001 0.805783 1.0000e-06 0.999999
## gamma1 0.160263 0.518386 3.0916e-01 0.757201
## shape 6.846286 1.576305 4.3432e+00 0.000014
##
## LogLikelihood : -12526.66
##
## Information Criteria
## ------------------------------------
##
## Akaike 6.8512
## Bayes 6.8698
## Shibata 6.8512
## Hannan-Quinn 6.8578
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 2.677 0.101802
## Lag[2*(p+q)+(p+q)-1][11] 7.694 0.004471
## Lag[4*(p+q)+(p+q)-1][19] 14.297 0.047316
## d.o.f=4
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.04388 0.8341
## Lag[2*(p+q)+(p+q)-1][8] 4.37548 0.4448
## Lag[4*(p+q)+(p+q)-1][14] 8.96507 0.2843
## d.o.f=3
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[4] 0.5777 0.500 2.000 0.4472
## ARCH Lag[6] 1.0872 1.461 1.711 0.7228
## ARCH Lag[8] 5.3995 2.368 1.583 0.2093
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 6.8901
## Individual Statistics:
## mu 0.002116
## ar1 0.437501
## ar2 0.437538
## ma1 0.080833
## ma2 0.123870
## omega 0.260444
## alpha1 0.238423
## beta1 0.441951
## beta2 0.452689
## gamma1 0.733919
## shape 0.801790
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 2.49 2.75 3.27
## Individual Statistic: 0.35 0.47 0.75
##
## Sign Bias Test
## ------------------------------------
## t-value prob sig
## Sign Bias 2.0673 0.038779 **
## Negative Sign Bias 2.8673 0.004164 ***
## Positive Sign Bias 0.9222 0.356494
## Joint Effect 9.1416 0.027466 **
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 39.46 0.003845
## 2 30 49.02 0.011502
## 3 40 56.59 0.033951
## 4 50 68.33 0.035292
##
##
## Elapsed time : 3.31423sp500.garch12st.spec <- ugarchspec(variance.model = list(model = "sGARCH", garchOrder = c(1, 2)),
mean.model = list(armaOrder = c(2, 2)),
distribution.model = "sstd")
sp500.garch12st.fit <- ugarchfit(spec = sp500.garch12st.spec, data = sp500.ts)
sp500.garch12st.fit##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : sGARCH(1,2)
## Mean Model : ARFIMA(2,0,2)
## Distribution : sstd
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## mu 4511.589387 18.487424 244.0356 0.000000
## ar1 0.577210 0.000146 3961.1984 0.000000
## ar2 0.423627 0.000132 3207.1844 0.000000
## ma1 0.374190 0.017358 21.5570 0.000000
## ma2 -0.019835 0.017808 -1.1138 0.265353
## omega 6.257336 1.578916 3.9631 0.000074
## alpha1 0.166132 0.025590 6.4921 0.000000
## beta1 0.832868 0.208335 3.9977 0.000064
## beta2 0.000000 0.186726 0.0000 1.000000
## skew 0.965547 0.022172 43.5487 0.000000
## shape 6.534946 0.722996 9.0387 0.000000
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## mu 4511.589387 2.257541 1998.4529 0.000000
## ar1 0.577210 0.000164 3523.5340 0.000000
## ar2 0.423627 0.000134 3164.5358 0.000000
## ma1 0.374190 0.015549 24.0646 0.000000
## ma2 -0.019835 0.018112 -1.0952 0.273450
## omega 6.257336 1.712301 3.6543 0.000258
## alpha1 0.166132 0.022671 7.3279 0.000000
## beta1 0.832868 0.273862 3.0412 0.002356
## beta2 0.000000 0.254292 0.0000 1.000000
## skew 0.965547 0.027579 35.0102 0.000000
## shape 6.534946 0.731827 8.9296 0.000000
##
## LogLikelihood : -16008.09
##
## Information Criteria
## ------------------------------------
##
## Akaike 8.7536
## Bayes 8.7722
## Shibata 8.7536
## Hannan-Quinn 8.7602
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.07103 0.7898
## Lag[2*(p+q)+(p+q)-1][11] 5.82406 0.6030
## Lag[4*(p+q)+(p+q)-1][19] 9.67054 0.5239
## d.o.f=4
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 11.04 0.0008937
## Lag[2*(p+q)+(p+q)-1][8] 13.97 0.0039916
## Lag[4*(p+q)+(p+q)-1][14] 16.93 0.0099874
## d.o.f=3
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[4] 0.055 0.500 2.000 0.8146
## ARCH Lag[6] 4.079 1.461 1.711 0.1810
## ARCH Lag[8] 5.870 2.368 1.583 0.1698
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 10.3589
## Individual Statistics:
## mu 0.01227
## ar1 0.37039
## ar2 0.37031
## ma1 0.09491
## ma2 0.04002
## omega 0.30336
## alpha1 2.66563
## beta1 1.94569
## beta2 1.78919
## skew 0.10463
## shape 1.33146
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 2.49 2.75 3.27
## Individual Statistic: 0.35 0.47 0.75
##
## Sign Bias Test
## ------------------------------------
## t-value prob sig
## Sign Bias 2.0701 0.03851 **
## Negative Sign Bias 0.4359 0.66292
## Positive Sign Bias 2.2619 0.02376 **
## Joint Effect 17.7301 0.00050 ***
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 49.95 0.0001335
## 2 30 59.39 0.0007344
## 3 40 74.67 0.0005078
## 4 50 91.69 0.0002110
##
##
## Elapsed time : 7.63024vni.garch12st.spec <- ugarchspec(variance.model = list(model = "gjrGARCH", garchOrder = c(1, 2)),
mean.model = list(armaOrder = c(2, 2)),
distribution.model = "sstd")
vni.garch12st.fit <- ugarchfit(spec = vni.garch12st.spec, data = vni.ts)
vni.garch12st.fit##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : gjrGARCH(1,2)
## Mean Model : ARFIMA(2,0,2)
## Distribution : sstd
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## mu 748.410564 15.691451 47.69544 0.000000
## ar1 1.133946 0.000213 5321.76710 0.000000
## ar2 -0.134287 0.000222 -604.66014 0.000000
## ma1 -0.021198 0.014757 -1.43645 0.150875
## ma2 0.008410 0.012597 0.66762 0.504373
## omega 1.020081 0.292106 3.49216 0.000479
## alpha1 0.001042 0.001816 0.57401 0.565962
## beta1 0.803899 0.128924 6.23543 0.000000
## beta2 0.089694 0.117364 0.76424 0.444725
## gamma1 0.199486 0.026008 7.67017 0.000000
## skew 1.120489 0.023532 47.61591 0.000000
## shape 5.579190 0.451879 12.34664 0.000000
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## mu 748.410564 17.741643 42.18384 0.000000
## ar1 1.133946 0.000224 5056.27223 0.000000
## ar2 -0.134287 0.000268 -500.89822 0.000000
## ma1 -0.021198 0.013511 -1.56892 0.116666
## ma2 0.008410 0.010667 0.78836 0.430489
## omega 1.020081 0.343871 2.96647 0.003012
## alpha1 0.001042 0.002030 0.51350 0.607599
## beta1 0.803899 0.081307 9.88720 0.000000
## beta2 0.089694 0.071098 1.26155 0.207112
## gamma1 0.199486 0.021008 9.49555 0.000000
## skew 1.120489 0.024816 45.15165 0.000000
## shape 5.579190 0.869132 6.41926 0.000000
##
## LogLikelihood : -12580.56
##
## Information Criteria
## ------------------------------------
##
## Akaike 6.8812
## Bayes 6.9015
## Shibata 6.8812
## Hannan-Quinn 6.8884
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 88.28 0
## Lag[2*(p+q)+(p+q)-1][11] 90.14 0
## Lag[4*(p+q)+(p+q)-1][19] 93.60 0
## d.o.f=4
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 903.3 0
## Lag[2*(p+q)+(p+q)-1][8] 903.3 0
## Lag[4*(p+q)+(p+q)-1][14] 903.4 0
## d.o.f=3
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[4] 0.0004037 0.500 2.000 0.984
## ARCH Lag[6] 0.0007283 1.461 1.711 1.000
## ARCH Lag[8] 0.0043078 2.368 1.583 1.000
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 13.1284
## Individual Statistics:
## mu 1.41807
## ar1 0.36968
## ar2 0.35804
## ma1 0.72150
## ma2 0.04017
## omega 0.38799
## alpha1 0.10349
## beta1 2.03388
## beta2 2.02167
## gamma1 3.03580
## skew 0.49796
## shape 0.59464
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 2.69 2.96 3.51
## Individual Statistic: 0.35 0.47 0.75
##
## Sign Bias Test
## ------------------------------------
## t-value prob sig
## Sign Bias 10.0293 2.266e-23 ***
## Negative Sign Bias 0.6314 5.278e-01
## Positive Sign Bias 48.1729 0.000e+00 ***
## Joint Effect 2323.3761 0.000e+00 ***
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 29.63 0.056722
## 2 30 48.18 0.014077
## 3 40 58.27 0.024232
## 4 50 79.23 0.004023
##
##
## Elapsed time : 2.824389sp500.garch12g.spec <- ugarchspec(variance.model = list(model = "gjrGARCH", garchOrder = c(1, 2)),
mean.model = list(armaOrder = c(2, 2)),
distribution.model = "ged")
sp500.garch12g.fit <- ugarchfit(spec = sp500.garch12g.spec, data = sp500.ts)
sp500.garch12g.fit##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : gjrGARCH(1,2)
## Mean Model : ARFIMA(2,0,2)
## Distribution : ged
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## mu 4512.695016 17.580961 256.680800 0.000000
## ar1 0.437120 0.000148 2953.334358 0.000000
## ar2 0.563851 0.000058 9771.409474 0.000000
## ma1 0.526454 0.017270 30.483268 0.000000
## ma2 -0.020979 0.018236 -1.150418 0.249972
## omega 4.036754 1.144918 3.525802 0.000422
## alpha1 0.008130 0.005080 1.600459 0.109497
## beta1 0.879751 0.220373 3.992102 0.000065
## beta2 0.000000 0.199778 0.000001 0.999999
## gamma1 0.222238 0.040123 5.538965 0.000000
## shape 1.434439 0.048047 29.855192 0.000000
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## mu 4512.695016 4.110325 1097.892411 0.000000
## ar1 0.437120 0.000120 3634.986613 0.000000
## ar2 0.563851 0.000060 9389.244655 0.000000
## ma1 0.526454 0.016868 31.210797 0.000000
## ma2 -0.020979 0.019951 -1.051509 0.293025
## omega 4.036754 1.394004 2.895798 0.003782
## alpha1 0.008130 0.006435 1.263403 0.206444
## beta1 0.879751 0.249686 3.523425 0.000426
## beta2 0.000000 0.229602 0.000001 0.999999
## gamma1 0.222238 0.038409 5.786049 0.000000
## shape 1.434439 0.052988 27.071037 0.000000
##
## LogLikelihood : -15918.52
##
## Information Criteria
## ------------------------------------
##
## Akaike 8.7047
## Bayes 8.7233
## Shibata 8.7046
## Hannan-Quinn 8.7113
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.132 0.7164
## Lag[2*(p+q)+(p+q)-1][11] 5.978 0.5013
## Lag[4*(p+q)+(p+q)-1][19] 10.001 0.4682
## d.o.f=4
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 8.425 0.003701
## Lag[2*(p+q)+(p+q)-1][8] 10.169 0.031841
## Lag[4*(p+q)+(p+q)-1][14] 12.250 0.083417
## d.o.f=3
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[4] 0.3872 0.500 2.000 0.5338
## ARCH Lag[6] 1.8791 1.461 1.711 0.5180
## ARCH Lag[8] 2.5227 2.368 1.583 0.6360
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 9.9228
## Individual Statistics:
## mu 0.004884
## ar1 0.396307
## ar2 0.396784
## ma1 0.133843
## ma2 0.021644
## omega 0.055688
## alpha1 0.682230
## beta1 0.569805
## beta2 0.548016
## gamma1 0.736351
## shape 0.272532
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 2.49 2.75 3.27
## Individual Statistic: 0.35 0.47 0.75
##
## Sign Bias Test
## ------------------------------------
## t-value prob sig
## Sign Bias 0.5780 0.5633
## Negative Sign Bias 1.3696 0.1709
## Positive Sign Bias 0.2191 0.8266
## Joint Effect 1.9253 0.5881
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 31.79 0.03298
## 2 30 42.07 0.05540
## 3 40 53.88 0.05685
## 4 50 73.47 0.01338
##
##
## Elapsed time : 7.720208vni.garch12g.spec <- ugarchspec(variance.model = list(model = "gjrGARCH", garchOrder = c(1, 2)),
mean.model = list(armaOrder = c(2, 2)),
distribution.model = "ged")
vni.garch12g.fit <- ugarchfit(spec = vni.garch12g.spec, data = vni.ts)
vni.garch12g.fit##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : gjrGARCH(1,2)
## Mean Model : ARFIMA(2,0,2)
## Distribution : ged
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## mu 1223.135591 4.225064 2.8950e+02 0.000000
## ar1 1.900977 0.000078 2.4363e+04 0.000000
## ar2 -0.900908 0.000036 -2.5268e+04 0.000000
## ma1 -0.795065 0.015457 -5.1438e+01 0.000000
## ma2 -0.072242 0.014847 -4.8659e+00 0.000001
## omega 1.039628 0.258389 4.0235e+00 0.000057
## alpha1 0.058373 0.009594 6.0842e+00 0.000000
## beta1 0.839266 0.172969 4.8521e+00 0.000001
## beta2 0.017770 0.154077 1.1533e-01 0.908185
## gamma1 0.167183 0.025663 6.5145e+00 0.000000
## shape 1.398808 0.045857 3.0503e+01 0.000000
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## mu 1223.135591 4.722735 2.5899e+02 0.000000
## ar1 1.900977 0.000078 2.4367e+04 0.000000
## ar2 -0.900908 0.000317 -2.8386e+03 0.000000
## ma1 -0.795065 0.018494 -4.2991e+01 0.000000
## ma2 -0.072242 0.016268 -4.4407e+00 0.000009
## omega 1.039628 0.382599 2.7173e+00 0.006582
## alpha1 0.058373 0.058698 9.9446e-01 0.320000
## beta1 0.839266 0.248494 3.3774e+00 0.000732
## beta2 0.017770 0.233743 7.6021e-02 0.939402
## gamma1 0.167183 0.116748 1.4320e+00 0.152144
## shape 1.398808 0.048476 2.8856e+01 0.000000
##
## LogLikelihood : -12525.92
##
## Information Criteria
## ------------------------------------
##
## Akaike 6.8508
## Bayes 6.8694
## Shibata 6.8508
## Hannan-Quinn 6.8574
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 2.633 0.1046326
