11.12 Modelos ARCH

11.12.1 ARCH

(R. F. Engle 1982) propõe o modelo ARCH (Autoregressive Conditional Heteroscedasticity), que permite modelar a variância condicional de uma série em função dos quadrados dos retornos passados. A variância inconstante ao longo do tempo é chamada heterocedasticidade.

O modelo ARCH(\(s\)) pode ser expresso por

\[\begin{equation} y_t = a_t \tag{11.35} \end{equation}\]

\[\begin{equation} a_t = \varepsilon_t \sqrt{h_t} \tag{11.36} \end{equation}\]

\[\begin{equation} h_t = \alpha_0 + \sum_{i=1}^s \alpha_i y_{t-i}^2 \tag{11.37} \end{equation}\]

\[\begin{equation} \varepsilon_t \sim WN(0,1) \tag{11.38} \end{equation}\]

Demonstra-se que \(a_t \sim WN(0,\sigma^2_a)\), i.e., \(a_t\) tem variância constante tal que

\[\begin{equation} \sigma^2_a = \frac{\alpha_0}{1-\sum_{i=1}^s \alpha_i} \tag{11.39} \end{equation}\]

Portanto \(a_t\) possui estacionariedade fraca ou de segunda ordem. Para mais detalhes recomenda-se (R. F. Engle 1982), (Shephard 1996) e (G. E. P. Box, Jenkins, and Reinsel 2008, 413).

A função tseries::garch() (Trapletti and Hornik 2024) ajusta um modelo de série temporal GARCH\((r,s)\) via máxima verossimilhança.

Exemplo 11.54 Adaptado da documentação de tseries::garch(), simulando e ajustando um modelo ARCH\((2)\) ou GARCH\((0,2)\).

library(tseries)
# número de observações simuladas, 100 primeiras descartadas
n <- 1100
# coeficientes ARCH(2): alfa_0, alfa_1 e alfa_2 
a <- c(0.1, 0.5, 0.2)
# \varepsilon_t, Eq. 11.37
set.seed(42); e <- rnorm(n) 
# vetor para conter a série temporal y simulada (double-precision vector)
y <- double(n)
# Eq. 11.34 e 11.38
set.seed(314); y[1:2] <- rnorm(2, sd = sqrt(a[1]/(1-a[2]-a[3])))
# gera processo ARCH(2)
for(i in 3:n){
  # Eq. 11.36
  ht <- a[1] + a[2]*y[i-1]^2 + a[3]*y[i-2]^2
  # Eqs. 11.34 e 11.35
  y[i] <- e[i]*sqrt(ht)
}
# descartando as 100 primeiras observações
y <- ts(y[101:n])
# ajusta ARCH(2) 
y_arch <- tseries::garch(y, order = c(0,2), trace = FALSE)
# testes de diagnóstico
summary(y_arch)

## 
## Call:
## tseries::garch(x = y, order = c(0, 2), trace = FALSE)
## 
## Model:
## GARCH(0,2)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -3.34285 -0.70636 -0.02587  0.63041  3.41272 
## 
## Coefficient(s):
##     Estimate  Std. Error  t value Pr(>|t|)    
## a0  0.095299    0.009129   10.439   <2e-16 ***
## a1  0.507829    0.060802    8.352   <2e-16 ***
## a2  0.225846    0.045248    4.991    6e-07 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Diagnostic Tests:
##  Jarque Bera Test
## 
## data:  Residuals
## X-squared = 1.5664, df = 2, p-value = 0.457
## 
## 
##  Box-Ljung test
## 
## data:  Squared.Residuals
## X-squared = 0.15934, df = 1, p-value = 0.6898

plot(y_arch, ask = FALSE)

11.12.2 GARCH

(Bollerslev 1986) estendeu o trabalho original de Engle, propondo o modelo GARCH (Generalized Autoregressive Conditional Heteroscedasticity), que permite que a variância condicional não dependa somente dos quadrados dos retornos passados, mas também da própria variância passada. O modelo GARCH\((r,s)\) é definido pelas Eq. (11.35), (11.36) e (11.38), com \(h_t\) dado pela Eq. (11.40).

