Trabajo final

Serie Temporal del Plomo

Preparación de los datos

Cargamos las librerías necesarias para este trabajo.

library(dplyr)

Warning: package 'dplyr' was built under R version 4.2.3

library(forecast)

Warning: package 'forecast' was built under R version 4.2.3

library(tseries)

Warning: package 'tseries' was built under R version 4.2.3

library(lmtest)

Warning: package 'lmtest' was built under R version 4.2.3

Warning: package 'zoo' was built under R version 4.2.3

library(readr)
library(FinTS)

Warning: package 'FinTS' was built under R version 4.2.3

library(rugarch)

Warning: package 'rugarch' was built under R version 4.2.3

Importamos la base de datos, seleccionamos la variable plomo y la establecemos como serie temporal.

datos <- read.csv("../data/raw/DATOS_XNT_PROD_TRAD.csv", header = TRUE, sep = ";" , dec = ".")
Serie_plomo <- ts(datos$XNT15, start = c(1994,1), end = c(2021,8), frequency = 12)

Especificación

Vamos a explorar visualmente los datos.

par(mfrow = c(2,1))
plot(Serie_plomo)
boxplot(Serie_plomo ~ cycle(Serie_plomo))

par(mfrow = c(1,1))

La serie temporal del plomo presenta tendencia y volatilidad en general. Aunque podríamos partir la serie en dos periodos, tomando como año de corte el año del 2005. Sin embargo, para este trabajo optamos por toda la serie temporal.

Identificación

¿La serie es o no estacionaria?

Realicemos la prueba de raíz unitaria de Dicker-Fuller aumentado.

adf.test(Serie_plomo,alternative="stationary")

    Augmented Dickey-Fuller Test

data:  Serie_plomo
Dickey-Fuller = -2.3561, Lag order = 6, p-value = 0.4264
alternative hypothesis: stationary

Dado que el p-value es mayor que 5%, entonces rechazamos la hipótesis nula de estacionariedad de la serie temporal. Por ende, la serie plomo no es estacionaria.

Transformación de la serie

Identifiquemos el valor óptimo de lambda para la transformación Box-Cox.

lamb.x15 <- BoxCox.lambda(Serie_plomo)
lamb.x15

[1] 0.1352101

Aplicamos la transformación Box-Cox con el valor óptimo de lambda.

tran_box <- function(xt,lamb.x){
  if (lamb.x == 0) {
    xt_box <- log(xt)
  } else {
    xt_box <- (xt^lamb.x - 1)/lamb.x
  }
}

Serie_plomo_boxCox <- tran_box(Serie_plomo, lamb.x15)
plot(Serie_plomo_boxCox)

Realizamos la prueba de raíz unitaria sobre esta serie transformada.

adf.test(Serie_plomo_boxCox,alternative="stationary")

    Augmented Dickey-Fuller Test

data:  Serie_plomo_boxCox
Dickey-Fuller = -1.8879, Lag order = 6, p-value = 0.6239
alternative hypothesis: stationary

Aún resulta ser no estacionaria.

Tomamos la primera diferencia de esta serie transformada.

D_Serie_plomo_boxCox = diff(Serie_plomo_boxCox, differences = 1)
plot(D_Serie_plomo_boxCox)

Evaluamos si es estacionaria.

adf.test(D_Serie_plomo_boxCox,alternative="stationary")

Warning in adf.test(D_Serie_plomo_boxCox, alternative = "stationary"): p-value
smaller than printed p-value

    Augmented Dickey-Fuller Test

data:  D_Serie_plomo_boxCox
Dickey-Fuller = -10.17, Lag order = 6, p-value = 0.01
alternative hypothesis: stationary

Ahora, esta nueva serie transformada y diferencia resulta ser estacionaria.

Funciones de autocorrelación

par(mfrow = c(2,1))
acf(D_Serie_plomo_boxCox, lag.max=34, main = "Función de autocorrelación")
pacf(D_Serie_plomo_boxCox, lag.max=34, main = "Función de autocorrelación parcial")

par(mfrow = c(1,1))

De acuerdo a las funciones de autocorrelación, \(q=1\) y \(p=4\).

Estimación

modelo_2 = arima(Serie_plomo_boxCox, order = c(0,1,1))
summary(modelo_2)

Call:
arima(x = Serie_plomo_boxCox, order = c(0, 1, 1))

Coefficients:
          ma1
      -0.8114
s.e.   0.0263

sigma^2 estimated as 0.4218:  log likelihood = -327.35,  aic = 658.69

Training set error measures:
                     ME      RMSE       MAE        MPE     MAPE      MASE
Training set 0.06699502 0.6484927 0.5184251 -0.5057398 11.08249 0.7253665
                   ACF1
Training set -0.1720489

coeftest(modelo_2)

z test of coefficients:

     Estimate Std. Error z value  Pr(>|z|)    
ma1 -0.811391   0.026328 -30.819 < 2.2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Probemos el autoarima.

modelo_auto = auto.arima(Serie_plomo_boxCox)
summary(modelo_auto)

Series: Serie_plomo_boxCox 
ARIMA(2,1,1)(2,0,0)[12] 

Coefficients:
          ar1      ar2      ma1    sar1    sar2
      -0.2990  -0.1106  -0.6781  0.0872  0.1358
s.e.   0.0852   0.0755   0.0702  0.0548  0.0557

sigma^2 = 0.3975:  log likelihood = -315.31
AIC=642.63   AICc=642.89   BIC=665.44

Training set error measures:
                     ME      RMSE       MAE        MPE     MAPE      MASE
Training set 0.04632497 0.6247158 0.4928839 -0.7262001 10.48772 0.7049116
                     ACF1
Training set -0.006410278

coeftest(modelo_auto)

z test of coefficients:

      Estimate Std. Error z value  Pr(>|z|)    
ar1  -0.299014   0.085168 -3.5109 0.0004466 ***
ar2  -0.110563   0.075521 -1.4640 0.1431950    
ma1  -0.678100   0.070203 -9.6591 < 2.2e-16 ***
sar1  0.087189   0.054763  1.5921 0.1113601    
sar2  0.135843   0.055745  2.4369 0.0148155 *  
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Validación

Validación del modelo

Visualizamos el comportamiento de los residuos.

residuos_modelo2 = modelo_2$residuals
hist(residuos_modelo2)

boxplot(residuos_modelo2)

plot(residuos_modelo2)

Según el histograma, los residuos parecen distribuirse de manera normal. De acuerdo al gráfico de los residuos parece ser que la volatilidad se deba a los residuos.

Capturamos los residuos del modelo y realizamos una prueba de normalidad.

residuos_modelo2 = residuos_modelo2[!is.na(residuos_modelo2)]
jarque.bera.test(residuos_modelo2)

    Jarque Bera Test

data:  residuos_modelo2
X-squared = 5.8774, df = 2, p-value = 0.05293

#Funciones de autocorrelacion
par(mfrow = c(2,1))
acf(residuos_modelo2, lag.max=34, main = "Función de autocorrelación")
pacf(residuos_modelo2, lag.max=34, main = "Función de autocorrelación parcial")

par(mfrow = c(1,1))

Validación del modelo autoarima

Visualizamos el comportamiento de los residuos.

residuos_modelo_auto = modelo_auto$residuals
hist(residuos_modelo_auto)

boxplot(residuos_modelo_auto)

plot(residuos_modelo_auto)

Capturamos los residuos del modelo y realizamos una prueba de normalidad.

residuos_modelo_auto = residuos_modelo_auto[!is.na(residuos_modelo_auto)]
jarque.bera.test(residuos_modelo_auto)

    Jarque Bera Test

data:  residuos_modelo_auto
X-squared = 3.4096, df = 2, p-value = 0.1818

Este valor p significa que no hay suficiente evidencia estadística para rechazar la hipótesis nula a los niveles de significancia comunes (usualmente 0.05 o 0.01). Por lo tanto, con un valor p de 0.1818, puedes concluir que no se puede rechazar la hipótesis nula de que tus datos se distribuyen normalmente.

Ahora evaluamos las funciones de autocorrelación de los residuos.

par(mfrow = c(2,1))
acf(residuos_modelo_auto^2, lag.max=34, main = "Función de autocorrelación")
pacf(residuos_modelo_auto^2, lag.max=34, main = "Función de autocorrelación parcial")

par(mfrow = c(1,1))

Los correlogramas muestran que hay un comportamiento heterocedástico de los residuos.

