start_time <- Sys.time() # Execution timing document counter

Bayesian Estimation of the GARCH(1,1) Model with Student-t Innovations


Reprodução do modelo original do artigo de Ardia e Hoogerheide (2010). Neste post apresentaremos o uso do pacote bayesGARCH que fornece funções para a estimação Bayesiana do modelo GARCH(1,1) parcimonioso e efetivo com distribuição assimétrica de erros Student-t. O procedimento de estimativa é totalmente automático e, portanto, evita a tediosa tarefa de tunelamento de um algoritmo de amostragem MCMC. O uso do pacote é mostrado em uma aplicação empírica para log-retornos da commoditie milho e soja.


Introdução

Pesquisas sobre a mudança de volatilidade usando séries temporais modelos estão presentes desde o artigo pioneiro de Engle (1982). A partir daí, ARCH (modelos autoregressivos de heterocedasticidade condicional) e GARCH (ARCH Generalizado) cresceram rapidamente em uma rica família de modelos empíricos para previsão de volatilidade durante os anos 80. Esses modelos são difundidos e ferramentas essenciais em econometria financeira.

No modelo GARCH(\(p, q\)) introduzido por Bollerslev (1986), a variância condicional no tempo \(t\) do retorno logarítmico de \(y_t\) (de um ativo financeiro ou índice financeiro), denotado por \(h_t\), é postulado como uma função linear dos quadrados de \(q\) passados log-retornos e variâncias condicionais passadas \(p\). Mais precisamente:

\[ h_t = \alpha_0 + \sum_{i=1}^{q}\alpha_i y^{2}_{t-1} + \sum_{j=1}^{p}\beta_j h_{t-j}, \]

onde os parâmetros satisfazem as restrições: \(\alpha_i \geq 0\,\,(i=0,\cdots,q)\) e \(\beta_j \geq 0\,\,(j=1,\cdots,p)\) para garantir uma variância condicional positiva. Na maioria das aplicações empíricas verifica-se que a simples especificação \(p = q = 1\) é capaz de reproduzir a dinâmica da volatilidade dos dados financeiros. Isso tem levado a Modelo GARCH(1,1) para se tornar o modelo de trabalho (workhorese model) por acadêmicos e profissionais. Dado um modelo especificação para \(h_t\), os log-retornos são então modelados como \(y+t = \epsilon_t h_{t}^{1/2}\), onde \(\epsilon_t\) são perturbações i.d.d. Escolhas comuns para \(\epsilon_t\) são distribuições do tipo Normal ou Student-\(t\). A distribuição Student-\(t\) é particularmente útil uma vez que pode fornecer o excesso de curtose na distribuição condicional que é frequentemente encontrada em processos de séries temporais financeiras (ao contrário de modelos com distribuições normais).

Até recentemente, os modelos GARCH eram principalmente estimados usando a técnica clássica da máxima verossimilhança. Vários pacotes do R fornecem funções para sua estimativa; ver, por exemplo fGarch (Wuertz e Chalabi, 2009), rgarch (Ghalanos, 2010) e tseries (Trapletti and Hornik, 2009). A abordagem Bayesiana oferece uma alternativa atraente que permite obter resultados em pequenas amostras, estimativa robusta, discriminação de modelo, combinação de modelo e declarações probabilísticas em funções (possivelmente não lineares) dos parâmetros do modelo.

O pacote bayesGARCH (Ardia, 2007) implementa o procedimento de estimação Bayesiana descrita em Ardia (2008, capítulo 5) para o modelo GARCH(1,1) com distribuição de Student\(-t\). A abordagem, baseada sobre o trabalho de Nakatsuma (1998), consiste em uma Algoritmo Metropolis-Hastings (MH) onde as distribuições propostas são construídas a partir de processos ARMA nas observações ao quadrado. Esse A metodologia evita a tarefa demorada e difícil, especialmente para não especialistas, de escolher e ajustar um algoritmo de amostragem. O programa é escrito em R com algumas sub-rotinas implementadas em C para acelerar o procedimento de simulação. A validade do algoritmo, bem como a exatidão do código de computador foi verificado pelo método de Geweke (2004).

Modelagem, prioris e esquema MCMC

Um modelo GARCH(1,1) com distribuição assimétrica Student\(-t\) para os log-returnos \(\{y_t\}\) podem ser escritos via aumento de dados (ver Geweke, 1993) como

\[ y_t = \epsilon_t (\frac{v-2}{v}\omega_t h_t )^{1/2}\quad t = 1,\cdots,T \] \[ \epsilon_t \sim idd\,\,N(0,1) \]

\[ \omega_t \sim idd\,\,IG(\frac{v}{2},\frac{v}{2}) \]

\[ h_t = \alpha_0 +\alpha_1 y^{2}_{t-1}+\beta h_{t-1}, \] onde \(\alpha_0 >0, \alpha_1,\beta_1\geq 0\) e \(v>2\) seguem \(N(0,1)\) denota a distribuição normal padrão.\(IG\) a gama inversa. As restrições nos graus de liberdade dos parâmetros \(v\) garante que a variância condicional seja finita e as restrições nos parâmetros do modelo GARCH \(\alpha_0,\alpha_1\) e \(\beta\) garantem sua positividade. Ressaltamos o fato de que apenas restrições de positividade são implementadas no algoritmo Metropolis-Hastings; não são impostas condições de estacionaridade no procedimento de simulação.

Para escrevermos a função de verossimilhança, definimos o vetor \(y = (y_1,\cdots,y_T)', \omega = (\omega_1,\cdots,\omega_T)'\) e \(\alpha = (\alpha_0,\alpha_1)'\). Reagrupamos os parâmetros do modelo no vetor \(\psi = (\alpha,\beta,v)\). Em seguida, ao definir a matriz diagonal \(T\times T\)

\[ \sum = \sum(\psi,\omega)=diag(\{\omega_t \frac{v-2}{v}h_t(\alpha,\beta)\}^{T}_{t=1}) \]

onde \(h_t(\alpha,\beta)= \alpha_0+\alpha_1y^{2}_{t-1}+\beta h_{t-1}(\alpha,\beta)\) podemos expressar a verossimilhança de (\(\psi,\omega\)) como:

\[ L(\psi, \omega|y) \propto (det\sum)^{-1/2}exp[-\frac{1}{2}y'\sum^{-1}y] \] A abordagem bayesiana considera (\(\psi, \omega\)) como variáveis aleatórias que são caracterizadas pelas densidades à priori denotada por \(p(\psi,\omega)\). A prioiri é especificada com a ajuda de parâmetros chamados hiperparâmetros que são inicialmente considerados conhecidos e constantes. Além disso, dependendo da informação prévia do pesquisador, esta densidade pode ser mais ou menos informativa. Então, se acoplando a função de verossimilhança dos parâmetros do modelo com a densidade anterior, podemos transformar a densidade de probabilidade usando a regra de Bayes para obter a densidade à posteriori \(p(\psi,\omega|y)\) como:

\[ p(\psi, \omega|y) = \frac{L(\psi,\omega|y)p(\psi,\omega)}{\int L(\psi,\omega|y)p(\psi, \omega)d\psi d \omega} \] Esta posteriori é uma descrição quantitativa e probabilística do conhecimento sobre os parâmetros do modelo após observar os dados. Para uma excelente introdução à econometria bayesiana, remetemos o leitor para Koop (2003).

Usamos prioris truncadas normais nos parâmetros GARCH \(\alpha\) e \(\beta\)

\[ p(\alpha)\propto\phi_{N_2}(\alpha|\mu_\alpha,\Sigma_\alpha) 1 \{\alpha\in R_+^2\} \] \[ p(\beta)\propto\phi_{N_1}(\beta|\mu_\beta,\Sigma_\beta) 1 \{\beta\in R_+\} \] onde:

\(\mu_\bullet\) e \(\sum_\bullet\) são os hiperparâmetros;

\(1\{\bullet\}\) é a função indicadora;

\(\phi_{N_d}\) é a densidade normal \(n-\)dimensional;

A distribuição à priori do vetor condicional \(\lambda=(\lambda_1,...\lambda_T)'\) em \(v\) é encontrado observando que os componentes \(\lambda_t\) são independentes e identicamente distribuídos da gama invertida, que produz:

\[ p(\lambda|v)=\left(\frac{v}{2} \right)^{\frac{Tv}{2}}\left[ \Gamma\left(\frac{v}{2}\right)\right]^{-T}\left(\prod_{t=1}^T\lambda_t\right)^{-\frac{v}{2}-1}exp\left[-\frac{1}{2}\sum_{t=1}^T\frac{v}{\lambda_t}\right] \]

além disso, a distribuição a priori nos parâmetros dos graus de liberdade é uma exponencial traduzida com parâmetros \(\lambda^*>0\) e \(\delta\geq2\).

\[ p(v)=\lambda^*exp[-\lambda^*(v-\delta)]1\{v>\delta\} \] Seguimos Deschamps (2006) na escolha prévia na distribuição dos parâmetros de graus de liberdade. A distribuição é uma exponencial traduzida com parâmetros \(\lambda > 0\) e \(\delta \geq 2\)

\[ p(v)=\lambda^*exp[-\lambda^*(v-\delta)]1\{v>\delta\} \]

Para grandes valores de \(\lambda\), a distribuição de massa da priori é concentrada na vizinhança de \(\delta\) e uma restrição sobre os graus de liberdade pode ser imposta nesta forma. A normalidade dos erros é assumida quando \(\delta\) é escolhido grande. Como aponta Deschamps (2006), essa densidade prévia é útil por dois motivos. Primeiro, é potencialmente importante, por razões numéricas, limitar o parâmetro de graus de liberdade longe de dois para evitar a explosão da variância condicional. Em segundo lugar, podemos aproximar a normalidade dos erros enquanto mantemos uma priori que pode melhorar a convergência do amostrador.

A distribuição a priori conjunta é então formada assumindo a independência a priori entre os parâmetros \(p(\psi,\omega)=p(\alpha)p(\beta)p(\omega|v)p(v)\). A natureza recursiva da variância na equação GARCH(1,1) implica que a à posteriori conjunta e as densidades condicionais não podem ser expressas neste formato. Não existe nenhuma priori (conjugada) que possa remediar esta propriedade. Portanto, não podemos usar o amostrador de Gibbs simples e precisamos contar com uma estratégia de simulação de Monte Carlo Markov Chain (MCMC) mais elaborada para aproximar a densidade à posteriori.

A ideia de amostragem MCMC foi introduzida pela primeira vez por Metropolis et al. (1953) e foi posteriormente generalizado por Hastings (1970). A estratégia de amostragem baseia-se na construção de uma cadeia de Markov com realizações \((\psi^{[0]},\omega^{[0]}),\cdots,(\psi^{[j]},\omega^{[j]}),\cdots,\) no espaço de parâmetros. Em condições de regularidade apropriadas, resultados assintóticos garantem que como \(j\) tende a infinito (\(\psi^{[j]},\omega^{[j]}\)) tende em distribuição para uma variável aleatória cuja densidade é \(p(\psi,v | y)\). Assim, após descartar um burn-in dos primeiros sorteios, os valores realizados da cadeia pode ser usada para fazer inferência sobre a posteriori conjunta.

O amostrador MCMC implementado no pacote bayesGARCH é baseado na abordagem de Ardia (2008, capítulo 5), inspirado no trabalho anterior de Nakatsuma (1998). O algoritmo consiste em um algoritmo Metropolis-Hastings onde os parâmetros GARCH são atualizados por blocos (um bloco para \(\alpha\) e um bloco para \(\beta\)) enquanto o parâmetro de graus de liberdade é amostrado usando uma técnica de rejeição otimizada de uma densidade de fonte exponencial traduzida. Esta metodologia tem a vantagem de ser totalmente automática. Além disso, em nossa experiência, o algoritmo explora a domínio da posteriori conjunta de forma eficiente em comparação com abordagens naive de Metropolis-Hastings ou do amostrador Griddy-Gibb de Ritter and Tanner (1992).

