Homework5 - Modelagem das ações da VALE: uma abordagem ARMA-GARCH para volatilidade
Givanildo de Sousa Gramacho
2023-08-17
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.
library(BatchGetSymbols) ### pegar dados yahoo finance (preços diárias ativos negociados em bolsa)## Warning: package 'BatchGetSymbols' was built under R version 4.3.1## Carregando pacotes exigidos: rvest## Carregando pacotes exigidos: dplyr##
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## kurtosis, skewness#limpar o diretório
rm(list=ls())# para pegar vários, podemos usar o argumento abaixo:
tickers = c('VALE3.SA')
#### Sys.time() é a última data disponível
assets <- BatchGetSymbols(tickers,
first.date = '2019-03-03',
last.date = Sys.time(),
type.return = "log",
freq.data = "daily")## Warning: `BatchGetSymbols()` was deprecated in BatchGetSymbols 2.6.4.
## ℹ Please use `yfR::yf_get()` instead.
## ℹ 2022-05-01: Package BatchGetSymbols will soon be replaced by yfR. More
## details about the change is available at github
## <<www.github.com/msperlin/yfR> You can install yfR by executing:
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## remotes::install_github('msperlin/yfR')
## This warning is displayed once every 8 hours.
## Call `lifecycle::last_lifecycle_warnings()` to see where this warning was
## generated.##
## Running BatchGetSymbols for:
## tickers =VALE3.SA
## Downloading data for benchmark ticker
## ^GSPC | yahoo (1|1) | Not Cached | Saving cache
## VALE3.SA | yahoo (1|1) | Not Cached | Saving cache - Got 96% of valid prices | OK!vale3 <- assets[[2]]
#head(vale3)# Eliminando os valores NA
vale3= na.omit(vale3)
head(vale3)## price.open price.high price.low price.close volume price.adjusted
## 2 47.83 48.96 47.72 48.86 21112500 33.65979
## 3 48.26 48.89 47.65 48.85 25446000 33.65290
## 4 48.90 49.95 48.61 49.88 20201600 34.36247
## 5 49.80 50.33 49.55 49.97 15591300 34.42447
## 6 49.94 51.00 49.70 50.70 21999200 34.92736
## 7 50.80 50.93 50.20 50.71 17257400 34.93425
## ref.date ticker ret.adjusted.prices ret.closing.prices
## 2 2019-03-07 VALE3.SA 0.0167170108 0.0167169588
## 3 2019-03-08 VALE3.SA -0.0002048101 -0.0002047311
## 4 2019-03-11 VALE3.SA 0.0208658022 0.0208657950
## 5 2019-03-12 VALE3.SA 0.0018027871 0.0018027076
## 6 2019-03-13 VALE3.SA 0.0145028705 0.0145030759
## 7 2019-03-14 VALE3.SA 0.0001972283 0.0001971861daily_returns <- vale3 %>%
select(ref.date,ret.closing.prices)
date <- daily_returns %>%
select(ref.date) %>%
rename(date = ref.date) %>%
slice(-1)
daily_returns <- daily_returns %>%
select(ret.closing.prices) %>%
slice(-1)
## declarando como série temporal através do comando as.ts
daily_returns=as.ts(daily_returns)Dados e análises descritivas.
# Fazer uma análise visual da série, verificando os fatos estilizados.
# Função para plotar os gráficos para qualquer ação que deseja analisar.
plot_graficos <- function(acao) {
# Gráfico 1: preços
p <- ggplot(acao, aes(ref.date, price.close)) +
geom_line(color='#006600') +
labs(x = "", y = 'Preço Fechamento',
title = paste("Cotação Diária da", acao$ticker[1]),
subtitle = "Período: de 03/03/2019 a 11/08/2023",
caption = "Fonte: B3") +
theme_economist()
# Gráfico 2: retornos diários
daily_returns <- acao %>% select(ref.date, ticker, ret.closing.prices)
plot_returns <- ggplot(daily_returns) +
geom_line(aes(x = ref.date, y = ret.closing.prices), color='#006600') +
labs(x = "", y = 'Retornos',
title = paste("Retornos da", acao$ticker[1]),
subtitle = "Período: de 03/03/2019 a 11/08/2023",
caption = "Fonte: B3") +
theme_economist()
# Gráfico 3: Retornos absolutos (abs)
plot_volatility <- ggplot(daily_returns) +
geom_line(aes(x = ref.date, y = abs(ret.closing.prices)), color='#006600') +
labs(x = "", y = 'Retornos absolutos',
title = paste("Retornos abs da", acao$ticker[1]),
subtitle = "Período: de 03/03/2019 a 11/08/2023",
