Lista 4 - R
Giordano Bruno
2023-12-05
#BAsE DE DADOS:
library(GetBCBData)
cambio = gbcbd_get_series(id = 11753, first.date = "2008-01-01", last.date = Sys.Date())
cambio = cambio[,0:2]
cambio = data.frame(cambio, row.names = 1)
colnames(cambio) = c("Câmbio")
cambio = ts(cambio, start = 2008, frequency = 12)
plot(cambio)
#QUESTÃO A: Proponha um modelo GARCH para os dados. Estime utilizando a distribuicao normal e t-student.
library(rugarch)
#MODELO COM DISTRIBUIÇÃO NORMAL:
dist_norm = ugarchspec(mean.model = list(armaOrder =c(0 ,0)),
variance.model = list(model = "sGARCH",
garchOrder = c(1,1)),
distribution.model = "norm")
fit = ugarchfit(data = cambio, spec = dist_norm, out.sample = 20); fit##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : sGARCH(1,1)
## Mean Model : ARFIMA(0,0,0)
## Distribution : norm
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## mu 75.5224 0.64129 117.766922 0.000000
## omega 4.1093 1.84192 2.230972 0.025683
## alpha1 0.9990 0.11703 8.536177 0.000000
## beta1 0.0000 0.02498 0.000001 0.999999
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## mu 75.5224 1.971346 38.310057 0.000000
## omega 4.1093 1.485347 2.766535 0.005666
## alpha1 0.9990 0.076576 13.045916 0.000000
## beta1 0.0000 0.024231 0.000001 0.999999
##
## LogLikelihood : -691.8694
##
## Information Criteria
## ------------------------------------
##
## Akaike 8.2351
## Bayes 8.3092
## Shibata 8.2341
## Hannan-Quinn 8.2652
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 118.6 0
## Lag[2*(p+q)+(p+q)-1][2] 161.2 0
## Lag[4*(p+q)+(p+q)-1][5] 259.4 0
## d.o.f=0
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.4157 0.5191
## Lag[2*(p+q)+(p+q)-1][5] 0.5610 0.9479
## Lag[4*(p+q)+(p+q)-1][9] 2.1761 0.8830
## d.o.f=2
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[3] 0.007434 0.500 2.000 0.9313
## ARCH Lag[5] 0.279131 1.440 1.667 0.9458
## ARCH Lag[7] 1.469520 2.315 1.543 0.8275
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 4.1244
## Individual Statistics:
## mu 1.09679
## omega 0.03229
## alpha1 0.56846
## beta1 0.86458
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 1.07 1.24 1.6
## Individual Statistic: 0.35 0.47 0.75
##
## Sign Bias Test
## ------------------------------------
## t-value prob sig
## Sign Bias 0.5855 0.5590
## Negative Sign Bias 0.1489 0.8818
## Positive Sign Bias 0.7890 0.4312
## Joint Effect 1.1619 0.7622
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 376.2 3.832e-68
## 2 30 467.4 1.401e-80
## 3 40 456.1 1.509e-72
## 4 50 453.8 5.906e-67
##
##
## Elapsed time : 0.09539509#MODELO COM DISTRIBUIÇÃO T-STUDENT:
dist_t = ugarchspec(mean.model = list(armaOrder =c(0 ,0)),
variance.model = list(model = "sGARCH",
garchOrder = c(1,1)),
distribution.model = "sstd")
fit2 = ugarchfit(data = cambio, spec = dist_t, out.sample = 20); fit2##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : sGARCH(1,1)
## Mean Model : ARFIMA(0,0,0)
## Distribution : sstd
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## mu 97.22140 0.76277 127.4584 0.000000
## omega 16.44147 4.81211 3.4167 0.000634
## alpha1 0.97234 0.01309 74.2840 0.000000
## beta1 0.00000 0.15785 0.0000 1.000000
## skew 3.17646 0.53268 5.9632 0.000000
## shape 9.63767 8.93911 1.0781 0.280968
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## mu 97.22140 0.91817 105.88561 0.000000
