Homework 5
Marcelo Santos Carielo
ago/2023
Questão - Modelagem das ações da VALE: uma abordagem ARMA-GARCH
Seguindo o roteiro visto em aula, ajuste modelos GARCH(1, 1), EGARCH(1, 1) e GJR(1, 1) para as ações da VALE, usando as distribuições t-Student e Normal. Verifique qual é o melhor modelo dentre esses, verificando também os resíduos. Apresente previsões para a volatilidade condicional.
Resposta
Para este exercício, usaremos a série de retornos do VALE de
01/01/2019 até o dia de hoje (29-07-29). O
código abaixo coleta esses dados do Yahoo Finance.
library(rugarch)
library(BatchGetSymbols)
# define datas de início e fim
date_init <- "2019-01-01"
date_end <- "2023-07-29"
#date_end <- Sys.Date()
# coleta dados da VALE
tickers <- c("VALE3.SA")
assets <- BatchGetSymbols(tickers=tickers,
first.date=date_init,
last.date=date_end,
type.return="log", # log retorno
freq.data="daily")
assets <- assets[[2]]
vale <- assets %>%
filter(ticker=="VALE3.SA")Agora vemos um resumo estatístico e transformamos os dados para o formato de série temporal:
library(fBasics)
daily_returns_vale <- vale %>%
select(ref.date, ret.closing.prices)
basicStats(daily_returns_vale$ret.closing.prices)## X..daily_returns_vale.ret.closing.prices
## nobs 1137.000000
## NAs 1.000000
## Minimum -0.281822
## Maximum 0.193574
## 1. Quartile -0.013279
## 3. Quartile 0.012907
## Mean 0.000247
## Median 0.000000
## Sum 0.280463
## SE Mean 0.000784
## LCL Mean -0.001292
## UCL Mean 0.001786
## Variance 0.000699
## Stdev 0.026431
## Skewness -0.775017
## Kurtosis 17.355477date <- daily_returns_vale %>%
select(ref.date) %>%
rename(date=ref.date) %>%
slice(-1)
daily_returns_vale <- daily_returns_vale %>%
select(ret.closing.prices) %>%
slice(-1)
daily_returns_vale <- as.ts(daily_returns_vale)O resumo estatístico acima mostra que a curtose ficou maior do que 3, indicando que a série analisada possui cauda pesada. Além disso, também notamos que a média ficou por volta de zero. Estes resultados estão dentro do esperado (média zero e cauda pesada).
Vejamos os gráficos do preço diário e do log-retorno da série temporal da VALE.
library(ggplot2, quietly=TRUE)
library(gridExtra, quietly=TRUE)
g <- ggplot(data=assets) +
geom_line(mapping=aes(x=ref.date, y=price.close, color=ticker),
linewidth=0.8, na.rm=TRUE) +
geom_rect(aes(xmin=as.Date("2019-11-01"), xmax=as.Date("2020-10-01"),
ymin=25, ymax=70),
fill="transparent", linetype=2, color="brown", size=1.2) +
labs(x="", y="Preço de Fechamento",
title="Cotação Diária",
subtitle=paste("Período: de ", date_init, " a ", date_end, sep=""),
caption="Fonte: B3") +
theme_minimal()
g.returns <- ggplot(data=assets) +
geom_line(aes(x=ref.date, y=ret.closing.prices, color=ticker),
alpha=0.7, linewidth=0.4, na.rm=TRUE) +
geom_rect(aes(xmin=as.Date("2019-11-01"), xmax=as.Date("2020-10-01"),
ymin=-0.2, ymax=0.25),
fill="transparent", linetype=2, color="darkgreen", size=1.2) +
geom_rect(aes(xmin=as.Date("2019-01-01"), xmax=as.Date("2019-04-01"),
ymin=-0.33, ymax=0.12),
fill="transparent", linetype=2, color="darkblue", size=0.5) +
labs(x="" , y="Retornos",
title="Retorno Diário",
subtitle=paste("Período: de", date_init, " a ", date_end, sep=""),
caption="Fonte: B3") +
theme_minimal()
grid.arrange(g, g.returns, nrow=1, ncol=2)
O gráfico dos retornos mostra o aumento da volatilidade no período de início da pandemia no Brasil. Notamos ainda o início de 2019 também foi um período de alta volatilidade das ações da VALE.
A seguir vamos estimar modelos GARCH(1, 1), EGARCH(1, 1) e GJR(1, 1) para a série de retornos da VALE, usando as distribuições t-Student e Normal.