caption = "Fonte: B3") +
theme_economist()
# Gráfico 4: qqplot
qqplot <- ggplot(daily_returns, aes(sample = ret.closing.prices)) +
stat_qq() +
stat_qq_line() +
labs(x = "Teórico", y = 'Amostra',
title = "QQplot",
subtitle = paste("Retornos diários da", acao$ticker[1]),
caption = "Fonte: Elaborado a partir de dados da B3") +
theme_economist()
# Gráfico 5: Histograma
histogram <- ggplot(daily_returns) +
geom_histogram(aes(x = ret.closing.prices, y = after_stat(density)),
color = "white", fill = "Dark grey", linetype = "solid", alpha = 0.8, binwidth = 0.02) +
geom_density(aes(x = ret.closing.prices, y = after_stat(density)), color = "black") +
labs(x = "", y = 'Densidade',
title = "Histograma",
subtitle = paste("Retornos diários da", acao$ticker[1]),
caption = "Fonte: Elaborado a partir de dados da B3") +
theme_economist()
# Plotar os gráficos juntos
plot_grid(plot_returns, qqplot, plot_volatility, histogram, nrow = 2)
}# Gráfico 1: preços
p <- ggplot(vale3, aes(ref.date, price.close)) +
geom_line(color='#006600') +
labs(x = "", y = 'Preço Fechamento',
title = paste("Cotação Diária da", vale3$ticker[1]),
subtitle = "Período: de 03/03/2019 a 11/08/2023",
caption = "Fonte: B3") +
theme_economist()
p
plot_graficos(vale3)
# Calcular estatísticas descritivas para cada ação
estatisticas <- vale3 %>%
group_by(ticker) %>%
summarise(
media = mean(ret.closing.prices),
desvio_padrao = sd(ret.closing.prices),
variancia = var(ret.closing.prices),
curtose = kurtosis(ret.closing.prices),
assimetria = skewness(ret.closing.prices)
)
# Adicionar coluna indicando se a distribuição é normal ou de cauda pesada
estatisticas <- estatisticas %>%
mutate(distribuicao = ifelse(curtose > 3, "Cauda Pesada", "Normal"))
estatisticas## # A tibble: 1 × 7
## ticker media desvio_padrao variancia curtose assimetria distribuicao
## <chr> <dbl> <dbl> <dbl> <dbl> <dbl> <chr>
## 1 VALE3.SA 0.000216 0.0250 0.000624 11.3 0.298 Cauda Pesada# Função para calcular os graus de liberdade da distribuição t-Student
df_t_student <- function(data) {
n <- length(data)
df <- ifelse(n < 30, n - 1, 30)
return(df)
}
# Função para calcular estatísticas descritivas e plotar histograma
plot_estatisticas_histograma <- function(data, tickers) {
# Plotar histograma
grafico <- ggplot(data = data) +
geom_histogram(aes(x = ret.closing.prices, y = after_stat(density)),
color = "black", fill = "white") +
facet_wrap(~ticker, ncol = 2) +
geom_density(aes(x = ret.closing.prices),
color = "blue", fill = "lightblue", alpha = 0.5) +
stat_function(fun = dnorm,
args = list(mean = mean(data$ret.closing.prices), sd = sd(data$ret.closing.prices)),
color = "red", linetype = "dashed") +
stat_function(fun = dt,
args = list(df = df_t_student(data$ret.closing.prices)),
color = "green", linetype = "dotted") +
labs(x = "Retornos", y = "Densidade",
title = "Distribuição Normal e t-Student",
subtitle = "distribuicoes VALE3",
caption = "Fonte: Dados da Acao") +
theme_minimal()
print(grafico)
}
# Chamar a função para ações desejadas
plot_estatisticas_histograma(vale3, ticker)## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.
tsdisplay(daily_returns, lag.max = 50)
# Calculando os retornos ao quadrado
returns_squared <- daily_returns^2
# Calculando a função de autocorrelação parcial (FACP) dos retornos ao quadrado
pacf_returns_squared <- pacf(returns_squared, lag.max = 50, plot = TRUE)
# Exibindo os resultados da FACP
pacf_returns_squared##
## Partial autocorrelations of series 'returns_squared', by lag
##
## 1 2 3 4 5 6 7 8 9 10 11
## 0.539 0.066 0.376 -0.033 -0.121 -0.092 0.051 0.168 0.083 -0.035 -0.015
## 12 13 14 15 16 17 18 19 20 21 22
## -0.052 -0.059 0.021 -0.005 0.072 -0.002 -0.017 0.012 -0.043 -0.023 -0.001
## 23 24 25 26 27 28 29 30 31 32 33
## -0.028 0.088 -0.015 0.000 -0.022 0.005 0.023 -0.007 -0.004 0.035 -0.018
## 34 35 36 37 38 39 40 41 42 43 44
## 0.017 -0.018 -0.001 0.007 0.009 -0.018 0.029 -0.035 0.017 0.011 -0.045
## 45 46 47 48 49 50
## 0.009 -0.011 0.017 -0.016 0.013 -0.026Especificando os lags para o teste LM
lags = c(1,2,3,5,6,8,16)# Testando efeito ARCH: Teste LM