Heterocedasticidad

Identificación

Analizamos los residuos al cuadrado.

checkresiduals(modelo_auto)

    Ljung-Box test

data:  Residuals from ARIMA(2,1,1)(2,0,0)[12]
Q* = 21.437, df = 19, p-value = 0.3131

Model df: 5.   Total lags used: 24

Errores_cuadrado = resid(modelo_auto)^2

Indicamos que sea una serie de tiempo.

Serie_Errores_cuadrado <- ts(Errores_cuadrado, start = c(1994,1), end = c(2021,8), frequency = 12)
print(Serie_Errores_cuadrado)

              Jan          Feb          Mar          Apr          May
1994 8.825083e-06 4.024444e-02 1.253395e-03 1.471864e-01 1.270564e+00
1995 6.614774e-02 2.688251e-01 6.949149e-01 4.105068e-02 2.376022e-01
1996 1.811964e-01 4.743478e-01 9.453873e-04 5.524046e-01 1.332695e-01
1997 3.345305e-01 3.765696e-01 4.551632e-02 1.711462e-01 9.295233e-04
1998 1.169296e+00 1.468641e-02 5.899657e-01 1.495268e-03 1.896942e-02
1999 1.566131e+00 3.207233e-01 9.348788e-05 9.809146e-02 2.722979e-01
2000 1.395422e-03 5.575112e-01 6.953595e-01 6.645878e-02 3.216144e-01
2001 2.825342e-01 4.661894e-02 4.999611e-02 1.698966e-01 8.888091e-04
2002 9.917972e-02 1.675313e-02 2.635208e-01 1.207040e-01 2.114733e-01
2003 1.879075e-03 1.612431e+00 2.808369e-01 6.975479e-02 1.041209e+00
2004 1.218799e+00 1.273448e+00 7.501892e-01 1.249055e+00 2.346700e-02
2005 4.860391e-01 9.000632e-01 2.446237e+00 6.232073e-02 1.636292e-02
2006 3.182225e-01 2.085195e-01 9.335448e-02 1.483562e+00 7.433158e-03
2007 4.195750e-03 3.324688e-01 1.633146e-01 3.797582e-03 1.739742e-01
2008 7.666534e-01 2.244157e-01 1.040309e-01 6.131235e-04 3.120516e-02
2009 2.048392e+00 1.214144e-02 1.281780e+00 8.426437e-02 8.861281e-01
2010 4.329416e-02 4.272663e-01 5.231308e-01 2.906261e-01 4.444931e-01
2011 3.291230e-03 1.746469e-01 5.526611e-04 8.543515e-01 1.661042e+00
2012 6.489195e-01 1.434283e+00 1.385947e-02 2.893160e-01 1.249121e-01
2013 2.013479e+00 9.265711e-01 1.053469e+00 1.546321e-02 7.330080e-02
2014 1.578628e-01 4.806665e+00 4.708307e-02 1.884630e-02 3.878253e-02
2015 6.454511e-02 1.728557e-01 6.463158e-02 3.626115e-01 9.353012e-02
2016 2.337972e+00 5.925810e-01 1.995100e-01 5.785447e-02 2.777856e-05
2017 4.113327e-01 9.985676e-04 1.409161e+00 6.557746e-02 1.487650e-02
2018 1.899106e-02 4.289023e-05 1.466559e-01 2.611568e-02 1.215868e-01
2019 2.477108e-02 3.634477e-03 8.300501e-04 6.743738e-02 1.517375e-01
2020 1.663938e-02 1.467006e-03 3.105993e-02 7.004251e-01 6.099263e-01
2021 6.864445e-02 3.979423e-04 1.797476e-01 2.335375e-01 1.752703e+00
              Jun          Jul          Aug          Sep          Oct
1994 3.447020e-02 3.005274e-01 2.312610e-02 2.419328e-01 2.452580e-02
1995 6.408050e-01 7.253964e-01 4.063560e-02 2.185617e-03 1.566772e-03
1996 1.532564e-01 7.598455e-02 8.791211e-01 1.949520e-01 4.744201e-02
1997 8.676976e-02 9.452531e-03 1.520567e-03 6.595357e-03 2.285296e-01
1998 5.261544e-01 1.341044e-01 4.885417e-03 1.946850e-01 2.619521e-02
1999 1.625383e-01 1.814838e-03 1.358283e+00 1.392537e-01 1.327628e+00
2000 1.676657e-01 9.176032e-02 2.634954e-01 6.896724e-01 1.062494e+00
2001 2.284835e-01 1.346964e-02 5.503878e-03 4.452187e-03 5.258144e-02
2002 1.059794e+00 1.662591e-01 2.408379e-01 6.601447e-02 1.118810e-03
2003 1.265695e+00 1.852076e-01 1.646482e+00 1.359100e-02 4.180992e-01
2004 6.777530e-01 2.257012e-01 4.544355e-02 2.805319e-02 3.392562e-01
2005 1.315076e-02 7.652890e-02 4.717843e-01 2.598223e-04 1.813930e-01
2006 3.971588e-01 6.120360e-01 1.653679e-02 8.984458e-01 1.272011e+00
2007 3.183082e-02 1.055047e+00 4.354756e-01 3.592285e-01 5.924966e-01
2008 5.984612e-01 4.746537e-04 1.500940e-01 3.994908e-01 1.080939e-01
2009 9.885582e-01 1.207483e-01 1.400866e-01 8.445440e-01 4.127721e-01
2010 2.895186e-01 1.184524e-03 1.087646e-02 6.765764e-03 6.230866e-01
2011 2.310692e-01 5.499352e-01 3.159716e-01 2.074062e-01 8.870591e-02
2012 2.552921e-01 6.584762e-02 2.511086e-01 1.646498e-01 3.142328e-02
2013 7.330873e-02 2.869306e-02 4.971568e-01 2.565666e-01 4.255661e-01
2014 3.181062e-03 1.131758e+00 6.604675e-01 2.051388e-01 4.191605e-01
2015 1.940263e-02 6.219102e-01 1.526798e-01 7.052038e-02 1.343312e-01
2016 2.969063e-03 2.288348e-01 7.900553e-02 5.630602e-01 6.820749e-02
2017 2.082753e-01 5.632729e-02 1.386050e-01 3.901150e-01 1.906632e-01
2018 2.207238e-01 6.627297e-01 4.084611e-01 4.817538e-02 1.031621e-01
2019 1.630594e-01 1.602105e-01 5.823222e-02 8.058727e-02 2.751699e-01
2020 5.503419e-01 9.694237e-03 1.611402e+00 1.703908e-01 1.983250e-02
2021 1.091478e-01 2.219856e-01 6.444600e-01                          
              Nov          Dec
1994 5.946214e-01 1.028440e+00
1995 2.163990e-01 6.922266e-01
1996 1.751961e-02 1.627554e-02
1997 1.576371e-01 4.597916e-01
1998 3.758264e-02 5.928953e-01
1999 2.396269e-01 1.370542e-03
2000 1.724204e-01 2.138946e-02
2001 1.135316e+00 8.051752e-01
2002 3.507218e-01 8.894975e-01
2003 9.225909e-01 2.275598e-01
2004 2.735507e-01 3.578905e-02
2005 1.255500e-01 1.304803e+00
2006 1.026188e-02 4.235256e+00
2007 4.002932e-01 4.156399e-01
2008 1.374739e+00 1.000343e+00
2009 1.512642e+00 1.333438e+00
2010 2.343172e+00 5.629604e-01
2011 3.065836e+00 1.172939e-01
2012 5.876230e-02 8.646243e-02
2013 5.251270e-02 1.683282e-01
2014 5.960548e-01 5.675809e-03
2015 2.007938e-01 5.001929e-01
2016 5.019279e-01 3.268619e-02
2017 2.919417e-03 7.992851e-03
2018 5.921426e-01 1.069491e-03
2019 7.215070e-05 3.155599e-01
2020 3.754193e-04 4.420621e-02
2021                          

Visualizamos los residuos.

par(mfrow = c(2,1))
plot(Errores_cuadrado) 
boxplot(Errores_cuadrado)

par(mfrow = c(1,1))

Prueba de componente ARCH en el modelo

ModeloARCH1 = ArchTest(D_Serie_plomo_boxCox, lags = 1, demean = TRUE)
ModeloARCH1

    ARCH LM-test; Null hypothesis: no ARCH effects

data:  D_Serie_plomo_boxCox
Chi-squared = 32.35, df = 1, p-value = 1.288e-08

Concluímos la presencia de heterocedasticidad.