TESTE LM
A hipótese nula do Teste LM é que não há heterocedasticidade condicional (efeito ARCH). O código a seguir realiza o Teste LM para lags 1, 2, 3, 5, 10 e 15.
library(FinTS)
ArchTest(daily_returns_vale, lags=1,demean=TRUE)##
## ARCH LM-test; Null hypothesis: no ARCH effects
##
## data: daily_returns_vale
## Chi-squared = 52.697, df = 1, p-value = 3.891e-13ArchTest(daily_returns_vale, lags=2,demean=TRUE)##
## ARCH LM-test; Null hypothesis: no ARCH effects
##
## data: daily_returns_vale
## Chi-squared = 75.953, df = 2, p-value < 2.2e-16ArchTest(daily_returns_vale, lags=3,demean=TRUE)##
## ARCH LM-test; Null hypothesis: no ARCH effects
##
## data: daily_returns_vale
## Chi-squared = 94.783, df = 3, p-value < 2.2e-16ArchTest(daily_returns_vale, lags=5,demean=TRUE)##
## ARCH LM-test; Null hypothesis: no ARCH effects
##
## data: daily_returns_vale
## Chi-squared = 99.699, df = 5, p-value < 2.2e-16ArchTest(daily_returns_vale, lags=10,demean=TRUE)##
## ARCH LM-test; Null hypothesis: no ARCH effects
##
## data: daily_returns_vale
## Chi-squared = 103.6, df = 10, p-value < 2.2e-16ArchTest(daily_returns_vale, lags=20,demean=TRUE)##
## ARCH LM-test; Null hypothesis: no ARCH effects
##
## data: daily_returns_vale
## Chi-squared = 464.59, df = 20, p-value < 2.2e-16Em todos os casos acima não rejeitamos a hipótese nula, pois p < 0.05. Logo, a série das variâncias não é autocorrelacionada e uma boa opção para modelarmos os retornos da VALE é usarmos modelos da família ARCH.
GARCH(1, 1)
Estimamos um modelo GARCH(1, 1) para a série da VALE com o seguinte código:
garch.spec.student <- ugarchspec(variance.model=list(model="sGARCH",
garchOrder=c(1, 1)),
mean.model=list(armaOrder=c(1, 1),
include.mean=TRUE),
distribution.model="std")
garch.spec.normal <- ugarchspec(variance.model=list(model="sGARCH",
garchOrder=c(1, 1)),
mean.model=list(armaOrder=c(1, 1),
include.mean=TRUE),
distribution.model="norm")
garch.fit.vale.student <- ugarchfit(spec=garch.spec.student,
data=daily_returns_vale)
garch.fit.vale.normal <- ugarchfit(spec=garch.spec.normal,
data=daily_returns_vale)
garch.fit.vale.student##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : sGARCH(1,1)
## Mean Model : ARFIMA(1,0,1)
## Distribution : std
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## mu 0.000496 0.000539 0.92035 0.357391
## ar1 0.596578 0.220662 2.70358 0.006860
## ma1 -0.640059 0.210183 -3.04525 0.002325
## omega 0.000071 0.000022 3.16715 0.001539
## alpha1 0.105720 0.030508 3.46529 0.000530
## beta1 0.769408 0.057060 13.48420 0.000000
## shape 5.242010 0.770816 6.80060 0.000000
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## mu 0.000496 0.000576 0.86033 0.389605
## ar1 0.596578 0.139434 4.27856 0.000019
## ma1 -0.640059 0.131380 -4.87183 0.000001
## omega 0.000071 0.000019 3.64798 0.000264
## alpha1 0.105720 0.042408 2.49292 0.012670
## beta1 0.769408 0.054136 14.21252 0.000000
## shape 5.242010 1.006414 5.20860 0.000000
##
## LogLikelihood : 2711.734
##
## Information Criteria
## ------------------------------------
##
## Akaike -4.7619
## Bayes -4.7308
## Shibata -4.7619
## Hannan-Quinn -4.7501
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.1247 0.7240