## Lag[2*(p+q)+(p+q)-1][11] 8.459 0.0001271
## Lag[4*(p+q)+(p+q)-1][19] 15.396 0.0213571
## d.o.f=4
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.0141 0.9055
## Lag[2*(p+q)+(p+q)-1][8] 4.5114 0.4236
## Lag[4*(p+q)+(p+q)-1][14] 9.2023 0.2628
## d.o.f=3
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[4] 0.6098 0.500 2.000 0.4349
## ARCH Lag[6] 1.0041 1.461 1.711 0.7466
## ARCH Lag[8] 5.4588 2.368 1.583 0.2039
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 5.7335
## Individual Statistics:
## mu 0.001012
## ar1 0.459480
## ar2 0.460657
## ma1 0.435176
## ma2 0.054034
## omega 0.217455
## alpha1 0.268353
## beta1 0.470761
## beta2 0.482184
## gamma1 0.722402
## shape 1.410772
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 2.49 2.75 3.27
## Individual Statistic: 0.35 0.47 0.75
##
## Sign Bias Test
## ------------------------------------
## t-value prob sig
## Sign Bias 1.996 0.04604 **
## Negative Sign Bias 2.563 0.01043 **
## Positive Sign Bias 1.031 0.30248
## Joint Effect 7.729 0.05196 *
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 42.19 0.001673
## 2 30 55.10 0.002413
## 3 40 54.93 0.046759
## 4 50 71.04 0.021448
##
##
## Elapsed time : 3.377456sp500.garch12sg.spec <- ugarchspec(variance.model = list(model = "gjrGARCH", garchOrder = c(1, 2)),
mean.model = list(armaOrder = c(2, 2)),
distribution.model = "sged")
sp500.garch12sg.fit <- ugarchfit(spec = sp500.garch12sg.spec, data = sp500.ts)
sp500.garch12sg.fit##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : gjrGARCH(1,2)
## Mean Model : ARFIMA(2,0,2)
## Distribution : sged
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## mu 2387.516992 5.932445 402.45 0
## ar1 0.315555 0.000282 1120.54 0
## ar2 0.688043 0.000337 2042.70 0
## ma1 0.601693 0.001581 380.49 0
## ma2 -0.060528 0.000281 -215.11 0
## omega 1170.445183 10.661223 109.79 0
## alpha1 0.032331 0.000817 39.59 0
## beta1 0.209138 0.001394 150.07 0
## beta2 0.223622 0.001694 132.00 0
## gamma1 0.997320 0.007684 129.80 0
## skew 0.910536 0.001713 531.62 0
## shape 0.250141 0.001121 223.13 0
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## mu 2387.516992 192.184386 12.423 0
## ar1 0.315555 0.004979 63.380 0
## ar2 0.688043 0.004768 144.301 0
## ma1 0.601693 0.030958 19.436 0
## ma2 -0.060528 0.003101 -19.518 0
## omega 1170.445183 34.997144 33.444 0
## alpha1 0.032331 0.001144 28.261 0
## beta1 0.209138 0.004497 46.508 0
## beta2 0.223622 0.005237 42.701 0
## gamma1 0.997320 0.013916 71.665 0
## skew 0.910536 0.025180 36.160 0
## shape 0.250141 0.005674 44.085 0
##
## LogLikelihood : -17658.73
##
## Information Criteria
## ------------------------------------
##
## Akaike 9.6561
## Bayes 9.6765
## Shibata 9.6561
## Hannan-Quinn 9.6634
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 418.8 0
## Lag[2*(p+q)+(p+q)-1][11] 421.5 0
## Lag[4*(p+q)+(p+q)-1][19] 423.4 0
## d.o.f=4
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 929.2 0
## Lag[2*(p+q)+(p+q)-1][8] 929.3 0
## Lag[4*(p+q)+(p+q)-1][14] 929.3 0
## d.o.f=3
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[4] 0.0008790 0.500 2.000 0.9763
## ARCH Lag[6] 0.0009782 1.461 1.711 1.0000
## ARCH Lag[8] 0.0010089 2.368 1.583 1.0000
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 774.3619
## Individual Statistics:
## mu 0.33026
## ar1 0.93365
## ar2 0.42401
## ma1 0.64562
## ma2 0.02035
## omega 0.01888
## alpha1 0.18304
## beta1 0.01904
## beta2 0.01778
## gamma1 0.02677
## skew 0.90638
## shape 0.10491
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 2.69 2.96 3.51
## Individual Statistic: 0.35 0.47 0.75
##
## Sign Bias Test
## ------------------------------------
## t-value prob sig
## Sign Bias 7.8533 5.284e-15 ***
## Negative Sign Bias 0.1174 9.065e-01
## Positive Sign Bias 51.8813 0.000e+00 ***
## Joint Effect 2693.7564 0.000e+00 ***
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 1986 0
## 2 30 2134 0
## 3 40 2159 0
## 4 50 2183 0
##
##
## Elapsed time : 5.202625vni.garch12sg.spec <- ugarchspec(variance.model = list(model = "gjrGARCH", garchOrder = c(1, 2)),
mean.model = list(armaOrder = c(2, 2)),
distribution.model = "sged")
vni.garch12sg.fit <- ugarchfit(spec = vni.garch12sg.spec, data = vni.ts)
vni.garch12sg.fit##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : gjrGARCH(1,2)
## Mean Model : ARFIMA(2,0,2)
## Distribution : sged
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## mu 1223.585987 4.276012 2.8615e+02 0.000000
## ar1 1.898020 0.000066 2.8922e+04 0.000000
## ar2 -0.897971 0.000025 -3.5485e+04 0.000000
## ma1 -0.795932 0.019323 -4.1190e+01 0.000000
## ma2 -0.071099 0.019441 -3.6572e+00 0.000255
## omega 0.978234 0.233210 4.1947e+00 0.000027
## alpha1 0.057496 0.006471 8.8852e+00 0.000000
## beta1 0.818884 0.127756 6.4098e+00 0.000000
## beta2 0.045716 0.114957 3.9768e-01 0.690865
## gamma1 0.147570 0.018621 7.9250e+00 0.000000
## skew 1.089189 0.020525 5.3068e+01 0.000000
## shape 1.399463 0.045651 3.0656e+01 0.000000
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## mu 1223.585987 4.506727 271.50211 0.000000
## ar1 1.898020 0.000219 8668.26720 0.000000
## ar2 -0.897971 0.000392 -2290.03643 0.000000
## ma1 -0.795932 0.026654 -29.86149 0.000000
## ma2 -0.071099 0.027838 -2.55403 0.010649
## omega 0.978234 0.328406 2.97874 0.002894
## alpha1 0.057496 0.071414 0.80511 0.420755
## beta1 0.818884 0.154899 5.28655 0.000000
## beta2 0.045716 0.155413 0.29416 0.768636
## gamma1 0.147570 0.119696 1.23287 0.217623
## skew 1.089189 0.035243 30.90529 0.000000
## shape 1.399463 0.049394 28.33282 0.000000
##
## LogLikelihood : -12516.64
##
## Information Criteria
## ------------------------------------
##
## Akaike 6.8463
## Bayes 6.8666
## Shibata 6.8462
## Hannan-Quinn 6.8535
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 2.262 0.1326238
## Lag[2*(p+q)+(p+q)-1][11] 8.072 0.0008517
## Lag[4*(p+q)+(p+q)-1][19] 15.055 0.0275390
## d.o.f=4
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.2016 0.6534
## Lag[2*(p+q)+(p+q)-1][8] 6.0652 0.2286
## Lag[4*(p+q)+(p+q)-1][14] 10.6230 0.1585
## d.o.f=3
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[4] 0.2687 0.500 2.000 0.6042
## ARCH Lag[6] 0.9603 1.461 1.711 0.7592
## ARCH Lag[8] 4.9439 2.368 1.583 0.2549
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 6.9955
## Individual Statistics:
## mu 0.001066
## ar1 0.394983
## ar2 0.397180
## ma1 0.302903
## ma2 0.032782
## omega 0.230194
## alpha1 0.374771
## beta1 0.655210
## beta2 0.666246
## gamma1 1.016335
## skew 0.444767
## shape 1.677972
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 2.69 2.96 3.51
## Individual Statistic: 0.35 0.47 0.75
##
## Sign Bias Test
## ------------------------------------
## t-value prob sig
## Sign Bias 2.0927 0.036447 **
## Negative Sign Bias 3.0495 0.002309 ***
## Positive Sign Bias 0.8552 0.392495
## Joint Effect 10.0750 0.017939 **
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 24.99 0.16072
## 2 30 39.44 0.09347
## 3 40 42.56 0.32054
## 4 50 46.39 0.57942
##
##
## Elapsed time : 11.67726sp500.garch21n.spec <- ugarchspec(variance.model = list(model = "gjrGARCH", garchOrder = c(2, 1)),
mean.model = list(armaOrder = c(2, 2)),
distribution.model = "norm")
sp500.garch21n.fit <- ugarchfit(spec = sp500.garch21n.spec, data = sp500.ts)
sp500.garch21n.fit##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : gjrGARCH(2,1)
## Mean Model : ARFIMA(2,0,2)
## Distribution : norm
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## mu 4530.247424 11.533098 3.9280e+02 0.000000
## ar1 0.066261 0.000148 4.4921e+02 0.000000
## ar2 0.935067 0.000016 5.8964e+04 0.000000
## ma1 0.889206 0.008375 1.0617e+02 0.000000
## ma2 -0.042741 0.007065 -6.0495e+00 0.000000
## omega 5.140983 0.813395 6.3204e+00 0.000000
## alpha1 0.000000 0.017002 5.0000e-06 0.999996
## alpha2 0.011091 0.017469 6.3488e-01 0.525508
## beta1 0.862552 0.009036 9.5456e+01 0.000000
## gamma1 0.084617 0.029073 2.9105e+00 0.003609
## gamma2 0.166097 0.033164 5.0083e+00 0.000001
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## mu 4530.247424 6.184481 7.3252e+02 0.000000
## ar1 0.066261 0.000140 4.7325e+02 0.000000
## ar2 0.935067 0.000018 5.1023e+04 0.000000
## ma1 0.889206 0.010421 8.5327e+01 0.000000
## ma2 -0.042741 0.006781 -6.3028e+00 0.000000
## omega 5.140983 1.057501 4.8614e+00 0.000001
## alpha1 0.000000 0.014423 6.0000e-06 0.999995
## alpha2 0.011091 0.015010 7.3889e-01 0.459973
## beta1 0.862552 0.013487 6.3956e+01 0.000000
## gamma1 0.084617 0.029960 2.8244e+00 0.004737
## gamma2 0.166097 0.033980 4.8881e+00 0.000001
##
## LogLikelihood : -15954.02
##
## Information Criteria
## ------------------------------------
##
## Akaike 8.7241
## Bayes 8.7427
## Shibata 8.7240
## Hannan-Quinn 8.7307
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.001313 0.9711
## Lag[2*(p+q)+(p+q)-1][11] 5.471141 0.8080
## Lag[4*(p+q)+(p+q)-1][19] 9.328526 0.5825
## d.o.f=4
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.6505 0.4199
## Lag[2*(p+q)+(p+q)-1][8] 2.9184 0.7005
## Lag[4*(p+q)+(p+q)-1][14] 5.0034 0.7691
## d.o.f=3
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[4] 0.08424 0.500 2.000 0.7716
## ARCH Lag[6] 2.51763 1.461 1.711 0.3870
## ARCH Lag[8] 3.30552 2.368 1.583 0.4879
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 3.6085
## Individual Statistics:
## mu 0.001259
## ar1 0.435581
## ar2 0.431627
## ma1 0.275165
## ma2 0.163226
## omega 0.027320
## alpha1 0.547217
## alpha2 0.638633
## beta1 0.519554
## gamma1 0.561350
## gamma2 0.532688
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 2.49 2.75 3.27
## Individual Statistic: 0.35 0.47 0.75
##
## Sign Bias Test
## ------------------------------------
## t-value prob sig
## Sign Bias 1.44813 0.1477
## Negative Sign Bias 0.43234 0.6655
## Positive Sign Bias 0.01123 0.9910
## Joint Effect 4.59769 0.2037
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 84.79 2.744e-10
## 2 30 100.11 9.378e-10
## 3 40 109.77 1.185e-08
## 4 50 122.19 3.487e-08
##
##
## Elapsed time : 3.10709vni.garch21n.spec <- ugarchspec(variance.model = list(model = "gjrGARCH", garchOrder = c(2, 1)),
mean.model = list(armaOrder = c(2, 2)),
distribution.model = "norm")
vni.garch21n.fit <- ugarchfit(spec = vni.garch21n.spec, data = vni.ts)
vni.garch21n.fit##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : gjrGARCH(2,1)
## Mean Model : ARFIMA(2,0,2)
## Distribution : norm
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## mu 1218.842025 4.037886 301.851524 0.000000
## ar1 0.124920 0.000504 248.010105 0.000000
## ar2 0.876331 0.000549 1595.878117 0.000000
## ma1 0.984997 0.017115 57.552137 0.000000
## ma2 0.101461 0.016815 6.033881 0.000000
## omega 1.044425 0.211839 4.930274 0.000001
## alpha1 0.062208 0.021490 2.894689 0.003795
## alpha2 0.000000 0.022992 0.000005 0.999996
## beta1 0.854676 0.014617 58.470056 0.000000
## gamma1 0.170705 0.039829 4.285974 0.000018
## gamma2 -0.006474 0.039471 -0.164008 0.869725
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## mu 1218.842025 5.982188 203.745175 0.000000
## ar1 0.124920 0.000326 383.474050 0.000000
## ar2 0.876331 0.000559 1567.687665 0.000000
## ma1 0.984997 0.016688 59.025271 0.000000
## ma2 0.101461 0.015327 6.619671 0.000000
## omega 1.044425 0.273803 3.814518 0.000136
## alpha1 0.062208 0.029862 2.083168 0.037236
## alpha2 0.000000 0.034513 0.000004 0.999997
## beta1 0.854676 0.020135 42.447024 0.000000
## gamma1 0.170705 0.050077 3.408868 0.000652
## gamma2 -0.006474 0.047873 -0.135226 0.892433
##
## LogLikelihood : -12587.33
##
## Information Criteria
## ------------------------------------
##
## Akaike 6.8843
## Bayes 6.9030
## Shibata 6.8843
## Hannan-Quinn 6.8910
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 3.221 0.0726777
