Séries Temporais com R: Exemplo ARCH com stocks

Licença

This work is licensed under the Creative Commons Attribution-ShareAlike 4.0 International License. To view a copy of this license, visit http://creativecommons.org/licenses/by-sa/4.0/ or send a letter to Creative Commons, PO Box 1866, Mountain View, CA 94042, USA.

License: CC BY-SA 4.0

Citação

Sugestão de citação: FIGUEIREDO, Adriano Marcos Rodrigues. Séries Temporais com R: Exemplo ARCH com stocks. Campo Grande-MS,Brasil: RStudio/Rpubs, 2019. Disponível em http://rpubs.com/amrofi/stocks_garch.

1 Apresentação

Este artigo procura ilustrar os procedimentos para trabalhar com ações de uma empresa no âmbito dos métodos de séries temporais, com modelos da família ARCH. É baseado e adaptado a partir do script do Prof. Pedro Costa Ferreira (FGV) disponível em: https://github.com/pedrocostaferreira/SeriesTemporais/blob/master/Aula8_GARCHmodel_PETR4.R.

2 Dados

O primeiro passo para tal é baixar os dados da empresa desejada. Usaremos como exemplo a Petrobras (PETR4), a Usiminas (USIM5) e a Bovespa (BVSP). O pacote quantmod nos auxiliará a obter os dados eletronicamente. O pesquisador deve conhecer os tickers do papel desejado.

if (!require(quantmod)) { 
  install.packages('quantmod') 
  library(quantmod) 
}
if (!require(fpp2)) { 
  install.packages('fpp2') 
  library(fpp2) 
}
if (!require(Metrics)) { 
  install.packages('Metrics') 
  library(Metrics) 
}
if (!require(stockPortfolio)) { 
  install.packages('stockPortfolio') 
  library(stockPortfolio) 
}
if (!require(rugarch)) { 
  install.packages('rugarch') 
  library(rugarch)
}

2.1 Petrobrás - PETR4.SA

require(quantmod)
getSymbols('PETR4.SA',src='yahoo') # pelo {quantmod}

## [1] "PETR4.SA"

head(PETR4.SA)

##            PETR4.SA.Open PETR4.SA.High PETR4.SA.Low PETR4.SA.Close
## 2007-01-02        25.000        25.225       24.880         25.160
## 2007-01-03        25.080        25.200       24.005         24.395
## 2007-01-04        24.250        24.375       23.700         23.850
## 2007-01-05        23.600        23.995       22.550         23.125
## 2007-01-08        23.250        23.570       22.900         23.395
## 2007-01-09        22.985        23.195       22.305         22.680
##            PETR4.SA.Volume PETR4.SA.Adjusted
## 2007-01-02        10244800          19.29404
## 2007-01-03        19898600          18.70740
## 2007-01-04        21060200          18.28946
## 2007-01-05        24864000          17.73349
## 2007-01-08        19440200          17.94054
## 2007-01-09        25883600          17.39224

tail(PETR4.SA)

##            PETR4.SA.Open PETR4.SA.High PETR4.SA.Low PETR4.SA.Close
## 2019-09-25         27.03         27.35        26.78          27.34
## 2019-09-26         27.40         27.70        27.23          27.70
## 2019-09-27         27.55         27.95        27.52          27.66
## 2019-09-30         27.56         27.64        27.40          27.55
## 2019-10-01         27.60         27.73        27.36          27.51
## 2019-10-02         27.22         27.26        26.61          26.72
##            PETR4.SA.Volume PETR4.SA.Adjusted
## 2019-09-25        29038400             27.34
## 2019-09-26        33705900             27.70
## 2019-09-27        31886800             27.66
## 2019-09-30        23143400             27.55
## 2019-10-01        32423200             27.51
## 2019-10-02        51261200             26.72

PETR4.SA.A <- PETR4.SA$PETR4.SA.Adjusted
autoplot(PETR4.SA.A)

chartSeries(PETR4.SA)

2.2 Usiminas - USIM5.SA

getSymbols('USIM5.SA',src='yahoo') # pelo {quantmod}

## [1] "USIM5.SA"

head(USIM5.SA)

##            USIM5.SA.Open USIM5.SA.High USIM5.SA.Low USIM5.SA.Close
## 2007-01-02       17.7089       18.1444      17.4733        17.9556
## 2007-01-03       17.7800       17.8444      17.1556        17.3333
## 2007-01-04       17.3311       17.3311      16.8911        17.1222
## 2007-01-05       16.9978       17.0667      16.2444        16.2444
## 2007-01-08       16.5111       16.7333      16.1156        16.4000
## 2007-01-09       16.4511       16.6533      15.8467        16.1556
##            USIM5.SA.Volume USIM5.SA.Adjusted
## 2007-01-02         2535750          15.86188
## 2007-01-03         6224850          15.31215
## 2007-01-04         4378500          15.12566
## 2007-01-05         4972500          14.35022
## 2007-01-08         5725800          14.48767
## 2007-01-09         7182450          14.27177

tail(USIM5.SA)

##            USIM5.SA.Open USIM5.SA.High USIM5.SA.Low USIM5.SA.Close
## 2019-09-25          7.89          8.09         7.77           8.05
## 2019-09-26          8.10          8.17         7.96           8.04
## 2019-09-27          8.05          8.10         7.90           8.00
## 2019-09-30          7.98          7.98         7.81           7.81
## 2019-10-01          7.68          7.82         7.65           7.74
## 2019-10-02          7.65          7.68         7.46           7.48
##            USIM5.SA.Volume USIM5.SA.Adjusted
## 2019-09-25        12112600              8.05
## 2019-09-26         8803100              8.04
## 2019-09-27         7618800              8.00
## 2019-09-30        12398600              7.81
## 2019-10-01        16711400              7.74
## 2019-10-02         9671600              7.48

USIM5.SA.A <- USIM5.SA$USIM5.SA.Adjusted
autoplot(USIM5.SA.A)

chartSeries(USIM5.SA)

2.3 Bovespa - ^BVSP

getSymbols('^BVSP',src='yahoo')  # pelo {quantmod}

## [1] "^BVSP"

chartSeries(BVSP)

2.4 Limpeza dos dados

require(stockPortfolio)

head(PETR4.SA)