## Lag[2*(p+q)+(p+q)-1][5] 1.0140 1.0000
## Lag[4*(p+q)+(p+q)-1][9] 3.1502 0.8666
## d.o.f=2
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.02059 0.8859
## Lag[2*(p+q)+(p+q)-1][5] 0.11148 0.9977
## Lag[4*(p+q)+(p+q)-1][9] 0.16299 1.0000
## d.o.f=2
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[3] 0.03452 0.500 2.000 0.8526
## ARCH Lag[5] 0.09693 1.440 1.667 0.9874
## ARCH Lag[7] 0.12091 2.315 1.543 0.9990
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 2.3093
## Individual Statistics:
## mu 0.3600
## ar1 0.4824
## ma1 0.4180
## omega 0.1927
## alpha1 0.2895
## beta1 0.2317
## shape 0.5620
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 1.69 1.9 2.35
## Individual Statistic: 0.35 0.47 0.75
##
## Sign Bias Test
## ------------------------------------
## t-value prob sig
## Sign Bias 1.0079 0.3137
## Negative Sign Bias 0.7736 0.4393
## Positive Sign Bias 0.5772 0.5639
## Joint Effect 1.1813 0.7575
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 18.72 0.4750
## 2 30 31.25 0.3536
## 3 40 32.10 0.7752
## 4 50 45.78 0.6045
##
##
## Elapsed time : 0.2241921garch.fit.vale.normal##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : sGARCH(1,1)
## Mean Model : ARFIMA(1,0,1)
## Distribution : norm
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## mu 0.000170 0.000637 0.26735 0.789201
## ar1 0.610005 0.343612 1.77527 0.075853
## ma1 -0.645701 0.330404 -1.95427 0.050669
## omega 0.000036 0.000016 2.26934 0.023248
## alpha1 0.049497 0.011640 4.25237 0.000021
## beta1 0.890526 0.032739 27.20103 0.000000
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## mu 0.000170 0.000840 0.20276 0.839323
## ar1 0.610005 0.289281 2.10869 0.034971
## ma1 -0.645701 0.277037 -2.33074 0.019767
## omega 0.000036 0.000048 0.76215 0.445968
## alpha1 0.049497 0.048221 1.02646 0.304676
## beta1 0.890526 0.119050 7.48029 0.000000
##
## LogLikelihood : 2605.142
##
## Information Criteria
## ------------------------------------
##
## Akaike -4.5760
## Bayes -4.5494
## Shibata -4.5760
## Hannan-Quinn -4.5659
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.004179 0.9485
## Lag[2*(p+q)+(p+q)-1][5] 0.799785 1.0000
## Lag[4*(p+q)+(p+q)-1][9] 2.880865 0.9079
## d.o.f=2
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.04009 0.8413
## Lag[2*(p+q)+(p+q)-1][5] 0.04955 0.9995
## Lag[4*(p+q)+(p+q)-1][9] 0.07334 1.0000
## d.o.f=2
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[3] 0.01063 0.500 2.000 0.9179
## ARCH Lag[5] 0.01547 1.440 1.667 0.9991
## ARCH Lag[7] 0.03805 2.315 1.543 0.9999
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 1.8809
## Individual Statistics:
## mu 0.1492
## ar1 0.5450
## ma1 0.5115
## omega 0.2021
## alpha1 0.1488
## beta1 0.1658
##
## 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 1.0315 0.3025
## Negative Sign Bias 1.6740 0.0944 *
## Positive Sign Bias 0.1906 0.8489
## Joint Effect 2.8967 0.4078
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 63.01 1.279e-06
## 2 30 62.52 2.969e-04
## 3 40 80.27 1.112e-04
## 4 50 86.54 7.540e-04
##
##
## Elapsed time : 0.1061714#infocriteria(garch.fit.vale.normal)
#infocriteria(garch.fit.vale.student)
options(repr.plot.width=15, repr.plot.height=15)