## Lag[2*(p+q)+(p+q)-1][11] 8.269 0.0003318
## Lag[4*(p+q)+(p+q)-1][19] 14.709 0.0354223
## d.o.f=4
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.0004813 0.9825
## Lag[2*(p+q)+(p+q)-1][8] 4.4134539 0.4388
## Lag[4*(p+q)+(p+q)-1][14] 8.9445177 0.2862
## d.o.f=3
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[4] 0.8703 0.500 2.000 0.3509
## ARCH Lag[6] 1.2323 1.461 1.711 0.6822
## ARCH Lag[8] 5.2797 2.368 1.583 0.2205
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 4.3207
## Individual Statistics:
## mu 0.0008132
## ar1 0.4163943
## ar2 0.4454755
## ma1 0.6180847
## ma2 0.2437415
## omega 0.2376982
## alpha1 0.5218082
## alpha2 0.5236632
## beta1 0.7749167
## gamma1 1.1999614
## gamma2 0.9023305
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 2.49 2.75 3.27
## Individual Statistic: 0.35 0.47 0.75
##
## Sign Bias Test
## ------------------------------------
## t-value prob sig
## Sign Bias 2.145 0.03200 **
## Negative Sign Bias 2.507 0.01222 **
## Positive Sign Bias 1.257 0.20900
## Joint Effect 8.043 0.04514 **
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 79.09 2.664e-09
## 2 30 104.67 1.726e-10
## 3 40 120.46 3.029e-10
## 4 50 127.57 6.354e-09
##
##
## Elapsed time : 2.103434sp500.garch21t.spec <- ugarchspec(variance.model = list(model = "gjrGARCH", garchOrder = c(2, 1)),
mean.model = list(armaOrder = c(2, 2)),
distribution.model = "std")
sp500.garch21t.fit <- ugarchfit(spec = sp500.garch21t.spec, data = sp500.ts)
sp500.garch21t.fit##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : gjrGARCH(2,1)
## Mean Model : ARFIMA(2,0,2)
## Distribution : std
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## mu 4512.514168 4.540982 9.9373e+02 0.000000
## ar1 1.963222 0.000004 5.4725e+05 0.000000
## ar2 -0.963201 0.000003 -2.7676e+05 0.000000
## ma1 -1.000182 0.000063 -1.5770e+04 0.000000
## ma2 0.028482 0.000062 4.6225e+02 0.000000
## omega 4.838084 0.932054 5.1908e+00 0.000000
## alpha1 0.000000 0.016265 1.0000e-06 0.999999
## alpha2 0.000000 0.016270 1.0000e-06 0.999999
## beta1 0.866598 0.010412 8.3232e+01 0.000000
## gamma1 0.085173 0.031127 2.7363e+00 0.006214
## gamma2 0.179632 0.035663 5.0369e+00 0.000000
## shape 9.150693 0.445537 2.0539e+01 0.000000
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## mu 4512.514168 4.711474 9.5777e+02 0.000000
## ar1 1.963222 0.000004 5.1342e+05 0.000000
## ar2 -0.963201 0.000004 -2.5962e+05 0.000000
## ma1 -1.000182 0.000072 -1.3816e+04 0.000000
## ma2 0.028482 0.000237 1.2020e+02 0.000000
## omega 4.838084 1.194961 4.0487e+00 0.000051
## alpha1 0.000000 0.013078 1.0000e-06 0.999999
## alpha2 0.000000 0.013081 1.0000e-06 0.999999
## beta1 0.866598 0.014421 6.0094e+01 0.000000
## gamma1 0.085173 0.029934 2.8453e+00 0.004437
## gamma2 0.179632 0.035465 5.0651e+00 0.000000
## shape 9.150693 0.476273 1.9213e+01 0.000000
##
## LogLikelihood : -15917.46
##
## Information Criteria
## ------------------------------------
##
## Akaike 8.7046
## Bayes 8.7250
## Shibata 8.7046
## Hannan-Quinn 8.7119
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.145 0.7033
## Lag[2*(p+q)+(p+q)-1][11] 3.949 0.9999
## Lag[4*(p+q)+(p+q)-1][19] 7.495 0.8583
## d.o.f=4
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.8228 0.3644
## Lag[2*(p+q)+(p+q)-1][8] 3.0460 0.6771
## Lag[4*(p+q)+(p+q)-1][14] 5.4190 0.7161
## d.o.f=3
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[4] 0.2334 0.500 2.000 0.6290
## ARCH Lag[6] 2.4600 1.461 1.711 0.3976
## ARCH Lag[8] 2.9776 2.368 1.583 0.5477
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 5.5101
## Individual Statistics:
## mu 0.003508
## ar1 0.659633
## ar2 0.659229
## ma1 0.107550
## ma2 0.082419
## omega 0.062891
## alpha1 1.217564
## alpha2 1.207468
## beta1 0.645511
## gamma1 0.839898
## gamma2 0.812640
## shape 0.249580
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 2.69 2.96 3.51
## Individual Statistic: 0.35 0.47 0.75
##
## Sign Bias Test
## ------------------------------------
## t-value prob sig
## Sign Bias 0.8549 0.3926
## Negative Sign Bias 0.5182 0.6043
## Positive Sign Bias 0.2932 0.7694
## Joint Effect 2.7232 0.4363
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 57.88 8.335e-06
## 2 30 64.84 1.490e-04
## 3 40 75.21 4.395e-04
## 4 50 94.23 1.099e-04
##
##
## Elapsed time : 24.35371vni.garch21t.spec <- ugarchspec(variance.model = list(model = "gjrGARCH", garchOrder = c(2, 1)),
mean.model = list(armaOrder = c(2, 2)),
distribution.model = "std")
vni.garch21t.fit <- ugarchfit(spec = vni.garch21t.spec, data = vni.ts)
vni.garch21t.fit##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : gjrGARCH(2,1)
## Mean Model : ARFIMA(2,0,2)
## Distribution : std
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## mu 1219.300378 1.792216 6.8033e+02 0.000000
## ar1 1.984254 0.000060 3.3197e+04 0.000000
## ar2 -0.984243 0.000027 -3.6147e+04 0.000000
## ma1 -0.878449 0.000083 -1.0626e+04 0.000000
## ma2 -0.097584 0.000362 -2.6985e+02 0.000000
## omega 1.041264 0.246707 4.2206e+00 0.000024
## alpha1 0.055776 0.024602 2.2672e+00 0.023380
## alpha2 0.000000 0.026297 1.0000e-06 0.999999
## beta1 0.864241 0.015878 5.4431e+01 0.000000
## gamma1 0.171315 0.046324 3.6982e+00 0.000217
## gamma2 -0.013351 0.047554 -2.8075e-01 0.778902
## shape 6.843260 0.760487 8.9985e+00 0.000000
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## mu 1219.300378 4.769493 255.64568 0.000000
## ar1 1.984254 0.000219 9071.67648 0.000000
## ar2 -0.984243 0.001141 -862.57714 0.000000
## ma1 -0.878449 0.000259 -3387.88817 0.000000
## ma2 -0.097584 0.002589 -37.68648 0.000000
## omega 1.041264 0.496261 2.09822 0.035886
## alpha1 0.055776 0.173581 0.32133 0.747962
## alpha2 0.000000 0.149287 0.00000 1.000000
## beta1 0.864241 0.109632 7.88310 0.000000
## gamma1 0.171315 0.293521 0.58366 0.559452
## gamma2 -0.013351 0.109438 -0.12199 0.902904
## shape 6.843260 1.633679 4.18886 0.000028
##
## LogLikelihood : -12526.62
##
## Information Criteria
## ------------------------------------
##
## Akaike 6.8517
## Bayes 6.8721
## Shibata 6.8517
## Hannan-Quinn 6.8589
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 2.709 0.099770
## Lag[2*(p+q)+(p+q)-1][11] 7.708 0.004214
## Lag[4*(p+q)+(p+q)-1][19] 14.319 0.046600
## d.o.f=4
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.02155 0.8833
## Lag[2*(p+q)+(p+q)-1][8] 4.47545 0.4291
## Lag[4*(p+q)+(p+q)-1][14] 9.02539 0.2787
## d.o.f=3
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[4] 0.535 0.500 2.000 0.4645
## ARCH Lag[6] 1.099 1.461 1.711 0.7194
## ARCH Lag[8] 5.347 2.368 1.583 0.2142
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 7.0507
## Individual Statistics:
## mu 0.002049
## ar1 0.439362
## ar2 0.439402
## ma1 0.081329
## ma2 0.123238
## omega 0.259492
## alpha1 0.232813
## alpha2 0.221741
## beta1 0.429736
## gamma1 0.706674
## gamma2 0.477694
## shape 0.793012
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 2.69 2.96 3.51
## Individual Statistic: 0.35 0.47 0.75
##
## Sign Bias Test
## ------------------------------------
## t-value prob sig
## Sign Bias 2.0648 0.039009 **
## Negative Sign Bias 2.6903 0.007171 ***
## Positive Sign Bias 0.8916 0.372660
## Joint Effect 8.1935 0.042177 **
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 36.74 0.008543
## 2 30 48.75 0.012259
## 3 40 56.04 0.037775
## 4 50 66.72 0.046828
##
##
## Elapsed time : 4.139568sp500.garch21st.spec <- ugarchspec(variance.model = list(model = "gjrGARCH", garchOrder = c(2, 1)),
mean.model = list(armaOrder = c(2, 2)),
distribution.model = "sstd")
sp500.garch21st.fit <- ugarchfit(spec = sp500.garch21st.spec, data = sp500.ts)
sp500.garch21st.fit##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : gjrGARCH(2,1)
## Mean Model : ARFIMA(2,0,2)
## Distribution : sstd
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## mu 4511.811221 5.874581 7.6802e+02 0.000000
## ar1 1.961280 0.000064 3.0429e+04 0.000000
## ar2 -0.961255 0.000032 -2.9881e+04 0.000000
## ma1 -1.002120 0.000998 -1.0039e+03 0.000000
## ma2 0.032125 0.002959 1.0857e+01 0.000000
## omega 5.203112 1.067883 4.8724e+00 0.000001
## alpha1 0.016133 0.021426 7.5297e-01 0.451466
## alpha2 0.000001 0.021708 2.3000e-05 0.999981
## beta1 0.850031 0.012606 6.7429e+01 0.000000
## gamma1 0.078596 0.034166 2.3004e+00 0.021426
## gamma2 0.191429 0.039238 4.8787e+00 0.000001
## skew 0.955653 0.020499 4.6619e+01 0.000000
## shape 8.804441 1.215401 7.2441e+00 0.000000
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## mu 4511.811221 17.822930 2.5315e+02 0.000000
## ar1 1.961280 0.001109 1.7693e+03 0.000000
## ar2 -0.961255 0.000552 -1.7412e+03 0.000000
## ma1 -1.002120 0.003307 -3.0306e+02 0.000000
## ma2 0.032125 0.007176 4.4768e+00 0.000008
## omega 5.203112 1.323796 3.9304e+00 0.000085
## alpha1 0.016133 0.019735 8.1749e-01 0.413651
## alpha2 0.000001 0.028986 1.8000e-05 0.999986
## beta1 0.850031 0.023092 3.6810e+01 0.000000
## gamma1 0.078596 0.032532 2.4160e+00 0.015693
## gamma2 0.191429 0.050143 3.8176e+00 0.000135
## skew 0.955653 0.022960 4.1622e+01 0.000000
## shape 8.804441 1.305289 6.7452e+00 0.000000
##
## LogLikelihood : -15914.25
##
## Information Criteria
## ------------------------------------
##
## Akaike 8.7034
## Bayes 8.7255
## Shibata 8.7034
## Hannan-Quinn 8.7113
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.01323 0.9084
## Lag[2*(p+q)+(p+q)-1][11] 3.49024 1.0000
## Lag[4*(p+q)+(p+q)-1][19] 6.94448 0.9131
## d.o.f=4
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 2.469 0.1161
## Lag[2*(p+q)+(p+q)-1][8] 6.062 0.2290
## Lag[4*(p+q)+(p+q)-1][14] 8.285 0.3526
## d.o.f=3
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[4] 0.02282 0.500 2.000 0.8799
## ARCH Lag[6] 3.23304 1.461 1.711 0.2750
## ARCH Lag[8] 3.88193 2.368 1.583 0.3931
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 5.3897
## Individual Statistics:
## mu 0.003346
## ar1 0.711196
## ar2 0.713058
## ma1 0.115639
## ma2 0.095219
## omega 0.049233
## alpha1 0.612297
## alpha2 0.719609
## beta1 0.529697
## gamma1 0.635610
## gamma2 0.601728
## skew 0.295636
## shape 0.250568
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 2.89 3.15 3.69
## Individual Statistic: 0.35 0.47 0.75
##
## Sign Bias Test
## ------------------------------------
## t-value prob sig
## Sign Bias 1.363053 0.1729
## Negative Sign Bias 0.006613 0.9947
## Positive Sign Bias 0.758367 0.4483
## Joint Effect 5.021751 0.1702
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 46.33 0.0004453
## 2 30 61.93 0.0003528
## 3 40 67.28 0.0032644
## 4 50 88.33 0.0004876
##
##
## Elapsed time : 11.40944vni.garch21st.spec <- ugarchspec(variance.model = list(model = "gjrGARCH", garchOrder = c(2, 1)),
mean.model = list(armaOrder = c(2, 2)),
distribution.model = "sstd")
vni.garch21st.fit <- ugarchfit(spec = vni.garch21st.spec, data = vni.ts)
vni.garch21st.fit##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : gjrGARCH(2,1)
## Mean Model : ARFIMA(2,0,2)
## Distribution : sstd
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## mu 1219.453640 1.885154 6.4687e+02 0.000000
## ar1 1.983118 0.000054 3.6801e+04 0.000000
## ar2 -0.983110 0.000023 -4.3232e+04 0.000000
## ma1 -0.880075 0.000119 -7.3657e+03 0.000000
## ma2 -0.094950 0.000554 -1.7148e+02 0.000000
## omega 0.958893 0.232099 4.1314e+00 0.000036
## alpha1 0.052474 0.023734 2.2109e+00 0.027043
## alpha2 0.000000 0.025242 0.0000e+00 1.000000
## beta1 0.874142 0.014605 5.9852e+01 0.000000
## gamma1 0.154531 0.043941 3.5167e+00 0.000437
## gamma2 -0.014717 0.045025 -3.2685e-01 0.743782
## skew 1.098107 0.023413 4.6902e+01 0.000000
## shape 6.863541 0.758472 9.0492e+00 0.000000
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## mu 1219.453640 2.914076 418.47006 0.000000