Aplicação em retornos de preços de commodities

Utilizaremos o método de estimação bayesiana para dados diários de dois ativos negociados na CBOT, a saber: (https://finance.yahoo.com/commodities/)

  • Milho (em US$/bushel) à futuro (ticker ZC=F)

  • Soja (em US$/bushel) à futuro (ticker ZS=F)

Utilizamos as séries temporais com início em 01-01-2010 até a data mais recente, ou seja, o último dia de negociação disponível (comando Sys.Date() = 2022-04-26)

library(quantmod)

Milho <- getSymbols("ZC=F", auto.assign = FALSE,
                    from = "2010-01-01", end = Sys.Date())

Soja  <- getSymbols("ZS=F", auto.assign = FALSE,
                    from = "2010-01-01", end = Sys.Date()) 

library(fpp3)
library(ggplot2)

fech_Milho <- autoplot(Cl(Milho)) + xlab("") +
  theme(axis.title.y = element_text(size = 7, angle = 90)) +
  theme(plot.title = element_text(size = 9, face = "bold")) +
  ylab("Fechamento (CORN futures)")

ret_Milho <- autoplot(diff(log(Cl(Milho)))) + xlab("") +
  theme(axis.title.y = element_text(size = 7, angle = 90)) +
  theme(plot.title = element_text(size = 9, face = "bold")) +
  ylab("log retornos do fechamento (CORN futures)")

hist_Milho <- ggplot(Milho, aes(x = diff(log(Cl(Milho))) )) + 
 geom_histogram(aes(y = ..density..), colour = "black", fill = "white")+
 geom_density(alpha = .2, fill = "#FF6666") + xlab("") +
  theme(axis.title.y = element_text(size = 7, angle = 90)) +
  theme(plot.title = element_text(size = 7, face = "bold")) +
  ylab("log retornos (CORN futures)")
  
fech_Soja <- autoplot(Cl(Soja)) + xlab("") +
  theme(axis.title.y = element_text(size = 7, angle = 90)) +
  theme(plot.title = element_text(size = 9, face = "bold")) +
  ylab("Fechamento (Soybean futures)")

ret_Soja <- autoplot(diff(log(Cl(Soja)))) + xlab("") +
  theme(axis.title.y = element_text(size = 7, angle = 90)) +
  theme(plot.title = element_text(size = 9, face = "bold")) +
  ylab("log retornos do fechamento (Soybean futures)")

hist_Soja <- ggplot(Soja, aes(x = diff(log(Cl(Soja))) )) + 
 geom_histogram(aes(y = ..density..), colour = "black", fill = "white")+
 geom_density(alpha = .2, fill = "#FF6666") + xlab("") +
  theme(axis.title.y = element_text(size = 7, angle = 90)) +
  theme(plot.title = element_text(size = 7, face = "bold")) +
  ylab("log retornos (Soybean futures)")

library(patchwork)

fech_Milho + ret_Milho + hist_Milho

fech_Soja + ret_Soja + hist_Soja

Criamos primeiramente dois dataframes somente com os vetores de cada commoditie:

retornos_Milho <- diff(log(Cl(Milho))) %>%
  as.data.frame() %>%
  drop_na() %>%
  rename("log_ret_milho" = `ZC=F.Close`) %>% 
  glimpse()
Rows: 3,089
Columns: 1
$ log_ret_milho <dbl> 0.0005971932, 0.0071386383, -0.0101281771, 0.0130876348,~
retornos_Milho <- retornos_Milho[1:nrow(retornos_Milho),] 
retornos_Milho <- 100*as.vector(retornos_Milho)

retornos_Soja <- diff(log(Cl(Soja))) %>%
  as.data.frame() %>%
  drop_na() %>%
  rename("log_ret_soja" = `ZS=F.Close`) %>% 
  glimpse()
Rows: 3,092
Columns: 1
$ log_ret_soja <dbl> 0.0026168684, -0.0016644874, -0.0316719329, -0.0046780831~
retornos_Soja<- retornos_Soja[1:nrow(retornos_Soja),] 
retornos_Soja <- 100*as.vector(retornos_Soja)

Então rodamos o modelo GARCH(1,1) bayesiano via Monte Carlo Markov Chain. Como distribuição à priori para a estimativa Bayesiana, tomamos os valores padrão do pacote bayesGARCH, que são prioris difusas. Geramos duas cadeias para 20000 passos cada uma definindo os valores dos parâmetros de controle n.chain = 2 e l.chain = 20000.

library(bayesGARCH)

set.seed(1234)