plot(garch.fit.vale.student, which="all")##
## please wait...calculating quantiles...
plot(garch.fit.vale.normal, which="all")##
## please wait...calculating quantiles...
EGARCH (Exponential GARCH)
Agora vamos estimar um modelo EGARCH(1, 1) para a mesma série de retornos:
#?ugarchspec
egarch.spec.student <- ugarchspec(variance.model=list(model="eGARCH",
garchOrder=c(1, 1)),
mean.model=list(armaOrder=c(1, 1),
include.mean=TRUE),
distribution.model="std")
egarch.spec.normal <- ugarchspec(variance.model=list(model="eGARCH",
garchOrder=c(1, 1)),
mean.model=list(armaOrder=c(1, 1),
include.mean=TRUE),
distribution.model="norm")
egarch.fit.vale.student <- ugarchfit(spec=egarch.spec.student,
data=daily_returns_vale)
egarch.fit.vale.normal <- ugarchfit(spec=egarch.spec.normal,
data=daily_returns_vale)
egarch.fit.vale.student##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : eGARCH(1,1)
## Mean Model : ARFIMA(1,0,1)
## Distribution : std
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## mu 0.000305 0.000543 0.56107 0.574753
## ar1 0.585531 0.043632 13.41988 0.000000
## ma1 -0.627989 0.042096 -14.91799 0.000000
## omega -0.556195 0.189119 -2.94098 0.003272
## alpha1 -0.074640 0.026644 -2.80136 0.005089
## beta1 0.926352 0.024950 37.12790 0.000000
## gamma1 0.154126 0.031232 4.93487 0.000001
## shape 5.229301 0.780191 6.70259 0.000000
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## mu 0.000305 0.000580 0.52468 0.599807
## ar1 0.585531 0.013036 44.91564 0.000000
## ma1 -0.627989 0.013346 -47.05568 0.000000
## omega -0.556195 0.278914 -1.99414 0.046137
## alpha1 -0.074640 0.032250 -2.31439 0.020646
## beta1 0.926352 0.036339 25.49216 0.000000
## gamma1 0.154126 0.050332 3.06220 0.002197
## shape 5.229301 1.081064 4.83718 0.000001
##
## LogLikelihood : 2715.315
##
## Information Criteria
## ------------------------------------
##
## Akaike -4.7664
## Bayes -4.7309
## Shibata -4.7665
## Hannan-Quinn -4.7530
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.126 0.7226
## Lag[2*(p+q)+(p+q)-1][5] 1.241 0.9998
## Lag[4*(p+q)+(p+q)-1][9] 3.243 0.8504
## d.o.f=2
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.004729 0.9452
## Lag[2*(p+q)+(p+q)-1][5] 0.046540 0.9996
## Lag[4*(p+q)+(p+q)-1][9] 0.083633 1.0000
## d.o.f=2
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[3] 0.01103 0.500 2.000 0.9164
## ARCH Lag[5] 0.04657 1.440 1.667 0.9955
## ARCH Lag[7] 0.06438 2.315 1.543 0.9998
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 2.2821
## Individual Statistics:
## mu 0.2706
## ar1 0.3954
## ma1 0.3488
## omega 0.1363
## alpha1 0.1485
## beta1 0.1308
## gamma1 0.3031
## shape 0.4737
##
## 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 1.0903 0.2758
## Negative Sign Bias 0.8171 0.4141
## Positive Sign Bias 0.2752 0.7832
## Joint Effect 1.3735 0.7118
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 15.48 0.69171
## 2 30 42.61 0.04945
## 3 40 37.03 0.56011
## 4 50 45.87 0.60095
##
##
## Elapsed time : 0.229763egarch.fit.vale.normal##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : eGARCH(1,1)
## Mean Model : ARFIMA(1,0,1)
## Distribution : norm
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## mu -0.00012 0.000671 -0.1786 0.858252
## ar1 0.67105 0.033818 19.8428 0.000000
## ma1 -0.70271 0.032188 -21.8312 0.000000
## omega -0.24657 0.001904 -129.5082 0.000000
## alpha1 -0.02485 0.009842 -2.5247 0.011578
## beta1 0.96597 0.000877 1101.2948 0.000000
## gamma1 0.10437 0.008632 12.0906 0.000000
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## mu -0.00012 0.000873 -0.13718 0.89089
## ar1 0.67105 0.013876 48.36205 0.00000
## ma1 -0.70271 0.016219 -43.32565 0.00000
## omega -0.24657 0.020645 -11.94335 0.00000
## alpha1 -0.02485 0.039845 -0.62366 0.53285
## beta1 0.96597 0.002272 425.17142 0.00000
## gamma1 0.10437 0.015429 6.76430 0.00000
##
## LogLikelihood : 2606.869
##
## Information Criteria
## ------------------------------------
##
## Akaike -4.5772
## Bayes -4.5462
## Shibata -4.5773
## Hannan-Quinn -4.5655
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.05975 0.8069
## Lag[2*(p+q)+(p+q)-1][5] 0.90935 1.0000
## Lag[4*(p+q)+(p+q)-1][9] 2.97274 0.8947
## d.o.f=2
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.08376 0.7723
## Lag[2*(p+q)+(p+q)-1][5] 0.23415 0.9899
## Lag[4*(p+q)+(p+q)-1][9] 0.26442 0.9998
## d.o.f=2
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[3] 0.04135 0.500 2.000 0.8389
## ARCH Lag[5] 0.04185 1.440 1.667 0.9962
## ARCH Lag[7] 0.04899 2.315 1.543 0.9999
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 1.7098
## Individual Statistics:
## mu 0.1806
## ar1 0.4022
## ma1 0.3859
## omega 0.2511
## alpha1 0.2119
## beta1 0.2535
## gamma1 0.2199
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 1.69 1.9 2.35
## Individual Statistic: 0.35 0.47 0.75
##
## Sign Bias Test
## ------------------------------------
## t-value prob sig
## Sign Bias 1.09801 0.27243
## Negative Sign Bias 1.84678 0.06504 *
## Positive Sign Bias 0.02883 0.97700
## Joint Effect 3.52288 0.31781
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 60.30 3.465e-06
## 2 30 67.38 6.861e-05
## 3 40 98.01 5.514e-07
## 4 50 100.71 1.948e-05
##
##
## Elapsed time : 0.1337895#infocriteria(egarch.fit.vale.normal)
#infocriteria(egarch.fit.vale.student)
options(repr.plot.width=15, repr.plot.height=15)