\[\begin{equation} h_t = \alpha_0 + \sum_{i=1}^s \alpha_i y_{t-i}^2 + \sum_{i=1}^r \beta_i h_{t-i} \tag{11.40} \end{equation}\]

Para esta formulação \(a_t\) também possui variância constante, definida pela Eq. (11.41).

\[\begin{equation} \sigma^2_a = \frac{\alpha_0}{1-\sum_{i=1}^s \alpha_i - \sum_{i=1}^r \beta_i} \tag{11.41} \end{equation}\]

Exemplo 11.55 Adaptado da documentação de tseries::garch(), simulando e ajustando um modelo GARCH(1,1). Lembre-se que “[t]he [a]utocovariances [d]o [n]ot [a]lways [d]ecrease” (Francq and Zakoian 2010, 51).

library(tseries)
# número de observações simuladas, 100 primeiras descartadas
n <- 1100
# coeficientes ARCH(1): alfa_0 e alfa_1
a <- c(0.1, 0.5)
# coeficiente GARCH(s,1): beta_1
b <- c(0.3)
# \varepsilon_t, Eq. 11.37
set.seed(42); e <- rnorm(n)
# vetor para conter a série temporal y simulada (double-precision vector)
y <- double(n)
# vetor para conter a série temporal de ht
h <- double(n) 
# Eqs. 11.34 e 11.40
set.seed(314); y[1] <- rnorm(1, sd = sqrt(a[1]/(1-a[2]-b[1])))
# gera processo GARCH(1,1)
for(i in 2:n){
  # Eq. 11.39
  h[i] <- a[1] + a[2]*y[i-1]^2 + b[1]*h[i-1] 
  # Eqs. 11.34 e 11.35
  y[i] <- e[i]*sqrt(h[i])
}
# descartando as 100 primeiras observações
y <- ts(y[101:n])
# ajusta GARCH(1,1)
fit_garch_tseries <- tseries::garch(y, order = c(1,1), trace = FALSE)
# testes de diagnóstico
summary(fit_garch_tseries)

## 
## Call:
## tseries::garch(x = y, order = c(1, 1), trace = FALSE)
## 
## Model:
## GARCH(1,1)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -3.01545 -0.70722 -0.02598  0.63435  3.22918 
## 
## Coefficient(s):
##     Estimate  Std. Error  t value Pr(>|t|)    
## a0   0.07349     0.01238    5.935 2.94e-09 ***
## a1   0.47695     0.05420    8.801  < 2e-16 ***
## b1   0.39325     0.04371    8.997  < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Diagnostic Tests:
##  Jarque Bera Test
## 
## data:  Residuals
## X-squared = 0.6907, df = 2, p-value = 0.708
## 
## 
##  Box-Ljung test
## 
## data:  Squared.Residuals
## X-squared = 0.77497, df = 1, p-value = 0.3787

plot(fit_garch_tseries, ask = FALSE)

A função bayesforecast::garch() (Alonzo and Cruz 2020) implementa um modelo GARCH\((s,k,h)\) baseado em (Ardia and Hoogerheide 2010) e (Fonseca et al. 2019). As três componentes \((s,k,h) \equiv (s,r,h)\) são a ordem do ARCH (\(s\)), a ordem do GARCH (\(k \equiv r\)) e a ordem do MGARCH (\(h\)). Para mais detalhes veja (Engle Robert and Kevin 2001) e (R. Engle 2002).