## omega 16.44147 6.90362 2.38157 0.017239
## alpha1 0.97234 0.41132 2.36394 0.018082
## beta1 0.00000 0.22636 0.00000 1.000000
## skew 3.17646 1.13886 2.78916 0.005284
## shape 9.63767 23.10696 0.41709 0.676613
##
## LogLikelihood : -675.0958
##
## Information Criteria
## ------------------------------------
##
## Akaike 8.0603
## Bayes 8.1714
## Shibata 8.0579
## Hannan-Quinn 8.1054
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 148.2 0
## Lag[2*(p+q)+(p+q)-1][2] 212.2 0
## Lag[4*(p+q)+(p+q)-1][5] 383.4 0
## d.o.f=0
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 33.87 5.898e-09
## Lag[2*(p+q)+(p+q)-1][5] 34.70 1.483e-09
## Lag[4*(p+q)+(p+q)-1][9] 35.34 1.465e-08
## d.o.f=2
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[3] 0.07553 0.500 2.000 0.7834
## ARCH Lag[5] 0.36727 1.440 1.667 0.9219
## ARCH Lag[7] 0.66478 2.315 1.543 0.9611
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 16.9992
## Individual Statistics:
## mu 0.7723
## omega 0.5180
## alpha1 0.1197
## beta1 1.3889
## skew 0.1006
## shape 1.4265
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 1.49 1.68 2.12
## Individual Statistic: 0.35 0.47 0.75
##
## Sign Bias Test
## ------------------------------------
## t-value prob sig
## Sign Bias 3.1327 0.002052 ***
## Negative Sign Bias 2.8974 0.004278 ***
## Positive Sign Bias 0.1457 0.884353
## Joint Effect 13.6182 0.003474 ***
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 211.0 2.183e-34
## 2 30 205.1 2.058e-28
## 3 40 218.5 7.990e-27
## 4 50 245.1 7.046e-28
##
##
## Elapsed time : 0.1993411#QUESTÃO B: Proponha um modelo GJR para os dados. Estime utilizando a distribuição normal e t-student.
library(rugarch)
#MODELO COM DISTRIBUIÇÃO NORMAL:
gjr_norm = ugarchspec(mean.model = list(armaOrder =c(0 ,0)),
variance.model = list(model = "gjrGARCH",
garchOrder = c(2,2)),
distribution.model = "norm")
fit3 = ugarchfit(data = cambio, spec = gjr_norm, out.sample = 20); fit3 ##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : gjrGARCH(2,2)
## Mean Model : ARFIMA(0,0,0)
## Distribution : norm
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## mu 76.960673 0.194913 394.85 0
## omega 1.990244 0.006826 291.56 0
## alpha1 0.331061 0.000963 343.65 0
## alpha2 0.029331 0.000105 279.78 0
## beta1 0.611169 0.001699 359.69 0
## beta2 0.071633 0.000231 310.74 0
## gamma1 0.608275 0.001685 361.04 0
## gamma2 -0.743058 0.001969 -377.36 0
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## mu 76.960673 4.034399 19.0761 0.000000
## omega 1.990244 0.025168 79.0793 0.000000
## alpha1 0.331061 0.095603 3.4629 0.000534
## alpha2 0.029331 0.001076 27.2545 0.000000
## beta1 0.611169 0.144171 4.2392 0.000022
## beta2 0.071633 0.063562 1.1270 0.259745
## gamma1 0.608275 0.143623 4.2352 0.000023
## gamma2 -0.743058 0.142508 -5.2141 0.000000
##
## LogLikelihood : -701.998
##
## Information Criteria
## ------------------------------------
##
## Akaike 8.4023
## Bayes 8.5505
## Shibata 8.3981
## Hannan-Quinn 8.4625
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 144.3 0
## Lag[2*(p+q)+(p+q)-1][2] 200.4 0
## Lag[4*(p+q)+(p+q)-1][5] 337.2 0
## d.o.f=0
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 52.13 5.196e-13