##            PETR4.SA.Open PETR4.SA.High PETR4.SA.Low PETR4.SA.Close
## 2007-01-02        25.000        25.225       24.880         25.160
## 2007-01-03        25.080        25.200       24.005         24.395
## 2007-01-04        24.250        24.375       23.700         23.850
## 2007-01-05        23.600        23.995       22.550         23.125
## 2007-01-08        23.250        23.570       22.900         23.395
## 2007-01-09        22.985        23.195       22.305         22.680
##            PETR4.SA.Volume PETR4.SA.Adjusted
## 2007-01-02        10244800          19.29404
## 2007-01-03        19898600          18.70740
## 2007-01-04        21060200          18.28946
## 2007-01-05        24864000          17.73349
## 2007-01-08        19440200          17.94054
## 2007-01-09        25883600          17.39224

tail(PETR4.SA)

##            PETR4.SA.Open PETR4.SA.High PETR4.SA.Low PETR4.SA.Close
## 2019-09-25         27.03         27.35        26.78          27.34
## 2019-09-26         27.40         27.70        27.23          27.70
## 2019-09-27         27.55         27.95        27.52          27.66
## 2019-09-30         27.56         27.64        27.40          27.55
## 2019-10-01         27.60         27.73        27.36          27.51
## 2019-10-02         27.22         27.26        26.61          26.72
##            PETR4.SA.Volume PETR4.SA.Adjusted
## 2019-09-25        29038400             27.34
## 2019-09-26        33705900             27.70
## 2019-09-27        31886800             27.66
## 2019-09-30        23143400             27.55
## 2019-10-01        32423200             27.51
## 2019-10-02        51261200             26.72

PETR4.SA.Close <- PETR4.SA$PETR4.SA.Close
plot(PETR4.SA.Close)

# Ver que tem outliers devido aos NAs e valores inconsistentes que necessitam correção 
# primeiro mudarei os zeros por NA, depois uso locf para tirar os NA
PETR4_clean<-is.na(PETR4.SA.Close) <- PETR4.SA.Close == 0 
PETR4_clean<-na.locf(PETR4.SA.Close, fromLast = FALSE, coredata = NULL)
max(PETR4_clean)

## [1] 52.51

min(PETR4_clean)

## [1] 4.2

USIM5.SA.Close <- USIM5.SA$USIM5.SA.Close
plot(USIM5.SA.Close)

USIM5_clean<-is.na(USIM5.SA.Close) <- USIM5.SA.Close == 0 
USIM5_clean<-na.locf(USIM5.SA.Close, fromLast = FALSE, coredata = NULL)
max(USIM5_clean)

## [1] 47.3

min(USIM5_clean)

## [1] 0.85

BVSP.Close <- BVSP$BVSP.Close
BVSP_clean<-is.na(BVSP.Close) <- BVSP.Close == 0 
BVSP_clean<-na.locf(BVSP.Close, fromLast = FALSE, coredata = NULL)
max(BVSP_clean)

## [1] 105817

min(BVSP_clean)

## [1] 29435

plot(BVSP_clean)

dados_basicos<-cbind.xts(BVSP_clean,PETR4_clean)
library(dygraphs)
dygraph(window(PETR4_clean,start = "2010-01-01"), 
        main="Valor da PETR4.SA após limpeza")

dygraph(window(USIM5_clean,start = "2010-01-01"), 
        main="Valor da USIM5.SA após limpeza")

dygraph(window(BVSP_clean,start = "2010-01-01"), 
        main="Valor da BVSP após limpeza")

3 Retornos diários

r_PETR4 <- dailyReturn(PETR4_clean,type = "log")
plot(r_PETR4)

hist(r_PETR4,100)

max(r_PETR4)

## [1] 0.152217

min(r_PETR4)

## [1] -0.17149

# ainda tem um outlier bem forte negativo

r_USIM5 <- dailyReturn(USIM5_clean,type = "log")
plot(r_USIM5)

hist(r_USIM5,100)

max(r_USIM5)

## [1] 0.3008923

min(r_USIM5)

## [1] -0.1759781

r_BVSP<-dailyReturn(BVSP_clean,type = "log")
plot(r_BVSP)

hist(r_BVSP,100)

max(r_BVSP)

## [1] 0.1367661

min(r_BVSP)

## [1] -0.1209605

## vamos encurtar um pouco a ST para termos uma informação mais relevante

r_PETR4_pos2010 <- window(r_PETR4, start = "2010-01-01")
plot(r_PETR4_pos2010)

hist(r_PETR4_pos2010,100)

max(r_PETR4_pos2010)    

## [1] 0.1508583

min(r_PETR4_pos2010)   

## [1] -0.17149

r_USIM5_pos2010 <- window(r_USIM5, start = "2010-01-01")
plot(r_USIM5_pos2010)

hist(r_USIM5_pos2010,100)

max(r_USIM5_pos2010)

## [1] 0.3008923

min(r_USIM5_pos2010)

## [1] -0.1759781

r_BVSP_pos2010<-window(r_BVSP,start = "2010-01-01")
plot(r_BVSP_pos2010)

hist(r_BVSP_pos2010,100)

max(r_BVSP_pos2010)

## [1] 0.06388665

min(r_BVSP_pos2010)

## [1] -0.09210685

4 Autocorrelações nos retornos

4.1 PETR4

# Correlograma de r_PETR4_pos2010
BETS::corrgram(r_PETR4_pos2010, lag.max = 36, mode = "bartlett", style="normal", knit = T)

# Função de autocorrelação parcial de r_PETR4_pos2010
BETS::corrgram(r_PETR4_pos2010, lag.max = 36, type = "partial", style="normal", knit = T)

# opção nativa para ACF e PACF
acf(r_PETR4_pos2010) # observar alteração na escala do eixo vertical

pacf(r_PETR4_pos2010)

4.2 USIM5

# Correlograma de r_USIM5_pos2010
BETS::corrgram(r_USIM5_pos2010, lag.max = 36, mode = "bartlett", style="normal", knit = T)

# Função de autocorrelação parcial de r_USIM5_pos2010
BETS::corrgram(r_USIM5_pos2010, lag.max = 36, type = "partial", style="normal", knit = T)

4.3 BVSP

# Correlograma de r_BVSP_pos2010
BETS::corrgram(r_BVSP_pos2010, lag.max = 36, mode = "bartlett", style="normal", knit = T)

# Função de autocorrelação parcial de r_BVSP_pos2010
BETS::corrgram(r_BVSP_pos2010, lag.max = 36, type = "partial", style="normal", knit = T)

5 Não-normalidade

5.1 PETR4

# Fat tails of returns

library(moments);library(normtest)
# Curtose
kurtosis(r_PETR4_pos2010)

## daily.returns 
##      7.020553

kurtosis.norm.test(r_PETR4_pos2010)