plot(egarch.fit.vale.student, which="all")##
## please wait...calculating quantiles...
plot(egarch.fit.vale.normal, which="all")##
## please wait...calculating quantiles...
GRJ - GARCH
Agora vamos estimar um modelo GJR(1, 1) para a mesma série de retornos:
#https://search.r-project.org/CRAN/refmans/rugarch/html/ugarchspec-methods.html
gjr_garch.spec.student <- ugarchspec(variance.model=list(model="gjrGARCH",
garchOrder=c(1, 1)),
mean.model=list(armaOrder=c(1, 1),
include.mean=TRUE),
distribution.model="std")
gjr_garch.spec.normal <- ugarchspec(variance.model=list(model="gjrGARCH",
garchOrder=c(1, 1)),
mean.model=list(armaOrder=c(1, 1),
include.mean=TRUE),
distribution.model="norm")
gjr_garch.fit.vale.student <- ugarchfit(spec=gjr_garch.spec.student,
data=daily_returns_vale)
gjr_garch.fit.vale.normal <- ugarchfit(spec=gjr_garch.spec.normal,
data=daily_returns_vale)
gjr_garch.fit.vale.student##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : gjrGARCH(1,1)
## Mean Model : ARFIMA(1,0,1)
## Distribution : std
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## mu 0.000284 0.000545 0.52207 0.601619
## ar1 0.608930 0.218656 2.78487 0.005355
## ma1 -0.650897 0.208302 -3.12478 0.001779
## omega 0.000080 0.000024 3.38724 0.000706
## alpha1 0.037005 0.028160 1.31413 0.188804
## beta1 0.745671 0.060236 12.37906 0.000000
## gamma1 0.160741 0.062026 2.59151 0.009556
## shape 5.408759 0.808587 6.68915 0.000000
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## mu 0.000284 0.000579 0.49159 0.623007
## ar1 0.608930 0.140526 4.33321 0.000015
## ma1 -0.650897 0.133325 -4.88204 0.000001
## omega 0.000080 0.000025 3.14669 0.001651
## alpha1 0.037005 0.028117 1.31610 0.188140
## beta1 0.745671 0.067246 11.08868 0.000000
## gamma1 0.160741 0.078331 2.05206 0.040163
## shape 5.408759 1.091120 4.95707 0.000001
##
## LogLikelihood : 2716.723
##
## Information Criteria
## ------------------------------------
##
## Akaike -4.7689
## Bayes -4.7334
## Shibata -4.7690
## Hannan-Quinn -4.7555
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.3006 0.5835
## Lag[2*(p+q)+(p+q)-1][5] 1.3976 0.9992
## Lag[4*(p+q)+(p+q)-1][9] 3.4091 0.8194
## d.o.f=2
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.03556 0.8504
## Lag[2*(p+q)+(p+q)-1][5] 0.15472 0.9955
## Lag[4*(p+q)+(p+q)-1][9] 0.22065 0.9999
## d.o.f=2
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[3] 0.0506 0.500 2.000 0.8220
## ARCH Lag[5] 0.1137 1.440 1.667 0.9842
## ARCH Lag[7] 0.1404 2.315 1.543 0.9987
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 2.3625
## Individual Statistics:
## mu 0.2865
## ar1 0.3715
## ma1 0.3228
## omega 0.1798
## alpha1 0.3211
## beta1 0.2676
## gamma1 0.2977
## shape 0.5389
##
## 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 0.9648 0.3349
## Negative Sign Bias 0.3262 0.7443
## Positive Sign Bias 0.4071 0.6840
## Joint Effect 0.9644 0.8099
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 18.72 0.4750
## 2 30 31.68 0.3343
## 3 40 37.45 0.5406
## 4 50 48.60 0.4895
##
##
## Elapsed time : 0.4059262gjr_garch.fit.vale.normal##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : gjrGARCH(1,1)
## Mean Model : ARFIMA(1,0,1)
## Distribution : norm
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## mu 0.000075 0.000640 0.11696 0.906894