ModeloARCH2 = ArchTest(D_Serie_plomo_boxCox, lags = 2, demean = TRUE)
ModeloARCH2

    ARCH LM-test; Null hypothesis: no ARCH effects

data:  D_Serie_plomo_boxCox
Chi-squared = 34.804, df = 2, p-value = 2.77e-08

Concluímos la presencia de heterocedasticidad.

ModeloARCH3 = ArchTest(D_Serie_plomo_boxCox, lags = 3, demean = TRUE)
ModeloARCH3

    ARCH LM-test; Null hypothesis: no ARCH effects

data:  D_Serie_plomo_boxCox
Chi-squared = 35.158, df = 3, p-value = 1.128e-07

Concluímos la presencia de heterocedasticidad.

ModeloARCH4 = ArchTest(D_Serie_plomo_boxCox, lags = 4, demean = TRUE)
ModeloARCH4

    ARCH LM-test; Null hypothesis: no ARCH effects

data:  D_Serie_plomo_boxCox
Chi-squared = 37.411, df = 4, p-value = 1.482e-07

Concluímos la presencia de heterocedasticidad.

En general, concluímos que la serie temporal del plomo presenta heterocedasticidad.

Estimación

Plnateamiento del modelo ARCH / GARCH

Tomamos como punto de partida a:

ugarch0 = ugarchspec()
ugarch0

*---------------------------------*
*       GARCH Model Spec          *
*---------------------------------*

Conditional Variance Dynamics   
------------------------------------
GARCH Model     : sGARCH(1,1)
Variance Targeting  : FALSE 

Conditional Mean Dynamics
------------------------------------
Mean Model      : ARFIMA(1,0,1)
Include Mean        : TRUE 
GARCH-in-Mean       : FALSE 

Conditional Distribution
------------------------------------
Distribution    :  norm 
Includes Skew   :  FALSE 
Includes Shape  :  FALSE 
Includes Lambda :  FALSE 

Modelo GARCH (0,1)

ugarch01 = ugarchspec(mean.model = list(armaOrder = c(0,1)))
ugfit01 = ugarchfit(spec = ugarch01, data = D_Serie_plomo_boxCox)
ugfit01

*---------------------------------*
*          GARCH Model Fit        *
*---------------------------------*

Conditional Variance Dynamics   
-----------------------------------
GARCH Model : sGARCH(1,1)
Mean Model  : ARFIMA(0,0,1)
Distribution    : norm 

Optimal Parameters
------------------------------------
        Estimate  Std. Error  t value Pr(>|t|)
mu      0.009657    0.006217   1.5532  0.12038
ma1    -0.826875    0.026231 -31.5228  0.00000
omega   0.006634    0.007570   0.8764  0.38081
alpha1  0.025464    0.016137   1.5780  0.11456
beta1   0.958633    0.022998  41.6832  0.00000

Robust Standard Errors:
        Estimate  Std. Error  t value Pr(>|t|)
mu      0.009657    0.006011   1.6066 0.108147
ma1    -0.826875    0.017672 -46.7910 0.000000
omega   0.006634    0.005155   1.2870 0.198101
alpha1  0.025464    0.015097   1.6868 0.091651
beta1   0.958633    0.014491  66.1560 0.000000

LogLikelihood : -323.4115 

Information Criteria
------------------------------------
                   
Akaike       1.9844
Bayes        2.0418
Shibata      1.9839
Hannan-Quinn 2.0073

Weighted Ljung-Box Test on Standardized Residuals
------------------------------------
                        statistic   p-value
Lag[1]                      8.285 3.996e-03
Lag[2*(p+q)+(p+q)-1][2]     8.538 6.319e-08
Lag[4*(p+q)+(p+q)-1][5]    10.659 5.655e-04
d.o.f=1
H0 : No serial correlation

Weighted Ljung-Box Test on Standardized Squared Residuals
------------------------------------
                        statistic p-value
Lag[1]                    0.04123  0.8391
Lag[2*(p+q)+(p+q)-1][5]   0.84588  0.8935
Lag[4*(p+q)+(p+q)-1][9]   1.53999  0.9522
d.o.f=2

Weighted ARCH LM Tests
------------------------------------
            Statistic Shape Scale P-Value
ARCH Lag[3]    0.2293 0.500 2.000  0.6320
ARCH Lag[5]    1.5923 1.440 1.667  0.5684
ARCH Lag[7]    1.8783 2.315 1.543  0.7431

Nyblom stability test
------------------------------------
Joint Statistic:  0.9311
Individual Statistics:             
mu     0.1535
ma1    0.2354
omega  0.2405
alpha1 0.1672
beta1  0.2202

Asymptotic Critical Values (10% 5% 1%)
Joint Statistic:         1.28 1.47 1.88
Individual Statistic:    0.35 0.47 0.75

Sign Bias Test
------------------------------------
                   t-value   prob sig
Sign Bias           0.7345 0.4631    
Negative Sign Bias  0.1926 0.8474    
Positive Sign Bias  0.5729 0.5671    
Joint Effect        1.1290 0.7701    


Adjusted Pearson Goodness-of-Fit Test:
------------------------------------
  group statistic p-value(g-1)
1    20     25.44      0.14671
2    30     24.20      0.71919
3    40     53.89      0.05669
4    50     51.63      0.37145


Elapsed time : 0.09644699 

Modelo GARCH (0,2)

ugarch02 = ugarchspec(mean.model = list(armaOrder = c(0,2)))
ugfit02 = ugarchfit(spec = ugarch02, data = D_Serie_plomo_boxCox)
ugfit02

*---------------------------------*
*          GARCH Model Fit        *
*---------------------------------*

Conditional Variance Dynamics   
-----------------------------------
GARCH Model : sGARCH(1,1)
Mean Model  : ARFIMA(0,0,2)
Distribution    : norm 

Optimal Parameters
------------------------------------
        Estimate  Std. Error   t value Pr(>|t|)
mu      0.010060    0.007187   1.39974  0.16159
ma1    -0.994372    0.054167 -18.35756  0.00000
ma2     0.199573    0.053482   3.73156  0.00019
omega   0.006903    0.007457   0.92565  0.35463
alpha1  0.023320    0.015818   1.47425  0.14041
beta1   0.959672    0.023234  41.30393  0.00000

Robust Standard Errors:
        Estimate  Std. Error  t value Pr(>|t|)
mu      0.010060    0.006890   1.4600 0.144292
ma1    -0.994372    0.057946 -17.1602 0.000000
ma2     0.199573    0.054301   3.6753 0.000238
omega   0.006903    0.005238   1.3178 0.187565
alpha1  0.023320    0.014253   1.6361 0.101813
beta1   0.959672    0.015920  60.2820 0.000000

LogLikelihood : -317.0428 

Information Criteria
------------------------------------
                   
Akaike       1.9519
Bayes        2.0208
Shibata      1.9513
Hannan-Quinn 1.9794

Weighted Ljung-Box Test on Standardized Residuals
------------------------------------
                        statistic p-value
Lag[1]                    0.00522  0.9424
Lag[2*(p+q)+(p+q)-1][5]   0.63198  1.0000
Lag[4*(p+q)+(p+q)-1][9]   1.43287  0.9978
d.o.f=2
H0 : No serial correlation

Weighted Ljung-Box Test on Standardized Squared Residuals
------------------------------------
                        statistic p-value
Lag[1]                     0.2052  0.6505
Lag[2*(p+q)+(p+q)-1][5]    1.2548  0.7998
Lag[4*(p+q)+(p+q)-1][9]    2.7599  0.7979
d.o.f=2

Weighted ARCH LM Tests
------------------------------------
            Statistic Shape Scale P-Value
ARCH Lag[3]    0.0367 0.500 2.000  0.8481
ARCH Lag[5]    2.0805 1.440 1.667  0.4537
ARCH Lag[7]    3.1926 2.315 1.543  0.4780