## ar1 1.983118 0.000542 3661.18507 0.000000
## ar2 -0.983110 0.001234 -796.47436 0.000000
## ma1 -0.880075 0.000140 -6293.30501 0.000000
## ma2 -0.094950 0.004441 -21.38215 0.000000
## omega 0.958893 0.320255 2.99416 0.002752
## alpha1 0.052474 0.165341 0.31737 0.750965
## alpha2 0.000000 0.155411 0.00000 1.000000
## beta1 0.874142 0.114253 7.65092 0.000000
## gamma1 0.154531 0.283205 0.54565 0.585306
## gamma2 -0.014717 0.120828 -0.12180 0.903060
## skew 1.098107 0.114617 9.58065 0.000000
## shape 6.863541 2.527410 2.71564 0.006615
##
## LogLikelihood : -12517.67
##
## Information Criteria
## ------------------------------------
##
## Akaike 6.8474
## Bayes 6.8694
## Shibata 6.8473
## Hannan-Quinn 6.8552
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 2.524 0.112095
## Lag[2*(p+q)+(p+q)-1][11] 7.528 0.008678
## Lag[4*(p+q)+(p+q)-1][19] 14.295 0.047393
## d.o.f=4
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.2771 0.5986
## Lag[2*(p+q)+(p+q)-1][8] 5.8758 0.2478
## Lag[4*(p+q)+(p+q)-1][14] 10.3311 0.1767
## d.o.f=3
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[4] 0.2606 0.500 2.000 0.6097
## ARCH Lag[6] 1.0976 1.461 1.711 0.7199
## ARCH Lag[8] 4.9721 2.368 1.583 0.2518
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 7.6805
## Individual Statistics:
## mu 0.002086
## ar1 0.379103
## ar2 0.378511
## ma1 0.085510
## ma2 0.177549
## omega 0.259267
## alpha1 0.306979
## alpha2 0.306219
## beta1 0.583038
## gamma1 0.951737
## gamma2 0.707423
## skew 0.357835
## shape 0.759841
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 2.89 3.15 3.69
## Individual Statistic: 0.35 0.47 0.75
##
## Sign Bias Test
## ------------------------------------
## t-value prob sig
## Sign Bias 2.2874 0.022229 **
## Negative Sign Bias 3.2087 0.001345 ***
## Positive Sign Bias 0.7376 0.460785
## Joint Effect 11.0326 0.011551 **
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 20.40 0.3706
## 2 30 36.77 0.1522
## 3 40 46.16 0.2003
## 4 50 58.06 0.1760
##
##
## Elapsed time : 8.577899sp500.garch21g.spec <- ugarchspec(variance.model = list(model = "gjrGARCH", garchOrder = c(2, 1)),
mean.model = list(armaOrder = c(2, 2)),
distribution.model = "ged")
sp500.garch21g.fit <- ugarchfit(spec = sp500.garch21g.spec, data = sp500.ts)
sp500.garch21g.fit##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : gjrGARCH(2,1)
## Mean Model : ARFIMA(2,0,2)
## Distribution : ged
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## mu 4514.600702 19.327654 2.3358e+02 0.000000
## ar1 0.920055 0.000002 4.4059e+05 0.000000
## ar2 0.080614 0.000079 1.0156e+03 0.000000
## ma1 0.044240 0.015882 2.7856e+00 0.005344
## ma2 -0.010295 0.017087 -6.0248e-01 0.546852
## omega 4.631098 0.984096 4.7059e+00 0.000003
## alpha1 0.000000 0.018530 0.0000e+00 1.000000
## alpha2 0.000000 0.018578 8.0000e-06 0.999993
## beta1 0.869272 0.010829 8.0274e+01 0.000000
## gamma1 0.087312 0.033066 2.6406e+00 0.008277
## gamma2 0.172143 0.037531 4.5867e+00 0.000005
## shape 1.445142 0.048624 2.9721e+01 0.000000
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## mu 4514.600702 3.280290 1.3763e+03 0.000000
## ar1 0.920055 0.000002 4.5713e+05 0.000000
## ar2 0.080614 0.000069 1.1765e+03 0.000000
## ma1 0.044240 0.015166 2.9172e+00 0.003532
## ma2 -0.010295 0.018438 -5.5835e-01 0.576607
## omega 4.631098 1.136933 4.0733e+00 0.000046
## alpha1 0.000000 0.014209 0.0000e+00 1.000000
## alpha2 0.000000 0.014249 1.1000e-05 0.999991
## beta1 0.869272 0.014293 6.0818e+01 0.000000
## gamma1 0.087312 0.029998 2.9106e+00 0.003608
## gamma2 0.172143 0.034032 5.0583e+00 0.000000
## shape 1.445142 0.053462 2.7031e+01 0.000000
##
## LogLikelihood : -15909.47
##
## Information Criteria
## ------------------------------------
##
## Akaike 8.7003
## Bayes 8.7206
## Shibata 8.7002
## Hannan-Quinn 8.7075
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.2746 0.6003
## Lag[2*(p+q)+(p+q)-1][11] 6.2648 0.3205
## Lag[4*(p+q)+(p+q)-1][19] 10.3754 0.4072
## d.o.f=4
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.815 0.3666
## Lag[2*(p+q)+(p+q)-1][8] 2.911 0.7019
## Lag[4*(p+q)+(p+q)-1][14] 5.162 0.7492
## d.o.f=3
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[4] 0.3459 0.500 2.000 0.5565
## ARCH Lag[6] 2.4530 1.461 1.711 0.3989
## ARCH Lag[8] 3.0856 2.368 1.583 0.5276
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 4.4281
## Individual Statistics:
## mu 0.01000
## ar1 0.38469
## ar2 0.38455
## ma1 0.19619
## ma2 0.05095
## omega 0.04644
## alpha1 1.08591
## alpha2 1.09107
## beta1 0.64871
## gamma1 0.89067
## gamma2 0.83977
## shape 0.24996
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 2.69 2.96 3.51
## Individual Statistic: 0.35 0.47 0.75
##
## Sign Bias Test
## ------------------------------------
## t-value prob sig
## Sign Bias 1.0777 0.2812
## Negative Sign Bias 0.5318 0.5949
## Positive Sign Bias 0.1495 0.8812
## Joint Effect 3.3840 0.3361
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 25.69 0.13894
## 2 30 47.98 0.01476
## 3 40 54.16 0.05395
## 4 50 57.13 0.19866
##
##
## Elapsed time : 11.36991vni.garch21g.spec <- ugarchspec(variance.model = list(model = "gjrGARCH", garchOrder = c(2, 1)),
mean.model = list(armaOrder = c(2, 2)),
distribution.model = "ged")
vni.garch21g.fit <- ugarchfit(spec = vni.garch21g.spec, data = vni.ts)
vni.garch21g.fit##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : gjrGARCH(2,1)
## Mean Model : ARFIMA(2,0,2)
## Distribution : ged
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## mu 1223.578842 6.902576 177.264088 0.000000
## ar1 0.747202 0.000310 2413.331429 0.000000
## ar2 0.253604 0.000262 968.255645 0.000000
## ma1 0.358871 0.016439 21.830854 0.000000
## ma2 0.037875 0.016558 2.287377 0.022174
## omega 1.009283 0.242815 4.156592 0.000032
## alpha1 0.060065 0.025966 2.313211 0.020711
## alpha2 0.000000 0.027708 0.000000 1.000000
## beta1 0.859651 0.016930 50.777808 0.000000
## gamma1 0.163147 0.047136 3.461220 0.000538
## gamma2 -0.004579 0.046898 -0.097639 0.922219
## shape 1.413327 0.046345 30.495541 0.000000
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## mu 1223.578842 2.446500 500.134462 0.000000
## ar1 0.747202 0.000383 1952.648546 0.000000
## ar2 0.253604 0.000144 1757.428274 0.000000
## ma1 0.358871 0.015401 23.302395 0.000000
## ma2 0.037875 0.015621 2.424542 0.015328
## omega 1.009283 0.252785 3.992658 0.000065
## alpha1 0.060065 0.029332 2.047767 0.040583
## alpha2 0.000000 0.034048 0.000000 1.000000
## beta1 0.859651 0.019545 43.983784 0.000000
## gamma1 0.163147 0.049390 3.303275 0.000956
## gamma2 -0.004579 0.046737 -0.097976 0.921952
## shape 1.413327 0.049534 28.532262 0.000000
##
## LogLikelihood : -12530.49
##
## Information Criteria
## ------------------------------------
##
## Akaike 6.8538
## Bayes 6.8742
## Shibata 6.8538
## Hannan-Quinn 6.8611
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 2.392 0.12193
## Lag[2*(p+q)+(p+q)-1][11] 6.886 0.07553
## Lag[4*(p+q)+(p+q)-1][19] 13.121 0.10187
## d.o.f=4
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.007623 0.9304
## Lag[2*(p+q)+(p+q)-1][8] 4.401258 0.4407
## Lag[4*(p+q)+(p+q)-1][14] 8.913342 0.2892
## d.o.f=3
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[4] 0.6846 0.500 2.000 0.4080
## ARCH Lag[6] 1.1056 1.461 1.711 0.7176
## ARCH Lag[8] 5.3392 2.368 1.583 0.2149
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 6.609
## Individual Statistics:
## mu 0.002428
## ar1 0.590140
## ar2 0.606988
## ma1 1.151685
## ma2 0.034444
## omega 0.215938
## alpha1 0.239996
## alpha2 0.229869
## beta1 0.421812
## gamma1 0.647444
## gamma2 0.448009
## shape 1.447849
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 2.69 2.96 3.51
## Individual Statistic: 0.35 0.47 0.75
##
## Sign Bias Test
## ------------------------------------
## t-value prob sig
## Sign Bias 1.7181 0.08587 *
## Negative Sign Bias 2.4624 0.01385 **
## Positive Sign Bias 0.9847 0.32485
## Joint Effect 7.0340 0.07082 *
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 40.85 0.002525
## 2 30 49.74 0.009637
## 3 40 60.96 0.013779
## 4 50 56.94 0.203564
##
##
## Elapsed time : 4.916411sp500.garch21sg.spec <- ugarchspec(variance.model = list(model = "gjrGARCH", garchOrder = c(2, 1)),
mean.model = list(armaOrder = c(2, 2)),
distribution.model = "sged")
sp500.garch21sg.fit <- ugarchfit(spec = sp500.garch21sg.spec, data = sp500.ts)
sp500.garch21sg.fit##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : gjrGARCH(2,1)
## Mean Model : ARFIMA(2,0,2)
## Distribution : sged
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## mu 2389.565485 19.114346 125.014 0
## ar1 0.324063 0.000484 670.021 0
## ar2 0.675108 0.000488 1383.404 0
## ma1 0.609406 0.002660 229.084 0
## ma2 -0.026511 0.000311 -85.121 0
## omega 1159.601716 15.430153 75.152 0
## alpha1 0.025254 0.000819 30.824 0
## alpha2 0.025210 0.000849 29.689 0
## beta1 0.732574 0.006470 113.235 0
## gamma1 0.192466 0.006050 31.814 0
## gamma2 0.189916 0.005913 32.120 0
## skew 1.020073 0.002635 387.147 0
## shape 0.372747 0.003576 104.228 0
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## mu 2389.565485 76.384454 31.283 0
## ar1 0.324063 0.002166 149.594 0
## ar2 0.675108 0.002023 333.642 0
## ma1 0.609406 0.010691 57.000 0
## ma2 -0.026511 0.000649 -40.853 0
## omega 1159.601716 28.809653 40.250 0
## alpha1 0.025254 0.001648 15.327 0
## alpha2 0.025210 0.001880 13.413 0
## beta1 0.732574 0.010886 67.293 0
## gamma1 0.192466 0.010148 18.966 0
## gamma2 0.189916 0.007377 25.745 0
## skew 1.020073 0.018079 56.422 0
## shape 0.372747 0.006067 61.442 0
##
## LogLikelihood : -16988.45
##
## Information Criteria
## ------------------------------------
##
## Akaike 9.2904
## Bayes 9.3124
## Shibata 9.2904
## Hannan-Quinn 9.2983
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 527.6 0
## Lag[2*(p+q)+(p+q)-1][11] 529.5 0
## Lag[4*(p+q)+(p+q)-1][19] 530.3 0
## d.o.f=4
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 926.1 0
## Lag[2*(p+q)+(p+q)-1][8] 926.1 0
## Lag[4*(p+q)+(p+q)-1][14] 926.1 0
## d.o.f=3
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[4] 0.0001270 0.500 2.000 0.991
## ARCH Lag[6] 0.0001310 1.461 1.711 1.000
## ARCH Lag[8] 0.0001355 2.368 1.583 1.000
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 512.3123
## Individual Statistics:
## mu 0.3781
## ar1 0.4829
## ar2 0.2949
## ma1 0.9490
## ma2 0.4429
## omega 6.1525
## alpha1 0.1445
## alpha2 0.1446
## beta1 4.5441
## gamma1 0.3230
## gamma2 0.5028
## skew 0.3057
## shape 14.1557
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 2.89 3.15 3.69
## Individual Statistic: 0.35 0.47 0.75
##
## Sign Bias Test
## ------------------------------------
## t-value prob sig
## Sign Bias 8.4012 6.244e-17 ***
## Negative Sign Bias 0.2164 8.287e-01
## Positive Sign Bias 51.5834 0.000e+00 ***
## Joint Effect 2663.1364 0.000e+00 ***
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 654.5 1.541e-126
## 2 30 663.0 5.117e-121
## 3 40 682.3 1.876e-118
## 4 50 675.8 4.030e-111
##
##
## Elapsed time : 5.470672vni.garch21sg.spec <- ugarchspec(variance.model = list(model = "gjrGARCH", garchOrder = c(2, 1)),
mean.model = list(armaOrder = c(2, 2)),
distribution.model = "sged")
vni.garch21sg.fit <- ugarchfit(spec = vni.garch21sg.spec, data = vni.ts)
vni.garch21sg.fit##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : gjrGARCH(2,1)
## Mean Model : ARFIMA(2,0,2)
## Distribution : sged
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## mu 1223.562788 4.280244 2.8586e+02 0.000000
## ar1 1.898630 0.000067 2.8402e+04 0.000000
## ar2 -0.898581 0.000026 -3.4672e+04 0.000000
## ma1 -0.796724 0.019112 -4.1688e+01 0.000000
## ma2 -0.071062 0.019204 -3.7004e+00 0.000215
## omega 0.930940 0.222686 4.1805e+00 0.000029
## alpha1 0.054946 0.023856 2.3032e+00 0.021266
## alpha2 0.000000 0.024962 1.0000e-06 0.999999
## beta1 0.871235 0.014657 5.9440e+01 0.000000
## gamma1 0.152088 0.042864 3.5482e+00 0.000388
## gamma2 -0.012357 0.043678 -2.8292e-01 0.777240
## skew 1.089195 0.020503 5.3124e+01 0.000000
## shape 1.399340 0.045667 3.0642e+01 0.000000
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## mu 1223.562788 4.547900 2.6904e+02 0.000000