## 
##  Kurtosis test for normality
## 
## data:  r_PETR4_pos2010
## T = 7.0206, p-value < 2.2e-16

# assimetria
skewness(r_PETR4_pos2010)

## daily.returns 
##    -0.1465884

skewness.norm.test(r_PETR4_pos2010)

## 
##  Skewness test for normality
## 
## data:  r_PETR4_pos2010
## T = -0.14659, p-value = 0.003

# Jarque-Bera
jb.norm.test(r_PETR4_pos2010)

## 
##  Jarque-Bera test for normality
## 
## data:  r_PETR4_pos2010
## JB = 1639.3, p-value < 2.2e-16

5.2 USIM5

# Fat tails of returns

# Curtose
kurtosis(r_USIM5_pos2010)

## daily.returns 
##      8.463703

kurtosis.norm.test(r_USIM5_pos2010)

## 
##  Kurtosis test for normality
## 
## data:  r_USIM5_pos2010
## T = 8.4637, p-value < 2.2e-16

# assimetria
skewness(r_USIM5_pos2010)

## daily.returns 
##      0.573222

skewness.norm.test(r_USIM5_pos2010)

## 
##  Skewness test for normality
## 
## data:  r_USIM5_pos2010
## T = 0.57322, p-value < 2.2e-16

# Jarque-Bera
jb.norm.test(r_USIM5_pos2010)

## 
##  Jarque-Bera test for normality
## 
## data:  r_USIM5_pos2010
## JB = 3143.9, p-value < 2.2e-16

5.3 BVSP

# Fat tails of returns

# Curtose
kurtosis(r_BVSP_pos2010)

## daily.returns 
##      4.899375

kurtosis.norm.test(r_BVSP_pos2010)

## 
##  Kurtosis test for normality
## 
## data:  r_BVSP_pos2010
## T = 4.8994, p-value < 2.2e-16

# assimetria
skewness(r_BVSP_pos2010)

## daily.returns 
##    -0.1551278

skewness.norm.test(r_BVSP_pos2010)

## 
##  Skewness test for normality
## 
## data:  r_BVSP_pos2010
## T = -0.15513, p-value = 0.001

# Jarque-Bera
jb.norm.test(r_BVSP_pos2010)

## 
##  Jarque-Bera test for normality
## 
## data:  r_BVSP_pos2010
## JB = 373.63, p-value < 2.2e-16

6 Estacionariedade

6.1 PETR4

require(urca)
# Level
adf.none <- ur.df(y = r_PETR4_pos2010, 
                  type = c("none"), 
                  lags = 24, selectlags = "AIC")
summary(adf.none)

## 
## ############################################### 
## # Augmented Dickey-Fuller Test Unit Root Test # 
## ############################################### 
## 
## Test regression none 
## 
## 
## Call:
## lm(formula = z.diff ~ z.lag.1 - 1 + z.diff.lag)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -0.16983 -0.01480  0.00010  0.01393  0.14985 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## z.lag.1     -0.9851123  0.0495661 -19.875   <2e-16 ***
## z.diff.lag1 -0.0296744  0.0456357  -0.650   0.5156    
## z.diff.lag2 -0.0106835  0.0408067  -0.262   0.7935    
## z.diff.lag3  0.0006969  0.0355072   0.020   0.9843    
## z.diff.lag4 -0.0146721  0.0291019  -0.504   0.6142    
## z.diff.lag5  0.0397461  0.0204257   1.946   0.0518 .  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.02802 on 2390 degrees of freedom
## Multiple R-squared:  0.5117, Adjusted R-squared:  0.5105 
## F-statistic: 417.4 on 6 and 2390 DF,  p-value: < 2.2e-16
## 
## 
## Value of test-statistic is: -19.8747 
## 
## Critical values for test statistics: 
##       1pct  5pct 10pct
## tau1 -2.58 -1.95 -1.62

BETS::corrgram(adf.none@res, lag.max = 36, mode = "bartlett", style="normal", knit = T)

6.2 USIM5

require(urca)
# Level
adf.none <- ur.df(y = r_USIM5_pos2010, 
                  type = c("none"), 
                  lags = 24, selectlags = "AIC")
summary(adf.none)

## 
## ############################################### 
## # Augmented Dickey-Fuller Test Unit Root Test # 
## ############################################### 
## 
## Test regression none 
## 
## 
## Call:
## lm(formula = z.diff ~ z.lag.1 - 1 + z.diff.lag)
## 
## Residuals:
##       Min        1Q    Median        3Q       Max 
## -0.173882 -0.019786 -0.001275  0.016939  0.298904 
## 
## Coefficients:
##            Estimate Std. Error t value Pr(>|t|)    
## z.lag.1    -0.95329    0.02791 -34.154   <2e-16 ***
## z.diff.lag  0.02215    0.02044   1.084    0.278    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.03475 on 2394 degrees of freedom
## Multiple R-squared:  0.4665, Adjusted R-squared:  0.466 
## F-statistic:  1047 on 2 and 2394 DF,  p-value: < 2.2e-16
## 
## 
## Value of test-statistic is: -34.1543 
## 
## Critical values for test statistics: 
##       1pct  5pct 10pct
## tau1 -2.58 -1.95 -1.62

BETS::corrgram(adf.none@res, lag.max = 36, mode = "bartlett", style="normal", knit = T)

6.3 BVSP

require(urca)
# Level
adf.none <- ur.df(y = r_BVSP_pos2010, 
                  type = c("none"), 
                  lags = 24, selectlags = "AIC")
summary(adf.none)

## 
## ############################################### 
## # Augmented Dickey-Fuller Test Unit Root Test # 
## ############################################### 
## 
## Test regression none 
## 
## 
## Call:
## lm(formula = z.diff ~ z.lag.1 - 1 + z.diff.lag)
## 
## Residuals:
##       Min        1Q    Median        3Q       Max 
## -0.092200 -0.008152  0.000256  0.008651  0.063727 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## z.lag.1    -1.013743   0.029031 -34.920   <2e-16 ***
## z.diff.lag  0.006919   0.020449   0.338    0.735    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.01414 on 2394 degrees of freedom
## Multiple R-squared:  0.503,  Adjusted R-squared:  0.5026 
## F-statistic:  1211 on 2 and 2394 DF,  p-value: < 2.2e-16
## 
## 
## Value of test-statistic is: -34.9196 
## 
## Critical values for test statistics: 
##       1pct  5pct 10pct
## tau1 -2.58 -1.95 -1.62

BETS::corrgram(adf.none@res, lag.max = 36, mode = "bartlett", style="normal", knit = T)