library(tseries)
library(bayesforecast)
# número de observações simuladas, 100 primeiras descartadas
n <- 1100
# coeficientes ARCH(1): alfa_0 e alfa_1
a <- c(0.1, 0.5)
# coeficiente GARCH(s,1): beta_1
b <- c(0.3)
# \varepsilon_t, Eq. 11.37
set.seed(42); e <- rnorm(n)
# vetor para conter a série temporal y simulada (double-precision vector)
y <- double(n)
# vetor para conter a série temporal de ht
h <- double(n)
# Eqs. 11.34 e 11.40
set.seed(314); y[1] <- rnorm(1, sd = sqrt(a[1]/(1-a[2]-b[1])))
# gera processo GARCH(1,1)
for(i in 2:n){
  # Eq. 11.39
  h[i] <- a[1] + a[2]*y[i-1]^2 + b[1]*h[i-1]
  # Eqs. 11.34 e 11.35
  y[i] <- e[i]*sqrt(h[i])
}
# descartando as 100 primeiras observações
y <- ts(y[101:n])
# ajusta GARCH(1,1)
fit_garch_bayes <- bayesforecast::garch(y, order = c(1,1,0))
fit_garch_varstan <- bayesforecast::varstan(fit_garch_bayes, iter = 500, chains = 1)

## 
## SAMPLING FOR MODEL 'tgarch' NOW (CHAIN 1).
## Chain 1: 
## Chain 1: Gradient evaluation took 0.002672 seconds
## Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 26.72 seconds.
## Chain 1: Adjust your expectations accordingly!
## Chain 1: 
## Chain 1: 
## Chain 1: Iteration:   1 / 500 [  0%]  (Warmup)
## Chain 1: Iteration:  50 / 500 [ 10%]  (Warmup)
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## Chain 1: Iteration: 500 / 500 [100%]  (Sampling)
## Chain 1: 
## Chain 1:  Elapsed Time: 0.618 seconds (Warm-up)
## Chain 1:                0.408 seconds (Sampling)
## Chain 1:                1.026 seconds (Total)
## Chain 1:

# análise de resíduos
check_residuals(fit_garch_varstan)

# projetando 48 passos
autoplot(bayesforecast::forecast(fit_garch_varstan, h = 48))

Exercício 11.32 Considere o modelo GARCH\((r,s)\) e GARCH\((s,r,h)\).

No Exemplo 11.54 verifique a equivalência entre os modelos ARCH\((2)\), GARCH\((0,2)\) de tseries e GARCH\((2,0,0)\) de bayesforecast.
Veja https://asael697.r-universe.dev/bayesforecast.
Veja https://cran.r-universe.dev/bayesforecast/doc/manual.html.
Veja https://cran.r-project.org/web/packages/bayesforecast/readme/README.html.

11.12.3 IGARCH

R. F. Engle and Bollerslev (1986) propõem o modelo IGARCH, “uma nova classe de modelos definidos para serem integrados em variância” a partir da restrição \(\sum_{i=1}^{s} \alpha_i + \sum_{i=1}^{r} \beta_i = 1\) na Eq. (11.40).

11.12.4 EGARCH

Nelson (1991) propõe o modelo EGARCH com o objetivo de “acomodar a relação assimétrica entre os retornos das ações e as mudanças na volatilidade”. Seguindo a notação de Vrontos, Dellaportas, and Politis (2000),

\[\begin{equation} \log h_t = \alpha_0 + \sum_{i=1}^s \theta_i a_{t-i} + \gamma_i (|a_{t-i}|-E|a_{t-i}|) + \sum_{i=1}^r \beta_i h_{t-i} \tag{11.42} \end{equation}\]

\[\begin{equation} E|a_{t-i}| = \frac{\Gamma(\frac{2}{\nu})}{[\Gamma(\frac{1}{\nu}) \Gamma(\frac{3}{\nu})]^{1/2}} \tag{11.43} \end{equation}\]

11.12.5 ARMA-GARCH

Exemplo 11.56 Considere a série temporal r unlist(tsdl::meta_tsdl[20,2][[1]][[1]]) apresentada por r tsdl::meta_tsdl[20,1], disponível em tsdl::tsdl[[20]]²².

library(forecast)
library(bayesforecast)
# ajustando um modelo ARIMA automático
y <- tsdl::tsdl[[20]]
# ARIMA(2,1,2)
forecast::auto.arima(y)