## Lag[2*(p+q)+(p+q)-1][11] 75.44 0.000e+00
## Lag[4*(p+q)+(p+q)-1][19] 90.15 0.000e+00
## d.o.f=4
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[5] 2.713 0.500 2.000 0.099560
## ARCH Lag[7] 5.140 1.473 1.746 0.112985
## ARCH Lag[9] 15.285 2.402 1.619 0.001716
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 9.576
## Individual Statistics:
## mu 0.6797
## omega 0.6805
## alpha1 0.6806
## alpha2 0.6805
## beta1 0.6770
## beta2 0.6796
## gamma1 0.6805
## gamma2 0.6792
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 1.89 2.11 2.59
## Individual Statistic: 0.35 0.47 0.75
##
## Sign Bias Test
## ------------------------------------
## t-value prob sig
## Sign Bias 2.8690 0.004659 ***
## Negative Sign Bias 2.6889 0.007910 ***
## Positive Sign Bias 0.1826 0.855335
## Joint Effect 9.4614 0.023745 **
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 245.3 2.740e-41
## 2 30 256.1 3.261e-38
## 3 40 268.6 4.554e-36
## 4 50 270.9 1.762e-32
##
##
## Elapsed time : 0.626899#MODELO COM DISTRIBUIÇÃO T-STUDENT:
gjr_t = ugarchspec(mean.model = list(armaOrder =c(0 ,0)),
variance.model = list(model = "gjrGARCH",
garchOrder = c(3,3)),
distribution.model = "sstd")
fit4 = ugarchfit(data = cambio, spec = gjr_t, out.sample = 30); fit4##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : gjrGARCH(3,3)
## Mean Model : ARFIMA(0,0,0)
## Distribution : sstd
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## mu 98.179244 0.245085 400.5931 0.000000
## omega 8.857862 0.028703 308.6065 0.000000
## alpha1 0.595140 0.002041 291.6560 0.000000
## alpha2 0.000016 0.000009 1.8240 0.068158
## alpha3 0.000013 0.000012 1.0532 0.292247
## beta1 0.101955 0.000338 301.6046 0.000000
## beta2 0.221331 0.000721 307.1210 0.000000
## beta3 0.000036 0.000000 238.7305 0.000000
## gamma1 0.824905 0.003006 274.4388 0.000000
## gamma2 -0.999949 0.003238 -308.7860 0.000000
## gamma3 0.273366 0.000981 278.8018 0.000000
## skew 3.507146 0.551088 6.3640 0.000000
## shape 7.342478 1.359707 5.4000 0.000000
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## mu 98.179244 2.0254e+02 0.484744 0.627858
## omega 8.857862 1.3063e+02 0.067811 0.945936
## alpha1 0.595140 2.3224e+00 0.256255 0.797754
## alpha2 0.000016 3.8791e-02 0.000404 0.999678
## alpha3 0.000013 6.3687e-02 0.000202 0.999839
## beta1 0.101955 8.3314e-01 0.122374 0.902603
## beta2 0.221331 4.0640e+00 0.054461 0.956568
## beta3 0.000036 9.9300e-04 0.036612 0.970794
## gamma1 0.824905 2.1567e+00 0.382488 0.702099
## gamma2 -0.999949 5.7922e-01 -1.726367 0.084281
## gamma3 0.273366 2.8932e-02 9.448601 0.000000
## skew 3.507146 1.0658e+03 0.003291 0.997374
## shape 7.342478 7.5527e+02 0.009722 0.992243
##
## LogLikelihood : -613.5045
##
## Information Criteria
## ------------------------------------
##
## Akaike 7.8806
## Bayes 8.1315
## Shibata 7.8685
## Hannan-Quinn 7.9825
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 130.8 0
## Lag[2*(p+q)+(p+q)-1][2] 187.1 0
## Lag[4*(p+q)+(p+q)-1][5] 336.3 0
## d.o.f=0
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 25.81 3.764e-07
## Lag[2*(p+q)+(p+q)-1][17] 31.82 1.340e-05
## Lag[4*(p+q)+(p+q)-1][29] 35.48 2.987e-04
## d.o.f=6
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[7] 1.076 0.500 2.000 0.2996