7 Clusters de volatilidade

7.1 PETR4

ts.plot(r_PETR4_pos2010)
abline(h = mean(r_PETR4_pos2010),col=2, lwd=2)

vol_PETR4<-r_PETR4_pos2010^2
plot(vol_PETR4)

7.2 USIM5

vol_USIM5<-r_USIM5_pos2010^2
plot(vol_USIM5)

7.3 BVSP

vol_BVSP<-r_BVSP_pos2010^2
plot(vol_BVSP)

8 ARCH Effects

# teste para efeitos ARCH no {FinTS}
library(FinTS)
ArchTest(r_PETR4_pos2010)

## 
##  ARCH LM-test; Null hypothesis: no ARCH effects
## 
## data:  r_PETR4_pos2010
## Chi-squared = 256.67, df = 12, p-value < 2.2e-16

ArchTest(r_USIM5_pos2010)

## 
##  ARCH LM-test; Null hypothesis: no ARCH effects
## 
## data:  r_USIM5_pos2010
## Chi-squared = 246.59, df = 12, p-value < 2.2e-16

ArchTest(r_BVSP_pos2010)

## 
##  ARCH LM-test; Null hypothesis: no ARCH effects
## 
## data:  r_BVSP_pos2010
## Chi-squared = 101.46, df = 12, p-value = 2.878e-16

9 Modelos ARCH - PETR4

9.1 Modeling PETR4 return GARCH (1,1)

library(rugarch)

garch11.spec = ugarchspec(mean.model = list(armaOrder = c(2,2),
                          include.mean=TRUE), 
                          variance.model = list(garchOrder = c(1,1), 
                          model = "sGARCH"))

garch.fit = ugarchfit(garch11.spec,
                      data = r_PETR4_pos2010*100,
                      fit.control=list(scale=TRUE), 
                      distribution.model = "norm")

## Warning in arima(data, order = c(modelinc[2], 0, modelinc[3]), include.mean
## = modelinc[1], : possible convergence problem: optim gave code = 1

print(garch.fit)

## 
## *---------------------------------*
## *          GARCH Model Fit        *
## *---------------------------------*
## 
## Conditional Variance Dynamics    
## -----------------------------------
## GARCH Model  : sGARCH(1,1)
## Mean Model   : ARFIMA(2,0,2)
## Distribution : norm 
## 
## Optimal Parameters
## ------------------------------------
##         Estimate  Std. Error     t value Pr(>|t|)
## mu     -0.004017    0.046794   -0.085844 0.931590
## ar1    -0.985290    0.018915  -52.091570 0.000000
## ar2    -0.958289    0.007288 -131.483257 0.000000
## ma1     0.976372    0.014279   68.375909 0.000000
## ma2     0.968387    0.002128  455.115294 0.000000
## omega   0.150207    0.037971    3.955829 0.000076
## alpha1  0.067363    0.010621    6.342152 0.000000
## beta1   0.912420    0.014085   64.778347 0.000000
## 
## Robust Standard Errors:
##         Estimate  Std. Error     t value Pr(>|t|)
## mu     -0.004017    0.050307   -0.079849 0.936358
## ar1    -0.985290    0.018066  -54.537987 0.000000
## ar2    -0.958289    0.006828 -140.351691 0.000000
## ma1     0.976372    0.014108   69.206055 0.000000
## ma2     0.968387    0.001307  740.982340 0.000000
## omega   0.150207    0.066019    2.275213 0.022893
## alpha1  0.067363    0.018990    3.547369 0.000389
## beta1   0.912420    0.022872   39.891705 0.000000
## 
## LogLikelihood : -5668.525 
## 
## Information Criteria
## ------------------------------------
##                    
## Akaike       4.6894
## Bayes        4.7085
## Shibata      4.6894
## Hannan-Quinn 4.6964
## 
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
##                          statistic p-value
## Lag[1]                     0.07085  0.7901
## Lag[2*(p+q)+(p+q)-1][11]   3.17928  1.0000
## Lag[4*(p+q)+(p+q)-1][19]   6.26215  0.9590
## d.o.f=4
## H0 : No serial correlation
## 
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
##                         statistic p-value
## Lag[1]                     0.2584  0.6112
## Lag[2*(p+q)+(p+q)-1][5]    0.6413  0.9339
## Lag[4*(p+q)+(p+q)-1][9]    1.2063  0.9761
## d.o.f=2
## 
## Weighted ARCH LM Tests
## ------------------------------------
##             Statistic Shape Scale P-Value
## ARCH Lag[3]    0.1169 0.500 2.000  0.7324
## ARCH Lag[5]    0.8240 1.440 1.667  0.7858
## ARCH Lag[7]    1.0154 2.315 1.543  0.9110
## 
## Nyblom stability test
## ------------------------------------
## Joint Statistic:  2.6619
## Individual Statistics:              
## mu     0.53254
## ar1    0.06217
## ar2    0.07813
## ma1    0.03444
## ma2    0.05402
## omega  0.13723
## alpha1 0.47822
## beta1  0.25534
## 
## 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           0.2121 0.8321    
## Negative Sign Bias  0.2131 0.8312    
## Positive Sign Bias  0.2699 0.7873    
## Joint Effect        0.5302 0.9122    
## 
## 
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
##   group statistic p-value(g-1)
## 1    20     78.79    3.009e-09
## 2    30     93.96    8.852e-09
## 3    40    101.61    1.744e-07
## 4    50    112.02    7.694e-07
## 
## 
## Elapsed time : 1.320244

coef(garch.fit)

##           mu          ar1          ar2          ma1          ma2 
## -0.004016979 -0.985289871 -0.958289407  0.976372108  0.968386599 
##        omega       alpha1        beta1 
##  0.150206770  0.067362982  0.912420154

####### Diagnóstico #######################

# 1 Time SeriesPlot
# 2 Conditional Standard Deviation Plot
# 3 Series Plot with 2 Conditional SD Superimposed
# 4 Autocorrelation function Plot of Observations
# 5 Autocorrelation function Plot of Squared Observations
# 6 Cross Correlation Plot
# 7 Residuals Plot
# 8 Conditional Standard Deviations Plot
# 9 Standardized Residuals Plot
# 10 ACF Plot of Standardized Residuals
# 11 ACF Plot of Squared Standardized Residuals
# 12 Cross Correlation Plot between $r^2$ and r
# 13 Quantile-Quantile Plot of Standardized Residuals

plot(garch.fit, which = "all")

## 
## please wait...calculating quantiles...

plot(garch.fit, which = 1)

plot(garch.fit, which = 2)