Nyblom stability test
------------------------------------
Joint Statistic:  1.2068
Individual Statistics:              
mu     0.11081
ma1    0.32076
ma2    0.06215
omega  0.19118
alpha1 0.14871
beta1  0.17799

Asymptotic Critical Values (10% 5% 1%)
Joint Statistic:         1.49 1.68 2.12
Individual Statistic:    0.35 0.47 0.75

Sign Bias Test
------------------------------------
                   t-value    prob sig
Sign Bias            2.021 0.04411  **
Negative Sign Bias   1.093 0.27526    
Positive Sign Bias   1.137 0.25639    
Joint Effect         4.085 0.25246    


Adjusted Pearson Goodness-of-Fit Test:
------------------------------------
  group statistic p-value(g-1)
1    20     13.23       0.8266
2    30     27.82       0.5275
3    40     31.42       0.8009
4    50     33.20       0.9591


Elapsed time : 0.0791769 

Modelo GARCH (0,3)

ugarch03 = ugarchspec(mean.model = list(armaOrder = c(0,3)))
ugfit03 = ugarchfit(spec = ugarch03, data = D_Serie_plomo_boxCox)
ugfit03

*---------------------------------*
*          GARCH Model Fit        *
*---------------------------------*

Conditional Variance Dynamics   
-----------------------------------
GARCH Model : sGARCH(1,1)
Mean Model  : ARFIMA(0,0,3)
Distribution    : norm 

Optimal Parameters
------------------------------------
        Estimate  Std. Error   t value Pr(>|t|)
mu      0.009986    0.007130   1.40060 0.161335
ma1    -0.998133    0.057503 -17.35795 0.000000
ma2     0.213962    0.084785   2.52359 0.011616
ma3    -0.012619    0.056749  -0.22236 0.824034
omega   0.006962    0.007531   0.92453 0.355212
alpha1  0.023730    0.016111   1.47291 0.140774
beta1   0.959089    0.023501  40.81061 0.000000

Robust Standard Errors:
        Estimate  Std. Error  t value Pr(>|t|)
mu      0.009986    0.006895   1.4482 0.147554
ma1    -0.998133    0.059362 -16.8142 0.000000
ma2     0.213962    0.090169   2.3729 0.017649
ma3    -0.012619    0.060989  -0.2069 0.836086
omega   0.006962    0.005348   1.3018 0.192978
alpha1  0.023730    0.015266   1.5545 0.120067
beta1   0.959089    0.016655  57.5860 0.000000

LogLikelihood : -317.0596 

Information Criteria
------------------------------------
                   
Akaike       1.9581
Bayes        2.0385
Shibata      1.9572
Hannan-Quinn 1.9901

Weighted Ljung-Box Test on Standardized Residuals
------------------------------------
                         statistic p-value
Lag[1]                   0.0001023  0.9919
Lag[2*(p+q)+(p+q)-1][8]  1.2486126  1.0000
Lag[4*(p+q)+(p+q)-1][14] 4.0148393  0.9736
d.o.f=3
H0 : No serial correlation

Weighted Ljung-Box Test on Standardized Squared Residuals
------------------------------------
                        statistic p-value
Lag[1]                     0.1953  0.6585
Lag[2*(p+q)+(p+q)-1][5]    1.2253  0.8069
Lag[4*(p+q)+(p+q)-1][9]    2.7848  0.7939
d.o.f=2

Weighted ARCH LM Tests
------------------------------------
            Statistic Shape Scale P-Value
ARCH Lag[3]   0.02798 0.500 2.000  0.8671
ARCH Lag[5]   2.04544 1.440 1.667  0.4613
ARCH Lag[7]   3.23763 2.315 1.543  0.4699

Nyblom stability test
------------------------------------
Joint Statistic:  1.3755
Individual Statistics:              
mu     0.11278
ma1    0.31865
ma2    0.06431
ma3    0.14812
omega  0.19207
alpha1 0.14822
beta1  0.17747

Asymptotic Critical Values (10% 5% 1%)
Joint Statistic:         1.69 1.9 2.35
Individual Statistic:    0.35 0.47 0.75

Sign Bias Test
------------------------------------
                   t-value    prob sig
Sign Bias           1.8514 0.06501   *
Negative Sign Bias  0.9903 0.32277    
Positive Sign Bias  1.0592 0.29028    
Joint Effect        3.4282 0.33020    


Adjusted Pearson Goodness-of-Fit Test:
------------------------------------
  group statistic p-value(g-1)
1    20     14.44       0.7576
2    30     26.92       0.5763
3    40     40.84       0.3894
4    50     33.50       0.9555


Elapsed time : 0.1280332 

Modelo GARCH (0,4)

ugarch04 = ugarchspec(mean.model = list(armaOrder = c(0,4)))
ugfit04 = ugarchfit(spec = ugarch04, data = D_Serie_plomo_boxCox)
ugfit04

*---------------------------------*
*          GARCH Model Fit        *
*---------------------------------*

Conditional Variance Dynamics   
-----------------------------------
GARCH Model : sGARCH(1,1)
Mean Model  : ARFIMA(0,0,4)
Distribution    : norm 

Optimal Parameters
------------------------------------
        Estimate  Std. Error   t value Pr(>|t|)
mu      0.009932    0.006857   1.44848 0.147484
ma1    -0.992990    0.056461 -17.58706 0.000000
ma2     0.201948    0.080330   2.51399 0.011938
ma3     0.049616    0.078187   0.63458 0.525703
ma4    -0.063896    0.055766  -1.14579 0.251881
omega   0.006949    0.007670   0.90602 0.364928
alpha1  0.022488    0.016090   1.39759 0.162235
beta1   0.960239    0.023906  40.16755 0.000000

Robust Standard Errors:
        Estimate  Std. Error   t value Pr(>|t|)
mu      0.009932    0.006678   1.48734 0.136925
ma1    -0.992990    0.058311 -17.02927 0.000000
ma2     0.201948    0.086046   2.34697 0.018927
ma3     0.049616    0.079027   0.62784 0.530111
ma4    -0.063896    0.055635  -1.14849 0.250765
omega   0.006949    0.005384   1.29061 0.196839
alpha1  0.022488    0.015586   1.44282 0.149070
beta1   0.960239    0.016793  57.18020 0.000000

LogLikelihood : -316.4289 

Information Criteria
------------------------------------
                   
Akaike       1.9603
Bayes        2.0522
Shibata      1.9592
Hannan-Quinn 1.9969

Weighted Ljung-Box Test on Standardized Residuals
------------------------------------
                         statistic p-value
Lag[1]                    0.001635  0.9677
Lag[2*(p+q)+(p+q)-1][11]  1.301535  1.0000
Lag[4*(p+q)+(p+q)-1][19]  5.666099  0.9819
d.o.f=4
H0 : No serial correlation

Weighted Ljung-Box Test on Standardized Squared Residuals
------------------------------------
                        statistic p-value
Lag[1]                     0.1761  0.6747
Lag[2*(p+q)+(p+q)-1][5]    1.3297  0.7817
Lag[4*(p+q)+(p+q)-1][9]    2.9565  0.7661
d.o.f=2

Weighted ARCH LM Tests
------------------------------------
            Statistic Shape Scale P-Value
ARCH Lag[3]    0.1805 0.500 2.000  0.6710
ARCH Lag[5]    2.2894 1.440 1.667  0.4108
ARCH Lag[7]    3.4861 2.315 1.543  0.4267

Nyblom stability test
------------------------------------
Joint Statistic:  1.77
Individual Statistics:              
mu     0.12647
ma1    0.31704
ma2    0.06312
ma3    0.13194
ma4    0.35910
omega  0.20227
alpha1 0.15233
beta1  0.18671

Asymptotic Critical Values (10% 5% 1%)
Joint Statistic:         1.89 2.11 2.59
Individual Statistic:    0.35 0.47 0.75

Sign Bias Test
------------------------------------
                   t-value    prob sig
Sign Bias            2.203 0.02831  **
Negative Sign Bias   1.217 0.22458    
Positive Sign Bias   1.230 0.21977    
Joint Effect         4.853 0.18291    


Adjusted Pearson Goodness-of-Fit Test:
------------------------------------
  group statistic p-value(g-1)
1    20     16.13       0.6486
2    30     18.03       0.9436
3    40     35.28       0.6401
4    50     38.94       0.8478