## ar1 0.734034 0.225233 3.25900 0.001118
## ma1 -0.763016 0.213002 -3.58220 0.000341
## omega 0.000161 0.000074 2.18110 0.029176
## alpha1 0.011273 0.029757 0.37882 0.704821
## beta1 0.651912 0.151948 4.29036 0.000018
## gamma1 0.167394 0.077494 2.16009 0.030766
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## mu 0.000075 0.000696 0.10756 0.914341
## ar1 0.734034 0.204238 3.59401 0.000326
## ma1 -0.763016 0.204093 -3.73858 0.000185
## omega 0.000161 0.000213 0.75387 0.450925
## alpha1 0.011273 0.038657 0.29161 0.770588
## beta1 0.651912 0.399272 1.63275 0.102521
## gamma1 0.167394 0.207350 0.80730 0.419494
##
## LogLikelihood : 2608.582
##
## Information Criteria
## ------------------------------------
##
## Akaike -4.5802
## Bayes -4.5492
## Shibata -4.5803
## Hannan-Quinn -4.5685
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.03582 0.8499
## Lag[2*(p+q)+(p+q)-1][5] 0.83674 1.0000
## Lag[4*(p+q)+(p+q)-1][9] 3.17762 0.8619
## d.o.f=2
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.003614 0.9521
## Lag[2*(p+q)+(p+q)-1][5] 0.052858 0.9995
## Lag[4*(p+q)+(p+q)-1][9] 0.071041 1.0000
## d.o.f=2
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[3] 0.007498 0.500 2.000 0.9310
## ARCH Lag[5] 0.025949 1.440 1.667 0.9981
## ARCH Lag[7] 0.033925 2.315 1.543 0.9999
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 1.7393
## Individual Statistics:
## mu 0.2307
## ar1 0.3445
## ma1 0.3312
## omega 0.2209
## alpha1 0.2057
## beta1 0.2095
## gamma1 0.3954
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 1.69 1.9 2.35
## Individual Statistic: 0.35 0.47 0.75
##
## Sign Bias Test
## ------------------------------------
## t-value prob sig
## Sign Bias 0.82333 0.4105
## Negative Sign Bias 0.79740 0.4254
## Positive Sign Bias 0.07324 0.9416
## Joint Effect 1.09455 0.7784
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 71.85 4.515e-08
## 2 30 77.26 2.898e-06
## 3 40 86.96 1.630e-05
## 4 50 97.27 4.934e-05
##
##
## Elapsed time : 0.4223883#infocriteria(gjr_garch.fit.vale.normal)
#infocriteria(gjr_garch.fit.vale.student)
options(repr.plot.width=15, repr.plot.height=15)
plot(gjr_garch.fit.vale.student, which="all")##
## please wait...calculating quantiles...
plot(gjr_garch.fit.vale.normal, which="all")##
## please wait...calculating quantiles...
Coeficientes de persistência e half-life
Agora vamos calcular os coeficientes de persistência e half-life para cada um dos modelos ajustados acima.
Primeiramente, calculamos os coeficientes de persistência:
paste("garch.normal:", persistence(garch.fit.vale.normal))## [1] "garch.normal: 0.940023085058988"paste("garch.student:", persistence(garch.fit.vale.student))## [1] "garch.student: 0.875127179811051"paste("egarch.normal:", persistence(egarch.fit.vale.normal))## [1] "egarch.normal: 0.965965583402948"paste("egarch.student:", persistence(egarch.fit.vale.student))## [1] "egarch.student: 0.926351723034298"paste("gjr_garch.normal:", persistence(gjr_garch.fit.vale.normal))## [1] "gjr_garch.normal: 0.746881897221491"paste("gjr_garch.student:", persistence(gjr_garch.fit.vale.student))## [1] "gjr_garch.student: 0.863046388509446"Os valores acima indicam que haverá maior pesistência dos choques no
caso de usarmos o modelo EGARCH(1, 1) com distribuição Normal
(egarch.normal). Ou seja, escolhendo este modelo haverá uma
maior persistência da volatilidade.