## 
## please wait...calculating quantiles...

plot(garch.fit, which = 3)

plot(garch.fit, which = 4)

plot(garch.fit, which = 5)

plot(garch.fit, which = 6)

plot(garch.fit, which = 7)

plot(garch.fit, which = 8)

####### plot in-sample and forecast 100-steps-ahead ############

# plot in-sample volatility estimates
hist <- ts(r_PETR4_pos2010,
           start=c(2010,1,1),
           frequency = 252)
fitted <- ts(garch.fit@fit$fitted.values,
             start=c(2010,1,1),
             frequency = 252)
ts.plot(r_PETR4_pos2010,fitted,col=c("red","blue"))

##### Obtain fitted volatility series
v1=sigma(garch.fit)
ts.plot(as.numeric(v1))

# Time Series Prediction (unconditional)
previsao <- ugarchforecast(garch.fit,n.ahead = 50)
plot(previsao,which = 1)

#fpm(previsao)

# Sigma Prediction (unconditional)
previsao <- ugarchforecast(garch.fit,n.ahead = 50)
plot(previsao,which = 3)

9.1.1 Modeling PETR4 return - Standard GARCH(1,1) sGARCH

require(rugarch)

garch11.spec1 = ugarchspec(variance.model = list(garchOrder = c(1,1),
                                                 model="sGARCH"),
                           mean.model = list(armaOrder=c(2,2),
                                             include.mean=TRUE))
fit1 = ugarchfit(data = r_PETR4_pos2010*100,
                 spec = garch11.spec1)
show(fit1)

## 
## *---------------------------------*
## *          GARCH Model Fit        *
## *---------------------------------*
## 
## Conditional Variance Dynamics    
## -----------------------------------
## GARCH Model  : sGARCH(1,1)
## Mean Model   : ARFIMA(2,0,2)
## Distribution : norm 
## 
## Optimal Parameters
## ------------------------------------
##         Estimate  Std. Error     t value Pr(>|t|)
## mu     -0.004019    0.046794   -0.085887 0.931556
## ar1    -0.985289    0.018912  -52.097504 0.000000
## ar2    -0.958291    0.007288 -131.485536 0.000000
## ma1     0.976371    0.014278   68.384556 0.000000
## ma2     0.968388    0.002128  455.154466 0.000000
## omega   0.150206    0.037973    3.955657 0.000076
## alpha1  0.067361    0.010622    6.341904 0.000000
## beta1   0.912423    0.014085   64.777460 0.000000
## 
## Robust Standard Errors:
##         Estimate  Std. Error    t value Pr(>|t|)
## mu     -0.004019    0.050306   -0.07989 0.936325
## ar1    -0.985289    0.018063  -54.54621 0.000000
## ar2    -0.958291    0.006828 -140.35482 0.000000
## ma1     0.976371    0.014106   69.21756 0.000000
## ma2     0.968388    0.001307  740.99916 0.000000
## omega   0.150206    0.066024    2.27501 0.022905
## alpha1  0.067361    0.018990    3.54715 0.000389
## beta1   0.912423    0.022874   39.88937 0.000000
## 
## LogLikelihood : -5668.525 
## 
## Information Criteria
## ------------------------------------
##                    
## Akaike       4.6894
## Bayes        4.7085
## Shibata      4.6894
## Hannan-Quinn 4.6964
## 
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
##                          statistic p-value
## Lag[1]                     0.07085  0.7901
## Lag[2*(p+q)+(p+q)-1][11]   3.17929  1.0000
## Lag[4*(p+q)+(p+q)-1][19]   6.26215  0.9590
## d.o.f=4
## H0 : No serial correlation
## 
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
##                         statistic p-value
## Lag[1]                     0.2584  0.6112
## Lag[2*(p+q)+(p+q)-1][5]    0.6413  0.9339
## Lag[4*(p+q)+(p+q)-1][9]    1.2063  0.9761
## d.o.f=2
## 
## Weighted ARCH LM Tests
## ------------------------------------
##             Statistic Shape Scale P-Value
## ARCH Lag[3]    0.1169 0.500 2.000  0.7324
## ARCH Lag[5]    0.8241 1.440 1.667  0.7858
## ARCH Lag[7]    1.0154 2.315 1.543  0.9110
## 
## Nyblom stability test
## ------------------------------------
## Joint Statistic:  2.6618
## Individual Statistics:              
## mu     0.53252
## ar1    0.06218
## ar2    0.07813
## ma1    0.03444
## ma2    0.05402
## omega  0.13727
## alpha1 0.47826
## beta1  0.25537
## 
## 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           0.2121 0.8321    
## Negative Sign Bias  0.2132 0.8312    
## Positive Sign Bias  0.2698 0.7873    
## Joint Effect        0.5302 0.9122    
## 
## 
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
##   group statistic p-value(g-1)
## 1    20     78.79    3.009e-09
## 2    30     93.96    8.852e-09
## 3    40    101.61    1.744e-07
## 4    50    112.02    7.694e-07
## 
## 
## Elapsed time : 1.13735

##### Obtendo serie de volatility ajustada
v1=sigma(fit1)
ts.plot(as.numeric(v1)) 

# Previsao da Serie temporal (incondicional)
previsao.fit1 <- ugarchforecast(fit1,n.ahead = 50)
plot(previsao.fit1, which = 1)

# Sigma Prediction (unconditional)
plot(previsao.fit1, which = 3)

# Acuracia
require(Metrics)
predicted<-ts(fitted(fit1))
actual<-ts(r_PETR4_pos2010*100)
(smape<-smape(actual, predicted))

## [1] 1.789236

(rmse<-rmse(actual, predicted))

## [1] 2.795217

9.1.2 Modeling PETR4 return - ARCH-M model with volatility in mean

require(rugarch)

archm.spec1 = ugarchspec(variance.model = list(model="sGARCH"),
                          mean.model = list(armaOrder=c(2,2),
                          archm=TRUE))
fit.ARCHM = ugarchfit(data = r_PETR4_pos2010*100,
                      spec = archm.spec1)
show(fit.ARCHM)