Elapsed time : 0.156929 

Modelo GARCH (1,1)

ugarch11 = ugarchspec(mean.model = list(armaOrder = c(1,1)))
ugfit11 = ugarchfit(spec = ugarch11, data = D_Serie_plomo_boxCox)
ugfit11

*---------------------------------*
*          GARCH Model Fit        *
*---------------------------------*

Conditional Variance Dynamics   
-----------------------------------
GARCH Model : sGARCH(1,1)
Mean Model  : ARFIMA(1,0,1)
Distribution    : norm 

Optimal Parameters
------------------------------------
        Estimate  Std. Error   t value Pr(>|t|)
mu      0.009751    0.006818   1.43016 0.152671
ar1    -0.225950    0.063675  -3.54850 0.000387
ma1    -0.761593    0.040378 -18.86139 0.000000
omega   0.006686    0.007181   0.93112 0.351792
alpha1  0.025485    0.015938   1.59904 0.109812
beta1   0.958066    0.022668  42.26425 0.000000

Robust Standard Errors:
        Estimate  Std. Error  t value Pr(>|t|)
mu      0.009751    0.006571   1.4839 0.137836
ar1    -0.225950    0.060184  -3.7543 0.000174
ma1    -0.761593    0.032027 -23.7795 0.000000
omega   0.006686    0.005188   1.2888 0.197465
alpha1  0.025485    0.014365   1.7741 0.076053
beta1   0.958066    0.015800  60.6355 0.000000

LogLikelihood : -317.4172 

Information Criteria
------------------------------------
                   
Akaike       1.9542
Bayes        2.0231
Shibata      1.9535
Hannan-Quinn 1.9817

Weighted Ljung-Box Test on Standardized Residuals
------------------------------------
                        statistic p-value
Lag[1]                     0.0508  0.8217
Lag[2*(p+q)+(p+q)-1][5]    1.4665  0.9985
Lag[4*(p+q)+(p+q)-1][9]    2.3031  0.9677
d.o.f=2
H0 : No serial correlation

Weighted Ljung-Box Test on Standardized Squared Residuals
------------------------------------
                        statistic p-value
Lag[1]                     0.2032  0.6521
Lag[2*(p+q)+(p+q)-1][5]    1.1546  0.8238
Lag[4*(p+q)+(p+q)-1][9]    2.6946  0.8081
d.o.f=2

Weighted ARCH LM Tests
------------------------------------
            Statistic Shape Scale P-Value
ARCH Lag[3]   0.01337 0.500 2.000  0.9080
ARCH Lag[5]   1.90820 1.440 1.667  0.4919
ARCH Lag[7]   3.11174 2.315 1.543  0.4927

Nyblom stability test
------------------------------------
Joint Statistic:  1.0279
Individual Statistics:             
mu     0.1207
ar1    0.3345
ma1    0.3564
omega  0.1972
alpha1 0.1509
beta1  0.1811

Asymptotic Critical Values (10% 5% 1%)
Joint Statistic:         1.49 1.68 2.12
Individual Statistic:    0.35 0.47 0.75

Sign Bias Test
------------------------------------
                   t-value   prob sig
Sign Bias           0.8957 0.3711    
Negative Sign Bias  0.4881 0.6258    
Positive Sign Bias  0.4861 0.6272    
Joint Effect        0.8037 0.8486    


Adjusted Pearson Goodness-of-Fit Test:
------------------------------------
  group statistic p-value(g-1)
1    20     13.83       0.7933
2    30     22.38       0.8040
3    40     38.67       0.4849
4    50     43.47       0.6960


Elapsed time : 0.09967899 

Modelo GARCH (1,2)

ugarch12 = ugarchspec(mean.model = list(armaOrder = c(1,2)))
ugfit12 = ugarchfit(spec = ugarch12, data = D_Serie_plomo_boxCox)
ugfit12

*---------------------------------*
*          GARCH Model Fit        *
*---------------------------------*

Conditional Variance Dynamics   
-----------------------------------
GARCH Model : sGARCH(1,1)
Mean Model  : ARFIMA(1,0,2)
Distribution    : norm 

Optimal Parameters
------------------------------------
        Estimate  Std. Error  t value Pr(>|t|)
mu      0.010016    0.007150  1.40089 0.161248
ar1    -0.037757    0.217066 -0.17394 0.861909
ma1    -0.959073    0.211598 -4.53254 0.000006
ma2     0.170786    0.175976  0.97051 0.331793
omega   0.006887    0.007426  0.92742 0.353709
alpha1  0.023616    0.015930  1.48249 0.138209
beta1   0.959412    0.023210 41.33580 0.000000

Robust Standard Errors:
        Estimate  Std. Error  t value Pr(>|t|)
mu      0.010016    0.006884  1.45486  0.14571
ar1    -0.037757    0.180183 -0.20955  0.83402
ma1    -0.959073    0.186571 -5.14053  0.00000
ma2     0.170786    0.148025  1.15376  0.24860
omega   0.006887    0.005259  1.30959  0.19034
alpha1  0.023616    0.014707  1.60576  0.10833
beta1   0.959412    0.016189 59.26458  0.00000

LogLikelihood : -317.0279 

Information Criteria
------------------------------------
                   
Akaike       1.9579
Bayes        2.0383
Shibata      1.9570
Hannan-Quinn 1.9899

Weighted Ljung-Box Test on Standardized Residuals
------------------------------------
                         statistic p-value
Lag[1]                    0.001225  0.9721
Lag[2*(p+q)+(p+q)-1][8]   1.225885  1.0000
Lag[4*(p+q)+(p+q)-1][14]  3.987657  0.9748
d.o.f=3
H0 : No serial correlation

Weighted Ljung-Box Test on Standardized Squared Residuals
------------------------------------
                        statistic p-value
Lag[1]                     0.2014  0.6536
Lag[2*(p+q)+(p+q)-1][5]    1.2422  0.8029
Lag[4*(p+q)+(p+q)-1][9]    2.7800  0.7947
d.o.f=2

Weighted ARCH LM Tests
------------------------------------
            Statistic Shape Scale P-Value
ARCH Lag[3]    0.0323 0.500 2.000  0.8574
ARCH Lag[5]    2.0675 1.440 1.667  0.4565
ARCH Lag[7]    3.2241 2.315 1.543  0.4723

Nyblom stability test
------------------------------------
Joint Statistic:  1.3507
Individual Statistics:              
mu     0.11179
ar1    0.45415
ma1    0.33206
ma2    0.06465
omega  0.19151
alpha1 0.14876
beta1  0.17790

Asymptotic Critical Values (10% 5% 1%)
Joint Statistic:         1.69 1.9 2.35
Individual Statistic:    0.35 0.47 0.75

Sign Bias Test
------------------------------------
                   t-value   prob sig
Sign Bias           1.8345 0.0675   *
Negative Sign Bias  0.9783 0.3286    
Positive Sign Bias  1.0518 0.2937    
Joint Effect        3.3656 0.3386    


Adjusted Pearson Goodness-of-Fit Test:
------------------------------------
  group statistic p-value(g-1)
1    20     14.92       0.7276
2    30     26.37       0.6056
3    40     37.22       0.5514
4    50     37.43       0.8863


Elapsed time : 0.131995 

Modelo GARCH (1,3)

ugarch13 = ugarchspec(mean.model = list(armaOrder = c(1,3)))
ugfit13 = ugarchfit(spec = ugarch13, data = D_Serie_plomo_boxCox)
ugfit13

*---------------------------------*
*          GARCH Model Fit        *
*---------------------------------*

Conditional Variance Dynamics   
-----------------------------------
GARCH Model : sGARCH(1,1)
Mean Model  : ARFIMA(1,0,3)
Distribution    : norm 

Optimal Parameters
------------------------------------
        Estimate  Std. Error  t value Pr(>|t|)
mu      0.010347    0.007322  1.41303 0.157646
ar1    -0.845299    0.177174 -4.77102 0.000002
ma1    -0.139790    0.179874 -0.77715 0.437068
ma2    -0.665335    0.162677 -4.08991 0.000043
ma3     0.189409    0.055373  3.42062 0.000625
omega   0.007103    0.008057  0.88157 0.378008
alpha1  0.020880    0.016400  1.27314 0.202968
beta1   0.961545    0.025225 38.11927 0.000000