Por outro lado, escolhendo o GJR(1, 1) com distribuição Normal
(gjr_garch.normal) haverá uma menor persistência da
volatilidade.
Calculamos os coeficientes de half-life com os códigos abaixo:
paste("garch.normal:", halflife(garch.fit.vale.normal))## [1] "garch.normal: 11.2067535303504"paste("garch.student:", halflife(garch.fit.vale.student))## [1] "garch.student: 5.19654908031971"paste("egarch.normal:", halflife(egarch.fit.vale.normal))## [1] "egarch.normal: 20.0174924840579"paste("egarch.student:", halflife(egarch.fit.vale.student))## [1] "egarch.student: 9.06059514996847"paste("gjr_garch.normal:", halflife(gjr_garch.fit.vale.normal))## [1] "gjr_garch.normal: 2.37502632831976"paste("gjr_garch.student:", halflife(gjr_garch.fit.vale.student))## [1] "gjr_garch.student: 4.70610406178515"Pelos valores acima, notamos que com a escolha do modelo GRJ(1, 1)
com distribuição Normal (gjr_garch.normal), modelo
correspondente ao menor valor de half-time, teremos uma menor
quantidade de dias (cerca de 2 dias) para o choque ser dissipado pela
metade.
Por outro lado, escolhendo o EGARCH(1, 1) com distribuição Normal
(egarch.normal), levará mais dias para que um choque se
dissipe pela metade. De fato, os cálculos indicam que demorará cerca de
11 dias para que isso ocorra.
Critério de Informação
Agora iremos calcular os critérios de informação de Akaike, Bayesiano (Schwarz), Shibata e Hannan-Quinn para dos nossos modelos:
print("garch.fit.vale.normal:")## [1] "garch.fit.vale.normal:"infocriteria(garch.fit.vale.normal)##
## Akaike -4.575955
## Bayes -4.549360
## Shibata -4.576010
## Hannan-Quinn -4.565910print("garch.fit.vale.student:")## [1] "garch.fit.vale.student:"infocriteria(garch.fit.vale.student)##
## Akaike -4.761855
## Bayes -4.730828
## Shibata -4.761930
## Hannan-Quinn -4.750136print("egarch.fit.vale.normal:")## [1] "egarch.fit.vale.normal:"infocriteria(egarch.fit.vale.normal)##
## Akaike -4.577233
## Bayes -4.546206
## Shibata -4.577309
## Hannan-Quinn -4.565514print("egarch.fit.vale.student:")## [1] "egarch.fit.vale.student:"infocriteria(egarch.fit.vale.student)##
## Akaike -4.766400
## Bayes -4.730940
## Shibata -4.766498
## Hannan-Quinn -4.753006print("gjr_garch.fit.vale.normal:")## [1] "gjr_garch.fit.vale.normal:"infocriteria(gjr_garch.fit.vale.normal)##
## Akaike -4.580250
## Bayes -4.549223
## Shibata -4.580325
## Hannan-Quinn -4.568531print("gjr_garch.fit.vale.student:")## [1] "gjr_garch.fit.vale.student:"infocriteria(gjr_garch.fit.vale.student)##
## Akaike -4.768878
## Bayes -4.733419
## Shibata -4.768977
## Hannan-Quinn -4.755485Critério de Informação - Resumo
| Modelo | Akaike | Bayes | Shibata | Hannan-Quinn |
|---|---|---|---|---|
| garch.fit.vale.normal | -4.575955 | -4.549360 | -4.576010 | -4.565910 |
| garch.fit.vale.student | -4.761855 | -4.730828 | -4.761930 | -4.750136 |
| egarch.fit.vale.normal | -4.577233 | -4.546206 | -4.577309 | -4.565514 |
| egarch.fit.vale.student | -4.766400 | -4.730940 | -4.766498 | -4.753006 |
| gjr_garch.fit.vale.normal | -4.580250 | -4.549223 | -4.580325 | -4.568531 |
| gjr_garch.fit.vale.student | -4.768878 | -4.733419 | -4.768977 | -4.755485 |
Como na tabela acima o modelo GJR(1,1) com t-Student obteve os
menores valores para cada um dos critérios de informação apresentados,
escolhemos este modelo (gjr_garch.fit.vale.student) para
prosseguirmos as análises.