## 
## *---------------------------------*
## *          GARCH Model Fit        *
## *---------------------------------*
## 
## Conditional Variance Dynamics    
## -----------------------------------
## GARCH Model  : sGARCH(1,1)
## Mean Model   : ARFIMA(2,0,2)
## Distribution : norm 
## 
## Optimal Parameters
## ------------------------------------
##         Estimate  Std. Error    t value Pr(>|t|)
## mu     -0.223266    0.249215   -0.89588 0.370317
## ar1    -0.985050    0.018590  -52.98885 0.000000
## ar2    -0.958114    0.007641 -125.39286 0.000000
## ma1     0.976096    0.014128   69.08948 0.000000
## ma2     0.968240    0.001861  520.26670 0.000000
## archm   0.096585    0.108002    0.89428 0.371171
## omega   0.148369    0.036026    4.11843 0.000038
## alpha1  0.067416    0.010026    6.72382 0.000000
## beta1   0.912657    0.013156   69.37126 0.000000
## 
## Robust Standard Errors:
##         Estimate  Std. Error    t value Pr(>|t|)
## mu     -0.223266    0.379508   -0.58830 0.556327
## ar1    -0.985050    0.017523  -56.21581 0.000000
## ar2    -0.958114    0.007364 -130.10034 0.000000
## ma1     0.976096    0.013747   71.00535 0.000000
## ma2     0.968240    0.001994  485.59167 0.000000
## archm   0.096585    0.162790    0.59331 0.552975
## omega   0.148369    0.060939    2.43471 0.014904
## alpha1  0.067416    0.017523    3.84735 0.000119
## beta1   0.912657    0.020433   44.66523 0.000000
## 
## LogLikelihood : -5667.747 
## 
## Information Criteria
## ------------------------------------
##                    
## Akaike       4.6896
## Bayes        4.7111
## Shibata      4.6896
## Hannan-Quinn 4.6974
## 
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
##                          statistic p-value
## Lag[1]                     0.07173  0.7888
## Lag[2*(p+q)+(p+q)-1][11]   3.18335  1.0000
## Lag[4*(p+q)+(p+q)-1][19]   6.30959  0.9565
## 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][5]    0.6500  0.9323
## Lag[4*(p+q)+(p+q)-1][9]    1.2001  0.9764
## d.o.f=2
## 
## Weighted ARCH LM Tests
## ------------------------------------
##             Statistic Shape Scale P-Value
## ARCH Lag[3]    0.1225 0.500 2.000  0.7263
## ARCH Lag[5]    0.7879 1.440 1.667  0.7968
## ARCH Lag[7]    0.9551 2.315 1.543  0.9208
## 
## Nyblom stability test
## ------------------------------------
## Joint Statistic:  2.5833
## Individual Statistics:              
## mu     0.35822
## ar1    0.06154
## ar2    0.07742
## ma1    0.03441
## ma2    0.05302
## archm  0.25095
## omega  0.15317
## alpha1 0.48685
## beta1  0.26630
## 
## 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           0.2478 0.8043    
## Negative Sign Bias  0.3999 0.6893    
## Positive Sign Bias  0.5247 0.5998    
## Joint Effect        0.5269 0.9129    
## 
## 
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
##   group statistic p-value(g-1)
## 1    20     86.65    1.294e-10
## 2    30    100.13    9.334e-10
## 3    40    111.89    5.802e-09
## 4    50    130.45    2.516e-09
## 
## 
## Elapsed time : 2.779408

##### Obtendo serie de volatility ajustada
v2=sigma(fit.ARCHM)
ts.plot(as.numeric(v2)) 

#Previsao da Serie temporal (incondicional)
previsao.fitARCHM <- ugarchforecast(fit.ARCHM,n.ahead = 50)
plot(previsao.fitARCHM, which = 1)

# Sigma prediction (unconditional)
plot(previsao.fitARCHM, which = 3)

# Acuracia
predicted2<-ts(fitted(fit.ARCHM))
actual<-ts(r_PETR4_pos2010*100)
(smape<-smape(actual, predicted2))

## [1] 1.777007

(rmse<-rmse(actual, predicted2))

## [1] 2.796718

9.1.3 Modeling PETR4 return - Exponential GARCH Model (eGARCH)

require(rugarch)

egarch.spec1 = ugarchspec(variance.model = list(model="eGARCH"),
                         mean.model = list(armaOrder=c(2,2)))
fit.eGARCH = ugarchfit(data = r_PETR4_pos2010*100,
                       spec = egarch.spec1)
show(fit.eGARCH)

## 
## *---------------------------------*
## *          GARCH Model Fit        *
## *---------------------------------*
## 
## Conditional Variance Dynamics    
## -----------------------------------
## GARCH Model  : eGARCH(1,1)
## Mean Model   : ARFIMA(2,0,2)
## Distribution : norm 
## 
## Optimal Parameters
## ------------------------------------
##         Estimate  Std. Error   t value Pr(>|t|)
## mu     -0.038900    0.023925   -1.6259 0.103971
## ar1    -1.565549    0.014215 -110.1304 0.000000
## ar2    -0.814752    0.018522  -43.9890 0.000000
## ma1     1.531517    0.017078   89.6767 0.000000
## ma2     0.770265    0.019807   38.8885 0.000000
## omega   0.045785    0.001775   25.7917 0.000000
## alpha1 -0.019580    0.008831   -2.2171 0.026614
## beta1   0.978611    0.000295 3315.0089 0.000000
## gamma1  0.105257    0.002885   36.4840 0.000000
## 
## Robust Standard Errors:
##         Estimate  Std. Error   t value Pr(>|t|)
## mu     -0.038900    0.014114   -2.7562 0.005848
## ar1    -1.565549    0.007061 -221.7303 0.000000
## ar2    -0.814752    0.009108  -89.4591 0.000000
## ma1     1.531517    0.010536  145.3614 0.000000
## ma2     0.770265    0.011969   64.3525 0.000000
## omega   0.045785    0.004137   11.0659 0.000000
## alpha1 -0.019580    0.013921   -1.4065 0.159567
## beta1   0.978611    0.001093  894.9690 0.000000
## gamma1  0.105257    0.008134   12.9396 0.000000
## 
## LogLikelihood : -5678.681 
## 
## Information Criteria
## ------------------------------------
##                    
## Akaike       4.6986
## Bayes        4.7202
## Shibata      4.6986
## Hannan-Quinn 4.7065
## 
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
##                          statistic p-value
## Lag[1]                      0.8364  0.3604
## Lag[2*(p+q)+(p+q)-1][11]    2.8615  1.0000
## Lag[4*(p+q)+(p+q)-1][19]    6.9755  0.9104
## d.o.f=4
## H0 : No serial correlation
## 
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
##                         statistic p-value
## Lag[1]                     0.2433  0.6218
## Lag[2*(p+q)+(p+q)-1][5]    1.4689  0.7476
## Lag[4*(p+q)+(p+q)-1][9]    2.1531  0.8860
## d.o.f=2
## 
## Weighted ARCH LM Tests
## ------------------------------------
##             Statistic Shape Scale P-Value
## ARCH Lag[3]   0.07238 0.500 2.000  0.7879
## ARCH Lag[5]   1.26723 1.440 1.667  0.6556
## ARCH Lag[7]   1.45763 2.315 1.543  0.8298
## 
## Nyblom stability test
## ------------------------------------
## Joint Statistic:  3.1609
## Individual Statistics:              
## mu     0.77736
## ar1    0.12391
## ar2    0.18947
## ma1    0.12145
## ma2    0.19087
## omega  0.15022
## alpha1 0.05083
## beta1  0.36133
## gamma1 0.10045
## 
## 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           0.5072 0.6121    
## Negative Sign Bias  1.0756 0.2822    
## Positive Sign Bias  0.1293 0.8971    
## Joint Effect        1.1747 0.7591    
## 
## 
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
##   group statistic p-value(g-1)
## 1    20     74.32    1.732e-08
## 2    30     74.80    6.504e-06
## 3    40     87.00    1.607e-05
## 4    50    103.23    9.700e-06
## 
## 
## Elapsed time : 1.930895