## ar1 1.898630 0.000210 9.0375e+03 0.000000
## ar2 -0.898581 0.000389 -2.3093e+03 0.000000
## ma1 -0.796724 0.026601 -2.9951e+01 0.000000
## ma2 -0.071062 0.027974 -2.5403e+00 0.011075
## omega 0.930940 0.230219 4.0437e+00 0.000053
## alpha1 0.054946 0.043584 1.2607e+00 0.207418
## alpha2 0.000000 0.052232 1.0000e-06 1.000000
## beta1 0.871235 0.030230 2.8820e+01 0.000000
## gamma1 0.152088 0.073882 2.0585e+00 0.039541
## gamma2 -0.012357 0.054105 -2.2840e-01 0.819338
## skew 1.089195 0.035340 3.0820e+01 0.000000
## shape 1.399340 0.049033 2.8539e+01 0.000000
##
## LogLikelihood : -12516.64
##
## Information Criteria
## ------------------------------------
##
## Akaike 6.8468
## Bayes 6.8688
## Shibata 6.8468
## Hannan-Quinn 6.8546
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 2.243 0.1342534
## Lag[2*(p+q)+(p+q)-1][11] 8.057 0.0009122
## Lag[4*(p+q)+(p+q)-1][19] 15.052 0.0276148
## d.o.f=4
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.2613 0.6093
## Lag[2*(p+q)+(p+q)-1][8] 6.0446 0.2307
## Lag[4*(p+q)+(p+q)-1][14] 10.5862 0.1607
## d.o.f=3
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[4] 0.2703 0.500 2.000 0.6032
## ARCH Lag[6] 0.9622 1.461 1.711 0.7586
## ARCH Lag[8] 4.9724 2.368 1.583 0.2518
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 7.1958
## Individual Statistics:
## mu 0.0009744
## ar1 0.3950649
## ar2 0.3972570
## ma1 0.3012919
## ma2 0.0328392
## omega 0.2302632
## alpha1 0.3716181
## alpha2 0.3722467
## beta1 0.6501475
## gamma1 0.9966982
## gamma2 0.7561154
## skew 0.4434325
## shape 1.6738556
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 2.89 3.15 3.69
## Individual Statistic: 0.35 0.47 0.75
##
## Sign Bias Test
## ------------------------------------
## t-value prob sig
## Sign Bias 2.0882 0.036853 **
## Negative Sign Bias 3.0025 0.002696 ***
## Positive Sign Bias 0.7869 0.431390
## Joint Effect 9.7099 0.021200 **
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 24.52 0.17679
## 2 30 39.74 0.08832
## 3 40 42.75 0.31308
## 4 50 47.76 0.52345
##
##
## Elapsed time : 13.54061sp500.garch22n.spec <- ugarchspec(variance.model = list(model = "gjrGARCH", garchOrder = c(2, 2)),
mean.model = list(armaOrder = c(2, 2)),
distribution.model = "norm")
sp500.garch22n.fit <- ugarchfit(spec = sp500.garch22n.spec, data = sp500.ts)
sp500.garch22n.fit##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : gjrGARCH(2,2)
## Mean Model : ARFIMA(2,0,2)
## Distribution : norm
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## mu 4521.733767 6.779027 6.6702e+02 0.000000
## ar1 0.023500 0.000151 1.5530e+02 0.000000
## ar2 0.977855 0.000015 6.5958e+04 0.000000
## ma1 0.932885 0.000006 1.5431e+05 0.000000
## ma2 -0.048984 0.000153 -3.2045e+02 0.000000
## omega 5.083530 1.163755 4.3682e+00 0.000013
## alpha1 0.000000 0.017172 2.0000e-06 0.999999
## alpha2 0.010257 0.017649 5.8115e-01 0.561142
## beta1 0.863729 0.192003 4.4985e+00 0.000007
## beta2 0.000001 0.169899 4.0000e-06 0.999997
## gamma1 0.083865 0.029217 2.8704e+00 0.004099
## gamma2 0.166162 0.047598 3.4910e+00 0.000481
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## mu 4521.733767 5.867173 7.7068e+02 0.000000
## ar1 0.023500 0.000144 1.6295e+02 0.000000
## ar2 0.977855 0.000017 5.7566e+04 0.000000
## ma1 0.932885 0.000007 1.3901e+05 0.000000
## ma2 -0.048984 0.000110 -4.4724e+02 0.000000
## omega 5.083530 1.338705 3.7973e+00 0.000146
## alpha1 0.000000 0.014812 2.0000e-06 0.999999
## alpha2 0.010257 0.015503 6.6161e-01 0.508221
## beta1 0.863729 0.179516 4.8114e+00 0.000001
## beta2 0.000001 0.159425 4.0000e-06 0.999997
## gamma1 0.083865 0.030447 2.7545e+00 0.005879
## gamma2 0.166162 0.039875 4.1671e+00 0.000031
##
## LogLikelihood : -15953.17
##
## Information Criteria
## ------------------------------------
##
## Akaike 8.7241
## Bayes 8.7445
## Shibata 8.7241
## Hannan-Quinn 8.7314
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.0005226 0.9818
## Lag[2*(p+q)+(p+q)-1][11] 5.6902919 0.6876
## Lag[4*(p+q)+(p+q)-1][19] 9.2956324 0.5881
## d.o.f=4
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.6611 0.4162
## Lag[2*(p+q)+(p+q)-1][11] 3.8401 0.7565
## Lag[4*(p+q)+(p+q)-1][19] 6.3855 0.8415
## d.o.f=4
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[5] 2.581 0.500 2.000 0.1081
## ARCH Lag[7] 3.108 1.473 1.746 0.3047
## ARCH Lag[9] 3.652 2.402 1.619 0.4516
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 4.3903
## Individual Statistics:
## mu 0.004119
## ar1 0.438264
## ar2 0.440777
## ma1 0.079109
## ma2 0.086722
## omega 0.029198
## alpha1 0.519980
## alpha2 0.607179
## beta1 0.504490
## beta2 0.514011
## gamma1 0.531415
## gamma2 0.504743
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 2.69 2.96 3.51
## Individual Statistic: 0.35 0.47 0.75
##
## Sign Bias Test
## ------------------------------------
## t-value prob sig
## Sign Bias 1.44593 0.1483
## Negative Sign Bias 0.40901 0.6826
## Positive Sign Bias 0.02257 0.9820
## Joint Effect 4.54081 0.2087
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 87.96 7.620e-11
## 2 30 98.36 1.787e-09
## 3 40 113.95 2.881e-09
## 4 50 124.26 1.817e-08
##
##
## Elapsed time : 3.927836vni.garch22n.spec <- ugarchspec(variance.model = list(model = "gjrGARCH", garchOrder = c(2, 2)),
mean.model = list(armaOrder = c(2, 2)),
distribution.model = "norm")
vni.garch22n.fit <- ugarchfit(spec = vni.garch22n.spec, data = vni.ts)
vni.garch22n.fit##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : gjrGARCH(2,2)
## Mean Model : ARFIMA(2,0,2)
## Distribution : norm
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## mu 1.2188e+03 4.106767 296.77656 0.000000
## ar1 1.2622e-01 0.000501 251.89842 0.000000
## ar2 8.7504e-01 0.000545 1605.76789 0.000000
## ma1 9.8317e-01 0.017195 57.17657 0.000000
## ma2 1.0096e-01 0.016869 5.98510 0.000000
## omega 1.2915e+00 1.436546 0.89902 0.368640
## alpha1 5.8146e-02 0.024964 2.32918 0.019849
## alpha2 1.9323e-02 0.101511 0.19035 0.849033
## beta1 6.2919e-01 1.284789 0.48972 0.624329
## beta2 1.9090e-01 1.091209 0.17494 0.861127
## gamma1 1.7021e-01 0.038068 4.47120 0.000008
## gamma2 3.2677e-02 0.220377 0.14828 0.882124
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## mu 1.2188e+03 6.183553 197.10224 0.000000
## ar1 1.2622e-01 0.000325 387.82994 0.000000
## ar2 8.7504e-01 0.000555 1577.90260 0.000000
## ma1 9.8317e-01 0.017744 55.40850 0.000000
## ma2 1.0096e-01 0.015979 6.31820 0.000000
## omega 1.2915e+00 2.137541 0.60419 0.545715
## alpha1 5.8146e-02 0.043857 1.32581 0.184903
## alpha2 1.9323e-02 0.169194 0.11421 0.909075
## beta1 6.2919e-01 1.845090 0.34101 0.733097
## beta2 1.9090e-01 1.559156 0.12244 0.902554
## gamma1 1.7021e-01 0.048164 3.53394 0.000409
## gamma2 3.2677e-02 0.315604 0.10354 0.917536
##
## LogLikelihood : -12587.36
##
## Information Criteria
## ------------------------------------
##
## Akaike 6.8849
## Bayes 6.9052
## Shibata 6.8849
## Hannan-Quinn 6.8921
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 3.143 0.076263
## Lag[2*(p+q)+(p+q)-1][11] 8.194 0.000478
## Lag[4*(p+q)+(p+q)-1][19] 14.653 0.036855
## d.o.f=4
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.01965 0.8885
## Lag[2*(p+q)+(p+q)-1][11] 6.60748 0.3581
## Lag[4*(p+q)+(p+q)-1][19] 12.47266 0.2292
## d.o.f=4
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[5] 0.1352 0.500 2.000 0.7131
## ARCH Lag[7] 2.4384 1.473 1.746 0.4150
## ARCH Lag[9] 5.7421 2.402 1.619 0.1961
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 4.4759
## Individual Statistics:
## mu 0.0008131
## ar1 0.4182387
## ar2 0.4475445
## ma1 0.6317757
## ma2 0.2525198
## omega 0.2384029
## alpha1 0.5078484
## alpha2 0.5084258
## beta1 0.7627681
## beta2 0.7753708
## gamma1 1.2465095
## gamma2 0.8809470
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 2.69 2.96 3.51
## Individual Statistic: 0.35 0.47 0.75
##
## Sign Bias Test
## ------------------------------------
## t-value prob sig
## Sign Bias 2.119 0.03415 **
## Negative Sign Bias 2.551 0.01078 **
## Positive Sign Bias 1.157 0.24731
## Joint Effect 8.019 0.04563 **
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 78.03 4.050e-09
## 2 30 104.26 2.012e-10
## 3 40 118.40 6.200e-10
## 4 50 128.33 4.972e-09
##
##
## Elapsed time : 2.391605sp500.garch22t.spec <- ugarchspec(variance.model = list(model = "gjrGARCH", garchOrder = c(2, 2)),
mean.model = list(armaOrder = c(2, 2)),
distribution.model = "std")
sp500.garch22t.fit <- ugarchfit(spec = sp500.garch22t.spec, data = sp500.ts)
sp500.garch22t.fit##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : gjrGARCH(2,2)
## Mean Model : ARFIMA(2,0,2)
## Distribution : std
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## mu 4512.155012 5.552777 8.1259e+02 0.000000
## ar1 1.961486 0.000062 3.1552e+04 0.000000
## ar2 -0.961463 0.000031 -3.0926e+04 0.000000
## ma1 -1.001546 0.000315 -3.1746e+03 0.000000
## ma2 0.030993 0.002033 1.5247e+01 0.000000
## omega 4.966491 1.222676 4.0620e+00 0.000049
## alpha1 0.010762 0.020370 5.2832e-01 0.597277
## alpha2 0.000000 0.020528 0.0000e+00 1.000000
## beta1 0.857392 0.126765 6.7636e+00 0.000000
## beta2 0.000905 0.109584 8.2580e-03 0.993411
## gamma1 0.078682 0.033164 2.3725e+00 0.017668
## gamma2 0.181201 0.045653 3.9691e+00 0.000072
## shape 8.939273 1.227263 7.2839e+00 0.000000
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## mu 4512.155012 11.025141 4.0926e+02 0.000000
## ar1 1.961486 0.001017 1.9296e+03 0.000000
## ar2 -0.961463 0.000510 -1.8862e+03 0.000000
## ma1 -1.001546 0.000524 -1.9099e+03 0.000000
## ma2 0.030993 0.001718 1.8040e+01 0.000000
## omega 4.966491 1.496031 3.3198e+00 0.000901
## alpha1 0.010762 0.017921 6.0050e-01 0.548173
## alpha2 0.000000 0.029155 0.0000e+00 1.000000
## beta1 0.857392 0.267083 3.2102e+00 0.001326
## beta2 0.000905 0.247127 3.6620e-03 0.997078
## gamma1 0.078682 0.030373 2.5905e+00 0.009583
## gamma2 0.181201 0.068866 2.6312e+00 0.008508
## shape 8.939273 1.221006 7.3212e+00 0.000000
##
## LogLikelihood : -15916.46
##
## Information Criteria
## ------------------------------------
##
## Akaike 8.7046
## Bayes 8.7267
## Shibata 8.7046
## Hannan-Quinn 8.7125
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.03585 0.8498
## Lag[2*(p+q)+(p+q)-1][11] 3.64088 1.0000
## Lag[4*(p+q)+(p+q)-1][19] 7.13814 0.8957
## d.o.f=4
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 1.801 0.1796
## Lag[2*(p+q)+(p+q)-1][11] 5.585 0.4938
## Lag[4*(p+q)+(p+q)-1][19] 8.288 0.6430
## d.o.f=4
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[5] 3.230 0.500 2.000 0.0723
## ARCH Lag[7] 3.562 1.473 1.746 0.2455
## ARCH Lag[9] 3.921 2.402 1.619 0.4092
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 5.6204
## Individual Statistics:
## mu 0.00315
## ar1 0.65308
## ar2 0.65394
## ma1 0.09920
## ma2 0.07643
## omega 0.03598
## alpha1 0.87824
## alpha2 0.94279
## beta1 0.67996
## beta2 0.67535
## gamma1 0.91804
## gamma2 0.87004
## shape 0.31501
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 2.89 3.15 3.69
## Individual Statistic: 0.35 0.47 0.75
##
## Sign Bias Test
## ------------------------------------
## t-value prob sig
## Sign Bias 1.1618 0.2454
## Negative Sign Bias 0.2710 0.7864
## Positive Sign Bias 0.5718 0.5675
## Joint Effect 4.1009 0.2508
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 57.43 9.790e-06
## 2 30 66.34 9.429e-05
## 3 40 71.69 1.096e-03
## 4 50 88.11 5.144e-04
##
##
## Elapsed time : 31.40015vni.garch22t.spec <- ugarchspec(variance.model = list(model = "gjrGARCH", garchOrder = c(2, 2)),
mean.model = list(armaOrder = c(2, 2)),
distribution.model = "std")
vni.garch22t.fit <- ugarchfit(spec = vni.garch22t.spec, data = vni.ts)
vni.garch22t.fit##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : gjrGARCH(2,2)