MCMC_Milho <- bayesGARCH(retornos_Milho, control = list(l.chain = 20000, n.chain = 2))
chain:  1  iteration:  10  parameters:  0.2834 0.1499 0.7641 70.024 
chain:  1  iteration:  20  parameters:  0.2349 0.1496 0.7746 62.0036 
chain:  1  iteration:  30  parameters:  0.2218 0.1341 0.7833 45.8531 
chain:  1  iteration:  40  parameters:  0.2572 0.1308 0.7777 36.6532 
chain:  1  iteration:  50  parameters:  0.2091 0.1325 0.8016 26.041 
chain:  1  iteration:  60  parameters:  0.1679 0.1002 0.8393 18.5343 
chain:  1  iteration:  70  parameters:  0.1115 0.0925 0.8627 10.4317 
chain:  1  iteration:  80  parameters:  0.103 0.0881 0.8679 8.8841 
chain:  1  iteration:  90  parameters:  0.1028 0.0868 0.8744 7.264 
chain:  1  iteration:  100  parameters:  0.1147 0.0709 0.882 6.8884 
chain:  1  iteration:  110  parameters:  0.0961 0.0854 0.8746 6.6569 
chain:  1  iteration:  120  parameters:  0.0892 0.0765 0.8915 5.7539 
chain:  1  iteration:  130  parameters:  0.0888 0.0775 0.8898 5.139 
chain:  1  iteration:  140  parameters:  0.0941 0.0853 0.8941 5.305 
chain:  1  iteration:  150  parameters:  0.0714 0.0826 0.9026 4.6809 
chain:  1  iteration:  160  parameters:  0.0859 0.0753 0.9008 4.5694 
chain:  1  iteration:  170  parameters:  0.0665 0.0796 0.9006 4.6893 
chain:  1  iteration:  180  parameters:  0.1055 0.067 0.9011 4.7506 
chain:  1  iteration:  190  parameters:  0.0928 0.0879 0.8861 5.0691 
chain:  1  iteration:  200  parameters:  0.0947 0.0766 0.8921 4.9254 
chain:  1  iteration:  210  parameters:  0.0651 0.0935 0.8993 4.8459 
chain:  1  iteration:  220  parameters:  0.0908 0.0859 0.8852 4.9372 
chain:  1  iteration:  230  parameters:  0.0927 0.087 0.8834 5.0606 
chain:  1  iteration:  240  parameters:  0.1046 0.089 0.8807 5.5168 
chain:  1  iteration:  250  parameters:  0.1179 0.0793 0.8794 5.0083 
chain:  1  iteration:  260  parameters:  0.0928 0.0932 0.8802 5.0139 
chain:  1  iteration:  270  parameters:  0.1072 0.0865 0.8789 5.1325 
chain:  1  iteration:  280  parameters:  0.0862 0.0928 0.8921 5.0894 
chain:  1  iteration:  290  parameters:  0.0922 0.076 0.8916 4.895 
chain:  1  iteration:  300  parameters:  0.1092 0.0803 0.8874 4.777 
chain:  1  iteration:  310  parameters:  0.0801 0.0858 0.8922 4.6943 
chain:  1  iteration:  320  parameters:  0.0708 0.0719 0.9094 4.4654 
chain:  1  iteration:  330  parameters:  0.0774 0.0822 0.8982 4.6237 
chain:  1  iteration:  340  parameters:  0.1063 0.0835 0.8908 4.4699 
chain:  1  iteration:  350  parameters:  0.0738 0.0911 0.898 4.2016 
chain:  1  iteration:  360  parameters:  0.0831 0.0773 0.9059 4.3299 
chain:  1  iteration:  370  parameters:  0.1031 0.0805 0.8962 4.5646 
chain:  1  iteration:  380  parameters:  0.0748 0.0771 0.9072 4.3374 
chain:  1  iteration:  390  parameters:  0.0918 0.0751 0.9039 4.3042 
chain:  1  iteration:  400  parameters:  0.076 0.0733 0.9047 4.285 
chain:  1  iteration:  410  parameters:  0.1082 0.0832 0.8918 4.17 
chain:  1  iteration:  420  parameters:  0.0597 0.0688 0.9154 4.3889 
chain:  1  iteration:  430  parameters:  0.0785 0.0776 0.9058 4.6749 
chain:  1  iteration:  440  parameters:  0.0857 0.0695 0.8984 5.1658 
chain:  1  iteration:  450  parameters:  0.0974 0.0629 0.8989 4.9409 
chain:  1  iteration:  460  parameters:  0.0896 0.0958 0.8846 4.7544 
chain:  1  iteration:  470  parameters:  0.1011 0.0613 0.909 5.1185 
chain:  1  iteration:  480  parameters:  0.0756 0.0734 0.9026 5.1224 
chain:  1  iteration:  490  parameters:  0.0957 0.0816 0.8878 4.97 
chain:  1  iteration:  500  parameters:  0.1196 0.0808 0.8804 5.125 
chain:  1  iteration:  510  parameters:  0.11 0.0967 0.8758 4.8785 
chain:  1  iteration:  520  parameters:  0.1502 0.0916 0.8605 4.9012 
chain:  1  iteration:  530  parameters:  0.1119 0.0798 0.8882 4.4794 
chain:  1  iteration:  540  parameters:  0.088 0.0854 0.8981 4.2949 
chain:  1  iteration:  550  parameters:  0.0762 0.0763 0.9057 4.5297 
chain:  1  iteration:  560  parameters:  0.0805 0.0656 0.9099 4.7068 
chain:  1  iteration:  570  parameters:  0.0539 0.0701 0.9225 4.4694 
chain:  1  iteration:  580  parameters:  0.0764 0.0781 0.9068 4.5983 
chain:  1  iteration:  590  parameters:  0.0831 0.0836 0.8996 4.522 
chain:  1  iteration:  600  parameters:  0.0734 0.0876 0.8948 5.1564 
chain:  1  iteration:  610  parameters:  0.0857 0.0759 0.891 5.5712 
chain:  1  iteration:  620  parameters:  0.1132 0.0767 0.8872 5.5512 
chain:  1  iteration:  630  parameters:  0.0993 0.0699 0.8914 5.3545 
chain:  1  iteration:  640  parameters:  0.0984 0.0751 0.8915 5.0646 
chain:  1  iteration:  650  parameters:  0.0823 0.0691 0.9027 5.1476 
chain:  1  iteration:  660  parameters:  0.0796 0.0816 0.8943 5.1965 
chain:  1  iteration:  670  parameters:  0.1023 0.0762 0.8885 4.8072 
chain:  1  iteration:  680  parameters:  0.1071 0.0823 0.8844 4.7254 
chain:  1  iteration:  690  parameters:  0.0837 0.085 0.8983 4.5278 
chain:  1  iteration:  700  parameters:  0.0904 0.0828 0.8875 4.6235 
chain:  1  iteration:  710  parameters:  0.0979 0.092 0.8837 4.422 
chain:  1  iteration:  720  parameters:  0.0843 0.0805 0.9037 4.3239 
chain:  1  iteration:  730  parameters:  0.0954 0.077 0.8988 4.2733 
chain:  1  iteration:  740  parameters:  0.0734 0.0752 0.902 4.6588 
chain:  1  iteration:  750  parameters:  0.0791 0.0866 0.8915 5.1469 
chain:  1  iteration:  760  parameters:  0.0634 0.0819 0.9045 5.0256 
chain:  1  iteration:  770  parameters:  0.0875 0.0909 0.8783 5.3447 
chain:  1  iteration:  780  parameters:  0.0765 0.0872 0.8864 5.3564 
chain:  1  iteration:  790  parameters:  0.0677 0.0791 0.8972 5.4362 
chain:  1  iteration:  800  parameters:  0.1066 0.0992 0.8643 5.304 
chain:  1  iteration:  810  parameters:  0.1092 0.0877 0.8761 5.5799 
chain:  1  iteration:  820  parameters:  0.0812 0.0739 0.8952 5.3912 
chain:  1  iteration:  830  parameters:  0.0812 0.0678 0.9061 5.3654 
chain:  1  iteration:  840  parameters:  0.0674 0.0692 0.9058 5.7856 
chain:  1  iteration:  850  parameters:  0.0704 0.0667 0.9092 5.6355 
chain:  1  iteration:  860  parameters:  0.0504 0.0725 0.9094 6.2453 
chain:  1  iteration:  870  parameters:  0.054 0.0846 0.8996 5.3565 
chain:  1  iteration:  880  parameters:  0.0676 0.0783 0.9004 5.377 
chain:  1  iteration:  890  parameters:  0.0871 0.0719 0.8949 5.5708 
chain:  1  iteration:  900  parameters:  0.0725 0.089 0.8906 4.9703 
chain:  1  iteration:  910  parameters:  0.0914 0.0762 0.8937 4.6245 
chain:  1  iteration:  920  parameters:  0.0798 0.0794 0.894 4.5314 
chain:  1  iteration:  930  parameters:  0.0902 0.0884 0.8898 4.3239 
chain:  1  iteration:  940  parameters:  0.0761 0.1044 0.8911 4.2774 
chain:  1  iteration:  950  parameters:  0.134 0.0911 0.8758 4.1244 
chain:  1  iteration:  960  parameters:  0.0895 0.1007 0.8901 4.5043 
chain:  1  iteration:  970  parameters:  0.1063 0.0666 0.8985 4.7134 
chain:  1  iteration:  980  parameters:  0.0858 0.0833 0.8893 5.329 
chain:  1  iteration:  990  parameters:  0.0839 0.0716 0.8988 5.4582 
chain:  1  iteration:  1000  parameters:  0.078 0.0709 0.9013 5.9528 
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chain:  2  iteration:  18360  parameters:  0.06 0.0708 0.9111 5.1737 
chain:  2  iteration:  18370  parameters:  0.0943 0.0595 0.9051 5.075 
chain:  2  iteration:  18380  parameters:  0.082 0.0771 0.8946 5.2534 
chain:  2  iteration:  18390  parameters:  0.0689 0.0713 0.9038 5.3238 
chain:  2  iteration:  18400  parameters:  0.066 0.0701 0.9098 5.3114 
chain:  2  iteration:  18410  parameters:  0.0671 0.0817 0.8947 5.5548 
chain:  2  iteration:  18420  parameters:  0.1155 0.0804 0.8775 5.4974 
chain:  2  iteration:  18430  parameters:  0.1044 0.0848 0.8866 5.4705 
chain:  2  iteration:  18440  parameters:  0.1241 0.0856 0.8755 4.9745 
chain:  2  iteration:  18450  parameters:  0.085 0.0948 0.8794 4.9159 
chain:  2  iteration:  18460  parameters:  0.1168 0.0804 0.8844 4.9016 
chain:  2  iteration:  18470  parameters:  0.103 0.075 0.8935 4.8207 
chain:  2  iteration:  18480  parameters:  0.0958 0.0779 0.8931 4.9064 
chain:  2  iteration:  18490  parameters:  0.0971 0.0709 0.9018 4.6476 
chain:  2  iteration:  18500  parameters:  0.0702 0.0645 0.9139 4.6544 
chain:  2  iteration:  18510  parameters:  0.091 0.0804 0.8976 4.5307 
chain:  2  iteration:  18520  parameters:  0.0916 0.0694 0.9057 4.745 
chain:  2  iteration:  18530  parameters:  0.0817 0.064 0.9134 4.7665 
chain:  2  iteration:  18540  parameters:  0.085 0.0666 0.9046 4.7825 
chain:  2  iteration:  18550  parameters:  0.0691 0.0801 0.9022 4.9746 
chain:  2  iteration:  18560  parameters:  0.0756 0.078 0.9049 4.6021 
chain:  2  iteration:  18570  parameters:  0.0745 0.0712 0.9131 4.7997 
chain:  2  iteration:  18580  parameters:  0.064 0.0702 0.9161 5.0282 
chain:  2  iteration:  18590  parameters:  0.0552 0.0605 0.9252 4.6283 
chain:  2  iteration:  18600  parameters:  0.0631 0.0697 0.9142 4.7964 
chain:  2  iteration:  18610  parameters:  0.0627 0.0646 0.919 4.7418 
chain:  2  iteration:  18620  parameters:  0.0751 0.0604 0.9185 4.7681 
chain:  2  iteration:  18630  parameters:  0.0864 0.0565 0.9113 5.0033 
chain:  2  iteration:  18640  parameters:  0.0967 0.0703 0.898 4.8133 
chain:  2  iteration:  18650  parameters:  0.0645 0.0883 0.9036 4.953 
chain:  2  iteration:  18660  parameters:  0.0644 0.0744 0.9114 4.4117 
chain:  2  iteration:  18670  parameters:  0.0793 0.0663 0.9076 4.7061 
chain:  2  iteration:  18680  parameters:  0.0748 0.0724 0.9092 4.6034 
chain:  2  iteration:  18690  parameters:  0.0647 0.0789 0.9091 4.3376 
chain:  2  iteration:  18700  parameters:  0.0756 0.0725 0.9123 4.3819 
chain:  2  iteration:  18710  parameters:  0.0659 0.0683 0.9152 4.2665 
chain:  2  iteration:  18720  parameters:  0.0858 0.0766 0.9087 4.3448 
chain:  2  iteration:  18730  parameters:  0.0657 0.0625 0.9192 4.8251 
chain:  2  iteration:  18740  parameters:  0.0776 0.0616 0.9178 4.9987 
chain:  2  iteration:  18750  parameters:  0.0758 0.0736 0.9032 4.8934 
chain:  2  iteration:  18760  parameters:  0.0928 0.0735 0.9041 4.5906 
chain:  2  iteration:  18770  parameters:  0.0597 0.0802 0.9025 4.9172 
chain:  2  iteration:  18780  parameters:  0.0547 0.0786 0.9063 4.9055 
chain:  2  iteration:  18790  parameters:  0.0764 0.0624 0.9123 4.6944 
chain:  2  iteration:  18800  parameters:  0.0764 0.0495 0.9183 5.2084 
chain:  2  iteration:  18810  parameters:  0.0565 0.0575 0.926 4.9762 
chain:  2  iteration:  18820  parameters:  0.0614 0.0535 0.9249 5.0445 
chain:  2  iteration:  18830  parameters:  0.0528 0.0558 0.9248 5.0295 
chain:  2  iteration:  18840  parameters:  0.0607 0.0527 0.9247 5.4927 
chain:  2  iteration:  18850  parameters:  0.0701 0.0679 0.9084 5.3798 
chain:  2  iteration:  18860  parameters:  0.0772 0.0645 0.9044 5.6377 
chain:  2  iteration:  18870  parameters:  0.0669 0.0761 0.9008 5.8002 
chain:  2  iteration:  18880  parameters:  0.0896 0.0625 0.9026 5.6121 
chain:  2  iteration:  18890  parameters:  0.0789 0.0773 0.8971 6.1277 
chain:  2  iteration:  18900  parameters:  0.0891 0.0746 0.8942 5.5714 
chain:  2  iteration:  18910  parameters:  0.0997 0.0778 0.8841 5.3878 
chain:  2  iteration:  18920  parameters:  0.109 0.0895 0.8708 5.0676 
chain:  2  iteration:  18930  parameters:  0.1116 0.1045 0.8662 5.0252 
chain:  2  iteration:  18940  parameters:  0.1091 0.0939 0.8766 5.2246 
chain:  2  iteration:  18950  parameters:  0.1145 0.0899 0.8771 5.3757 
chain:  2  iteration:  18960  parameters:  0.11 0.0928 0.8684 5.3579 
chain:  2  iteration:  18970  parameters:  0.1664 0.0884 0.8655 5.0263 
chain:  2  iteration:  18980  parameters:  0.0997 0.1245 0.852 5.1537 
chain:  2  iteration:  18990  parameters:  0.164 0.0936 0.8536 5.2756 
chain:  2  iteration:  19000  parameters:  0.1214 0.088 0.8704 5.2994 
chain:  2  iteration:  19010  parameters:  0.1181 0.1024 0.8656 5.3975 
chain:  2  iteration:  19020  parameters:  0.1093 0.0955 0.8686 5.1294 
chain:  2  iteration:  19030  parameters:  0.1031 0.101 0.8788 4.6554 
chain:  2  iteration:  19040  parameters:  0.1196 0.0754 0.8908 4.6802 
chain:  2  iteration:  19050  parameters:  0.1037 0.0774 0.897 4.78 
chain:  2  iteration:  19060  parameters:  0.074 0.072 0.9034 4.8383 
chain:  2  iteration:  19070  parameters:  0.076 0.0723 0.9082 4.501 
chain:  2  iteration:  19080  parameters:  0.0755 0.0519 0.9251 4.4119 
chain:  2  iteration:  19090  parameters:  0.0438 0.0536 0.9338 4.2618 
chain:  2  iteration:  19100  parameters:  0.0716 0.066 0.9153 4.5366 
chain:  2  iteration:  19110  parameters:  0.0546 0.0756 0.9104 4.5983 
chain:  2  iteration:  19120  parameters:  0.0916 0.0768 0.8941 4.9954 
chain:  2  iteration:  19130  parameters:  0.1087 0.0759 0.8923 4.9962 
chain:  2  iteration:  19140  parameters:  0.1268 0.0958 0.865 5.2063 