## ARCH Lag[9] 3.788 1.485 1.796 0.2348
## ARCH Lag[11] 4.197 2.440 1.677 0.3994
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 25.9836
## Individual Statistics:
## mu 0.2640
## omega 0.3272
## alpha1 0.3270
## alpha2 0.3272
## alpha3 0.3267
## beta1 0.3270
## beta2 0.3271
## beta3 0.3270
## gamma1 0.3270
## gamma2 0.3273
## gamma3 0.3270
## skew 0.1527
## shape 0.6086
##
## 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 3.0984 0.002313 ***
## Negative Sign Bias 1.3587 0.176232
## Positive Sign Bias 0.3405 0.733936
## Joint Effect 15.5240 0.001419 ***
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 183.3 7.035e-29
## 2 30 228.0 8.974e-33
## 3 40 232.2 2.573e-29
## 4 50 224.9 2.274e-24
##
##
## Elapsed time : 4.681827#QUESTÃO C: Compare a persistencia estimada em cada um dos casos.
library(rugarch)
#QUSTÃO A:
ugarchforecast(fit, n.ahead = 10)##
## *------------------------------------*
## * GARCH Model Forecast *
## *------------------------------------*
## Model: sGARCH
## Horizon: 10
## Roll Steps: 0
## Out of Sample: 10
##
## 0-roll forecast [T0=jan 2022]:
## Series Sigma
## T+1 75.52 70.94
## T+2 75.52 70.93
## T+3 75.52 70.93
## T+4 75.52 70.92
## T+5 75.52 70.92
## T+6 75.52 70.91
## T+7 75.52 70.90
## T+8 75.52 70.90
## T+9 75.52 70.89
## T+10 75.52 70.88ugarchforecast(fit2, n.ahead = 10)##
## *------------------------------------*
## * GARCH Model Forecast *
## *------------------------------------*
## Model: sGARCH
## Horizon: 10
## Roll Steps: 0
## Out of Sample: 10
##
## 0-roll forecast [T0=jan 2022]:
## Series Sigma
## T+1 97.22 48.73
## T+2 97.22 48.22
## T+3 97.22 47.73
## T+4 97.22 47.23
## T+5 97.22 46.75
## T+6 97.22 46.28
## T+7 97.22 45.82
## T+8 97.22 45.36
## T+9 97.22 44.91
## T+10 97.22 44.47#QUESTÃO B:
ugarchforecast(fit3, n.ahead = 10)##
## *------------------------------------*
## * GARCH Model Forecast *
## *------------------------------------*
## Model: gjrGARCH
## Horizon: 10
## Roll Steps: 0
## Out of Sample: 10
##
## 0-roll forecast [T0=jan 2022]:
## Series Sigma
## T+1 76.96 73.87
## T+2 76.96 85.65
## T+3 76.96 87.57
## T+4 76.96 87.03
## T+5 76.96 85.84
## T+6 76.96 84.48
## T+7 76.96 83.08
## T+8 76.96 81.70
## T+9 76.96 80.33
## T+10 76.96 78.99ugarchforecast(fit4, n.ahead = 10)##
## *------------------------------------*
## * GARCH Model Forecast *
## *------------------------------------*
## Model: gjrGARCH
## Horizon: 10
## Roll Steps: 0
## Out of Sample: 10
##
## 0-roll forecast [T0=mar 2021]:
## Series Sigma
## T+1 98.18 49.38
## T+2 98.18 57.94
## T+3 98.18 55.54
## T+4 98.18 53.07
## T+5 98.18 52.42
## T+6 98.18 52.18
## T+7 98.18 51.72
## T+8 98.18 51.15
## T+9 98.18 50.60
## T+10 98.18 50.08Conclusão: O modelo com distribuição normal apresentar uma previsão da valotilidade maior quando comparado com o modelo com distribuição t-student.