## Mean Model : ARFIMA(2,0,2)
## Distribution : std
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## mu 1219.143438 1.839663 6.6270e+02 0e+00
## ar1 1.983689 0.000059 3.3399e+04 0e+00
## ar2 -0.983678 0.000027 -3.6483e+04 0e+00
## ma1 -0.877354 0.000177 -4.9507e+03 0e+00
## ma2 -0.097851 0.000803 -1.2179e+02 0e+00
## omega 1.928727 0.410810 4.6949e+00 3e-06
## alpha1 0.065349 0.010726 6.0925e+00 0e+00
## alpha2 0.039266 0.008697 4.5146e+00 6e-06
## beta1 0.000000 0.136867 2.0000e-06 1e+00
## beta2 0.745700 0.117655 6.3380e+00 0e+00
## gamma1 0.142742 0.022113 6.4552e+00 0e+00
## gamma2 0.154628 0.020016 7.7251e+00 0e+00
## shape 6.828221 0.757430 9.0150e+00 0e+00
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## mu 1219.143438 3.007329 4.0539e+02 0.000000
## ar1 1.983689 0.000249 7.9595e+03 0.000000
## ar2 -0.983678 0.001133 -8.6794e+02 0.000000
## ma1 -0.877354 0.000244 -3.5897e+03 0.000000
## ma2 -0.097851 0.004522 -2.1638e+01 0.000000
## omega 1.928727 1.737706 1.1099e+00 0.267030
## alpha1 0.065349 0.324992 2.0108e-01 0.840637
## alpha2 0.039266 0.209468 1.8746e-01 0.851304
## beta1 0.000000 0.451974 1.0000e-06 1.000000
## beta2 0.745700 0.528753 1.4103e+00 0.158452
## gamma1 0.142742 0.393554 3.6270e-01 0.716829
## gamma2 0.154628 0.422898 3.6564e-01 0.714634
## shape 6.828221 1.793925 3.8063e+00 0.000141
##
## LogLikelihood : -12526.23
##
## Information Criteria
## ------------------------------------
##
## Akaike 6.8520
## Bayes 6.8741
## Shibata 6.8520
## Hannan-Quinn 6.8599
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 2.871 0.090168
## Lag[2*(p+q)+(p+q)-1][11] 7.831 0.002509
## Lag[4*(p+q)+(p+q)-1][19] 14.382 0.044633
## d.o.f=4
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.00365 0.9518
## Lag[2*(p+q)+(p+q)-1][11] 7.61791 0.2501
## Lag[4*(p+q)+(p+q)-1][19] 13.46868 0.1666
## d.o.f=4
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[5] 0.2775 0.500 2.000 0.5984
## ARCH Lag[7] 2.6132 1.473 1.746 0.3833
## ARCH Lag[9] 6.5691 2.402 1.619 0.1363
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 7.1512
## Individual Statistics:
## mu 0.002064
## ar1 0.436608
## ar2 0.436638
## ma1 0.080862
## ma2 0.117221
## omega 0.254955
## alpha1 0.209311
## alpha2 0.208421
## beta1 0.431104
## beta2 0.432693
## gamma1 0.621783
## gamma2 0.516039
## shape 0.814375
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 2.89 3.15 3.69
## Individual Statistic: 0.35 0.47 0.75
##
## Sign Bias Test
## ------------------------------------
## t-value prob sig
## Sign Bias 2.092 0.036513 **
## Negative Sign Bias 3.123 0.001806 ***
## Positive Sign Bias 1.136 0.255905
## Joint Effect 11.043 0.011498 **
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 39.42 0.003895
## 2 30 38.82 0.105166
## 3 40 56.83 0.032380
## 4 50 56.94 0.203564
##
##
## Elapsed time : 5.039516sp500.garch22st.spec <- ugarchspec(variance.model = list(model = "gjrGARCH", garchOrder = c(2, 2)),
mean.model = list(armaOrder = c(2, 2)),
distribution.model = "sstd")
sp500.garch22st.fit <- ugarchfit(spec = sp500.garch22st.spec, data = sp500.ts)
vni.garch22t.fit##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : gjrGARCH(2,2)
## Mean Model : ARFIMA(2,0,2)
## Distribution : std
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## mu 1219.143438 1.839663 6.6270e+02 0e+00
## ar1 1.983689 0.000059 3.3399e+04 0e+00
## ar2 -0.983678 0.000027 -3.6483e+04 0e+00
## ma1 -0.877354 0.000177 -4.9507e+03 0e+00
## ma2 -0.097851 0.000803 -1.2179e+02 0e+00
## omega 1.928727 0.410810 4.6949e+00 3e-06
## alpha1 0.065349 0.010726 6.0925e+00 0e+00
## alpha2 0.039266 0.008697 4.5146e+00 6e-06
## beta1 0.000000 0.136867 2.0000e-06 1e+00
## beta2 0.745700 0.117655 6.3380e+00 0e+00
## gamma1 0.142742 0.022113 6.4552e+00 0e+00
## gamma2 0.154628 0.020016 7.7251e+00 0e+00
## shape 6.828221 0.757430 9.0150e+00 0e+00
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## mu 1219.143438 3.007329 4.0539e+02 0.000000
## ar1 1.983689 0.000249 7.9595e+03 0.000000
## ar2 -0.983678 0.001133 -8.6794e+02 0.000000
## ma1 -0.877354 0.000244 -3.5897e+03 0.000000
## ma2 -0.097851 0.004522 -2.1638e+01 0.000000
## omega 1.928727 1.737706 1.1099e+00 0.267030
## alpha1 0.065349 0.324992 2.0108e-01 0.840637
## alpha2 0.039266 0.209468 1.8746e-01 0.851304
## beta1 0.000000 0.451974 1.0000e-06 1.000000
## beta2 0.745700 0.528753 1.4103e+00 0.158452
## gamma1 0.142742 0.393554 3.6270e-01 0.716829
## gamma2 0.154628 0.422898 3.6564e-01 0.714634
## shape 6.828221 1.793925 3.8063e+00 0.000141
##
## LogLikelihood : -12526.23
##
## Information Criteria
## ------------------------------------
##
## Akaike 6.8520
## Bayes 6.8741
## Shibata 6.8520
## Hannan-Quinn 6.8599
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 2.871 0.090168
## Lag[2*(p+q)+(p+q)-1][11] 7.831 0.002509
## Lag[4*(p+q)+(p+q)-1][19] 14.382 0.044633
## d.o.f=4
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.00365 0.9518
## Lag[2*(p+q)+(p+q)-1][11] 7.61791 0.2501
## Lag[4*(p+q)+(p+q)-1][19] 13.46868 0.1666
## d.o.f=4
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[5] 0.2775 0.500 2.000 0.5984
## ARCH Lag[7] 2.6132 1.473 1.746 0.3833
## ARCH Lag[9] 6.5691 2.402 1.619 0.1363
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 7.1512
## Individual Statistics:
## mu 0.002064
## ar1 0.436608
## ar2 0.436638
## ma1 0.080862
## ma2 0.117221
## omega 0.254955
## alpha1 0.209311
## alpha2 0.208421
## beta1 0.431104
## beta2 0.432693
## gamma1 0.621783
## gamma2 0.516039
## shape 0.814375
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 2.89 3.15 3.69
## Individual Statistic: 0.35 0.47 0.75
##
## Sign Bias Test
## ------------------------------------
## t-value prob sig
## Sign Bias 2.092 0.036513 **
## Negative Sign Bias 3.123 0.001806 ***
## Positive Sign Bias 1.136 0.255905
## Joint Effect 11.043 0.011498 **
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 39.42 0.003895
## 2 30 38.82 0.105166
## 3 40 56.83 0.032380
## 4 50 56.94 0.203564
##
##
## Elapsed time : 5.039516vni.garch22st.spec <- ugarchspec(variance.model = list(model = "gjrGARCH", garchOrder = c(2, 2)),
mean.model = list(armaOrder = c(2, 2)),
distribution.model = "sstd")
vni.garch22st.fit <- ugarchfit(spec = vni.garch22st.spec, data = vni.ts)
vni.garch22st.fit##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : gjrGARCH(2,2)
## Mean Model : ARFIMA(2,0,2)
## Distribution : sstd
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## mu 1219.268699 1.925296 6.3329e+02 0e+00
## ar1 1.982495 0.000052 3.8333e+04 0e+00
## ar2 -0.982487 0.000021 -4.6771e+04 0e+00
## ma1 -0.877728 0.000118 -7.4637e+03 0e+00
## ma2 -0.096392 0.000566 -1.7031e+02 0e+00
## omega 1.784366 0.376873 4.7347e+00 2e-06
## alpha1 0.063777 0.008721 7.3128e+00 0e+00
## alpha2 0.035231 0.005685 6.1973e+00 0e+00
## beta1 0.000000 0.172203 1.0000e-06 1e+00
## beta2 0.762955 0.150823 5.0586e+00 0e+00
## gamma1 0.121216 0.020631 5.8754e+00 0e+00
## gamma2 0.143307 0.020570 6.9669e+00 0e+00
## skew 1.100062 0.023408 4.6995e+01 0e+00
## shape 6.853733 0.755330 9.0738e+00 0e+00
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## mu 1219.268699 2.099324 580.79107 0.000000
## ar1 1.982495 0.000639 3100.23168 0.000000
## ar2 -0.982487 0.001265 -776.95277 0.000000
## ma1 -0.877728 0.000151 -5825.59372 0.000000
## ma2 -0.096392 0.003393 -28.40761 0.000000
## omega 1.784366 1.848626 0.96524 0.334425
## alpha1 0.063777 0.348509 0.18300 0.854799
## alpha2 0.035231 0.174639 0.20173 0.840125
## beta1 0.000000 0.915109 0.00000 1.000000
## beta2 0.762955 0.951265 0.80204 0.422528
## gamma1 0.121216 0.399803 0.30319 0.761745
## gamma2 0.143307 0.488289 0.29349 0.769149
## skew 1.100062 0.120648 9.11798 0.000000
## shape 6.853733 2.717496 2.52208 0.011666
##
## LogLikelihood : -12516.97
##
## Information Criteria
## ------------------------------------
##
## Akaike 6.8475
## Bayes 6.8713
## Shibata 6.8475
## Hannan-Quinn 6.8560
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 2.900 0.088584
## Lag[2*(p+q)+(p+q)-1][11] 7.882 0.002012
## Lag[4*(p+q)+(p+q)-1][19] 14.583 0.038745
## d.o.f=4
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.1927 0.66068
## Lag[2*(p+q)+(p+q)-1][11] 9.2849 0.12878
## Lag[4*(p+q)+(p+q)-1][19] 15.0883 0.09461
## d.o.f=4
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[5] 0.4861 0.500 2.000 0.4857
## ARCH Lag[7] 2.7999 1.473 1.746 0.3517
## ARCH Lag[9] 6.4405 2.402 1.619 0.1444
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 7.727
## Individual Statistics:
## mu 0.002066
## ar1 0.377233
## ar2 0.376594
## ma1 0.082207
## ma2 0.169833
## omega 0.255385
## alpha1 0.271778
## alpha2 0.292828
## beta1 0.587681
## beta2 0.596195
## gamma1 0.814500
## gamma2 0.748718
## skew 0.383927
## shape 0.778063
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 3.08 3.34 3.9
## Individual Statistic: 0.35 0.47 0.75
##
## Sign Bias Test
## ------------------------------------
## t-value prob sig
## Sign Bias 2.317 0.0205719 **
## Negative Sign Bias 3.770 0.0001659 ***
## Positive Sign Bias 1.019 0.3082169
## Joint Effect 15.255 0.0016112 ***
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 19.56 0.4213
## 2 30 32.23 0.3099
## 3 40 40.77 0.3927
## 4 50 52.24 0.3492
##
##
## Elapsed time : 8.122767sp500.garch22g.spec <- ugarchspec(variance.model = list(model = "gjrGARCH", garchOrder = c(2, 2)),
mean.model = list(armaOrder = c(2, 2)),
distribution.model = "ged")
sp500.garch22g.fit <- ugarchfit(spec = sp500.garch22g.spec, data = sp500.ts)
sp500.garch22g.fit##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : gjrGARCH(2,2)
## Mean Model : ARFIMA(2,0,2)
## Distribution : ged
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## mu 4514.850196 19.249130 2.3455e+02 0.000000
## ar1 0.979204 0.000002 5.8067e+05 0.000000
## ar2 0.021427 0.000076 2.8112e+02 0.000000
## ma1 -0.015621 0.015753 -9.9158e-01 0.321404
## ma2 -0.010320 0.015729 -6.5608e-01 0.511771
## omega 5.076447 1.498021 3.3888e+00 0.000702
## alpha1 0.000072 0.017963 4.0080e-03 0.996802
## alpha2 0.000000 0.017378 0.0000e+00 1.000000
## beta1 0.744350 0.275832 2.6986e+00 0.006964
## beta2 0.111294 0.246502 4.5149e-01 0.651635
## gamma1 0.091115 0.034877 2.6124e+00 0.008990
## gamma2 0.195454 0.058397 3.3470e+00 0.000817
## shape 1.443534 0.048530 2.9745e+01 0.000000
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## mu 4514.850196 3.278571 1.3771e+03 0.000000
## ar1 0.979204 0.000002 5.1128e+05 0.000000
## ar2 0.021427 0.000068 3.1636e+02 0.000000
## ma1 -0.015621 0.014926 -1.0465e+00 0.295320
## ma2 -0.010320 0.015646 -6.5957e-01 0.509527
## omega 5.076447 1.631391 3.1117e+00 0.001860
## alpha1 0.000072 0.013253 5.4330e-03 0.995665
## alpha2 0.000000 0.012820 0.0000e+00 1.000000
## beta1 0.744350 0.300458 2.4774e+00 0.013235
## beta2 0.111294 0.270364 4.1164e-01 0.680600
## gamma1 0.091115 0.034677 2.6275e+00 0.008601
## gamma2 0.195454 0.048916 3.9957e+00 0.000065
## shape 1.443534 0.053261 2.7103e+01 0.000000
##
## LogLikelihood : -15910.01
##
## Information Criteria
## ------------------------------------
##
## Akaike 8.7011
## Bayes 8.7231
## Shibata 8.7011
## Hannan-Quinn 8.7089
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.2314 0.6305
## Lag[2*(p+q)+(p+q)-1][11] 6.2610 0.3227
## Lag[4*(p+q)+(p+q)-1][19] 10.3612 0.4094
## d.o.f=4
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.9122 0.3395
## Lag[2*(p+q)+(p+q)-1][11] 4.3199 0.6847
## Lag[4*(p+q)+(p+q)-1][19] 7.0892 0.7742
## d.o.f=4
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[5] 2.095 0.500 2.000 0.1478
## ARCH Lag[7] 2.508 1.473 1.746 0.4020
## ARCH Lag[9] 3.182 2.402 1.619 0.5321
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 4.8966
## Individual Statistics:
## mu 0.01017
## ar1 0.38577
## ar2 0.38568
## ma1 0.18885
## ma2 0.05053
## omega 0.04798
## alpha1 1.05014
## alpha2 1.07084
## beta1 0.63580
## beta2 0.63110
## gamma1 0.86376
## gamma2 0.80687
## shape 0.24524
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 2.89 3.15 3.69