plot(fit.eGARCH, which = "all")

## 
## please wait...calculating quantiles...

##### Obtendo serie de volatility ajustada
v3=sigma(fit.eGARCH)
ts.plot(as.numeric(v3)) 

# previsao da serie temporal (incondicional)
previsao.eGARCH <- ugarchforecast(fit.eGARCH,n.ahead = 50)
plot(previsao.eGARCH, which = 1)

# Sigma Prediction (unconditional)
plot(previsao.eGARCH, which = 3)

predicted3<-ts(fitted(fit.eGARCH))
actual<-ts(r_PETR4_pos2010*100)
(smape<-smape(actual, predicted3))

## [1] 1.783881

(rmse<-rmse(actual, predicted3))

## [1] 2.793125

9.1.4 Modeling PETR4 return - GJR-GARCH ou TGARCH Model

require(rugarch)

gjrgarch.spec1 = ugarchspec(variance.model = list(model="gjrGARCH"),
                          mean.model = list(armaOrder=c(2,2)))
fit.gjrGARCH = ugarchfit(data = r_PETR4_pos2010*100,
                         spec = gjrgarch.spec1)
show(fit.gjrGARCH)

## 
## *---------------------------------*
## *          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     -0.011321    0.046875 -2.4152e-01 0.809150
## ar1    -1.141242    0.021288 -5.3609e+01 0.000000
## ar2    -0.169164    0.020825 -8.1233e+00 0.000000
## ma1     1.120346    0.000006  1.9543e+05 0.000000
## ma2     0.149273    0.000944  1.5809e+02 0.000000
## omega   0.162841    0.039829  4.0886e+00 0.000043
## alpha1  0.064034    0.011611  5.5150e+00 0.000000
## beta1   0.908395    0.014072  6.4556e+01 0.000000
## gamma1  0.011529    0.015907  7.2479e-01 0.468581
## 
## Robust Standard Errors:
##         Estimate  Std. Error     t value Pr(>|t|)
## mu     -0.011321    0.049275    -0.22976 0.818278
## ar1    -1.141242    0.020884   -54.64598 0.000000
## ar2    -0.169164    0.020492    -8.25529 0.000000
## ma1     1.120346    0.000022 50603.61000 0.000000
## ma2     0.149273    0.002635    56.64717 0.000000
## omega   0.162841    0.071329     2.28295 0.022433
## alpha1  0.064034    0.015650     4.09168 0.000043
## beta1   0.908395    0.023736    38.27078 0.000000
## gamma1  0.011529    0.028351     0.40665 0.684262
## 
## LogLikelihood : -5673.032 
## 
## Information Criteria
## ------------------------------------
##                    
## Akaike       4.6940
## Bayes        4.7155
## Shibata      4.6939
## Hannan-Quinn 4.7018
## 
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
##                          statistic p-value
## Lag[1]                     0.04268  0.8363
## Lag[2*(p+q)+(p+q)-1][11]   5.56393  0.7604
## Lag[4*(p+q)+(p+q)-1][19]   9.58575  0.5384
## d.o.f=4
## H0 : No serial correlation
## 
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
##                         statistic p-value
## Lag[1]                     0.3452  0.5568
## Lag[2*(p+q)+(p+q)-1][5]    0.8607  0.8903
## Lag[4*(p+q)+(p+q)-1][9]    1.5064  0.9550
## d.o.f=2
## 
## Weighted ARCH LM Tests
## ------------------------------------
##             Statistic Shape Scale P-Value
## ARCH Lag[3]    0.1887 0.500 2.000  0.6640
## ARCH Lag[5]    1.0771 1.440 1.667  0.7102
## ARCH Lag[7]    1.2857 2.315 1.543  0.8631
## 
## Nyblom stability test
## ------------------------------------
## Joint Statistic:  3.2096
## Individual Statistics:             
## mu     0.6058
## ar1    0.1241
## ar2    0.2190
## ma1    0.1223
## ma2    0.2209
## omega  0.1555
## alpha1 0.5081
## beta1  0.2784
## gamma1 0.7345
## 
## 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          0.01205 0.9904    
## Negative Sign Bias 0.14970 0.8810    
## Positive Sign Bias 0.34327 0.7314    
## Joint Effect       0.24681 0.9697    
## 
## 
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
##   group statistic p-value(g-1)
## 1    20     115.4    7.880e-16
## 2    30     129.5    1.158e-14
## 3    40     145.4    3.490e-14
## 4    50     152.3    1.626e-12
## 
## 
## Elapsed time : 2.773414

##### Obtendo serie de volatility ajustada
v4=sigma(fit.gjrGARCH)
ts.plot(as.numeric(v4)) 

# previsao da serie temporal (incondicional)
previsao.gjrGARCH <- ugarchforecast(fit.gjrGARCH,n.ahead = 50)
plot(previsao.gjrGARCH,which=1)

# Sigma Prediction (unconditional)
plot(previsao.gjrGARCH, which = 3)

predicted4<-ts(fitted(fit.gjrGARCH))
actual<-ts(r_PETR4_pos2010*100)
(smape<-smape(actual, predicted4))

## [1] 1.903512

(rmse<-rmse(actual, predicted4))