for (lag in lags) {
arch_test <- ArchTest(daily_returns, lags = lag, demean = TRUE)
print(paste("Resultado do Teste LM (", lag, "lags):"))
print(arch_test)
}## [1] "Resultado do Teste LM ( 1 lags):"
##
## ARCH LM-test; Null hypothesis: no ARCH effects
##
## data: daily_returns
## Chi-squared = 323.27, df = 1, p-value < 2.2e-16
##
## [1] "Resultado do Teste LM ( 2 lags):"
##
## ARCH LM-test; Null hypothesis: no ARCH effects
##
## data: daily_returns
## Chi-squared = 326.41, df = 2, p-value < 2.2e-16
##
## [1] "Resultado do Teste LM ( 3 lags):"
##
## ARCH LM-test; Null hypothesis: no ARCH effects
##
## data: daily_returns
## Chi-squared = 436.02, df = 3, p-value < 2.2e-16
##
## [1] "Resultado do Teste LM ( 5 lags):"
##
## ARCH LM-test; Null hypothesis: no ARCH effects
##
## data: daily_returns
## Chi-squared = 445.86, df = 5, p-value < 2.2e-16
##
## [1] "Resultado do Teste LM ( 6 lags):"
##
## ARCH LM-test; Null hypothesis: no ARCH effects
##
## data: daily_returns
## Chi-squared = 450.89, df = 6, p-value < 2.2e-16
##
## [1] "Resultado do Teste LM ( 8 lags):"
##
## ARCH LM-test; Null hypothesis: no ARCH effects
##
## data: daily_returns
## Chi-squared = 470.48, df = 8, p-value < 2.2e-16
##
## [1] "Resultado do Teste LM ( 16 lags):"
##
## ARCH LM-test; Null hypothesis: no ARCH effects
##
## data: daily_returns
## Chi-squared = 479.64, df = 16, p-value < 2.2e-16Configurar as ordens padrão para os modelos
# Configurar as ordens padrão para os modelos
orders <- c(1, 1)
# Configurações padrão para GARCH, EGARCH e GJR-GARCH com distribuição norm
garch_spec_norm <- ugarchspec(variance.model = list(model = "sGARCH", garchOrder = orders),
mean.model = list(armaOrder = c(0, 0), include.mean = FALSE),
distribution.model = "norm")
egarch_spec_norm <- ugarchspec(variance.model = list(model = "eGARCH", garchOrder = orders),
mean.model = list(armaOrder = c(0, 0), include.mean = FALSE),
distribution.model = "norm")
gjr_spec_norm <- ugarchspec(variance.model = list(model = "gjrGARCH", garchOrder = orders),
mean.model = list(armaOrder = c(0, 0), include.mean = FALSE),
distribution.model = "norm")
# Configurações padrão para GARCH, EGARCH e GJR-GARCH com distribuição std
garch_spec_std <- ugarchspec(variance.model = list(model = "sGARCH", garchOrder = orders),
mean.model = list(armaOrder = c(0, 0), include.mean = FALSE),
distribution.model = "std")
egarch_spec_std <- ugarchspec(variance.model = list(model = "eGARCH", garchOrder = orders),
mean.model = list(armaOrder = c(0, 0), include.mean = FALSE),
distribution.model = "std")
gjr_spec_std <- ugarchspec(variance.model = list(model = "gjrGARCH", garchOrder = orders),
mean.model = list(armaOrder = c(0, 0), include.mean = FALSE),
distribution.model = "std")Ajustando o modelo GARCH(1,1), EGARCH(1,1), GJR(1,1) para distribuição normal e t-student.
# Ajustando o modelo GARCH(1,1) com distribuição Normal e t-student aos retornos diários
garch_fit_norm <- ugarchfit(spec = garch_spec_norm, data = daily_returns[,1])
garch_fit_std <- ugarchfit(spec = garch_spec_std, data = daily_returns[,1])
# Ajustando o modelo EGARCH(1,1) com distribuição Normal e t-student aos retornos diários
egarch_fit_norm <- ugarchfit(spec = egarch_spec_norm, data = daily_returns[,1])
egarch_fit_std <- ugarchfit(spec = egarch_spec_std, data = daily_returns[,1])
# Ajustando o modelo GJR(1,1) com distribuição Normal e t-student aos retornos diários
gjr_fit_norm <- ugarchfit(spec = gjr_spec_norm, data = daily_returns[,1])
gjr_fit_std <- ugarchfit(spec = gjr_spec_std, data = daily_returns[,1])Exibindo os resultados modelo GARCH(1,1), EGARCH(1,1), GJR(1,1) para distribuição normal e t-student.