Robust Standard Errors:
        Estimate  Std. Error  t value Pr(>|t|)
mu      0.010347    0.007088   1.4599 0.144330
ar1    -0.845299    0.094567  -8.9387 0.000000
ma1    -0.139790    0.115530  -1.2100 0.226285
ma2    -0.665335    0.084815  -7.8446 0.000000
ma3     0.189409    0.051972   3.6444 0.000268
omega   0.007103    0.005488   1.2943 0.195560
alpha1  0.020880    0.016301   1.2809 0.200229
beta1   0.961545    0.017991  53.4471 0.000000

LogLikelihood : -316.7772 

Information Criteria
------------------------------------
                   
Akaike       1.9624
Bayes        2.0543
Shibata      1.9613
Hannan-Quinn 1.9991

Weighted Ljung-Box Test on Standardized Residuals
------------------------------------
                         statistic p-value
Lag[1]                     0.02783  0.8675
Lag[2*(p+q)+(p+q)-1][11]   1.68543  1.0000
Lag[4*(p+q)+(p+q)-1][19]   5.79123  0.9782
d.o.f=4
H0 : No serial correlation

Weighted Ljung-Box Test on Standardized Squared Residuals
------------------------------------
                        statistic p-value
Lag[1]                      0.173  0.6775
Lag[2*(p+q)+(p+q)-1][5]     1.341  0.7789
Lag[4*(p+q)+(p+q)-1][9]     2.739  0.8011
d.o.f=2

Weighted ARCH LM Tests
------------------------------------
            Statistic Shape Scale P-Value
ARCH Lag[3]   0.07763 0.500 2.000  0.7805
ARCH Lag[5]   2.27391 1.440 1.667  0.4138
ARCH Lag[7]   3.19721 2.315 1.543  0.4771

Nyblom stability test
------------------------------------
Joint Statistic:  1.7521
Individual Statistics:              
mu     0.11114
ar1    0.65376
ma1    0.86535
ma2    0.06299
ma3    0.22298
omega  0.19740
alpha1 0.15064
beta1  0.18425

Asymptotic Critical Values (10% 5% 1%)
Joint Statistic:         1.89 2.11 2.59
Individual Statistic:    0.35 0.47 0.75

Sign Bias Test
------------------------------------
                   t-value    prob sig
Sign Bias           1.8057 0.07189   *
Negative Sign Bias  0.9135 0.36166    
Positive Sign Bias  1.0711 0.28492    
Joint Effect        3.2671 0.35225    


Adjusted Pearson Goodness-of-Fit Test:
------------------------------------
  group statistic p-value(g-1)
1    20     15.65       0.6807
2    30     21.11       0.8549
3    40     44.47       0.2523
4    50     59.79       0.1390


Elapsed time : 0.133106 

Modelo GARCH (1,4)

ugarch14 = ugarchspec(mean.model = list(armaOrder = c(1,4)))
ugfit14 = ugarchfit(spec = ugarch14, data = D_Serie_plomo_boxCox)
ugfit14

*---------------------------------*
*          GARCH Model Fit        *
*---------------------------------*

Conditional Variance Dynamics   
-----------------------------------
GARCH Model : sGARCH(1,1)
Mean Model  : ARFIMA(1,0,4)
Distribution    : norm 

Optimal Parameters
------------------------------------
        Estimate  Std. Error  t value Pr(>|t|)
mu      0.010186    0.007067  1.44142  0.14947
ar1    -0.695216    0.929908 -0.74762  0.45469
ma1    -0.299613    0.934070 -0.32076  0.74839
ma2    -0.489851    0.921389 -0.53164  0.59497
ma3     0.186250    0.138820  1.34167  0.17970
ma4    -0.057204    0.085977 -0.66535  0.50583
omega   0.007056    0.008129  0.86790  0.38545
alpha1  0.021061    0.016512  1.27555  0.20211
beta1   0.961385    0.025428 37.80839  0.00000

Robust Standard Errors:
        Estimate  Std. Error  t value Pr(>|t|)
mu      0.010186    0.007027  1.44969  0.14714
ar1    -0.695216    2.393740 -0.29043  0.77149
ma1    -0.299613    2.413828 -0.12412  0.90122
ma2    -0.489851    2.378788 -0.20592  0.83685
ma3     0.186250    0.329677  0.56495  0.57211
ma4    -0.057204    0.173712 -0.32930  0.74193
omega   0.007056    0.005652  1.24824  0.21194
alpha1  0.021061    0.017016  1.23777  0.21580
beta1   0.961385    0.018732 51.32318  0.00000

LogLikelihood : -316.4512 

Information Criteria
------------------------------------
                   
Akaike       1.9665
Bayes        2.0699
Shibata      1.9650
Hannan-Quinn 2.0077

Weighted Ljung-Box Test on Standardized Residuals
------------------------------------
                         statistic p-value
Lag[1]                   0.0001623  0.9898
Lag[2*(p+q)+(p+q)-1][14] 3.0826708  1.0000
Lag[4*(p+q)+(p+q)-1][24] 7.9994739  0.9705
d.o.f=5
H0 : No serial correlation

Weighted Ljung-Box Test on Standardized Squared Residuals
------------------------------------
                        statistic p-value
Lag[1]                     0.1665  0.6832
Lag[2*(p+q)+(p+q)-1][5]    1.3046  0.7878
Lag[4*(p+q)+(p+q)-1][9]    2.8402  0.7850
d.o.f=2

Weighted ARCH LM Tests
------------------------------------
            Statistic Shape Scale P-Value
ARCH Lag[3]   0.09564 0.500 2.000  0.7571
ARCH Lag[5]   2.24045 1.440 1.667  0.4205
ARCH Lag[7]   3.36065 2.315 1.543  0.4482

Nyblom stability test
------------------------------------
Joint Statistic:  2.3943
Individual Statistics:              
mu     0.12112
ar1    0.57946
ma1    0.69639
ma2    0.02841
ma3    0.14808
ma4    0.56082
omega  0.20442
alpha1 0.15297
beta1  0.18833

Asymptotic Critical Values (10% 5% 1%)
Joint Statistic:         2.1 2.32 2.82
Individual Statistic:    0.35 0.47 0.75

Sign Bias Test
------------------------------------
                   t-value    prob sig
Sign Bias            2.554 0.01111  **
Negative Sign Bias   1.476 0.14078    
Positive Sign Bias   1.400 0.16246    
Joint Effect         6.523 0.08876   *


Adjusted Pearson Goodness-of-Fit Test:
------------------------------------
  group statistic p-value(g-1)
1    20     16.98       0.5915
2    30     27.82       0.5275
3    40     37.46       0.5402
4    50     43.47       0.6960


Elapsed time : 0.177067 

Modelo GARCH (2,1)

ugarch21 = ugarchspec(mean.model = list(armaOrder = c(2,1)))
ugfit21 = ugarchfit(spec = ugarch21, data = D_Serie_plomo_boxCox)
ugfit21

*---------------------------------*
*          GARCH Model Fit        *
*---------------------------------*

Conditional Variance Dynamics   
-----------------------------------
GARCH Model : sGARCH(1,1)
Mean Model  : ARFIMA(2,0,1)
Distribution    : norm 

Optimal Parameters
------------------------------------
        Estimate  Std. Error   t value Pr(>|t|)
mu      0.010511    0.007609   1.38134 0.167174
ar1    -0.293959    0.084594  -3.47495 0.000511
ar2    -0.101256    0.076193  -1.32894 0.183867
ma1    -0.698508    0.068083 -10.25972 0.000000
omega   0.007180    0.008009   0.89646 0.370006
alpha1  0.021131    0.016064   1.31542 0.188370
beta1   0.961075    0.024798  38.75665 0.000000

Robust Standard Errors:
        Estimate  Std. Error  t value Pr(>|t|)
mu      0.010511    0.007233   1.4533 0.146152
ar1    -0.293959    0.087315  -3.3666 0.000761
ar2    -0.101256    0.081690  -1.2395 0.215151
ma1    -0.698508    0.057143 -12.2238 0.000000
omega   0.007180    0.005485   1.3089 0.190556
alpha1  0.021131    0.015529   1.3607 0.173600
beta1   0.961075    0.017531  54.8205 0.000000