## [1] 2.797725

9.1.5 Modeling PETR4 return TARCH (1,1) - fGARCH-TGARCH

# install.packages("rugarch")
require(rugarch)

tarch11.spec = ugarchspec(mean.model = list(armaOrder = c(2,2),include.mean=TRUE), 
                          variance.model = list(garchOrder = c(1,1), 
                                                model = "fGARCH",
                                                submodel = "TGARCH"))

tarch.fit = ugarchfit(tarch11.spec, 
                      data = r_PETR4_pos2010*100,
                      fit.control=list(scale=TRUE), 
                      distribution.model = "norm")

## Warning in arima(data, order = c(modelinc[2], 0, modelinc[3]), include.mean
## = modelinc[1], : possible convergence problem: optim gave code = 1

print(tarch.fit)

## 
## *---------------------------------*
## *          GARCH Model Fit        *
## *---------------------------------*
## 
## Conditional Variance Dynamics    
## -----------------------------------
## GARCH Model  : fGARCH(1,1)
## fGARCH Sub-Model : TGARCH
## Mean Model   : ARFIMA(2,0,2)
## Distribution : norm 
## 
## Optimal Parameters
## ------------------------------------
##         Estimate  Std. Error    t value Pr(>|t|)
## mu     -0.118193    0.000102 -1157.5506 0.000000
## ar1     0.028663    0.000144   199.2592 0.000000
## ar2     0.968479    0.000250  3878.9314 0.000000
## ma1    -0.046492    0.000021 -2247.9059 0.000000
## ma2    -0.959282    0.000116 -8256.0921 0.000000
## omega   0.055229    0.020688     2.6695 0.007596
## alpha1  0.054165    0.011493     4.7128 0.000002
## beta1   0.937825    0.016546    56.6806 0.000000
## eta11   0.280039    0.080800     3.4658 0.000529
## 
## Robust Standard Errors:
##         Estimate  Std. Error     t value Pr(>|t|)
## mu     -0.118193    0.000278  -424.84928  0.00000
## ar1     0.028663    0.000406    70.66863  0.00000
## ar2     0.968479    0.000826  1173.01505  0.00000
## ma1    -0.046492    0.000036 -1275.38364  0.00000
## ma2    -0.959282    0.001266  -757.72853  0.00000
## omega   0.055229    0.175839     0.31409  0.75345
## alpha1  0.054165    0.103890     0.52137  0.60211
## beta1   0.937825    0.146527     6.40035  0.00000
## eta11   0.280039    0.219619     1.27511  0.20227
## 
## LogLikelihood : -5669.496 
## 
## Information Criteria
## ------------------------------------
##                    
## Akaike       4.6910
## Bayes        4.7126
## Shibata      4.6910
## Hannan-Quinn 4.6989
## 
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
##                          statistic   p-value
## Lag[1]                      0.0754 7.836e-01
## Lag[2*(p+q)+(p+q)-1][11]    9.3746 6.877e-07
## Lag[4*(p+q)+(p+q)-1][19]   14.7994 3.319e-02
## d.o.f=4
## H0 : No serial correlation
## 
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
##                         statistic p-value
## Lag[1]                     0.1225  0.7264
## Lag[2*(p+q)+(p+q)-1][5]    1.8679  0.6500
## Lag[4*(p+q)+(p+q)-1][9]    3.0612  0.7488
## d.o.f=2
## 
## Weighted ARCH LM Tests
## ------------------------------------
##             Statistic Shape Scale P-Value
## ARCH Lag[3]    0.0478 0.500 2.000  0.8269
## ARCH Lag[5]    2.3523 1.440 1.667  0.3985
## ARCH Lag[7]    2.6710 2.315 1.543  0.5777
## 
## Nyblom stability test
## ------------------------------------
## Joint Statistic:  3.671
## Individual Statistics:              
## mu     0.02335
## ar1    0.02303
## ar2    0.02264
## ma1    0.02347
## ma2    0.02410
## omega  0.12439
## alpha1 0.51277
## beta1  0.21082
## eta11  0.04649
## 
## 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           1.4169 0.1566    
## Negative Sign Bias  0.1704 0.8647    
## Positive Sign Bias  0.8282 0.4076    
## Joint Effect        2.4047 0.4928    
## 
## 
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
##   group statistic p-value(g-1)
## 1    20     86.78    1.227e-10
## 2    30     99.11    1.357e-09
## 3    40    101.11    2.047e-07
## 4    50    119.25    8.662e-08
## 
## 
## Elapsed time : 3.399057

coef(tarch.fit)

##          mu         ar1         ar2         ma1         ma2       omega 
## -0.11819279  0.02866343  0.96847873 -0.04649185 -0.95928229  0.05522862 
##      alpha1       beta1       eta11 
##  0.05416522  0.93782515  0.28003892

####### Diagnóstico #######################
plot(tarch.fit, which = "all")

## 
## please wait...calculating quantiles...

####### plot in-sample and forecast 50-steps-ahead ############

# plot in-sample volatility estimates
hist <- ts(r_PETR4_pos2010,
           start=c(2010,1,1),
           frequency = 252)
fitted <- ts(tarch.fit@fit$fitted.values,
             start=c(2010,1,1),
             frequency = 252)
ts.plot(r_PETR4_pos2010,fitted,col=c("red","blue"))

# Time Series Prediction (unconditional)
previsao.tarch <- ugarchforecast(tarch.fit,n.ahead = 50)
plot(previsao.tarch,which = 1)

#fpm(previsao.tarch, summary = TRUE)

# Sigma Prediction (unconditional)
previsao.tarch <- ugarchforecast(tarch.fit,n.ahead = 50)
plot(previsao.tarch,which = 3)

predicted5<-ts(fitted(tarch.fit))
actual<-ts(r_PETR4_pos2010*100)
(smape<-smape(actual, predicted5))

## [1] 1.77058

(rmse<-rmse(actual, predicted5))

## [1] 2.795627

Referências

FERREIRA, Pedro Costa; SPERANZA, Talitha; COSTA, Jonatha (2018). BETS: Brazilian Economic Time Series. R package version 0.4.9. Disponível em: https://CRAN.R-project.org/package=BETS.

HYNDMAN, Rob. (2018). fpp2: Data for “Forecasting: Principles and Practice” (2nd Edition). R package version 2.3. Disponível em: https://CRAN.R-project.org/package=fpp2.

HYNDMAN, R.J.; ATHANASOPOULOS, G. (2018) Forecasting: principles and practice, 2nd edition, OTexts: Melbourne, Australia. Disponível em: https://otexts.com/fpp2/. Accessed on 12 Set 2019.

PERLIN, M.; RAMOS, H. (2016). GetHFData: A R Package for Downloading and Aggregating High Frequency Trading Data from Bovespa. Brazilian Review of Finance, V. 14, N. 3.