chain:  2  iteration:  19150  parameters:  0.1056 0.1087 0.8648 5.4542 
chain:  2  iteration:  19160  parameters:  0.1301 0.0745 0.878 5.2208 
chain:  2  iteration:  19170  parameters:  0.1052 0.0797 0.8832 4.9091 
chain:  2  iteration:  19180  parameters:  0.0942 0.0963 0.8819 5.1894 
chain:  2  iteration:  19190  parameters:  0.1225 0.0869 0.8754 5.2527 
chain:  2  iteration:  19200  parameters:  0.1004 0.0838 0.8827 5.3561 
chain:  2  iteration:  19210  parameters:  0.1024 0.0943 0.8748 5.2895 
chain:  2  iteration:  19220  parameters:  0.0955 0.108 0.8676 5.0578 
chain:  2  iteration:  19230  parameters:  0.1497 0.0886 0.8609 5.2739 
chain:  2  iteration:  19240  parameters:  0.1343 0.1068 0.8526 4.7302 
chain:  2  iteration:  19250  parameters:  0.0958 0.1163 0.8647 4.4508 
chain:  2  iteration:  19260  parameters:  0.1402 0.1008 0.8615 4.8979 
chain:  2  iteration:  19270  parameters:  0.1327 0.099 0.8622 4.7466 
chain:  2  iteration:  19280  parameters:  0.0871 0.1013 0.8786 4.9888 
chain:  2  iteration:  19290  parameters:  0.0859 0.1026 0.8798 4.8168 
chain:  2  iteration:  19300  parameters:  0.1155 0.0824 0.8794 4.8464 
chain:  2  iteration:  19310  parameters:  0.1209 0.105 0.8663 4.7344 
chain:  2  iteration:  19320  parameters:  0.1028 0.1017 0.8693 4.9376 
chain:  2  iteration:  19330  parameters:  0.1308 0.0956 0.8624 4.9493 
chain:  2  iteration:  19340  parameters:  0.0797 0.1114 0.8725 4.9848 
chain:  2  iteration:  19350  parameters:  0.1068 0.0911 0.8832 4.8597 
chain:  2  iteration:  19360  parameters:  0.0982 0.0773 0.8895 5.0336 
chain:  2  iteration:  19370  parameters:  0.0969 0.0735 0.9036 4.7845 
chain:  2  iteration:  19380  parameters:  0.085 0.0748 0.9039 4.8507 
chain:  2  iteration:  19390  parameters:  0.0686 0.0606 0.9204 4.5948 
chain:  2  iteration:  19400  parameters:  0.0454 0.064 0.9195 4.6921 
chain:  2  iteration:  19410  parameters:  0.0696 0.0583 0.921 4.7555 
chain:  2  iteration:  19420  parameters:  0.0681 0.0593 0.9193 4.2389 
chain:  2  iteration:  19430  parameters:  0.0762 0.058 0.9226 4.1802 
chain:  2  iteration:  19440  parameters:  0.0531 0.0721 0.9227 4.3202 
chain:  2  iteration:  19450  parameters:  0.0617 0.0649 0.9211 4.4559 
chain:  2  iteration:  19460  parameters:  0.0711 0.0673 0.9111 4.6927 
chain:  2  iteration:  19470  parameters:  0.0831 0.0618 0.9084 5.1709 
chain:  2  iteration:  19480  parameters:  0.0572 0.0752 0.9097 5.3507 
chain:  2  iteration:  19490  parameters:  0.0813 0.0598 0.9089 5.5106 
chain:  2  iteration:  19500  parameters:  0.0626 0.0647 0.9168 4.9875 
chain:  2  iteration:  19510  parameters:  0.0858 0.0673 0.9032 4.7815 
chain:  2  iteration:  19520  parameters:  0.0691 0.0725 0.9095 4.8082 
chain:  2  iteration:  19530  parameters:  0.0698 0.0816 0.9007 4.7442 
chain:  2  iteration:  19540  parameters:  0.1139 0.0688 0.8967 4.6735 
chain:  2  iteration:  19550  parameters:  0.0654 0.0756 0.9057 4.6676 
chain:  2  iteration:  19560  parameters:  0.0656 0.0746 0.9085 4.7106 
chain:  2  iteration:  19570  parameters:  0.0685 0.0795 0.9014 5.1065 
chain:  2  iteration:  19580  parameters:  0.0774 0.0595 0.9144 4.9946 
chain:  2  iteration:  19590  parameters:  0.0683 0.0621 0.9177 4.6996 
chain:  2  iteration:  19600  parameters:  0.0486 0.0731 0.9162 4.9376 
chain:  2  iteration:  19610  parameters:  0.0602 0.0699 0.9106 4.6984 
chain:  2  iteration:  19620  parameters:  0.0773 0.068 0.9087 4.6486 
chain:  2  iteration:  19630  parameters:  0.0759 0.0846 0.8983 4.7071 
chain:  2  iteration:  19640  parameters:  0.0665 0.0698 0.9078 5.0348 
chain:  2  iteration:  19650  parameters:  0.0719 0.0663 0.9126 4.8025 
chain:  2  iteration:  19660  parameters:  0.0682 0.055 0.9221 4.8526 
chain:  2  iteration:  19670  parameters:  0.0647 0.0655 0.9161 4.8774 
chain:  2  iteration:  19680  parameters:  0.0569 0.0605 0.9253 4.6749 
chain:  2  iteration:  19690  parameters:  0.0545 0.056 0.9264 4.6381 
chain:  2  iteration:  19700  parameters:  0.0605 0.058 0.9225 5.0844 
chain:  2  iteration:  19710  parameters:  0.066 0.0559 0.9237 4.6774 
chain:  2  iteration:  19720  parameters:  0.0882 0.0614 0.9068 4.8601 
chain:  2  iteration:  19730  parameters:  0.0872 0.072 0.8995 5.1965 
chain:  2  iteration:  19740  parameters:  0.0896 0.068 0.9033 5.3199 
chain:  2  iteration:  19750  parameters:  0.0876 0.0763 0.8968 5.2421 
chain:  2  iteration:  19760  parameters:  0.0676 0.0731 0.905 5.5137 
chain:  2  iteration:  19770  parameters:  0.0701 0.0843 0.8938 5.1208 
chain:  2  iteration:  19780  parameters:  0.0695 0.0695 0.9077 5.1248 
chain:  2  iteration:  19790  parameters:  0.0708 0.0628 0.9206 4.4887 
chain:  2  iteration:  19800  parameters:  0.0479 0.0619 0.9246 4.5673 
chain:  2  iteration:  19810  parameters:  0.0715 0.0656 0.9083 4.8377 
chain:  2  iteration:  19820  parameters:  0.0809 0.064 0.9148 4.5151 
chain:  2  iteration:  19830  parameters:  0.0646 0.0683 0.9225 4.7458 
chain:  2  iteration:  19840  parameters:  0.0635 0.0666 0.9165 4.2018 
chain:  2  iteration:  19850  parameters:  0.1012 0.051 0.916 4.3558 
chain:  2  iteration:  19860  parameters:  0.0867 0.0569 0.9189 4.4456 
chain:  2  iteration:  19870  parameters:  0.0765 0.0741 0.9078 4.658 
chain:  2  iteration:  19880  parameters:  0.0636 0.0682 0.9106 4.9904 
chain:  2  iteration:  19890  parameters:  0.0685 0.0674 0.9063 5.1229 
chain:  2  iteration:  19900  parameters:  0.0713 0.0539 0.9191 5.4463 
chain:  2  iteration:  19910  parameters:  0.0846 0.0591 0.9156 5.3782 
chain:  2  iteration:  19920  parameters:  0.065 0.0531 0.9258 4.8354 
chain:  2  iteration:  19930  parameters:  0.0444 0.058 0.9289 4.7607 
chain:  2  iteration:  19940  parameters:  0.0579 0.0582 0.9277 4.4915 
chain:  2  iteration:  19950  parameters:  0.0766 0.0536 0.9221 4.4521 
chain:  2  iteration:  19960  parameters:  0.0667 0.0604 0.9132 4.7368 
chain:  2  iteration:  19970  parameters:  0.0405 0.0656 0.9239 5.0802 
chain:  2  iteration:  19980  parameters:  0.0552 0.0533 0.9255 5.4606 
chain:  2  iteration:  19990  parameters:  0.0629 0.074 0.9145 4.9885 
chain:  2  iteration:  20000  parameters:  0.0696 0.0663 0.909 4.7552 
MCMC_Soja <- bayesGARCH(retornos_Soja, control = list(l.chain = 20000, n.chain = 2))
chain:  1  iteration:  10  parameters:  0.2223 0.1402 0.7399 76.5393 
chain:  1  iteration:  20  parameters:  0.1715 0.1372 0.7726 78.0532 
chain:  1  iteration:  30  parameters:  0.1433 0.1284 0.8 67.134 
chain:  1  iteration:  40  parameters:  0.1252 0.1268 0.8041 68.3097 
chain:  1  iteration:  50  parameters:  0.1426 0.1278 0.8018 57.5374 
chain:  1  iteration:  60  parameters:  0.1127 0.0876 0.8429 50.4549 
chain:  1  iteration:  70  parameters:  0.0937 0.093 0.8525 48.7928 
chain:  1  iteration:  80  parameters:  0.0831 0.098 0.8559 36.5728 
chain:  1  iteration:  90  parameters:  0.1037 0.0771 0.854 25.8239 
chain:  1  iteration:  100  parameters:  0.0909 0.0902 0.8555 22.5588 
chain:  1  iteration:  110  parameters:  0.0763 0.0771 0.8785 18.9709 
chain:  1  iteration:  120  parameters:  0.0848 0.0781 0.872 13.8316 
chain:  1  iteration:  130  parameters:  0.0855 0.0721 0.8675 11.0847 
chain:  1  iteration:  140  parameters:  0.061 0.0901 0.8782 9.1665 
chain:  1  iteration:  150  parameters:  0.059 0.0664 0.8964 7.6296 
chain:  1  iteration:  160  parameters:  0.0784 0.067 0.8873 6.2041 
chain:  1  iteration:  170  parameters:  0.0698 0.0746 0.8931 5.6943 
chain:  1  iteration:  180  parameters:  0.0561 0.0632 0.9015 6.1391 
chain:  1  iteration:  190  parameters:  0.0677 0.0682 0.8946 5.5495 
chain:  1  iteration:  200  parameters:  0.0616 0.0801 0.8938 4.8107 
chain:  1  iteration:  210  parameters:  0.0864 0.0674 0.8878 4.8842 
chain:  1  iteration:  220  parameters:  0.0929 0.0697 0.8824 4.8514 
chain:  1  iteration:  230  parameters:  0.09 0.0816 0.8842 4.9649 
chain:  1  iteration:  240  parameters:  0.0928 0.0732 0.8727 4.7344 
chain:  1  iteration:  250  parameters:  0.0467 0.0826 0.8942 4.9988 
chain:  1  iteration:  260  parameters:  0.0624 0.0795 0.8906 5.0819 
chain:  1  iteration:  270  parameters:  0.0759 0.0703 0.8924 5.1289 
chain:  1  iteration:  280  parameters:  0.1069 0.0534 0.8802 5.3769 
chain:  1  iteration:  290  parameters:  0.0727 0.067 0.8865 5.8408 
chain:  1  iteration:  300  parameters:  0.0685 0.0777 0.8868 6.396 
chain:  1  iteration:  310  parameters:  0.0656 0.0474 0.9057 6.2107 
chain:  1  iteration:  320  parameters:  0.044 0.0589 0.9144 5.9482 
chain:  1  iteration:  330  parameters:  0.0453 0.0641 0.9135 5.5029 
chain:  1  iteration:  340  parameters:  0.0632 0.0681 0.9065 5.24 
chain:  1  iteration:  350  parameters:  0.0587 0.0716 0.901 5.0176 
chain:  1  iteration:  360  parameters:  0.0569 0.0583 0.9126 4.9997 
chain:  1  iteration:  370  parameters:  0.0513 0.0522 0.9205 5.1795 
chain:  1  iteration:  380  parameters:  0.0344 0.0495 0.9329 5.3309 
chain:  1  iteration:  390  parameters:  0.0356 0.0474 0.933 5.7148 
chain:  1  iteration:  400  parameters:  0.0446 0.0549 0.9212 5.9664 
chain:  1  iteration:  410  parameters:  0.0392 0.0523 0.9265 5.7917 
chain:  1  iteration:  420  parameters:  0.0328 0.0499 0.9363 5.8854 
chain:  1  iteration:  430  parameters:  0.0518 0.0431 0.9246 5.7591 
chain:  1  iteration:  440  parameters:  0.0452 0.0604 0.9159 5.2608 
chain:  1  iteration:  450  parameters:  0.0602 0.0564 0.911 5.0827 
chain:  1  iteration:  460  parameters:  0.0528 0.0785 0.8987 5.0035 
chain:  1  iteration:  470  parameters:  0.0721 0.0681 0.8923 5.3856 
chain:  1  iteration:  480  parameters:  0.0508 0.059 0.9132 5.9896 
chain:  1  iteration:  490  parameters:  0.0619 0.0462 0.917 5.8238 
chain:  1  iteration:  500  parameters:  0.0591 0.0533 0.9155 5.1744 
chain:  1  iteration:  510  parameters:  0.0728 0.0498 0.9144 5.2638 
chain:  1  iteration:  520  parameters:  0.0667 0.0637 0.9015 5.0269 
chain:  1  iteration:  530  parameters:  0.0642 0.0707 0.9037 5.254 
chain:  1  iteration:  540  parameters:  0.0726 0.0676 0.8976 4.7564 
chain:  1  iteration:  550  parameters:  0.0614 0.0679 0.9077 4.4259 
chain:  1  iteration:  560  parameters:  0.069 0.0576 0.9024 4.665 
chain:  1  iteration:  570  parameters:  0.0901 0.0654 0.8852 4.8305 
chain:  1  iteration:  580  parameters:  0.0906 0.0645 0.8869 4.7614 
chain:  1  iteration:  590  parameters:  0.088 0.0861 0.8778 4.5563 
chain:  1  iteration:  600  parameters:  0.0603 0.0668 0.909 4.4095 
chain:  1  iteration:  610  parameters:  0.057 0.0628 0.9125 4.7275 
chain:  1  iteration:  620  parameters:  0.0715 0.0533 0.9118 4.8137 
chain:  1  iteration:  630  parameters:  0.053 0.0617 0.9159 4.7518 
chain:  1  iteration:  640  parameters:  0.0637 0.058 0.9111 4.5436 
chain:  1  iteration:  650  parameters:  0.0595 0.0504 0.9163 4.9318 
chain:  1  iteration:  660  parameters:  0.0545 0.0595 0.9145 5.646 
chain:  1  iteration:  670  parameters:  0.0377 0.0643 0.9083 6.0225 
chain:  1  iteration:  680  parameters:  0.0297 0.0672 0.9254 5.5409 
chain:  1  iteration:  690  parameters:  0.0447 0.0523 0.9234 5.8132 
chain:  1  iteration:  700  parameters:  0.0422 0.0566 0.92 5.2964 
chain:  1  iteration:  710  parameters:  0.0513 0.0644 0.916 5.1245 
chain:  1  iteration:  720  parameters:  0.0593 0.0648 0.914 4.6463 
chain:  1  iteration:  730  parameters:  0.0692 0.0645 0.9058 4.5975 
chain:  1  iteration:  740  parameters:  0.0722 0.0601 0.9065 4.7234 
chain:  1  iteration:  750  parameters:  0.0625 0.0583 0.9132 4.8253 
chain:  1  iteration:  760  parameters:  0.0469 0.0617 0.916 4.7287 
chain:  1  iteration:  770  parameters:  0.0521 0.0605 0.913 5.1427 
chain:  1  iteration:  780  parameters:  0.0392 0.0802 0.9104 4.8765 
chain:  1  iteration:  790  parameters:  0.0574 0.061 0.9094 5.0717 
chain:  1  iteration:  800  parameters:  0.0527 0.0687 0.9052 5.2012 
chain:  1  iteration:  810  parameters:  0.0509 0.0694 0.9045 5.0748 
chain:  1  iteration:  820  parameters:  0.0658 0.067 0.8976 5.2541 
chain:  1  iteration:  830  parameters:  0.0529 0.0682 0.9072 4.9265 
chain:  1  iteration:  840  parameters:  0.0658 0.0556 0.9112 4.8805 
chain:  1  iteration:  850  parameters:  0.0372 0.0747 0.9104 5.2862 
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chain:  1  iteration:  870  parameters:  0.0703 0.0564 0.9016 5.8531 
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chain:  2  iteration:  18180  parameters:  0.0643 0.0541 0.9128 5.0114 
chain:  2  iteration:  18190  parameters:  0.0545 0.0556 0.9137 5.473 
chain:  2  iteration:  18200  parameters:  0.0667 0.0566 0.9042 5.3658 
chain:  2  iteration:  18210  parameters:  0.044 0.0578 0.9211 5.2836 
chain:  2  iteration:  18220  parameters:  0.0515 0.0425 0.9276 5.1815 