# Exibindo os resultados GARCH(1,1) do ajuste com distribuição Normal e t-student
print("Resultado do ajuste GARCH(1,1) com distribuição Normal:")## [1] "Resultado do ajuste GARCH(1,1) com distribuição Normal:"garch_fit_norm##
## *---------------------------------*
## * 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|)
## omega 0.000061 0.000020 3.1370 0.001707
## alpha1 0.119540 0.027300 4.3788 0.000012
## beta1 0.767551 0.055123 13.9244 0.000000
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## omega 0.000061 0.000020 3.0493 0.002293
## alpha1 0.119540 0.045873 2.6059 0.009163
## beta1 0.767551 0.059724 12.8517 0.000000
##
## LogLikelihood : 2631.778
##
## Information Criteria
## ------------------------------------
##
## Akaike -4.7537
## Bayes -4.7401
## Shibata -4.7537
## Hannan-Quinn -4.7485
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.1968 0.6573
## Lag[2*(p+q)+(p+q)-1][2] 0.4148 0.7350
## Lag[4*(p+q)+(p+q)-1][5] 1.1513 0.8246
## d.o.f=0
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.8065 0.3691
## Lag[2*(p+q)+(p+q)-1][5] 3.3512 0.3466
## Lag[4*(p+q)+(p+q)-1][9] 4.9485 0.4370
## d.o.f=2
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[3] 0.6886 0.500 2.000 0.4066
## ARCH Lag[5] 0.8256 1.440 1.667 0.7853
## ARCH Lag[7] 1.2010 2.315 1.543 0.8788
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 0.3914
## Individual Statistics:
## omega 0.07219
## alpha1 0.11211
## beta1 0.06103
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 0.846 1.01 1.35
## Individual Statistic: 0.35 0.47 0.75
##
## Sign Bias Test
## ------------------------------------
## t-value prob sig
## Sign Bias 0.21569 0.829272
## Negative Sign Bias 2.74845 0.006086 ***
## Positive Sign Bias 0.08747 0.930313
## Joint Effect 10.10590 0.017687 **
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 41.78 0.001900
## 2 30 50.60 0.007776
## 3 40 61.16 0.013199
## 4 50 79.17 0.004079
##
##
## Elapsed time : 0.318609# Exibindo os resultados do ajuste com distribuição t-Student
print("Resultado do ajuste GARCH(1,1) com distribuição t-Student:")## [1] "Resultado do ajuste GARCH(1,1) com distribuição t-Student:"garch_fit_std##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : sGARCH(1,1)
## Mean Model : ARFIMA(0,0,0)
## Distribution : std
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## omega 0.000049 0.000020 2.4035 0.016240
## alpha1 0.093832 0.027995 3.3517 0.000803
## beta1 0.816838 0.056518 14.4528 0.000000
## shape 6.081206 1.119406 5.4325 0.000000
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## omega 0.000049 0.000019 2.5383 0.011138
## alpha1 0.093832 0.040503 2.3166 0.020523
## beta1 0.816838 0.059473 13.7347 0.000000
## shape 6.081206 1.029334 5.9079 0.000000
##
## LogLikelihood : 2656.329
##
## Information Criteria
## ------------------------------------
##
## Akaike -4.7963
## Bayes -4.7781
## Shibata -4.7963
## Hannan-Quinn -4.7894
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.2801 0.5967
## Lag[2*(p+q)+(p+q)-1][2] 0.5023 0.6925
## Lag[4*(p+q)+(p+q)-1][5] 1.2275 0.8064
## d.o.f=0
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 2.397 0.1216
## Lag[2*(p+q)+(p+q)-1][5] 4.819 0.1681
## Lag[4*(p+q)+(p+q)-1][9] 6.722 0.2236
## d.o.f=2
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[3] 1.776 0.500 2.000 0.1827
## ARCH Lag[5] 2.121 1.440 1.667 0.4451
## ARCH Lag[7] 2.537 2.315 1.543 0.6048
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 0.517
## Individual Statistics:
## omega 0.12555
## alpha1 0.12410
## beta1 0.09135
## shape 0.14617
##
## 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 0.4052 0.6854046
## Negative Sign Bias 3.4431 0.0005966 ***
## Positive Sign Bias 0.4039 0.6863538
## Joint Effect 14.6100 0.0021822 ***
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 17.47 0.5579
## 2 30 37.25 0.1398
## 3 40 38.59 0.4883
## 4 50 41.47 0.7690
##
##
## Elapsed time : 0.169718# Exibindo os resultados EGARCH(1,1) do ajuste com distribuição Normal e t-student
print("Resultado do ajuste EGARCH(1,1) com distribuição Normal:")## [1] "Resultado do ajuste EGARCH(1,1) com distribuição Normal:"egarch_fit_norm##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : eGARCH(1,1)
## Mean Model : ARFIMA(0,0,0)
## Distribution : norm
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## omega -0.625420 0.281461 -2.2220 0.026280
## alpha1 -0.099382 0.026797 -3.7087 0.000208
## beta1 0.916430 0.037184 24.6456 0.000000
## gamma1 0.204454 0.046242 4.4214 0.000010
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## omega -0.625420 0.450004 -1.3898 0.164586