## Individual Statistic: 0.35 0.47 0.75
##
## Sign Bias Test
## ------------------------------------
## t-value prob sig
## Sign Bias 1.1096 0.2673
## Negative Sign Bias 0.4661 0.6412
## Positive Sign Bias 0.1052 0.9162
## Joint Effect 3.2349 0.3568
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 30.09 0.05070
## 2 30 45.80 0.02454
## 3 40 62.19 0.01055
## 4 50 63.72 0.07705
##
##
## Elapsed time : 2.391156vni.garch22g.spec <- ugarchspec(variance.model = list(model = "gjrGARCH", garchOrder = c(2, 2)),
mean.model = list(armaOrder = c(2, 2)),
distribution.model = "ged")
vni.garch22g.fit <- ugarchfit(spec = vni.garch22g.spec, data = vni.ts)
vni.garch22g.fit##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : gjrGARCH(2,2)
## Mean Model : ARFIMA(2,0,2)
## Distribution : ged
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## mu 1223.149331 4.271623 2.8634e+02 0.000000
## ar1 1.897630 0.000079 2.4066e+04 0.000000
## ar2 -0.897558 0.000035 -2.5377e+04 0.000000
## ma1 -0.791730 0.015684 -5.0479e+01 0.000000
## ma2 -0.071525 0.015180 -4.7118e+00 0.000002
## omega 1.025523 0.729159 1.4064e+00 0.159592
## alpha1 0.057625 0.009135 6.3081e+00 0.000000
## alpha2 0.000004 0.065980 5.6000e-05 0.999955
## beta1 0.853337 0.688936 1.2386e+00 0.215483
## beta2 0.005678 0.583654 9.7280e-03 0.992238
## gamma1 0.168929 0.044409 3.8040e+00 0.000142
## gamma2 -0.004221 0.094055 -4.4883e-02 0.964201
## shape 1.398756 0.045853 3.0505e+01 0.000000
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## mu 1223.149331 4.793140 2.5519e+02 0.000000
## ar1 1.897630 0.000084 2.2700e+04 0.000000
## ar2 -0.897558 0.000316 -2.8417e+03 0.000000
## ma1 -0.791730 0.020572 -3.8486e+01 0.000000
## ma2 -0.071525 0.017701 -4.0407e+00 0.000053
## omega 1.025523 1.321079 7.7628e-01 0.437585
## alpha1 0.057625 0.016972 3.3953e+00 0.000686
## alpha2 0.000004 0.136646 2.7000e-05 0.999978
## beta1 0.853337 1.249222 6.8309e-01 0.494547
## beta2 0.005678 1.050054 5.4070e-03 0.995686
## gamma1 0.168929 0.066750 2.5308e+00 0.011382
## gamma2 -0.004221 0.186409 -2.2646e-02 0.981932
## shape 1.398756 0.048374 2.8916e+01 0.000000
##
## LogLikelihood : -12525.92
##
## Information Criteria
## ------------------------------------
##
## Akaike 6.8519
## Bayes 6.8739
## Shibata 6.8518
## Hannan-Quinn 6.8597
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 2.63 0.1048911
## Lag[2*(p+q)+(p+q)-1][11] 8.48 0.0001146
## Lag[4*(p+q)+(p+q)-1][19] 15.40 0.0212274
## d.o.f=4
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.01794 0.8935
## Lag[2*(p+q)+(p+q)-1][11] 7.01111 0.3117
## Lag[4*(p+q)+(p+q)-1][19] 12.85080 0.2037
## d.o.f=4
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[5] 0.2691 0.500 2.000 0.6040
## ARCH Lag[7] 2.3746 1.473 1.746 0.4271
## ARCH Lag[9] 6.2667 2.402 1.619 0.1560
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 7.2625
## Individual Statistics:
## mu 0.0009979
## ar1 0.4618352
## ar2 0.4631451
## ma1 0.4449275
## ma2 0.0546495
## omega 0.2174493
## alpha1 0.2674840
## alpha2 0.2598062
## beta1 0.4693510
## beta2 0.4807609
## gamma1 0.7158899
## gamma2 0.4986240
## shape 1.4076353
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 2.89 3.15 3.69
## Individual Statistic: 0.35 0.47 0.75
##
## Sign Bias Test
## ------------------------------------
## t-value prob sig
## Sign Bias 1.8651 0.06224 *
## Negative Sign Bias 2.4754 0.01335 **
## Positive Sign Bias 0.9561 0.33910
## Joint Effect 7.1004 0.06877 *
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 42.43 0.001552
## 2 30 53.36 0.003829
## 3 40 56.39 0.035287
## 4 50 64.81 0.064578
##
##
## Elapsed time : 9.079568sp500.garch22sg.spec <- ugarchspec(variance.model = list(model = "gjrGARCH", garchOrder = c(2, 2)),
mean.model = list(armaOrder = c(2, 2)),
distribution.model = "sged")
sp500.garch22sg.fit <- ugarchfit(spec = sp500.garch22sg.spec, data = sp500.ts)
sp500.garch22sg.fit##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : gjrGARCH(2,2)
## Mean Model : ARFIMA(2,0,2)
## Distribution : sged
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## mu 4511.720661 18.211612 247.738672 0.000000
## ar1 0.499187 0.000160 3126.345257 0.000000
## ar2 0.501776 0.000156 3206.523972 0.000000
## ma1 0.464337 0.015222 30.504743 0.000000
## ma2 -0.018451 0.015901 -1.160388 0.245891
## omega 4.647798 1.202910 3.863794 0.000112
## alpha1 0.000000 0.018465 0.000001 0.999999
## alpha2 0.000000 0.018600 0.000001 0.999999
## beta1 0.866787 0.203538 4.258596 0.000021
## beta2 0.000004 0.180853 0.000019 0.999984
## gamma1 0.088947 0.033562 2.650212 0.008044
## gamma2 0.178655 0.054184 3.297195 0.000977
## skew 0.975990 0.018809 51.889803 0.000000
## shape 1.445141 0.049728 29.061018 0.000000
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## mu 4511.720661 3.663553 1.2315e+03 0.000000
## ar1 0.499187 0.000010 4.7585e+04 0.000000
## ar2 0.501776 0.000011 4.4157e+04 0.000000
## ma1 0.464337 0.014201 3.2698e+01 0.000000
## ma2 -0.018451 0.015602 -1.1826e+00 0.236959
## omega 4.647798 1.227400 3.7867e+00 0.000153
## alpha1 0.000000 0.014125 1.0000e-06 0.999999
## alpha2 0.000000 0.014246 1.0000e-06 0.999999
## beta1 0.866787 0.158721 5.4611e+00 0.000000
## beta2 0.000004 0.141967 2.5000e-05 0.999980
## gamma1 0.088947 0.030969 2.8722e+00 0.004077
## gamma2 0.178655 0.039360 4.5390e+00 0.000006
## skew 0.975990 0.020818 4.6883e+01 0.000000
## shape 1.445141 0.058097 2.4875e+01 0.000000
##
## LogLikelihood : -15906.94
##
## Information Criteria
## ------------------------------------
##
## Akaike 8.7000
## Bayes 8.7237
## Shibata 8.6999
## Hannan-Quinn 8.7084
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.2082 0.6482
## Lag[2*(p+q)+(p+q)-1][11] 6.1414 0.3950
## Lag[4*(p+q)+(p+q)-1][19] 10.1486 0.4437
## d.o.f=4
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.8842 0.3471
## Lag[2*(p+q)+(p+q)-1][11] 4.0667 0.7230
## Lag[4*(p+q)+(p+q)-1][19] 6.7852 0.8044
## d.o.f=4
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[5] 2.456 0.500 2.000 0.1171
## ARCH Lag[7] 2.825 1.473 1.746 0.3477
## ARCH Lag[9] 3.387 2.402 1.619 0.4962
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 5.3554
## Individual Statistics:
## mu 0.008912
## ar1 0.454036
## ar2 0.454607
## ma1 0.184698
## ma2 0.049950
## omega 0.060212
## alpha1 0.897003
## alpha2 0.918152
## beta1 0.513471
## beta2 0.518580
## gamma1 0.637085
## gamma2 0.602256
## skew 0.414402
## shape 0.203069
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 3.08 3.34 3.9
## Individual Statistic: 0.35 0.47 0.75
##
## Sign Bias Test
## ------------------------------------
## t-value prob sig
## Sign Bias 1.0626 0.2880
## Negative Sign Bias 0.4130 0.6796
## Positive Sign Bias 0.2442 0.8071
## Joint Effect 3.1704 0.3661
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 25.55 0.14317
## 2 30 39.89 0.08583
## 3 40 49.88 0.11372
## 4 50 59.48 0.14509
##
##
## Elapsed time : 29.05758vni.garch22sg.spec <- ugarchspec(variance.model = list(model = "gjrGARCH", garchOrder = c(2, 2)),
mean.model = list(armaOrder = c(2, 2)),
distribution.model = "sged")
vni.garch22sg.fit <- ugarchfit(spec = vni.garch22sg.spec, data = vni.ts)
vni.garch22sg.fit##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : gjrGARCH(2,2)
## Mean Model : ARFIMA(2,0,2)
## Distribution : sged
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## mu 1.2256e+03 1.538726 7.9647e+02 0.000000
## ar1 1.9456e-02 0.000614 3.1670e+01 0.000000
## ar2 9.8145e-01 0.000650 1.5101e+03 0.000000
## ma1 1.0833e+00 0.000007 1.5897e+05 0.000000
## ma2 9.9930e-02 0.000152 6.5611e+02 0.000000
## omega 1.7323e+00 0.377514 4.5887e+00 0.000004
## alpha1 6.4154e-02 0.016709 3.8394e+00 0.000123
## alpha2 3.8922e-02 0.008111 4.7985e+00 0.000002
## beta1 0.0000e+00 0.166102 0.0000e+00 1.000000
## beta2 7.5884e-01 0.143322 5.2947e+00 0.000000
## gamma1 1.2063e-01 0.027905 4.3229e+00 0.000015
## gamma2 1.4179e-01 0.014616 9.7013e+00 0.000000
## skew 1.0953e+00 0.022265 4.9194e+01 0.000000
## shape 1.4005e+00 0.045734 3.0623e+01 0.000000
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## mu 1.2256e+03 1.085889 1.1286e+03 0.000000
## ar1 1.9456e-02 0.000356 5.4637e+01 0.000000
## ar2 9.8145e-01 0.000543 1.8079e+03 0.000000
## ma1 1.0833e+00 0.000009 1.2593e+05 0.000000
## ma2 9.9930e-02 0.000126 7.9506e+02 0.000000
## omega 1.7323e+00 0.464894 3.7262e+00 0.000194
## alpha1 6.4154e-02 0.019534 3.2842e+00 0.001023
## alpha2 3.8922e-02 0.029047 1.3400e+00 0.180254
## beta1 0.0000e+00 0.271419 0.0000e+00 1.000000
## beta2 7.5884e-01 0.239051 3.1744e+00 0.001501
## gamma1 1.2063e-01 0.030174 3.9978e+00 0.000064
## gamma2 1.4179e-01 0.040643 3.4886e+00 0.000486
## skew 1.0953e+00 0.022705 4.8241e+01 0.000000
## shape 1.4005e+00 0.048959 2.8606e+01 0.000000
##
## LogLikelihood : -12519.12
##
## Information Criteria
## ------------------------------------
##
## Akaike 6.8487
## Bayes 6.8724
## Shibata 6.8487
## Hannan-Quinn 6.8571
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 2.201 0.137893
## Lag[2*(p+q)+(p+q)-1][11] 7.961 0.001413
## Lag[4*(p+q)+(p+q)-1][19] 14.002 0.057849
## d.o.f=4
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.1951 0.6587
## Lag[2*(p+q)+(p+q)-1][11] 8.8268 0.1558
## Lag[4*(p+q)+(p+q)-1][19] 14.6111 0.1124
## d.o.f=4
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[5] 0.4212 0.500 2.000 0.5163
## ARCH Lag[7] 2.6255 1.473 1.746 0.3811
## ARCH Lag[9] 6.0737 2.402 1.619 0.1698
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 7.2295
## Individual Statistics:
## mu 0.0007026
## ar1 0.5016898
## ar2 0.5290166
## ma1 0.0622718
## ma2 0.0295469
## omega 0.2213487
## alpha1 0.2778797
## alpha2 0.3258190
## beta1 0.6254821
## beta2 0.6383940
## gamma1 0.8400894
## gamma2 0.7321655
## skew 0.3723595
## shape 1.7778885
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 3.08 3.34 3.9
## Individual Statistic: 0.35 0.47 0.75
##
## Sign Bias Test
## ------------------------------------
## t-value prob sig
## Sign Bias 1.7091 0.0875118 *
## Negative Sign Bias 3.4090 0.0006591 ***
## Positive Sign Bias 0.8867 0.3753221
## Joint Effect 12.7375 0.0052402 ***
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 20.62 0.3580
## 2 30 23.87 0.7354
## 3 40 28.68 0.8876
## 4 50 40.49 0.8014
##
##
## Elapsed time : 15.84051sp500.model.list <- list(
garch11n = sp500.garch11n.fit,
garch11t = sp500.garch11t.fit,
garch11st = sp500.garch11st.fit,
garch11g = sp500.garch11g.fit,
garch11sg = sp500.garch11sg.fit,
garch12n = sp500.garch12n.fit,
garch12t = sp500.garch12t.fit,
garch12st = sp500.garch12st.fit,
garch12g = sp500.garch12g.fit,
garch12sg = sp500.garch12sg.fit,
garch21n = sp500.garch21n.fit,
garch21t = sp500.garch21t.fit,
garch21st = sp500.garch21st.fit,
garch21g = sp500.garch21g.fit,
garch21sg = sp500.garch21sg.fit,
garch22n = sp500.garch22n.fit,
garch22t = sp500.garch22t.fit,
garch22st = sp500.garch22st.fit,
garch22g = sp500.garch22g.fit,
garch22sg = sp500.garch22sg.fit)
sp500.info.mat <- sapply(sp500.model.list, infocriteria)
rownames(sp500.info.mat) <- rownames(infocriteria(sp500.garch11n.fit))
sp500.info.mat## garch11n garch11t garch11st garch11g garch11sg garch12n garch12t
## Akaike 8.729106 8.710112 8.751429 8.704309 8.702952 8.729652 8.710764
## Bayes 8.744365 8.727066 8.768383 8.721263 8.721602 8.746607 8.729414
## Shibata 8.729094 8.710097 8.751414 8.704294 8.702934 8.729638 8.710746
## Hannan-Quinn 8.734539 8.716149 8.757466 8.710346 8.709593 8.735689 8.717405
## garch12st garch12g garch12sg garch21n garch21t garch21st garch21g
## Akaike 8.753600 8.704654 9.656139 8.724053 8.704621 8.703415 8.700259
## Bayes 8.772249 8.723304 9.676484 8.742703 8.724966 8.725456 8.720604
## Shibata 8.753582 8.704636 9.656117 8.724035 8.704600 8.703390 8.700237
## Hannan-Quinn 8.760240 8.711295 9.663383 8.730694 8.711866 8.711264 8.707503
## garch21sg garch22n garch22t garch22st garch22g garch22sg
## Akaike 9.290409 8.724134 8.704625 8.708318 8.701096 8.699966
## Bayes 9.312449 8.744479 8.726665 8.732054 8.723136 8.723701
## Shibata 9.290384 8.724113 8.704600 8.708289 8.701071 8.699937
## Hannan-Quinn 9.298257 8.731378 8.712473 8.716770 8.708944 8.708417aux <- which(sp500.info.mat == min(sp500.info.mat), arr.ind=TRUE)
auxg <- colnames(sp500.info.mat)[aux[,2]]