LogLikelihood : -317.6742 

Information Criteria
------------------------------------
                   
Akaike       1.9618
Bayes        2.0422
Shibata      1.9609
Hannan-Quinn 1.9938

Weighted Ljung-Box Test on Standardized Residuals
------------------------------------
                         statistic p-value
Lag[1]                   0.0004629  0.9828
Lag[2*(p+q)+(p+q)-1][8]  0.7796213  1.0000
Lag[4*(p+q)+(p+q)-1][14] 3.4857826  0.9905
d.o.f=3
H0 : No serial correlation

Weighted Ljung-Box Test on Standardized Squared Residuals
------------------------------------
                        statistic p-value
Lag[1]                     0.2403  0.6240
Lag[2*(p+q)+(p+q)-1][5]    1.3893  0.7671
Lag[4*(p+q)+(p+q)-1][9]    2.8927  0.7766
d.o.f=2

Weighted ARCH LM Tests
------------------------------------
            Statistic Shape Scale P-Value
ARCH Lag[3]    0.1036 0.500 2.000  0.7475
ARCH Lag[5]    2.2627 1.440 1.667  0.4160
ARCH Lag[7]    3.3196 2.315 1.543  0.4553

Nyblom stability test
------------------------------------
Joint Statistic:  1.0423
Individual Statistics:              
mu     0.10521
ar1    0.26764
ar2    0.09025
ma1    0.33217
omega  0.19383
alpha1 0.14957
beta1  0.18087

Asymptotic Critical Values (10% 5% 1%)
Joint Statistic:         1.69 1.9 2.35
Individual Statistic:    0.35 0.47 0.75

Sign Bias Test
------------------------------------
                   t-value    prob sig
Sign Bias            1.851 0.06514   *
Negative Sign Bias   1.023 0.30694    
Positive Sign Bias   1.066 0.28716    
Joint Effect         3.425 0.33058    


Adjusted Pearson Goodness-of-Fit Test:
------------------------------------
  group statistic p-value(g-1)
1    20     14.32       0.7649
2    30     24.92       0.6823
3    40     31.66       0.7919
4    50     33.80       0.9517


Elapsed time : 0.1504061 

Modelo GARCH (2,2)

ugarch22 = ugarchspec(mean.model = list(armaOrder = c(2,2)))
ugfit22 = ugarchfit(spec = ugarch22, data = D_Serie_plomo_boxCox)
ugfit22

*---------------------------------*
*          GARCH Model Fit        *
*---------------------------------*

Conditional Variance Dynamics   
-----------------------------------
GARCH Model : sGARCH(1,1)
Mean Model  : ARFIMA(2,0,2)
Distribution    : norm 

Optimal Parameters
------------------------------------
        Estimate  Std. Error  t value Pr(>|t|)
mu      0.010434    0.007355  1.41858 0.156021
ar1    -0.677456    0.469146 -1.44402 0.148733
ar2    -0.181261    0.104939 -1.72730 0.084114
ma1    -0.313505    0.472376 -0.66368 0.506897
ma2    -0.298671    0.372262 -0.80231 0.422373
omega   0.007044    0.008033  0.87687 0.380556
alpha1  0.020839    0.016192  1.28702 0.198088
beta1   0.961666    0.025094 38.32232 0.000000

Robust Standard Errors:
        Estimate  Std. Error  t value Pr(>|t|)
mu      0.010434    0.007069  1.47598 0.139950
ar1    -0.677456    0.361835 -1.87228 0.061168
ar2    -0.181261    0.075117 -2.41303 0.015820
ma1    -0.313505    0.377701 -0.83003 0.406519
ma2    -0.298671    0.297684 -1.00331 0.315709
omega   0.007044    0.005484  1.28456 0.198947
alpha1  0.020839    0.015981  1.30395 0.192251
beta1   0.961666    0.018076 53.20016 0.000000

LogLikelihood : -317.2459 

Information Criteria
------------------------------------
                   
Akaike       1.9652
Bayes        2.0571
Shibata      1.9641
Hannan-Quinn 2.0019

Weighted Ljung-Box Test on Standardized Residuals
------------------------------------
                         statistic p-value
Lag[1]                    0.001541  0.9687
Lag[2*(p+q)+(p+q)-1][11]  1.447172  1.0000
Lag[4*(p+q)+(p+q)-1][19]  5.565098  0.9845
d.o.f=4
H0 : No serial correlation

Weighted Ljung-Box Test on Standardized Squared Residuals
------------------------------------
                        statistic p-value
Lag[1]                     0.2443  0.6211
Lag[2*(p+q)+(p+q)-1][5]    1.4016  0.7641
Lag[4*(p+q)+(p+q)-1][9]    2.9202  0.7721
d.o.f=2

Weighted ARCH LM Tests
------------------------------------
            Statistic Shape Scale P-Value
ARCH Lag[3]    0.1159 0.500 2.000  0.7335
ARCH Lag[5]    2.2786 1.440 1.667  0.4129
ARCH Lag[7]    3.3624 2.315 1.543  0.4479

Nyblom stability test
------------------------------------
Joint Statistic:  1.7182
Individual Statistics:              
mu     0.11285
ar1    0.26945
ar2    0.07508
ma1    0.44421
ma2    0.04502
omega  0.20008
alpha1 0.15272
beta1  0.18584

Asymptotic Critical Values (10% 5% 1%)
Joint Statistic:         1.89 2.11 2.59
Individual Statistic:    0.35 0.47 0.75

Sign Bias Test
------------------------------------
                   t-value    prob sig
Sign Bias            2.125 0.03431  **
Negative Sign Bias   1.240 0.21598    
Positive Sign Bias   1.143 0.25404    
Joint Effect         4.522 0.21035    


Adjusted Pearson Goodness-of-Fit Test:
------------------------------------
  group statistic p-value(g-1)
1    20     16.37       0.6324
2    30     24.92       0.6823
3    40     31.42       0.8009
4    50     34.71       0.9388


Elapsed time : 0.1708441 

Modelo GARCH (2,3)

ugarch23 = ugarchspec(mean.model = list(armaOrder = c(2,3)))
ugfit23 = ugarchfit(spec = ugarch23, data = D_Serie_plomo_boxCox)
ugfit23

*---------------------------------*
*          GARCH Model Fit        *
*---------------------------------*

Conditional Variance Dynamics   
-----------------------------------
GARCH Model : sGARCH(1,1)
Mean Model  : ARFIMA(2,0,3)
Distribution    : norm 

Optimal Parameters
------------------------------------
        Estimate  Std. Error  t value Pr(>|t|)
mu      0.010182    0.006859  1.48455 0.137664
ar1     0.327643    0.185212  1.76901 0.076892
ar2    -0.662035    0.147796 -4.47939 0.000007
ma1    -1.286391    0.168931 -7.61489 0.000000
ma2     1.146133    0.250298  4.57907 0.000005
ma3    -0.601682    0.130058 -4.62627 0.000004
omega   0.007298    0.008448  0.86377 0.387715
alpha1  0.020709    0.016272  1.27263 0.203149
beta1   0.960832    0.025731 37.34088 0.000000

Robust Standard Errors:
        Estimate  Std. Error  t value Pr(>|t|)
mu      0.010182    0.006729   1.5131 0.130242
ar1     0.327643    0.222745   1.4709 0.141310
ar2    -0.662035    0.203244  -3.2573 0.001125
ma1    -1.286391    0.197606  -6.5099 0.000000
ma2     1.146133    0.347553   3.2977 0.000975
ma3    -0.601682    0.178842  -3.3643 0.000767
omega   0.007298    0.005900   1.2368 0.216147
alpha1  0.020709    0.015918   1.3009 0.193285
beta1   0.960832    0.017046  56.3674 0.000000

LogLikelihood : -316.8221 

Information Criteria
------------------------------------
                   
Akaike       1.9687
Bayes        2.0721
Shibata      1.9673
Hannan-Quinn 2.0099

Weighted Ljung-Box Test on Standardized Residuals
------------------------------------
                         statistic p-value
Lag[1]                      0.3275  0.5672
Lag[2*(p+q)+(p+q)-1][14]    3.7336  1.0000
Lag[4*(p+q)+(p+q)-1][24]    8.7711  0.9340
d.o.f=5
H0 : No serial correlation