## alpha1 -0.099382 0.039619 -2.5084 0.012128
## beta1 0.916430 0.059569 15.3843 0.000000
## gamma1 0.204454 0.083894 2.4371 0.014807
##
## LogLikelihood : 2639.125
##
## Information Criteria
## ------------------------------------
##
## Akaike -4.7651
## Bayes -4.7470
## Shibata -4.7652
## Hannan-Quinn -4.7583
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.1080 0.7425
## Lag[2*(p+q)+(p+q)-1][2] 0.2261 0.8383
## Lag[4*(p+q)+(p+q)-1][5] 1.0214 0.8548
## d.o.f=0
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 2.205 0.1376
## Lag[2*(p+q)+(p+q)-1][5] 5.434 0.1218
## Lag[4*(p+q)+(p+q)-1][9] 7.052 0.1949
## d.o.f=2
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[3] 2.770 0.500 2.000 0.09606
## ARCH Lag[5] 3.130 1.440 1.667 0.27148
## ARCH Lag[7] 3.584 2.315 1.543 0.41054
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 0.7066
## Individual Statistics:
## omega 0.10936
## alpha1 0.26934
## beta1 0.10610
## gamma1 0.08588
##
## 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 0.6233 0.53320
## Negative Sign Bias 2.5496 0.01092 **
## Positive Sign Bias 1.0791 0.28080
## Joint Effect 7.7094 0.05242 *
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 38.38 0.005314
## 2 30 48.05 0.014521
## 3 40 60.37 0.015659
## 4 50 68.68 0.033142
##
##
## Elapsed time : 0.105809# Exibindo os resultados EGARCH(1,1) do ajuste com distribuição t-Student
print("Resultado do ajuste EGARCH(1,1) com distribuição t-Student:")## [1] "Resultado do ajuste EGARCH(1,1) com distribuição t-Student:"egarch_fit_std##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : eGARCH(1,1)
## Mean Model : ARFIMA(0,0,0)
## Distribution : std
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## omega -0.457699 0.085984 -5.3231 0.000000
## alpha1 -0.087423 0.024094 -3.6285 0.000285
## beta1 0.939447 0.011225 83.6922 0.000000
## gamma1 0.169392 0.036155 4.6852 0.000003
## shape 6.249158 1.199525 5.2097 0.000000
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## omega -0.457699 0.080652 -5.6750 0.000000
## alpha1 -0.087423 0.027967 -3.1259 0.001773
## beta1 0.939447 0.010060 93.3865 0.000000
## gamma1 0.169392 0.062498 2.7104 0.006721
## shape 6.249158 1.094034 5.7120 0.000000
##
## LogLikelihood : 2660.706
##
## Information Criteria
## ------------------------------------
##
## Akaike -4.8024
## Bayes -4.7797
## Shibata -4.8024
## Hannan-Quinn -4.7938
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.1664 0.6834
## Lag[2*(p+q)+(p+q)-1][2] 0.2946 0.7987
## Lag[4*(p+q)+(p+q)-1][5] 1.0953 0.8377
## d.o.f=0
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 4.001 0.04546
## Lag[2*(p+q)+(p+q)-1][5] 7.646 0.03606
## Lag[4*(p+q)+(p+q)-1][9] 9.706 0.05794
## d.o.f=2
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[3] 4.247 0.500 2.000 0.03932
## ARCH Lag[5] 5.032 1.440 1.667 0.10103
## ARCH Lag[7] 5.605 2.315 1.543 0.17009
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 0.876
## Individual Statistics:
## omega 0.11513
## alpha1 0.28525
## beta1 0.11459
## gamma1 0.18476
## shape 0.08511
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 1.28 1.47 1.88
## Individual Statistic: 0.35 0.47 0.75
##
## Sign Bias Test
## ------------------------------------
## t-value prob sig
## Sign Bias 0.7436 0.457248
## Negative Sign Bias 3.1343 0.001768 ***
## Positive Sign Bias 1.2268 0.220173
## Joint Effect 11.4578 0.009491 ***
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 18.99 0.4574
## 2 30 30.20 0.4039
## 3 40 37.80 0.5246
## 4 50 36.59 0.9049
##
##
## Elapsed time : 0.207397# Exibindo os resultados GJR(1,1) do ajuste com distribuição Normal e t-student
print("Resultado do ajuste GJR(1,1) com distribuição Normal:")## [1] "Resultado do ajuste GJR(1,1) com distribuição Normal:"gjr_fit_norm##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : gjrGARCH(1,1)
## Mean Model : ARFIMA(0,0,0)
## Distribution : norm
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## omega 0.000052 0.000017 3.1085 0.001881
## alpha1 0.030092 0.021468 1.4017 0.161000
## beta1 0.801463 0.049689 16.1297 0.000000
## gamma1 0.152219 0.041915 3.6316 0.000282
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## omega 0.000052 0.000021 2.5101 0.012070
## alpha1 0.030092 0.027669 1.0876 0.276774
## beta1 0.801463 0.064279 12.4685 0.000000
## gamma1 0.152219 0.061437 2.4777 0.013225
##
## LogLikelihood : 2640.967
##
## Information Criteria
## ------------------------------------
##
## Akaike -4.7685
## Bayes -4.7504
## Shibata -4.7685
## Hannan-Quinn -4.7616
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.06897 0.7928