auxg## [1] "garch22sg"Mô hình biên phù hợp cho chuỗi S&P500 là garch22sg
vni.model.list <- list(
garch11n = vni.garch11n.fit,
garch11t = vni.garch11t.fit,
garch11st = vni.garch11st.fit,
garch11g = vni.garch11g.fit,
garch11sg = vni.garch11sg.fit,
garch12n = vni.garch12n.fit,
garch12t = vni.garch12t.fit,
garch12st = vni.garch12st.fit,
garch12g = vni.garch12g.fit,
garch12sg = vni.garch12sg.fit,
garch21n = vni.garch21n.fit,
garch21t = vni.garch21t.fit,
garch21st = vni.garch21st.fit,
garch21g = vni.garch21g.fit,
garch21sg = vni.garch21sg.fit,
garch22n = vni.garch22n.fit,
garch22t = vni.garch22t.fit,
garch22st = vni.garch22st.fit,
garch22g = vni.garch22g.fit,
garch22sg = vni.garch22sg.fit)
vni.info.mat <- sapply(vni.model.list, infocriteria)
rownames(vni.info.mat) <- rownames(infocriteria(vni.garch11n.fit))
vni.info.mat## garch11n garch11t garch11st garch11g garch11sg garch12n garch12t
## Akaike 6.883246 6.850635 6.857967 6.850233 6.845727 6.883789 6.851181
## Bayes 6.898504 6.867589 6.874921 6.867187 6.864376 6.900743 6.869831
## Shibata 6.883234 6.850620 6.857952 6.850218 6.845709 6.883774 6.851163
## Hannan-Quinn 6.888679 6.856672 6.864004 6.856270 6.852367 6.889826 6.857822
## garch12st garch12g garch12sg garch21n garch21t garch21st garch21g
## Akaike 6.881178 6.850777 6.846251 6.884331 6.851705 6.847362 6.853821
## Bayes 6.901523 6.869426 6.866596 6.902980 6.872050 6.869402 6.874166
## Shibata 6.881156 6.850759 6.846230 6.884313 6.851684 6.847337 6.853799
## Hannan-Quinn 6.888422 6.857417 6.853496 6.890971 6.858949 6.855210 6.861065
## garch21sg garch22n garch22t garch22st garch22g garch22sg
## Akaike 6.846797 6.884895 6.852040 6.847523 6.851868 6.848698
## Bayes 6.868837 6.905240 6.874081 6.871259 6.873909 6.872434
## Shibata 6.846771 6.884873 6.852015 6.847494 6.851843 6.848669
## Hannan-Quinn 6.854645 6.892139 6.859888 6.855975 6.859716 6.857150vniaux <- which(vni.info.mat == min(vni.info.mat), arr.ind=TRUE)
vniauxg <- colnames(vni.info.mat)[aux[,2]]
vniauxg## [1] "garch22sg"Mô hình biên phù hợp cho chuỗi Vni index là garch22sg
vni.res <- residuals(vni.garch22sg.fit)/sigma(vni.garch22sg.fit)
fitdist(distribution = "sged", vni.res, control = list())## $pars
## mu sigma skew shape
## -0.02475599 0.99949333 1.08509791 1.39735826
##
## $convergence
## [1] 0
##
## $values
## [1] 5197.403 5124.526 5124.526
##
## $lagrange
## [1] 0
##
## $hessian
## [,1] [,2] [,3] [,4]
## [1,] 4178.02434 -520.039 -1021.64804 -36.29203
## [2,] -520.03898 5066.006 -200.95104 399.76697
## [3,] -1021.64804 -200.951 2268.82990 -59.39997
## [4,] -36.29203 399.767 -59.39997 501.39130
##
## $ineqx0
## NULL
##
## $nfuneval
## [1] 98
##
## $outer.iter
## [1] 2
##
## $elapsed
## Time difference of 0.455797 secs
##
## $vscale
## [1] 1 1 1 1 1Xuất ra các tham số đưa vào biến v
v <-pdist(distribution = "sged", q = vni.res, mu = -0.02475599 , sigma = 0.99949333 , skew = 1.08509791, shape = 1.39735826)sp500.res <- residuals(sp500.garch22sg.fit)/sigma(sp500.garch22sg.fit)
fitdist(distribution = "sged", sp500.res, control = list())## $pars
## mu sigma skew shape
## -0.01968168 1.01400013 0.94747800 1.41937720
##
## $convergence
## [1] 0
##
## $values
## [1] 5237.140 5187.564 5187.564
##
## $lagrange
## [1] 0
##
## $hessian
## [,1] [,2] [,3] [,4]
## [1,] 4219.7004 118.5430 -1492.8854 113.1888
## [2,] 118.5430 4906.3742 414.0387 346.3273
## [3,] -1492.8854 414.0387 3659.7745 -183.5394
## [4,] 113.1888 346.3273 -183.5394 453.1738
##
## $ineqx0
## NULL
##
## $nfuneval
## [1] 89
##
## $outer.iter
## [1] 2
##
## $elapsed
## Time difference of 0.4215689 secs
##
## $vscale
## [1] 1 1 1 1 1Xuất ra các tham số đưa vào biến s
s <-pdist(distribution = "sged", q = sp500.res, mu = -0.01968168 , sigma = 1.01400013 , skew = 0.94747800, shape = 1.41937720)Kiểm định Anderson-Darling là một phương pháp thống kê được sử dụng để kiểm tra tính phân phối của một tập dữ liệu cụ thể dựa trên so sánh giữa hàm phân phối tích lũy (cumulative distribution function - CDF) của dữ liệu thực tế và hàm phân phối tích lũy của một phân phối giả định, thường là phân phối chuẩn (normal distribution) hoặc một phân phối khác. Mục tiêu của kiểm định Anderson-Darling là xác định xem tập dữ liệu có tuân theo phân phối giả định hay không.
Quá trình kiểm định Anderson-Darling bao gồm các bước sau:
Kiểm định Anderson-Darling thường được sử dụng để kiểm tra tính phân phối chuẩn (normality) của dữ liệu, nhưng nó cũng có thể được áp dụng cho các phân phối khác tùy thuộc vào giả định của bạn.
Kiểm định Cramér-von Mises là một phương pháp thống kê được sử dụng để kiểm tra xem một tập dữ liệu có tuân theo một phân phối cụ thể hay không, giống như kiểm định Anderson-Darling và kiểm định Kolmogorov-Smirnov. Tên gọi “Cramér-von Mises” xuất phát từ tên của hai nhà toán học người Thụy Điển, Harald Cramér và Carl von Mises, người đã đóng góp vào phương pháp này.
Quá trình kiểm định Cramér-von Mises thường bao gồm các bước sau:
Kiểm định Cramér-von Mises thường được sử dụng để kiểm tra tính phân phối của dữ liệu và được ưa chuộng trong thống kê và khoa học dữ liệu. Tuy nhiên, cũng giống như các kiểm định thống kê khác, nó có các giả định và hạn chế mà bạn cần phải xem xét khi áp dụng vào tập dữ liệu cụ thể.
Kiểm định Kolmogorov-Smirnov (KS test) là một phương pháp thống kê được sử dụng để kiểm tra tính phân phối của một tập dữ liệu so với một phân phối giả định hoặc một tập dữ liệu tham chiếu. Phương pháp này giúp xác định xem tập dữ liệu có phù hợp với phân phối đã giả định hay không.
Cách thức hoạt động của kiểm định KS test như sau:
Dựa trên giá trị p-value, bạn có thể kết luận về tính phân phối của tập dữ liệu. Nếu p-value nhỏ hơn một ngưỡng xác định trước (thường là 0.05), bạn có thể bác bỏ giả định rằng dữ liệu tuân theo phân phối đã giả định.
ad.test(s, "punif")##
## Anderson-Darling test of goodness-of-fit
## Null hypothesis: uniform distribution
## Parameters assumed to be fixed
##
## data: s
## An = 0.73702, p-value = 0.5287cvm.test(s, "punif")##
## Cramer-von Mises test of goodness-of-fit
## Null hypothesis: uniform distribution
## Parameters assumed to be fixed
##
## data: s
## omega2 = 0.10218, p-value = 0.5743ks.test(s, "punif")##
## Asymptotic one-sample Kolmogorov-Smirnov test
##
## data: s
## D = 0.013779, p-value = 0.4905
## alternative hypothesis: two-sidedad.test(v, "punif")##
## Anderson-Darling test of goodness-of-fit
## Null hypothesis: uniform distribution
## Parameters assumed to be fixed
##
## data: v
## An = 0.40688, p-value = 0.8419cvm.test(v, "punif")##
## Cramer-von Mises test of goodness-of-fit
## Null hypothesis: uniform distribution
## Parameters assumed to be fixed
##
## data: v
## omega2 = 0.050342, p-value = 0.8742ks.test(v, "punif")##
## Asymptotic one-sample Kolmogorov-Smirnov test
##
## data: v
## D = 0.0092824, p-value = 0.9107
## alternative hypothesis: two-sidedHàm copula lần đầu tiên được đưa ra bởi Sklar (1959), người đã đặt ra các nguyên tắc cơ bản của lý thuyết copula. Thuật ngữ “copula” có nguồn gốc từ chữ Latinh (copulare) có nghĩa là “liên kết” hoặc “kết nối”. Bài viết đầu tiên liên quan đến copula với sự phụ thuộc của các biến ngẫu nhiên là công trình của Schweizer và Wol (1981),trong đó các tác giả trình bày các tính chất cơ bản bất biến của copula dưới các phép biến đổi đơn điệu của các biến ngẫu nhiên. Copula là một hàm phân phối tích lũy đa biến được xác định trên một không gian n chiều sao cho mọi phân phối biên đều xác định trên đoạn [0,1] .
Copula là một công cụ để mô hình hóa sự phụ thuộc giữa các biến ngẫu nhiên vì nó mô tả mối tương quan bên trong chúng. Ngoài ra, hàm copula cho phép xây dựng hàm phân phối đa biến với các phân phối biên khác nhau một cách riêng biệt, với cấu trúc phụ thuộc giữa chúng được nắm bắt bởi copula. Tuy nhiên, một sự phát triển lớn mạnh trong lý thuyết copula đã bắt đầu khi khả năng ứng dụng của chúng tăng lên cùng với sự gia tăng khả năng tính toán thông qua các phần mềm hỗ trợ. Vì những đặc tính hữu ích này, copula đã trở thành công cụ rất phổ biến hiện nay được vận dụng để xác định sự phụ thuộc giữa các tài sản tài chính cũng như giữa tài sản tài chính với các tài sản thực
Định lý Sklar cho copula hai biến
Gọi \(Z_1\) và \(Z_2\) lần lượt là hai biến ngẫu nhiên đại diện cho phần dư chuẩn hóa của mỗi chuỗi lợi suất chứng khoán được trình bày trong phương trình :
\(z_t \mid \Omega_{t-i}=\sqrt{\frac{d f}{\sigma_t^2(d f-2)}} \varepsilon_t\)
Hàm phân phối biên có điều kiện của \(Z_1\) và \(Z_2\) được xác định như sau:
\(\begin{aligned} & F_1\left(z_1 \mid \Omega_{t-1}\right)=\operatorname{Pr}\left(Z_1 \leq z_1 \mid \Omega_{t-1}\right)=u \\ & F_2\left(z_2 \mid \Omega_{t-1}\right)=\operatorname{Pr}\left(Z_2 \leq z_2 \mid \Omega_{t-1}\right)=v\end{aligned}\)
trong đó \(Ω_{t-1}\) đại diện cho tập hợp các thông tin tại thời điểm \(t_1\) và hàm phân phối đồng thời có điều kiện:
\(H\left(z_1, z_2 \mid \Omega_{t-1}\right)=\operatorname{Pr}\left(Z_1 \leq z_1 ; Z_2 \leq z_2 \mid \Omega_{t-1}\right)\)
Khi đó, tồn tại một copula có điều kiện duy nhất C:\([0,1]^2\) -> [0,1] sao cho :
\(H\left(z_1, z_2 \mid \Omega_{t-1}\right)=C\left(u, v \mid \Omega_{t-1}\right)=C\left[F_1\left(z_1 \mid \Omega_{t-1}\right), F_2\left(z_2 \mid \Omega_{t-1}\right)\right]\) với biến u,v có phân phối đồng nhất trên [0,1]
Ngược lại, nếu C là một copula hai biến có điều kiện và \(F_1\)(.), \(F_2\)(.) là hai hàm phân phối đơn biến có điều kiện. Khi đó, hàm H là một hàm phân phối đồng thời có điều kiện với các hàm phân phối biên có điều kiện lần lượt là \(F_1\left(z_1 \mid \Omega_{t-1}\right), F_2\left(z_2 \mid \Omega_{t-1}\right)\)
Hàm copula Gumbel
Hàm copula Gumbel dùng để mô hình hóa cấu trúc phụ thuộc bất đối xứng nhưng tập trung đuôi trên. Copula Gumbel với hàm sinh \(\varphi(t)=(-\log t)^\gamma\) được Gumbel (1960) đề xuất có dạng:
\(C_G(u, v ; \alpha)=\exp \left\{-\left[(-\log u)^\alpha+(-\log v)^\alpha\right]^{1 / \alpha}\right\}, \alpha \in[1, \infty)\)
Khi α = 1, copula Gumbel thể hiện tính độc lập giữa phân phối biên u và v; khi α -> ∞, u và v phụ thuộc dương hoàn toàn.
Mối quan hệ giữa tham số Gumbel copula và Kendall tau (τ) được xác định bởi τ = 1-1/α. Hệ số phụ thuộc đuôi trên và đuôi dưới được xác định lần lượt bởi\(λ_u\)= 2-\(2^1/α\), \(λ_L\) =0
Hàm mật độ xác suất của copula Gumbel có công thức:
\(c(u, v)=\frac{C(u, v)}{u v} \times \frac{(\log u \log v)^{\alpha-1}}{\left((-\log u)^\alpha+(-\log u)^\alpha\right)^{2-\frac{1}{\alpha}}} \times\left[\left((-\log u)^\alpha+(-\log u)^\alpha\right)^{\frac{1}{\alpha}}+\alpha-1\right]\)
result <- BiCopSelect(s, v, familyset = 1:9, selectioncrit = "AIC", indeptest = FALSE, level = 0.05)
result## Bivariate copula: Gumbel (par = 1.01, tau = 0.01)Mô hình Copula phù hợp là Gumbel với par = 1.01 và tau = 0.01, đây là các tham số cụ thể của copula Gumbel được ước tính từ dữ liệu. Trong trường hợp này, “par” có giá trị 1.01 và “tau” có giá trị 0.01. Các giá trị này cung cấp thông tin về hình dạng cụ thể của copula Gumbel trong mô hình. Thường thì “par” sẽ ảnh hưởng đến độ dốc của mối quan hệ và “tau” có thể liên quan đến độ tương quan giữa các biến.
Stu <- BiCopEst(s, v, family = 4, method = "mle", se = TRUE, max.df = 10)
summary(Stu)## Family
## ------
## No: 4
## Name: Gumbel
##
## Parameter(s)
## ------------
## par: 1.01 (SE = 0.01)
##
## Dependence measures
## -------------------
## Kendall's tau: 0.01 (empirical = 0.01, p value = 0.34)
## Upper TD: 0.01
## Lower TD: 0
##
## Fit statistics
## --------------
## logLik: 0.83
## AIC: 0.34
## BIC: 6.54Từ kết quả ước lượng mô hình, ta có :