Resíduos
Vejamos a ACF dos resíduos:
options(repr.plot.width=25, repr.plot.height=15)
#plot(gjr_garch.fit.vale.student, which="all")
plot(gjr_garch.fit.vale.student, which=10)
Notemos que para lag = 29 temos uma autocorrelação
significativa para a série analisada. Para os demais lags os valores
exibidos no gráfico não são significativos.
Vejamos os resíduos ao quadrado:
options(repr.plot.width=25, repr.plot.height=15)
#plot(gjr_garch.fit.vale.student, which="all")
plot(gjr_garch.fit.vale.student, which=11)
Para os resíduos ao quadrado não notamos autocorrelação significativa para nenhum dos lags exibidos no gráfico.
Volatilidade condicional
A seguir temos o gráfico da volatilidade condicional para o modelo selecionado acima:
library(tbl2xts)
sigma <- sigma(gjr_garch.fit.vale.student) %>% xts_tbl()
colnames(sigma) <- c("date", "sigma")
sigma <- sigma %>% mutate(date=as.Date(date))
#sigma$date
ggplot(sigma) + geom_line(aes(x=date , y=sigma)) +
geom_rect(aes(xmin=as.Date("1-01-01"), xmax=as.Date("100-01-01"),
ymin=0, ymax=0.135),
fill="transparent", linetype=2, color="darkgreen", size=1.2) +
geom_rect(aes(xmin=as.Date("200-01-01"), xmax=as.Date("350-01-01"),
ymin=0, ymax=0.100),
fill="transparent", linetype=2, color="darkblue", size=0.5) +
theme_minimal()
Nos trechos destacados no gráfico acima temos um aumento na volatilidade. Os períodos correspondem ao início de 2019 e ao início da pandemia.
Previsões para volatilidade condicional
Usando o modelo GJR(1, 1) com t-Student
gjr_garch.fit.vale.student estimado acima, realizamos a
predição para volatilidade condicional 42 passos a frente:
garchf.1 <- ugarchforecast(gjr_garch.fit.vale.student, n.ahead=42)
garchf.1##
## *------------------------------------*
## * GARCH Model Forecast *
## *------------------------------------*
## Model: gjrGARCH
## Horizon: 42
## Roll Steps: 0
## Out of Sample: 0
##
## 0-roll forecast [T0=1136-01-01]:
## Series Sigma
## T+1 0.0019883 0.02686
## T+2 0.0013220 0.02651
## T+3 0.0009163 0.02620
## T+4 0.0006692 0.02593
## T+5 0.0005187 0.02569
## T+6 0.0004271 0.02549
## T+7 0.0003714 0.02531
## T+8 0.0003374 0.02516
## T+9 0.0003167 0.02503
## T+10 0.0003041 0.02491
## T+11 0.0002964 0.02481
## T+12 0.0002918 0.02472
## T+13 0.0002889 0.02465
## T+14 0.0002872 0.02458
## T+15 0.0002861 0.02453
## T+16 0.0002855 0.02448
## T+17 0.0002851 0.02443
## T+18 0.0002849 0.02440
## T+19 0.0002847 0.02437
## T+20 0.0002846 0.02434
## T+21 0.0002846 0.02432
## T+22 0.0002845 0.02429
## T+23 0.0002845 0.02428
## T+24 0.0002845 0.02426
## T+25 0.0002845 0.02425
## T+26 0.0002845 0.02424
## T+27 0.0002845 0.02423
## T+28 0.0002845 0.02422
## T+29 0.0002845 0.02421
## T+30 0.0002845 0.02421
## T+31 0.0002845 0.02420
## T+32 0.0002845 0.02420
## T+33 0.0002845 0.02419
## T+34 0.0002845 0.02419
## T+35 0.0002845 0.02419
## T+36 0.0002845 0.02418
## T+37 0.0002845 0.02418
## T+38 0.0002845 0.02418
## T+39 0.0002845 0.02418
## T+40 0.0002845 0.02418
## T+41 0.0002845 0.02417
## T+42 0.0002845 0.02417plot(garchf.1, which=1)
plot(garchf.1, which=3)
Pelos gráficos notamos que para os próximos 42 dias haverá um aumento na volatilidade, que tenderá a se estabilizar em um valor mais alto que o dos últimos dias. Por sua vez, a média continuará por volta de zero.
Referências
Materiais das aulas (profa. Andreza Palma)
CAP. 4 do livro “TSAY, Ruey S. An introduction to analysis of financial data with R. John Wiley & Sons, 2014.”