## Lag[2*(p+q)+(p+q)-1][2] 0.28107 0.8063
## Lag[4*(p+q)+(p+q)-1][5] 1.05412 0.8473
## d.o.f=0
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.4951 0.4816
## Lag[2*(p+q)+(p+q)-1][5] 3.2492 0.3634
## Lag[4*(p+q)+(p+q)-1][9] 4.4022 0.5221
## d.o.f=2
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[3] 0.4389 0.500 2.000 0.5077
## ARCH Lag[5] 0.5636 1.440 1.667 0.8647
## ARCH Lag[7] 0.8894 2.315 1.543 0.9309
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 0.5788
## Individual Statistics:
## omega 0.08956
## alpha1 0.08552
## beta1 0.06930
## gamma1 0.15315
##
## 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 0.0005076 0.9996
## Negative Sign Bias 1.5740641 0.1158
## Positive Sign Bias 0.9478807 0.3434
## Joint Effect 3.6064555 0.3072
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 39.75 0.003528
## 2 30 46.97 0.018767
## 3 40 58.70 0.022193
## 4 50 63.89 0.074919
##
##
## Elapsed time : 0.249527# Exibindo os resultados EGARCH(1,1) do ajuste com distribuição t-Student
print("Resultado do ajuste GJR(1,1) com distribuição t-Student:")## [1] "Resultado do ajuste GJR(1,1) com distribuição t-Student:"gjr_fit_std##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : gjrGARCH(1,1)
## Mean Model : ARFIMA(0,0,0)
## Distribution : std
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## omega 0.000044 0.000018 2.4485 0.014346
## alpha1 0.024837 0.022510 1.1034 0.269861
## beta1 0.833759 0.052266 15.9522 0.000000
## gamma1 0.129588 0.046128 2.8093 0.004965
## shape 6.491127 1.263675 5.1367 0.000000
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## omega 0.000044 0.000020 2.2154 0.026731
## alpha1 0.024837 0.024523 1.0128 0.311160
## beta1 0.833759 0.062840 13.2679 0.000000
## gamma1 0.129588 0.059553 2.1760 0.029554
## shape 6.491127 1.132248 5.7330 0.000000
##
## LogLikelihood : 2662.127
##
## Information Criteria
## ------------------------------------
##
## Akaike -4.8049
## Bayes -4.7823
## Shibata -4.8050
## Hannan-Quinn -4.7964
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.1047 0.7462
## Lag[2*(p+q)+(p+q)-1][2] 0.3305 0.7790
## Lag[4*(p+q)+(p+q)-1][5] 1.1031 0.8359
## d.o.f=0
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.9673 0.3254
## Lag[2*(p+q)+(p+q)-1][5] 3.4482 0.3313
## Lag[4*(p+q)+(p+q)-1][9] 4.7092 0.4733
## d.o.f=2
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[3] 0.8644 0.500 2.000 0.3525
## ARCH Lag[5] 1.0885 1.440 1.667 0.7068
## ARCH Lag[7] 1.4433 2.315 1.543 0.8327
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 0.6406
## Individual Statistics:
## omega 0.1343
## alpha1 0.1320
## beta1 0.1009
## gamma1 0.1950
## shape 0.1361
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 1.28 1.47 1.88
## Individual Statistic: 0.35 0.47 0.75
##
## Sign Bias Test
## ------------------------------------
## t-value prob sig
## Sign Bias 0.08094 0.93550
## Negative Sign Bias 2.02926 0.04267 **
## Positive Sign Bias 1.05273 0.29269
## Joint Effect 5.60912 0.13226
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 22.54 0.2584
## 2 30 33.51 0.2576
## 3 40 46.19 0.1996
## 4 50 44.54 0.6542
##
##
## Elapsed time : 0.3875811Calculando os critérios de informação
# Calculando os critérios de informação BIC
bics <- c(infocriteria(garch_fit_norm)[2], infocriteria(garch_fit_std)[2],
infocriteria(egarch_fit_norm)[2], infocriteria(egarch_fit_std)[2],
infocriteria(gjr_fit_norm)[2], infocriteria(gjr_fit_std)[2])
# Criando a tabela de resultados
results_table <- data.frame(Model = c("sGARCH Normal", "sGARCH t-Student",
"eGARCH Normal", "eGARCH t-Student",
"GJR-GARCH Normal", "GJR-GARCH t-Student"),
Bayes = bics)
# Ordenando a tabela pelo valor do BIC
results_table <- results_table[order(results_table$Bayes), ]
# Exibindo a tabela de resultados
print("Tabela de Resultados:")## [1] "Tabela de Resultados:"print(results_table)## Model Bayes
## 6 GJR-GARCH t-Student -4.782290
## 4 eGARCH t-Student -4.779720
## 2 sGARCH t-Student -4.778142
## 5 GJR-GARCH Normal -4.750362
## 3 eGARCH Normal -4.747030
## 1 sGARCH Normal -4.740082| Seleção das configurações do modelos | VALE3 |
|---|---|
| Especificações | Amostra completa usando (BIC) |
| GJR-GARCH t-Student | -4.782290 |
| eGARCH t-Student | -4.779720 |
| sGARCH t-Student | -4.778142 |
| GJR-GARCH Normal | -4.750362 |
| eGARCH Normal | -4.747030 |
| sGARCH Normal | -4.740082 |