chain:  2  iteration:  18230  parameters:  0.0423 0.0456 0.9314 5.1363 
chain:  2  iteration:  18240  parameters:  0.0381 0.049 0.933 4.9706 
chain:  2  iteration:  18250  parameters:  0.0345 0.0553 0.9312 4.7339 
chain:  2  iteration:  18260  parameters:  0.0421 0.0479 0.9318 4.5527 
chain:  2  iteration:  18270  parameters:  0.0589 0.0558 0.9111 5.2866 
chain:  2  iteration:  18280  parameters:  0.0506 0.0687 0.9053 5.2613 
chain:  2  iteration:  18290  parameters:  0.0474 0.0735 0.9013 5.195 
chain:  2  iteration:  18300  parameters:  0.0539 0.0636 0.9102 4.7558 
chain:  2  iteration:  18310  parameters:  0.0476 0.064 0.9106 4.4812 
chain:  2  iteration:  18320  parameters:  0.0796 0.0538 0.9036 4.6616 
chain:  2  iteration:  18330  parameters:  0.0635 0.0738 0.8976 4.8716 
chain:  2  iteration:  18340  parameters:  0.0694 0.0771 0.8958 4.7885 
chain:  2  iteration:  18350  parameters:  0.0403 0.0653 0.9192 5.0211 
chain:  2  iteration:  18360  parameters:  0.0727 0.0551 0.9124 4.9953 
chain:  2  iteration:  18370  parameters:  0.0393 0.0657 0.9174 5.4009 
chain:  2  iteration:  18380  parameters:  0.0385 0.0614 0.9169 5.132 
chain:  2  iteration:  18390  parameters:  0.0592 0.0485 0.9242 4.7587 
chain:  2  iteration:  18400  parameters:  0.0455 0.0605 0.9221 4.7156 
chain:  2  iteration:  18410  parameters:  0.0368 0.05 0.9306 4.8037 
chain:  2  iteration:  18420  parameters:  0.0386 0.0484 0.9367 4.4663 
chain:  2  iteration:  18430  parameters:  0.047 0.0488 0.9355 4.5347 
chain:  2  iteration:  18440  parameters:  0.0342 0.05 0.9379 4.5347 
chain:  2  iteration:  18450  parameters:  0.0428 0.0449 0.9339 4.4006 
chain:  2  iteration:  18460  parameters:  0.0345 0.0465 0.9399 4.9208 
chain:  2  iteration:  18470  parameters:  0.0453 0.0452 0.9276 5.8814 
chain:  2  iteration:  18480  parameters:  0.0385 0.0468 0.9318 6.0078 
chain:  2  iteration:  18490  parameters:  0.041 0.0526 0.9263 5.8468 
chain:  2  iteration:  18500  parameters:  0.0494 0.0609 0.9135 5.8719 
chain:  2  iteration:  18510  parameters:  0.0487 0.0473 0.9209 5.7212 
chain:  2  iteration:  18520  parameters:  0.0449 0.0618 0.9124 6.1983 
chain:  2  iteration:  18530  parameters:  0.0684 0.051 0.9055 6.5161 
chain:  2  iteration:  18540  parameters:  0.0616 0.0619 0.9067 5.9585 
chain:  2  iteration:  18550  parameters:  0.06 0.0494 0.9141 5.695 
chain:  2  iteration:  18560  parameters:  0.056 0.055 0.9112 5.2733 
chain:  2  iteration:  18570  parameters:  0.0532 0.0628 0.9118 5.5225 
chain:  2  iteration:  18580  parameters:  0.047 0.0616 0.9128 5.4953 
chain:  2  iteration:  18590  parameters:  0.0632 0.0519 0.9118 6.0663 
chain:  2  iteration:  18600  parameters:  0.047 0.0586 0.9202 6.2288 
chain:  2  iteration:  18610  parameters:  0.041 0.0545 0.9221 6.1483 
chain:  2  iteration:  18620  parameters:  0.0447 0.0534 0.9198 6.119 
chain:  2  iteration:  18630  parameters:  0.0648 0.0539 0.909 5.8298 
chain:  2  iteration:  18640  parameters:  0.061 0.062 0.9069 6.0107 
chain:  2  iteration:  18650  parameters:  0.0544 0.0637 0.9058 5.8906 
chain:  2  iteration:  18660  parameters:  0.0761 0.0614 0.8974 5.8174 
chain:  2  iteration:  18670  parameters:  0.0679 0.0583 0.9018 5.392 
chain:  2  iteration:  18680  parameters:  0.0488 0.0757 0.9027 5.0944 
chain:  2  iteration:  18690  parameters:  0.065 0.0595 0.9072 5.0184 
chain:  2  iteration:  18700  parameters:  0.0605 0.0569 0.9125 5.2247 
chain:  2  iteration:  18710  parameters:  0.0735 0.064 0.8994 4.5895 
chain:  2  iteration:  18720  parameters:  0.0567 0.0798 0.8892 4.6965 
chain:  2  iteration:  18730  parameters:  0.0806 0.0757 0.8882 4.9484 
chain:  2  iteration:  18740  parameters:  0.0893 0.0837 0.8735 4.9398 
chain:  2  iteration:  18750  parameters:  0.0869 0.0807 0.8761 4.8784 
chain:  2  iteration:  18760  parameters:  0.0887 0.0943 0.8733 4.5457 
chain:  2  iteration:  18770  parameters:  0.1026 0.079 0.8686 5.0787 
chain:  2  iteration:  18780  parameters:  0.0801 0.0993 0.8696 5.0319 
chain:  2  iteration:  18790  parameters:  0.0913 0.0883 0.8691 4.8907 
chain:  2  iteration:  18800  parameters:  0.0831 0.0692 0.8975 4.3993 
chain:  2  iteration:  18810  parameters:  0.0647 0.0774 0.8973 4.3107 
chain:  2  iteration:  18820  parameters:  0.066 0.0673 0.9095 4.3197 
chain:  2  iteration:  18830  parameters:  0.0551 0.0715 0.9062 4.4781 
chain:  2  iteration:  18840  parameters:  0.0575 0.0628 0.9141 5.1728 
chain:  2  iteration:  18850  parameters:  0.0583 0.0732 0.9012 4.9748 
chain:  2  iteration:  18860  parameters:  0.0595 0.0565 0.9163 4.924 
chain:  2  iteration:  18870  parameters:  0.0813 0.0479 0.9133 4.8399 
chain:  2  iteration:  18880  parameters:  0.07 0.0564 0.9092 5.0003 
chain:  2  iteration:  18890  parameters:  0.0532 0.0667 0.9139 4.868 
chain:  2  iteration:  18900  parameters:  0.0614 0.0657 0.9045 4.6762 
chain:  2  iteration:  18910  parameters:  0.0436 0.066 0.9173 4.901 
chain:  2  iteration:  18920  parameters:  0.0532 0.0516 0.9236 4.5185 
chain:  2  iteration:  18930  parameters:  0.0443 0.0667 0.9194 4.6334 
chain:  2  iteration:  18940  parameters:  0.0676 0.041 0.9207 5.0557 
chain:  2  iteration:  18950  parameters:  0.0455 0.0691 0.9121 5.4343 
chain:  2  iteration:  18960  parameters:  0.052 0.0697 0.9034 5.3675 
chain:  2  iteration:  18970  parameters:  0.0564 0.0585 0.9062 5.5252 
chain:  2  iteration:  18980  parameters:  0.0508 0.0517 0.9214 5.188 
chain:  2  iteration:  18990  parameters:  0.0577 0.0515 0.9173 5.261 
chain:  2  iteration:  19000  parameters:  0.0316 0.0598 0.9246 4.9256 
chain:  2  iteration:  19010  parameters:  0.0316 0.0558 0.9268 5.2778 
chain:  2  iteration:  19020  parameters:  0.0474 0.0494 0.9223 5.3836 
chain:  2  iteration:  19030  parameters:  0.0567 0.0557 0.9139 6.1424 
chain:  2  iteration:  19040  parameters:  0.0664 0.0546 0.9067 5.2866 
chain:  2  iteration:  19050  parameters:  0.0607 0.0632 0.9021 5.753 
chain:  2  iteration:  19060  parameters:  0.0777 0.0735 0.8817 5.2335 
chain:  2  iteration:  19070  parameters:  0.0869 0.0804 0.8806 5.2609 
chain:  2  iteration:  19080  parameters:  0.0705 0.0741 0.8897 4.8158 
chain:  2  iteration:  19090  parameters:  0.0845 0.0793 0.8818 4.8287 
chain:  2  iteration:  19100  parameters:  0.0715 0.0792 0.8814 4.8006 
chain:  2  iteration:  19110  parameters:  0.1052 0.0589 0.882 4.6377 
chain:  2  iteration:  19120  parameters:  0.0704 0.0727 0.887 4.9794 
chain:  2  iteration:  19130  parameters:  0.0829 0.0641 0.89 5.067 
chain:  2  iteration:  19140  parameters:  0.0574 0.0696 0.9047 5.0537 
chain:  2  iteration:  19150  parameters:  0.073 0.0575 0.8995 5.0105 
chain:  2  iteration:  19160  parameters:  0.069 0.0639 0.8963 5.2656 
chain:  2  iteration:  19170  parameters:  0.0845 0.058 0.8972 5.4886 
chain:  2  iteration:  19180  parameters:  0.0662 0.0667 0.895 5.4382 
chain:  2  iteration:  19190  parameters:  0.0618 0.062 0.9065 5.6228 
chain:  2  iteration:  19200  parameters:  0.0459 0.0691 0.91 5.6265 
chain:  2  iteration:  19210  parameters:  0.0548 0.0572 0.9123 5.8554 
chain:  2  iteration:  19220  parameters:  0.0384 0.0605 0.9174 5.832 
chain:  2  iteration:  19230  parameters:  0.0664 0.0519 0.9092 5.1288 
chain:  2  iteration:  19240  parameters:  0.0631 0.062 0.9101 4.7079 
chain:  2  iteration:  19250  parameters:  0.0506 0.0793 0.9074 4.5838 
chain:  2  iteration:  19260  parameters:  0.0834 0.0567 0.8961 4.6471 
chain:  2  iteration:  19270  parameters:  0.087 0.0638 0.8926 4.8185 
chain:  2  iteration:  19280  parameters:  0.059 0.0723 0.9049 4.8616 
chain:  2  iteration:  19290  parameters:  0.0688 0.0686 0.8973 4.878 
chain:  2  iteration:  19300  parameters:  0.0505 0.0766 0.9095 4.5141 
chain:  2  iteration:  19310  parameters:  0.0306 0.0625 0.9246 4.7904 
chain:  2  iteration:  19320  parameters:  0.0454 0.0524 0.9237 5.5437 
chain:  2  iteration:  19330  parameters:  0.0465 0.0569 0.9167 5.5877 
chain:  2  iteration:  19340  parameters:  0.0467 0.0505 0.923 5.8118 
chain:  2  iteration:  19350  parameters:  0.0502 0.0456 0.9235 5.5152 
chain:  2  iteration:  19360  parameters:  0.0483 0.0588 0.9161 5.4202 
chain:  2  iteration:  19370  parameters:  0.025 0.056 0.9331 5.2107 
chain:  2  iteration:  19380  parameters:  0.0551 0.0479 0.9239 5.0404 
chain:  2  iteration:  19390  parameters:  0.0647 0.0489 0.9204 4.8481 
chain:  2  iteration:  19400  parameters:  0.0509 0.057 0.9167 5.4887 
chain:  2  iteration:  19410  parameters:  0.076 0.0545 0.8957 5.7088 
chain:  2  iteration:  19420  parameters:  0.0753 0.0569 0.8975 6.0515 
chain:  2  iteration:  19430  parameters:  0.0673 0.0615 0.8969 6.6033 
chain:  2  iteration:  19440  parameters:  0.0489 0.0693 0.904 6.4781 
chain:  2  iteration:  19450  parameters:  0.0539 0.0565 0.9112 5.9641 
chain:  2  iteration:  19460  parameters:  0.0421 0.0614 0.9201 5.6838 
chain:  2  iteration:  19470  parameters:  0.0608 0.0479 0.919 5.6983 
chain:  2  iteration:  19480  parameters:  0.0538 0.0486 0.9223 5.628 
chain:  2  iteration:  19490  parameters:  0.0353 0.0654 0.9117 5.5963 
chain:  2  iteration:  19500  parameters:  0.0407 0.0646 0.9125 5.6357 
chain:  2  iteration:  19510  parameters:  0.0602 0.0631 0.9047 5.4145 
chain:  2  iteration:  19520  parameters:  0.0514 0.0538 0.9173 5.34 
chain:  2  iteration:  19530  parameters:  0.0611 0.047 0.9177 4.8775 
chain:  2  iteration:  19540  parameters:  0.0441 0.0626 0.9182 4.8223 
chain:  2  iteration:  19550  parameters:  0.0455 0.0623 0.9164 4.9568 
chain:  2  iteration:  19560  parameters:  0.0482 0.0565 0.9235 4.9531 
chain:  2  iteration:  19570  parameters:  0.0515 0.0479 0.9237 5.5819 
chain:  2  iteration:  19580  parameters:  0.0547 0.0441 0.9236 5.2552 
chain:  2  iteration:  19590  parameters:  0.0327 0.0518 0.9298 5.695 
chain:  2  iteration:  19600  parameters:  0.0518 0.0499 0.9175 5.815 
chain:  2  iteration:  19610  parameters:  0.0374 0.0609 0.919 5.5506 
chain:  2  iteration:  19620  parameters:  0.0308 0.0611 0.9251 5.262 
chain:  2  iteration:  19630  parameters:  0.0546 0.0459 0.9252 5.251 
chain:  2  iteration:  19640  parameters:  0.0359 0.0496 0.9337 4.819 
chain:  2  iteration:  19650  parameters:  0.0476 0.0564 0.9233 4.8469 
chain:  2  iteration:  19660  parameters:  0.0459 0.0519 0.9256 4.8325 
chain:  2  iteration:  19670  parameters:  0.0402 0.0555 0.9262 4.9373 
chain:  2  iteration:  19680  parameters:  0.0417 0.0522 0.9293 5.1578 
chain:  2  iteration:  19690  parameters:  0.0437 0.0454 0.9319 5.0896 
chain:  2  iteration:  19700  parameters:  0.0595 0.0507 0.9115 5.6578 
chain:  2  iteration:  19710  parameters:  0.0491 0.0576 0.9187 5.9749 
chain:  2  iteration:  19720  parameters:  0.0453 0.053 0.9211 5.394 
chain:  2  iteration:  19730  parameters:  0.0452 0.0422 0.9359 5.3169 
chain:  2  iteration:  19740  parameters:  0.0377 0.0491 0.9322 5.1255 
chain:  2  iteration:  19750  parameters:  0.0373 0.0567 0.9203 5.7887 
chain:  2  iteration:  19760  parameters:  0.0485 0.0583 0.9185 5.5817 
chain:  2  iteration:  19770  parameters:  0.0324 0.0628 0.9242 5.1093 
chain:  2  iteration:  19780  parameters:  0.0546 0.0627 0.9089 5.3231 
chain:  2  iteration:  19790  parameters:  0.0359 0.0713 0.9173 5.2071 
chain:  2  iteration:  19800  parameters:  0.0559 0.045 0.9211 5.629 
chain:  2  iteration:  19810  parameters:  0.0456 0.0542 0.921 5.9181 
chain:  2  iteration:  19820  parameters:  0.0598 0.0448 0.9144 6.2835 
chain:  2  iteration:  19830  parameters:  0.048 0.0531 0.9208 6.2116 
chain:  2  iteration:  19840  parameters:  0.0367 0.0629 0.9185 5.9657 
chain:  2  iteration:  19850  parameters:  0.0503 0.0507 0.922 5.9153 
chain:  2  iteration:  19860  parameters:  0.0465 0.0534 0.918 6.44 
chain:  2  iteration:  19870  parameters:  0.0356 0.0624 0.9217 5.8918 
chain:  2  iteration:  19880  parameters:  0.0549 0.0553 0.9186 4.8452 
chain:  2  iteration:  19890  parameters:  0.037 0.0629 0.9241 4.6196 
chain:  2  iteration:  19900  parameters:  0.047 0.0608 0.9228 4.483 
chain:  2  iteration:  19910  parameters:  0.0588 0.0622 0.9121 4.2364 
chain:  2  iteration:  19920  parameters:  0.0676 0.0602 0.9045 4.5675 
chain:  2  iteration:  19930  parameters:  0.0753 0.0652 0.9003 4.4636 
chain:  2  iteration:  19940  parameters:  0.0853 0.0644 0.8939 4.7828 
chain:  2  iteration:  19950  parameters:  0.0625 0.0698 0.9053 4.462 
chain:  2  iteration:  19960  parameters:  0.07 0.0679 0.9039 4.568 
chain:  2  iteration:  19970  parameters:  0.0601 0.0713 0.9071 4.4584 
chain:  2  iteration:  19980  parameters:  0.061 0.067 0.906 4.7316 
chain:  2  iteration:  19990  parameters:  0.0597 0.0687 0.9045 4.7262 
chain:  2  iteration:  20000  parameters:  0.0709 0.059 0.909 4.6507 