## Series: y 
## ARIMA(2,1,2) 
## 
## Coefficients:
##          ar1      ar2      ma1     ma2
##       1.3467  -0.3963  -1.7710  0.8103
## s.e.  0.0303   0.0287   0.0205  0.0194
## 
## sigma^2 = 243.8:  log likelihood = -11745.5
## AIC=23500.99   AICc=23501.01   BIC=23530.71

# ajustando ARMA(2,2)-GARCH(1,1) para a série com uma diferença via bayesforecast
fit_garch_bayes <- bayesforecast::garch(y, order = c(1,1,0), arma = c(2,2))
fit_garch_varstan <- bayesforecast::varstan(fit_garch_bayes, iter = 500, chains = 1)

## 
## SAMPLING FOR MODEL 'tgarch' NOW (CHAIN 1).
## Chain 1: 
## Chain 1: Gradient evaluation took 0.005778 seconds
## Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 57.78 seconds.
## Chain 1: Adjust your expectations accordingly!
## Chain 1: 
## Chain 1: 
## Chain 1: Iteration:   1 / 500 [  0%]  (Warmup)
## Chain 1: Iteration:  50 / 500 [ 10%]  (Warmup)
## Chain 1: Iteration: 100 / 500 [ 20%]  (Warmup)
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## Chain 1: Iteration: 500 / 500 [100%]  (Sampling)
## Chain 1: 
## Chain 1:  Elapsed Time: 22.069 seconds (Warm-up)
## Chain 1:                34.254 seconds (Sampling)
## Chain 1:                56.323 seconds (Total)
## Chain 1:

# análise de resíduos
check_residuals(fit_garch_varstan)

# projetando 48 passos
autoplot(bayesforecast::forecast(fit_garch_varstan, h = 48))

# # tomando uma diferença da série
# ydiff <- diff(y)
# fit_garch_bayes_diff <- bayesforecast::garch(ydiff, order = c(1,1,0), arma = c(2,2)) # ajusta ARMA(2,2)-GARCH(1,1)
# fit_garch_varstan_diff <- bayesforecast::varstan(fit_garch_bayes_diff, iter = 500, chains = 1)
# check_residuals(fit_garch_varstan_diff) # análise de resíduos
# bf <- bayesforecast::forecast(fit_garch_varstan_diff, h = 48)
#  # desfazendo as diferenças
# bf$x <- diffinv(bf$x, xi = y[1])
# bf$mean <- diffinv(bf$mean, xi = y[length(y)])
# autoplot(bf) # projetando 48 passos

A função rugarch::ugarchspec() (Galanos 2023) permite ajustar um modelo ARFIMA-GARCH, com variantes do modelo GARCH e variáveis externas (veja documentação).

# libs
library(forecast)
library(rugarch)
# ajustando um modelo ARIMA automático
y <- tsdl::tsdl[[37]]
# ARIMA(1,1,1)
(fit_arima <- forecast::auto.arima(y))

## Series: y 
## ARIMA(1,1,1) 
## 
## Coefficients:
##           ar1     ma1
##       -0.5322  0.9476
## s.e.   0.1389  0.0657
## 
## sigma^2 = 95.43:  log likelihood = -199.28
## AIC=404.57   AICc=405.05   BIC=410.54

# avaliando a série ao quadrado e os resíduos ao quadrado do ARIMA(1,1,1)
par(mfrow=c(2,2))
acf(y^2)
pacf(y^2)
acf(fit_arima$residuals^2)
pacf(fit_arima$residuals^2)

# tomando uma diferença da série
ydiff <- diff(y)
# ajustando ARMA-GARCH para a série com uma diferença via rugarch
spec <- ugarchspec(
  mean.model = list(armaOrder = c(2,2), arfima = FALSE),
  variance.model = list(garchOrder = c(0,1))
)
fit_garch <- ugarchfit(spec = spec, data = y, solver.control = list(trace=0))
fit_garch