Podemos analisar pelos critérios informações BIC os melhores modelos são GJR-GARCH t-Student, eGARCH t-Student, sGARCH t-Student por se tratar de um modelo t-student observado anteriormente com fatos estilizados, escolheremos o modelos GJR-GARCH t-Student e eGARCH t-Student
Parametros dos modelo escolhido GJR-GARCH t-Student e eGARCH t-Student para VALE3.SA
# Exibindo a tabela de coeficientes em formato LaTeX
Table <- xtable(gjr_fit_std@fit$matcoef)
# print(Table, type = "latex", comment = FALSE)
#Table
gjr_fit_std@fit$matcoef## Estimate Std. Error t value Pr(>|t|)
## omega 4.402175e-05 1.797923e-05 2.448478 1.434613e-02
## alpha1 2.483690e-02 2.250976e-02 1.103384 2.698606e-01
## beta1 8.337588e-01 5.226596e-02 15.952235 0.000000e+00
## gamma1 1.295883e-01 4.612847e-02 2.809292 4.965060e-03
## shape 6.491127e+00 1.263675e+00 5.136706 2.795963e-07# Exibindo a tabela de coeficientes em formato LaTeX
Table <- xtable(egarch_fit_std@fit$matcoef)
# print(Table, type = "latex", comment = FALSE)
#Table
egarch_fit_std@fit$matcoef## Estimate Std. Error t value Pr(>|t|)
## omega -0.45769858 0.08598381 -5.323079 1.020257e-07
## alpha1 -0.08742331 0.02409360 -3.628487 2.850868e-04
## beta1 0.93944671 0.01122502 83.692178 0.000000e+00
## gamma1 0.16939209 0.03615485 4.685183 2.797100e-06
## shape 6.24915808 1.19952507 5.209694 1.891528e-07Calculando e plotando a volatilidade condicional estimada
sigma <- sigma(gjr_fit_std) %>% xts_tbl()
colnames(sigma) <- c("date", "sigma")
sigma <- sigma %>% mutate(date = as.Date(date))
# Plotando a volatilidade condicional estimada
ggplot(sigma) + geom_line(aes(x = date , y = sigma)) + theme_solarized()
Calculando o parâmetro de persistência escolhido da ação VALE3 para periodo completo, dos melhores modelos eGARCH(1,1)-std vs GJR-GARCH(1,1)-std
# Calculando o parâmetro de persistência escolhido para periodo completo
print("Parâmetro de Persistência EGARCH(1,1) t-student:")## [1] "Parâmetro de Persistência EGARCH(1,1) t-student:"print(persistence(egarch_fit_std))## [1] 0.9394467print("Parâmetro de Persistência GJRGARCH(1,1) t-student:")## [1] "Parâmetro de Persistência GJRGARCH(1,1) t-student:"print(persistence(gjr_fit_std))## [1] 0.9233899# Calculando a persistência para cada modelo
persist_egarch <- persistence(egarch_fit_std)
persist_gjr <- persistence(gjr_fit_std)
# Calculando o half-life para cada modelo
half_life_egarch <- -log(2) / log(persist_egarch)
half_life_gjr <- -log(2) / log(persist_gjr)
# Criando a tabela com os resultados
results_table <- data.frame(Model = c("eGARCH(1,1) t-student", "GJRGARCH(1,1) t-student"),
Persistência = c(persist_egarch, persist_gjr),
Half_Life = c(half_life_egarch, half_life_gjr))
print("Tabela de Resultados Persistência e Half life:")## [1] "Tabela de Resultados Persistência e Half life:"print(results_table)## Model Persistência Half_Life
## 1 eGARCH(1,1) t-student 0.9394467 11.09671
## 2 GJRGARCH(1,1) t-student 0.9233899 8.69655Verificar os resíduos
plot(egarch_fit_std, which=11)
acf(residuals(egarch_fit_std))
pacf(residuals(egarch_fit_std))
Realizando previsões usando o modelo ajustado eGARCH(1,1) t-student
# Realizando previsões usando o modelo ajustado
n_ahead <- 10
egarch_std_forecast <- ugarchforecast(egarch_fit_std, n.ahead = n_ahead)
print(paste("Previsão da Volatilidade para", n_ahead, "passos à frente:"))## [1] "Previsão da Volatilidade para 10 passos à frente:"print(egarch_std_forecast)##
## *------------------------------------*
## * GARCH Model Forecast *
## *------------------------------------*
## Model: eGARCH
## Horizon: 10
## Roll Steps: 0
## Out of Sample: 0
##
## 0-roll forecast [T0=1106-01-01]:
## Series Sigma
## T+1 0 0.02532
## T+2 0 0.02516
## T+3 0 0.02501
## T+4 0 0.02488
## T+5 0 0.02475
## T+6 0 0.02463
## T+7 0 0.02452
## T+8 0 0.02441
## T+9 0 0.02431
## T+10 0 0.02422# Plotando previsões da volatilidade
p1 <- plot(egarch_fit_std, which = 1)
p1##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : eGARCH(1,1)
## Mean Model : ARFIMA(0,0,0)
## Distribution : std
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## omega -0.457699 0.085984 -5.3231 0.000000
## alpha1 -0.087423 0.024094 -3.6285 0.000285
## beta1 0.939447 0.011225 83.6922 0.000000
## gamma1 0.169392 0.036155 4.6852 0.000003
## shape 6.249158 1.199525 5.2097 0.000000
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## omega -0.457699 0.080652 -5.6750 0.000000