A função gera as cadeias de Markov via Monte Carlo como um objeto da classe mcmc do pacote coda (Plummer et al., 2010). Este pacote contém funções para pós-processamento da saída MCMC; veja Plummer et alli (2006) para uma introdução. Observe que o coda ainda não é carregado automaticamente com bayesGARCH.

Um gráfico de convergência das Cadeias de Markov via Monte Carlo (MCMC) (ou seja, um gráfico de iterações vs. valores amostrados) podem ser gerados usando a função traceplot; a saída é exibida nos gráficos a seguir:

plot(MCMC_Milho)

e da mesma maneira temos o Monte Carlo Markov Chain para os log-retornos dos preços da soja:

plot(MCMC_Soja)

A convergência do amostrador (usando o diagnóstico teste de Gelman e Rubin (1992)), taxas de aceitação e autocorrelações nas cadeias podem ser computadas do seguinte modo:

library(coda)

gelman.diag(MCMC_Milho)
Potential scale reduction factors:

       Point est. Upper C.I.
alpha0       1.00       1.01
alpha1       1.01       1.01
beta         1.00       1.01
nu           1.00       1.01

Multivariate psrf

1
1 - rejectionRate(MCMC_Milho)
   alpha0    alpha1      beta        nu 
0.9329216 0.9329216 0.9772489 1.0000000 
autocorr.diag(MCMC_Milho)
          alpha0    alpha1      beta        nu
Lag 0  1.0000000 1.0000000 1.0000000 1.0000000
Lag 1  0.8777122 0.8842316 0.9790031 0.9624513
Lag 5  0.6923933 0.7061891 0.9066346 0.8713121
Lag 10 0.6156425 0.6320716 0.8305203 0.7647324
Lag 50 0.2927899 0.3001936 0.4034588 0.1644005

E para a soja temos:

gelman.diag(MCMC_Soja)
Potential scale reduction factors:

       Point est. Upper C.I.
alpha0       1.00       1.00
alpha1       1.00       1.00
beta         1.00       1.00
nu           1.02       1.07

Multivariate psrf

1.01
1 - rejectionRate(MCMC_Soja)
   alpha0    alpha1      beta        nu 
0.9393220 0.9393220 0.9757238 1.0000000 
autocorr.diag(MCMC_Soja)
          alpha0    alpha1      beta        nu
Lag 0  1.0000000 1.0000000 1.0000000 1.0000000
Lag 1  0.9021817 0.8588541 0.9821917 0.9767121
Lag 5  0.7678302 0.6884803 0.9208822 0.9175472
Lag 10 0.7054303 0.6282169 0.8509719 0.8488941
Lag 50 0.3668334 0.3408356 0.4549799 0.4022810

O diagnóstico de convergência não mostra evidências para a estabilização até as 3000 primeiras iterações (apenas a segunda metade da cadeia é usada por padrão em gelman.diag) já que o fator de redução de escala é menor que 1.01 e 1 para ambas as séries de retornos; (ver Gelman e Rubin (1992) para detalhes). O algoritmo de amostragem MCMC atinge taxas muito altas de aceitação variando de 93% para o vetor \(\alpha\) a 97% para \(\beta\) sugerindo que as distribuições propostas estão próximas das condicionais completas. A técnica de rejeição usada para gerar \(ν\) permite que um novo valor seja desenhado em cada passo no algoritmo Metropolis-Hastings.