## 
## *---------------------------------*
## *          GARCH Model Fit        *
## *---------------------------------*
## 
## Conditional Variance Dynamics    
## -----------------------------------
## GARCH Model  : sGARCH(0,1)
## Mean Model   : ARFIMA(2,0,2)
## Distribution : norm 
## 
## Optimal Parameters
## ------------------------------------
##        Estimate  Std. Error   t value Pr(>|t|)
## mu     9.766934    0.729335  13.39155  0.00000
## ar1    0.973898    0.396209   2.45804  0.01397
## ar2    0.315425    0.525091   0.60071  0.54804
## ma1   -0.033162    0.048408  -0.68504  0.49332
## ma2   -1.091585    0.082780 -13.18655  0.00000
## omega  0.744145    5.849780   0.12721  0.89877
## beta1  0.999000    0.081919  12.19498  0.00000
## 
## Robust Standard Errors:
##        Estimate  Std. Error   t value Pr(>|t|)
## mu     9.766934     1.68699  5.789559 0.000000
## ar1    0.973898     1.37181  0.709935 0.477744
## ar2    0.315425     1.83388  0.171999 0.863438
## ma1   -0.033162     0.15540 -0.213397 0.831018
## ma2   -1.091585     0.29944 -3.645414 0.000267
## omega  0.744145    22.98703  0.032372 0.974175
## beta1  0.999000     0.32620  3.062507 0.002195
## 
## LogLikelihood : -190.7872 
## 
## Information Criteria
## ------------------------------------
##                    
## Akaike       7.1923
## Bayes        7.4477
## Shibata      7.1645
## Hannan-Quinn 7.2911
## 
## Weighted Ljung-Box Test on Standardized Residuals
## ------------------------------------
##                          statistic p-value
## Lag[1]                      0.5043  0.4776
## Lag[2*(p+q)+(p+q)-1][11]    4.4612  0.9972
## Lag[4*(p+q)+(p+q)-1][19]    8.3802  0.7387
## d.o.f=4
## H0 : No serial correlation
## 
## Weighted Ljung-Box Test on Standardized Squared Residuals
## ------------------------------------
##                         statistic p-value
## Lag[1]                      6.099 0.01352
## Lag[2*(p+q)+(p+q)-1][2]     7.204 0.01052
## Lag[4*(p+q)+(p+q)-1][5]     9.123 0.01541
## d.o.f=1
## 
## Weighted ARCH LM Tests
## ------------------------------------
##             Statistic Shape Scale P-Value
## ARCH Lag[2]     2.055 0.500 2.000  0.1517
## ARCH Lag[4]     3.418 1.397 1.611  0.2099
## ARCH Lag[6]     3.773 2.222 1.500  0.3403
## 
## Nyblom stability test
## ------------------------------------
## Joint Statistic:  13.3381
## Individual Statistics:             
## mu    0.03699
## ar1   0.03213
## ar2   0.03141
## ma1   0.03472
## ma2   0.06054
## omega 0.58637
## beta1 0.46954
## 
## 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           0.06206 0.950761    
## Negative Sign Bias  3.02333 0.003938 ***
## Positive Sign Bias  1.64253 0.106754    
## Joint Effect       14.34129 0.002476 ***
## 
## 
## Adjusted Pearson Goodness-of-Fit Test:
## ------------------------------------
##   group statistic p-value(g-1)
## 1    20     17.36       0.5652
## 2    30     25.18       0.6688
## 3    40     30.09       0.8463
## 4    50     44.09       0.6720
## 
## 
## Elapsed time : 0.101428

# previsão com o modelo
fcast <- ugarchforecast(fit_garch)
# plot(fcast, ask = FALSE)
# desfazendo a diferença
# fcast2 <- fcast
# fcast2@model$modeldata$data <- as.matrix(diffinv(fcast2@model$modeldata$data, xi = y[1]))
# fcast2@model$modeldata$data <- y[-1]
# fcast2@forecast$seriesFor <- as.matrix(diffinv(fcast2@forecast$seriesFor)[-1])
# plot(fcast2)

Exercício 11.33 Veja

References

Alonzo, Izhar Asael, and Cristian Cruz. 2020. “varstan: An R Package for Bayesian Time Series Models with Stan.” ARXIV Preprint.