## alpha1 -0.087423 0.027967 -3.1259 0.001773
## beta1 0.939447 0.010060 93.3865 0.000000
## gamma1 0.169392 0.062498 2.7104 0.006721
## shape 6.249158 1.094034 5.7120 0.000000
##
## LogLikelihood : 2660.706
##
## Information Criteria
## ------------------------------------
##
## Akaike -4.8024
## Bayes -4.7797
## Shibata -4.8024
## Hannan-Quinn -4.7938
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.1664 0.6834
## Lag[2*(p+q)+(p+q)-1][2] 0.2946 0.7987
## Lag[4*(p+q)+(p+q)-1][5] 1.0953 0.8377
## d.o.f=0
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 4.001 0.04546
## Lag[2*(p+q)+(p+q)-1][5] 7.646 0.03606
## Lag[4*(p+q)+(p+q)-1][9] 9.706 0.05794
## d.o.f=2
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[3] 4.247 0.500 2.000 0.03932
## ARCH Lag[5] 5.032 1.440 1.667 0.10103
## ARCH Lag[7] 5.605 2.315 1.543 0.17009
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 0.876
## Individual Statistics:
## omega 0.11513
## alpha1 0.28525
## beta1 0.11459
## gamma1 0.18476
## shape 0.08511
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 1.28 1.47 1.88
## Individual Statistic: 0.35 0.47 0.75
##
## Sign Bias Test
## ------------------------------------
## t-value prob sig
## Sign Bias 0.7436 0.457248
## Negative Sign Bias 3.1343 0.001768 ***
## Positive Sign Bias 1.2268 0.220173
## Joint Effect 11.4578 0.009491 ***
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 18.99 0.4574
## 2 30 30.20 0.4039
## 3 40 37.80 0.5246
## 4 50 36.59 0.9049
##
##
## Elapsed time : 0.207397p2 <- plot(egarch_fit_std, which = 3)
p2##
## *---------------------------------*
## * GARCH Model Fit *
## *---------------------------------*
##
## Conditional Variance Dynamics
## -----------------------------------
## GARCH Model : eGARCH(1,1)
## Mean Model : ARFIMA(0,0,0)
## Distribution : std
##
## Optimal Parameters
## ------------------------------------
## Estimate Std. Error t value Pr(>|t|)
## omega -0.457699 0.085984 -5.3231 0.000000
## alpha1 -0.087423 0.024094 -3.6285 0.000285
## beta1 0.939447 0.011225 83.6922 0.000000
## gamma1 0.169392 0.036155 4.6852 0.000003
## shape 6.249158 1.199525 5.2097 0.000000
##
## Robust Standard Errors:
## Estimate Std. Error t value Pr(>|t|)
## omega -0.457699 0.080652 -5.6750 0.000000
## alpha1 -0.087423 0.027967 -3.1259 0.001773
## beta1 0.939447 0.010060 93.3865 0.000000
## gamma1 0.169392 0.062498 2.7104 0.006721
## shape 6.249158 1.094034 5.7120 0.000000
##
## LogLikelihood : 2660.706
##
## Information Criteria
## ------------------------------------
##
## Akaike -4.8024
## Bayes -4.7797
## Shibata -4.8024
## Hannan-Quinn -4.7938
##
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 0.1664 0.6834
## Lag[2*(p+q)+(p+q)-1][2] 0.2946 0.7987
## Lag[4*(p+q)+(p+q)-1][5] 1.0953 0.8377
## d.o.f=0
## H0 : No serial correlation
##
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
## statistic p-value
## Lag[1] 4.001 0.04546
## Lag[2*(p+q)+(p+q)-1][5] 7.646 0.03606
## Lag[4*(p+q)+(p+q)-1][9] 9.706 0.05794
## d.o.f=2
##
## Weighted ARCH LM Tests
## ------------------------------------
## Statistic Shape Scale P-Value
## ARCH Lag[3] 4.247 0.500 2.000 0.03932
## ARCH Lag[5] 5.032 1.440 1.667 0.10103
## ARCH Lag[7] 5.605 2.315 1.543 0.17009
##
## Nyblom stability test
## ------------------------------------
## Joint Statistic: 0.876
## Individual Statistics:
## omega 0.11513
## alpha1 0.28525
## beta1 0.11459
## gamma1 0.18476
## shape 0.08511
##
## Asymptotic Critical Values (10% 5% 1%)
## Joint Statistic: 1.28 1.47 1.88
## Individual Statistic: 0.35 0.47 0.75
##
## Sign Bias Test
## ------------------------------------
## t-value prob sig
## Sign Bias 0.7436 0.457248
## Negative Sign Bias 3.1343 0.001768 ***
## Positive Sign Bias 1.2268 0.220173
## Joint Effect 11.4578 0.009491 ***
##
##
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
## group statistic p-value(g-1)
## 1 20 18.99 0.4574
## 2 30 30.20 0.4039
## 3 40 37.80 0.5246
## 4 50 36.59 0.9049
##
##
## Elapsed time : 0.207397Conclusão:
| VALE3.SA | ||
|---|---|---|
| Model | Persistência | Half_Life |
| eGARCH(1,1) t-student | 0.9394467 | 11.09 |
| GJRGARCH(1,1) t-student | 0.9233899 | 8.69 |
Comparando a tabela acima para modelo a um aumento de 0.923 para 0.939 na persistência dos choques os choques serão mais lentamente enfraquecidos ao longo do tempo. O cálculo da half-life mostra que no periodo inteiro conforme pedido no exercício da ação VALE3 modelo GJRGARCH-std 8.69 dias para dissipação de um choque e para o modelo eGARCH(1,1) t-student seria necessarios 11.09 dias para dissipação do choque de volatilidade.