As autocorrelações de um lag no range das cadeias vão de 0,87 para o parâmetro \(\alpha_1\) a 0,96 para o parâmetro \(v\) no milho e de 0.9 a 0.97 na soja. Usando a função formSmpl, descartamos as primeiras 10000 extrações da saída geral do MCMC como um burn in no período, mantendo apenas a cada segundo sorteio para diminuir a autocorrelação e mesclar as duas cadeias para obter um comprimento de amostra final de 10000.

smpl_Milho <- formSmpl(MCMC_Milho, l.bi = 10000, batch.size = 2)

n.chain:  2 
l.chain:  20000 
l.bi:  10000 
batch.size:  2 
smpl size:  10000 
smpl_Soja <- formSmpl(MCMC_Soja, l.bi = 10000, batch.size = 2)

n.chain:  2 
l.chain:  20000 
l.bi:  10000 
batch.size:  2 
smpl size:  10000 

Estatísticas básicas da posteriori podem ser facilmente obtidas com o método/comando de summary disponível para objetos mcmc.

summary(smpl_Milho)

Iterations = 1:10000
Thinning interval = 1 
Number of chains = 1 
Sample size per chain = 10000 

1. Empirical mean and standard deviation for each variable,
   plus standard error of the mean:

          Mean      SD  Naive SE Time-series SE
alpha0 0.08559 0.02327 0.0002327      0.0014817
alpha1 0.07628 0.01302 0.0001302      0.0008526
beta   0.89838 0.01649 0.0001649      0.0012750
nu     4.98773 0.45235 0.0045235      0.0288124

2. Quantiles for each variable:

          2.5%     25%     50%     75%  97.5%
alpha0 0.04911 0.06926 0.08225 0.09868 0.1395
alpha1 0.05485 0.06712 0.07486 0.08397 0.1056
beta   0.85943 0.88888 0.90060 0.90989 0.9246
nu     4.18372 4.66991 4.96372 5.26986 5.9622

E o output do modelo GARCH bayesiano para soja, fica:

summary(smpl_Soja)

Iterations = 1:10000
Thinning interval = 1 
Number of chains = 1 
Sample size per chain = 10000 

1. Empirical mean and standard deviation for each variable,
   plus standard error of the mean:

          Mean      SD  Naive SE Time-series SE
alpha0 0.05607 0.01560 0.0001560      0.0010556
alpha1 0.06024 0.01011 0.0001011      0.0006395
beta   0.91152 0.01488 0.0001488      0.0011339
nu     5.23259 0.47780 0.0047780      0.0299262

2. Quantiles for each variable:

          2.5%     25%     50%     75%   97.5%
alpha0 0.03167 0.04484 0.05391 0.06481 0.09304
alpha1 0.04288 0.05318 0.05942 0.06644 0.08292
beta   0.87656 0.90261 0.91333 0.92224 0.93639
nu     4.42763 4.89624 5.18296 5.52154 6.30022

As posterioris marginais são demonstradas nos gráficos para cada parâmetro do modelo:

df_smpl_Milho <- as.data.frame(smpl_Milho) %>% 
  mutate(alpha1_plus_beta = alpha1+beta)

df_smpl_Soja <- as.data.frame(smpl_Soja) %>% 
  mutate(alpha1_plus_beta = alpha1+beta)

a<-
ggplot(df_smpl_Milho, aes(x=alpha0 )) + 
 geom_histogram(aes(y=..density..), colour="black", fill="white")+
 geom_density(alpha=.2, fill="#FF6666")+ xlab("") +
  theme(axis.title.y = element_text(size = 7, angle = 90)) +
  theme(plot.title = element_text(size = 7, face = "bold")) +
  ylab("alpha0 Milho")

b<-
ggplot(df_smpl_Milho, aes(x=alpha1 )) + 
 geom_histogram(aes(y=..density..), colour="black", fill="white")+
 geom_density(alpha=.2, fill="#FF6666")+ xlab("") +
  theme(axis.title.y = element_text(size = 7, angle = 90)) +
  theme(plot.title = element_text(size = 7, face = "bold")) +
  ylab("alpha1 Milho")

c<-
ggplot(df_smpl_Milho, aes(x=beta )) + 
 geom_histogram(aes(y=..density..), colour="black", fill="white")+
 geom_density(alpha=.2, fill="#FF6666")+ xlab("") +
  theme(axis.title.y = element_text(size = 7, angle = 90)) +
  theme(plot.title = element_text(size = 7, face = "bold")) +
  ylab("beta Milho")

e<-
ggplot(df_smpl_Milho, aes(x=nu )) + 
 geom_histogram(aes(y=..density..), colour="black", fill="white")+
 geom_density(alpha=.2, fill="#FF6666") + xlab("") +
  theme(axis.title.y = element_text(size = 7, angle = 90)) +
  theme(plot.title = element_text(size = 7, face = "bold")) +
  ylab("nu Milho")

a+b

c+e

E para a soja:

a<-
ggplot(df_smpl_Soja, aes(x=alpha0 )) + 
 geom_histogram(aes(y=..density..), colour="black", fill="white")+
 geom_density(alpha=.2, fill="#FF6666")+ xlab("") +
  theme(axis.title.y = element_text(size = 7, angle = 90)) +
  theme(plot.title = element_text(size = 7, face = "bold")) +
  ylab("alpha0 Soja")

b<-
ggplot(df_smpl_Soja, aes(x=alpha1 )) + 
 geom_histogram(aes(y=..density..), colour="black", fill="white")+
 geom_density(alpha=.2, fill="#FF6666")+ xlab("") +
  theme(axis.title.y = element_text(size = 7, angle = 90)) +
  theme(plot.title = element_text(size = 7, face = "bold")) +
  ylab("alpha1 Soja")

c<-
ggplot(df_smpl_Soja, aes(x=beta )) + 
 geom_histogram(aes(y=..density..), colour="black", fill="white")+
 geom_density(alpha=.2, fill="#FF6666")+ xlab("") +
  theme(axis.title.y = element_text(size = 7, angle = 90)) +
  theme(plot.title = element_text(size = 7, face = "bold")) +
  ylab("beta Soja")

e<-
ggplot(df_smpl_Soja, aes(x=nu )) + 
 geom_histogram(aes(y=..density..), colour="black", fill="white")+
 geom_density(alpha=.2, fill="#FF6666") + xlab("") +
  theme(axis.title.y = element_text(size = 7, angle = 90)) +
  theme(plot.title = element_text(size = 7, face = "bold")) +
  ylab("nu Soja")

a+b

c+e

Notamos claramente a forma assimétrica dos histogramas; em especial para o parâmetro \(ν\).

Isso também se reflete nas diferenças entre as médias da posterioris e medianas. Esses resultados devem nos alertar contra o uso abusivo de justificativas assintóticas. No presente caso, mesmo um pouco mais de 3000 observações não são suficientes para justificar a simetria assintótica de uma aproximação normal para o estimador de parâmetros da distribuição.

Declarações probabilísticas sobre funções não lineares dos parâmetros do modelo podem ser obtidas diretamente por simulação da amostra posteriori conjunta. Em particular, podemos testar a condição de estacionaridade da covariância e estimar a densidade da variância incondicional quando esta condição for satisfeita. Sob a especificação GARCH(1,1), o processo é de covariância estacionária se \(\alpha_1 + \beta < 1\), como mostrado por Bollerslev (1986, página 310). O termo (\(\alpha_1\) + \(\beta\)) é o grau de persistência na autocorrelação ao quadrado que controla a intensidade do agrupamento no processo de volatilidade. Com um valor próximo de um, choques e variações passadas terão um impacto mais longo sobre a variância condicional futura.

Para fazer inferência sobre a persistência ao quadrado, simplesmente usamos a amostra à posteriori e geramos (\(\alpha^{[j]}_{1} + \beta^{[j]}\)) para cada sorteio \(\psi^{[j]}\) dentro a amostra à posteriori. A densidade da persistência da posteriori é plotada nos gráficos a seguir para ambas as séries de preços das commodities:

amaisb_milho <-
ggplot(df_smpl_Milho, aes(x=alpha1_plus_beta )) + 
 geom_histogram(aes(y=..density..), colour="black", fill="white")+
 geom_density(alpha=.2, fill="#FF6666")+
 geom_vline(aes(xintercept = 1))+ xlab("") +
  theme(axis.title.y = element_text(size = 7, angle = 90)) +
  theme(plot.title = element_text(size = 7, face = "bold")) +
  ylab("alpha1_plus_beta Milho") +
  ggtitle("Densidade da persistência na posteriori conjunta com base nas 10000 simulações")

amaisb_soja <-
ggplot(df_smpl_Soja, aes(x=alpha1_plus_beta )) + 
 geom_histogram(aes(y=..density..), colour="black", fill="white")+
 geom_density(alpha=.2, fill="#FF6666")+
 geom_vline(aes(xintercept = 1))+ xlab("") +
  theme(axis.title.y = element_text(size = 7, angle = 90)) +
  theme(plot.title = element_text(size = 7, face = "bold")) +
  ylab("alpha1_plus_beta Soja")

amaisb_milho + amaisb_soja

A assimetria de ambos os histogramas é calculada a seguir, lembrando que:

  • Assimetria negativa (á esquerda) = \(Moda>Mediana>Media\)

  • Assimetria positiva (á direita) = \(Moda<Mediana<Media\)

Calculamos que a assimetria da distribuição da soma dos parâmetros \(\alpha_1\) e \(\beta\) do milho são assimétricos à esquerda e a da soja à direita.

Mode <- function(x) {
  ux <- unique(x)
  ux[which.max(tabulate(match(x, ux)))]
}

moda_milho <- Mode(df_smpl_Milho$alpha1_plus_beta)
moda_milho
[1] 0.9796581
mediana_milho <- median(df_smpl_Milho$alpha1_plus_beta)
mediana_milho
[1] 0.9754111
media_milho <- mean(df_smpl_Milho$alpha1_plus_beta)
media_milho
[1] 0.9746609
#max_milho <- max(df_smpl_Milho$alpha1_plus_beta)
#max_milho

ifelse(moda_milho>mediana_milho & mediana_milho>media_milho, "Assimétrica á esquerda", "Assimétrica à direita")
[1] "Assimétrica á esquerda"

E no caso da soma de \(\alpha_1 + \beta\) para a soja, temos:

moda_soja <- Mode(df_smpl_Soja$alpha1_plus_beta)
moda_soja
[1] 0.9665299
mediana_soja <- median(df_smpl_Soja$alpha1_plus_beta)  
mediana_soja
[1] 0.9727276
media_soja <- mean(df_smpl_Soja$alpha1_plus_beta)
media_soja
[1] 0.9717605
#max_soja <- max(df_smpl_Soja$alpha1_plus_beta)  
#max_soja

ifelse(moda_soja > mediana_soja & mediana_soja > media_soja, "Assimétrica á esquerda", "Assimétrica à direita")
[1] "Assimétrica à direita"

Neste caso, a estacionaridade da covariância do processo é suportada pelos dados. A variância incondicional do modelo GARCH(1,1) é \(\alpha_0 / (1-\alpha_1 - \beta)\) dado que \(\alpha_1 + \beta < 1\).

df_smpl_Milho <- df_smpl_Milho %>%
  mutate(
    var.inc = 
      ifelse(
        alpha1+beta <1,
        alpha0 / (1 - alpha1 - beta),
        NA
        )
  )

df_smpl_Soja <- df_smpl_Soja %>%
  mutate(
    var.inc = 
      ifelse(
        alpha1+beta <1,
        alpha0 / (1 - alpha1 - beta),
        NA
        )
  )