Ardia, David, and Lennart F Hoogerheide. 2010. “Bayesian Estimation of the Garch (1, 1) Model with Student-t Innovations.” The R Journal 2 (2): 41–47. https://journal.r-project.org/archive/2010/RJ-2010-014/RJ-2010-014.pdf.

Bollerslev, Tim. 1986. “Generalized Autoregressive Conditional Heteroskedasticity.” Journal of Econometrics 31 (3): 307–27. https://citeseerx.ist.psu.edu/document?repid=rep1&type=pdf&doi=7da8bfa5295375c1141d797e80065a599153c19d.

Box, George E. P., Gwilym M. Jenkins, and Gregory C. Reinsel. 2008. Time Series Analysis: Forecasting and Control. 4th ed. John Wiley & Sons.

Engle, Robert. 2002. “Dynamic Conditional Correlation: A Simple Class of Multivariate Generalized Autoregressive Conditional Heteroskedasticity Models.” Journal of Business & Economic Statistics 20 (3): 339–50. https://www.bayes.city.ac.uk/__data/assets/pdf_file/0003/78960/Week7Engle_2002.pdf.

Engle, Robert F. 1982. “Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation.” Econometrica: Journal of the Econometric Society, 987–1007. http://www.econ.uiuc.edu/~econ508/Papers/engle82.pdf.

Engle, Robert F, and Tim Bollerslev. 1986. “Modelling the Persistence of Conditional Variances.” Econometric Reviews 5 (1): 1–50. https://doi.org/10.1080/07474938608800095.

Engle Robert, F, and Sheppard Kevin. 2001. “Theoretical and Empirical Properties of Dynamic Conditional Correlation Multivariate GARCH.” UCSD Working Paper NO. https://www.nber.org/system/files/working_papers/w8554/w8554.pdf.

Fonseca, T. C. O., V. S. Cerqueira, H. S. Migon, and C. A. C. Torres. 2019. “The Effects of Degrees of Freedom Estimation in the Asymmetric GARCH Model with Student-t Innovations.” https://arxiv.org/abs/1910.01398.

Francq, Christian, and Jean-Michel Zakoian. 2010. GARCH Models: Structure, Statistical Inference and Financial Applications. John Wiley & Sons.

Galanos, Alexios. 2023. Rugarch: Univariate GARCH Models. https://cran.r-project.org/package=rugarch.

Nelson, Daniel B. 1991. “Conditional Heteroskedasticity in Asset Returns: A New Approach.” Econometrica: Journal of the Econometric Society, 347–70. https://www.jstor.org/stable/pdf/2938260.pdf.

Shephard, Neil. 1996. “Statistical Aspects of ARCH and Stochastic Volatility.” In Time Series Models: In Econometrics, Finance and Other Fields, 1–68. Springer-Science+Business Media, B.V. https://citeseerx.ist.psu.edu/document?repid=rep1&type=pdf&doi=59c8b90f39410447614bcebbfedd985c4bcc59b2.

Trapletti, Adrian, and Kurt Hornik. 2024. Tseries: Time Series Analysis and Computational Finance. https://CRAN.R-project.org/package=tseries.

Vrontos, Ioannis D, Petros Dellaportas, and Dimitris N Politis. 2000. “Full Bayesian Inference for GARCH and EGARCH Models.” Journal of Business & Economic Statistics 18 (2): 187–98. https://www.tandfonline.com/doi/pdf/10.1080/07350015.2000.10524861?casa_token=gESFQI83c2sAAAAA:s-MSp0-EZZzxt_nhFIMZe4rmzCbjRuf380wfFAdo9Gt-XpmLhjNQGJbtkH3Ph9CuMS4BkJOmVqZTFg.

https://filipezabala.com/data/tsdl/tsdl_20.csv ↩︎