As médias dessas variâncias incondicionais para cada uma das commodities será de:

mean(df_smpl_Milho$var.inc, na.rm = TRUE) # Variância Incondicional parametros milho
[1] 3.640434
mean(df_smpl_Soja$var.inc, na.rm = TRUE) # Variância Incondicional parametros soja
[1] 2.054484

Podemos construir um intervalo confiável de 90% para essa médias de variâncias incondicionais:

ci_90_Milho <- quantile(df_smpl_Milho$var.inc, probs = c(0.05, 0.95), na.rm = TRUE)
ci_90_Milho
      5%      95% 
2.559016 5.535347 
ci_90_Soja <- quantile(df_smpl_Soja$var.inc, probs = c(0.05, 0.95), na.rm = TRUE)
ci_90_Soja
      5%      95% 
1.622352 2.705940 

Outras declarações probabilísticas sobre interessantes funções dos parâmetros do modelo podem ser obtidas usando a amostra à posteriori conjunta. De acordo com a especificação (1, início da descrição do modelo GARCH Bayesiano na metodologia), a curtose condicional é \(3(ν − 2)/(ν − 4)\) desde que \(ν > 4\). Usando a amostra à posteriori, temos como estimar a probabilidade à posteriori de existência para a curtose condicional para cada uma das commodities

df_smpl_Milho <- df_smpl_Milho %>%
  mutate(
    curt.inc = 
      ifelse(
        nu > 4,
        3*(nu-2) / (nu -4) ,
        NA
        )
  )

mean(df_smpl_Milho$curt.inc, na.rm = TRUE) # Curtose condicional nu milho
[1] 16.00723
median(df_smpl_Milho$curt.inc, na.rm = TRUE) # Mediana nu milho
[1] 9.21222
var(df_smpl_Milho$curt.inc, na.rm = TRUE) # Variância empírica nu milho
[1] 103366.2

E para a soja temos:

df_smpl_Soja <- df_smpl_Soja %>%
  mutate(
    curt.inc = 
      ifelse(
        nu > 4,
        3*(nu-2) / (nu -4) ,
        NA
        )
  )

mean(df_smpl_Soja$curt.inc, na.rm = TRUE) # Curtose condicional nu Soja
[1] 9.021393
median(df_smpl_Soja$curt.inc, na.rm = TRUE) # Mediana nu Soja
[1] 8.072012
var(df_smpl_Milho$curt.inc, na.rm = TRUE) # Variância empírica nu Soja
[1] 103366.2

Condicionalmente após a existência, a média à posteriori da curtose é 16 (milho), 9.02 (soja) a mediana é 9.21 (milho) e 8.07 (soja) e o intervalo confiável de 95% é dado a seguir, indicando caudas mais pesadas do que para uma distribuição normal.

ci_95_Milho <- quantile(df_smpl_Milho$curt.inc, probs = c(0.025, 0.975), na.rm = TRUE)
ci_95_Milho
     2.5%     97.5% 
 6.046734 31.413569 
ci_95_Soja <- quantile(df_smpl_Soja$curt.inc, probs = c(0.025, 0.975), na.rm = TRUE)
ci_95_Soja
    2.5%    97.5% 
 5.60845 17.03077 

A assimetria positiva da posteriori para a curtose condicional é causada por alguns valores muito grandes (o valor máximo simulado é NA (milho) e 394.464454 (soja)). Estes valores correspondem a empates com \(ν\) ligeiramente maior que 4.

Note que se algum pesquisador desejar lidar com grandes valores para a curtose condicional de antemão, então pode-se definir \(\delta > 4\) na priori para \(ν\). Por exemplo, a escolha \(\delta = 4,5\) garantiria que a curtose fosse menor que 15.

Restições na priori e perturbações normais

O parâmetro de controle addPriorConditions pode ser usado para impor qualquer tipo de restrição nos parâmetros do modelo durante a estimação. Por exemplo, para garantir a estimativa de uma covariância estacionária no modelo GARCH(1,1), a função deve ser definida como

addPriorConditions <- function(psi)
 psi[2] + psi[3] < 1

Finalmente, podemos impor a normalidade das perturbações de maneira direta, definindo os hiperparâmetros \(\lambda = 100\) e \(\delta = 500\) na função bayesGARCH.

Então em nossos dados, teríamos a seguinte especificação:

MCMC_Milho <- bayesGARCH(retornos_Milho, lambda = 100, delta = 500,
                         control = list(n.chain = 2, l.chain = 20000,
                         addPriorConditions = addPriorConditions))

MCMC_Soja <- bayesGARCH(retornos_Soja, lambda = 100, delta = 500,
                         control = list(n.chain = 2, l.chain = 20000,
                         addPriorConditions = addPriorConditions))

Conselho prático

A estratégia de estimativa implementada no pacote bayesGARCH é totalmente automática e não requer nenhum ajuste do amostrador MCMC. Isso é certamente um recurso atraente para os praticantes. A geração das cadeias de Markov é, no entanto, morosa em tempo consumido e se estimando o modelo ao longo de vários conjuntos de dados em uma base diária pode, portanto, levar uma quantidade significativa de tempo.

Neste caso, o algoritmo pode ser facilmente paralelizado, executando uma única cadeia em vários processadores. Isso pode ser facilmente alcançado com o pacote foreach (REvolution Computing, 2010), por exemplo. Além disso, quando a estimativa é repetida sobre séries temporais atualizadas (ou seja, séries temporais com mais observações recentes), é aconselhável iniciar o algoritmo utilizando a média da posteriori ou mediana dos parâmetros obtidos na etapa de estimação da priori. O impacto dos valores iniciais (fase de burn in) provavelmente será menor e, portanto, a convergência mais rápida.

Por fim, observe que, como qualquer algoritmo Metropolis-Hastings, o amostrador pode ficar preso em um determinado valor, de modo que a cadeia não se move mais. No entanto, o amostrador usa densidades candidatas feitas por Taylor que são especialmente construídas em cada etapa, então é quase impossível para este amostrador MCMC ficar preso em um dado valor para muitos sorteios subsequentes. Por exemplo, para nosso conjunto de dados ainda obtemos resultados das posterioris que são quase iguais aos resultados que obtidos para os valores iniciais com razoáveis padrões c(0.01,0.1,0.7,20), mesmo se considerarmos os muito pobres valores iniciais c(0,1,0,01,0,4,50). No improvável caso esse mau comportamento ocorra, pode-se dimensionar os dados (para ter desvio padrão 1) ou executar o algoritmo com valores iniciais diferentes ou uma semente aleatória diferente.

 

 

 


Referências


Ardia, David (2008). Financial Risk Management with Bayesian Estimation of GARCH Models: Theory and Application. Lecture Notes in Economics and Mathematical Systems 612. Springer-Verlag, Berlin, Germany. ISBN 978-3-540-78656-6 doi:10.1007/978-3-540-78657-3

Ardia, David and Hoogerheide, Lennart F. (2010). Bayesian estimation of the GARCH(1,1) model with Student-t innovations. R Journal 2(2), pp.41-47 doi:10.32614/RJ-2010-014, Disponível em: journal.r-project.org/

Ardia, David (2009). Bayesian estimation of a Markov-switching threshold asymmetric GARCH model with Student-t innovations. Econometrics Journal 12(1), pp.105-126. doi:10.1111/j.1368-423X.2008.00253.x

T. Bollerslev. Generalized autoregressive conditional heteroskedasticity. Journal of Econometrics, 31(3): 307–327, Apr. 1986.

P. J. Deschamps. A flexible prior distribution for Markov switching autoregressions with Student-t errors. Journal of Econometrics, 133(1):153–190, July 2006.

R. F. Engle. Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation. Econometrica, 50(4):987–1008, July 1982.

A. Gelman and D. B. Rubin. Inference from iterative simulation using multiple sequences. Statistical Science, 7(4):457–472, Nov. 1992.

J. F. Geweke. Getting it right: Joint distribution tests of posterior simulators. Journal of the American Statistical Association, 99(467):799–804, Sept. 2004.

J. F. Geweke. Bayesian treatment of the independent Student-t linear model. Journal of Applied Econometrics, 8(S1):S19–S40, Dec. 1993.

A. Ghalanos. rgarch: Flexible GARCH modelling in R, 2010. URL http://r-forge.r-project.org/ projects/rgarch.

W. K. Hastings. Monte Carlo sampling methods using Markov chains and their applications. Biometrika, 57(1):97–109, Apr. 1970.

G. Koop. Bayesian Econometrics. Wiley-Interscience, London, UK, 2003. ISBN 0470845678.

N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, and E. Teller. Equations of state calculations by fast computing machines. Journal of Chemical Physics, 21(6):1087–1092, June 1953.

T. Nakatsuma. A Markov-chain sampling algorithm for GARCH models. Studies in Nonlinear Dynamics and Econometrics, 3(2):107–117, July 1998. URL http://www.bepress.com/snde/vol3/iss2/algorithm1/. Algorithm nr.1.

M. Plummer, N. Best, K. Cowles, and K. Vines. CODA: Convergence diagnosis and output analysis for MCMC. R News, 6(1):7–11, Mar. 2006.

M. Plummer, N. Best, K. Cowles, and K. Vines. coda: Output analysis and diagnostics for MCMC, 2010. URL http://CRAN.R-project.org/package=coda. R package version 0.13-5.

REvolution Computing. foreach: Foreach looping construct for R, 2009. URL http://CRAN.R-project.org/package=foreach.

C. Ritter and M. A. Tanner. Facilitating the Gibbs sampler: The Gibbs stopper and the Griddy-Gibbs sampler. Journal of the American Statistical Association, 87(419):861–868, Sept. 1992.

A. Trapletti and K. Hornik. tseries: Time series analysis and computational finance, 2009. URL http://CRAN.R-project.org/package=tseries.

D. Wuertz and Y. Chalabi. fGarch: Rmetrics - Autoregressive Conditional Heteroskedastic Modelling, 2009. URL http://CRAN.R-project.org/package=fGarch.

Pacotes do R

citation(package = "bayesGARCH")

To cite 'bayesGARCH' in publications use:

  Ardia, David (2008). Financial Risk Management with Bayesian
  Estimation of GARCH Models: Theory and Application. Lecture Notes in
  Economics and Mathematical Systems 612. Springer-Verlag, Berlin,
  Germany. ISBN 978-3-540-78656-6 doi:10.1007/978-3-540-78657-3

  Ardia, David and Hoogerheide, Lennart F. (2010). Bayesian estimation
  of the GARCH(1,1) model with Student-t innovations. R Journal 2(2),
  pp.41-47 doi:10.32614/RJ-2010-014

  Ardia, David (2009). Bayesian estimation of a Markov-switching
  threshold asymmetric GARCH model with Student-t innovations.
  Econometrics Journal 12(1), pp.105-126.
  doi:10.1111/j.1368-423X.2008.00253.x

To see these entries in BibTeX format, use 'print(<citation>,
bibtex=TRUE)', 'toBibtex(.)', or set
'options(citation.bibtex.max=999)'.
citation(package = "coda")

To cite package coda in publications use:

  Martyn Plummer, Nicky Best, Kate Cowles and Karen Vines (2006). CODA:
  Convergence Diagnosis and Output Analysis for MCMC, R News, vol 6,
  7-11

A BibTeX entry for LaTeX users is

  @Article{,
    title = {CODA: Convergence Diagnosis and Output Analysis for MCMC},
    author = {Martyn Plummer and Nicky Best and Kate Cowles and Karen Vines},
    journal = {R News},
    year = {2006},
    volume = {6},
    number = {1},
    pages = {7--11},
    url = {https://journal.r-project.org/archive/},
    pdf = {https://www.r-project.org/doc/Rnews/Rnews_2006-1.pdf},
  }

 

 


Tempo total de execução do documento:

end_time <- Sys.time()

end_time - start_time
Time difference of 6.096849 mins