Weighted Ljung-Box Test on Standardized Squared Residuals
------------------------------------
                        statistic p-value
Lag[1]                     0.0214  0.8837
Lag[2*(p+q)+(p+q)-1][5]    1.5697  0.7227
Lag[4*(p+q)+(p+q)-1][9]    3.4305  0.6863
d.o.f=2

Weighted ARCH LM Tests
------------------------------------
            Statistic Shape Scale P-Value
ARCH Lag[3]    0.5403 0.500 2.000  0.4623
ARCH Lag[5]    3.0908 1.440 1.667  0.2769
ARCH Lag[7]    4.2882 2.315 1.543  0.3069

Nyblom stability test
------------------------------------
Joint Statistic:  1.9038
Individual Statistics:              
mu     0.13580
ar1    0.11257
ar2    0.20693
ma1    0.25181
ma2    0.23450
ma3    0.06002
omega  0.21806
alpha1 0.15331
beta1  0.19999

Asymptotic Critical Values (10% 5% 1%)
Joint Statistic:         2.1 2.32 2.82
Individual Statistic:    0.35 0.47 0.75

Sign Bias Test
------------------------------------
                   t-value   prob sig
Sign Bias           1.6283 0.1044    
Negative Sign Bias  0.6426 0.5210    
Positive Sign Bias  1.2354 0.2176    
Joint Effect        2.9060 0.4063    


Adjusted Pearson Goodness-of-Fit Test:
------------------------------------
  group statistic p-value(g-1)
1    20     14.92       0.7276
2    30     27.10       0.5665
3    40     38.43       0.4959
4    50     47.10       0.5506


Elapsed time : 0.1782079 

Modelo GARCH (2,4)

ugarch24 = ugarchspec(mean.model = list(armaOrder = c(2,4)))
ugfit24 = ugarchfit(spec = ugarch24, data = D_Serie_plomo_boxCox)
ugfit24

*---------------------------------*
*          GARCH Model Fit        *
*---------------------------------*

Conditional Variance Dynamics   
-----------------------------------
GARCH Model : sGARCH(1,1)
Mean Model  : ARFIMA(2,0,4)
Distribution    : norm 

Optimal Parameters
------------------------------------
        Estimate  Std. Error    t value Pr(>|t|)
mu      0.009989    0.007261     1.3758  0.16889
ar1     0.755847    0.039504    19.1334  0.00000
ar2    -0.874435    0.037315   -23.4336  0.00000
ma1    -1.752885    0.001468 -1193.7879  0.00000
ma2     1.887314    0.000584  3230.8775  0.00000
ma3    -1.077209    0.026768   -40.2431  0.00000
ma4     0.174974    0.021961     7.9674  0.00000
omega   0.007865    0.008166     0.9632  0.33545
alpha1  0.022316    0.016110     1.3852  0.16599
beta1   0.957617    0.025511    37.5374  0.00000

Robust Standard Errors:
        Estimate  Std. Error    t value Pr(>|t|)
mu      0.009989    0.006933     1.4408  0.14965
ar1     0.755847    0.042403    17.8251  0.00000
ar2    -0.874435    0.037250   -23.4746  0.00000
ma1    -1.752885    0.001175 -1491.8900  0.00000
ma2     1.887314    0.000738  2556.1433  0.00000
ma3    -1.077209    0.026120   -41.2402  0.00000
ma4     0.174974    0.020168     8.6757  0.00000
omega   0.007865    0.006004     1.3101  0.19018
alpha1  0.022316    0.015660     1.4250  0.15416
beta1   0.957617    0.018207    52.5951  0.00000

LogLikelihood : -314.6303 

Information Criteria
------------------------------------
                   
Akaike       1.9615
Bayes        2.0764
Shibata      1.9598
Hannan-Quinn 2.0073

Weighted Ljung-Box Test on Standardized Residuals
------------------------------------
                         statistic p-value
Lag[1]                    0.000615  0.9802
Lag[2*(p+q)+(p+q)-1][17]  3.879752  1.0000
Lag[4*(p+q)+(p+q)-1][29] 10.006373  0.9715
d.o.f=6
H0 : No serial correlation

Weighted Ljung-Box Test on Standardized Squared Residuals
------------------------------------
                        statistic p-value
Lag[1]                    0.06964  0.7919
Lag[2*(p+q)+(p+q)-1][5]   1.28302  0.7930
Lag[4*(p+q)+(p+q)-1][9]   2.74459  0.8003
d.o.f=2

Weighted ARCH LM Tests
------------------------------------
            Statistic Shape Scale P-Value
ARCH Lag[3]   0.05843 0.500 2.000  0.8090
ARCH Lag[5]   2.33959 1.440 1.667  0.4010
ARCH Lag[7]   3.31734 2.315 1.543  0.4557

Nyblom stability test
------------------------------------
Joint Statistic:  1.7919
Individual Statistics:              
mu     0.11207
ar1    0.06687
ar2    0.06525
ma1    0.07417
ma2    0.06206
ma3    0.06885
ma4    0.10452
omega  0.17859
alpha1 0.13754
beta1  0.16406

Asymptotic Critical Values (10% 5% 1%)
Joint Statistic:         2.29 2.54 3.05
Individual Statistic:    0.35 0.47 0.75

Sign Bias Test
------------------------------------
                   t-value    prob sig
Sign Bias           1.6715 0.09558   *
Negative Sign Bias  0.8077 0.41986    
Positive Sign Bias  1.1762 0.24038    
Joint Effect        2.8750 0.41130    


Adjusted Pearson Goodness-of-Fit Test:
------------------------------------
  group statistic p-value(g-1)
1    20     16.13       0.6486
2    30     33.98       0.2398
3    40     39.88       0.4310
4    50     44.98       0.6367


Elapsed time : 0.20351 

Modelo GARCH (3,1)

ugarch31 = ugarchspec(mean.model = list(armaOrder = c(3,1)))
ugfit31 = ugarchfit(spec = ugarch31, data = D_Serie_plomo_boxCox)
ugfit31

*---------------------------------*
*          GARCH Model Fit        *
*---------------------------------*

Conditional Variance Dynamics   
-----------------------------------
GARCH Model : sGARCH(1,1)
Mean Model  : ARFIMA(3,0,1)
Distribution    : norm 

Optimal Parameters
------------------------------------
        Estimate  Std. Error   t value Pr(>|t|)
mu      0.009831    0.007114   1.38202 0.166966
ar1    -0.237161    0.096101  -2.46783 0.013593
ar2    -0.037521    0.095097  -0.39455 0.693173
ar3     0.060043    0.076529   0.78458 0.432700
ma1    -0.755241    0.074716 -10.10814 0.000000
omega   0.006904    0.007760   0.88962 0.373667
alpha1  0.021788    0.016081   1.35483 0.175473
beta1   0.961056    0.024249  39.63247 0.000000

Robust Standard Errors:
        Estimate  Std. Error   t value Pr(>|t|)
mu      0.009831    0.006873   1.43043 0.152592
ar1    -0.237161    0.097891  -2.42271 0.015405
ar2    -0.037521    0.100967  -0.37161 0.710181
ar3     0.060043    0.069660   0.86196 0.388713
ma1    -0.755241    0.060686 -12.44499 0.000000
omega   0.006904    0.005358   1.28842 0.197599
alpha1  0.021788    0.015812   1.37790 0.168235
beta1   0.961056    0.017370  55.32952 0.000000

LogLikelihood : -316.6729 

Information Criteria
------------------------------------
                   
Akaike       1.9618
Bayes        2.0537
Shibata      1.9606
Hannan-Quinn 1.9984

Weighted Ljung-Box Test on Standardized Residuals
------------------------------------
                         statistic p-value
Lag[1]                    0.002717  0.9584
Lag[2*(p+q)+(p+q)-1][11]  1.321887  1.0000
Lag[4*(p+q)+(p+q)-1][19]  5.506516  0.9859
d.o.f=4
H0 : No serial correlation

Weighted Ljung-Box Test on Standardized Squared Residuals
------------------------------------
                        statistic p-value
Lag[1]                     0.2138  0.6438
Lag[2*(p+q)+(p+q)-1][5]    1.3679  0.7723
Lag[4*(p+q)+(p+q)-1][9]    2.9671  0.7644
d.o.f=2

Weighted ARCH LM Tests
------------------------------------
            Statistic Shape Scale P-Value
ARCH Lag[3]    0.1394 0.500 2.000  0.7089
ARCH Lag[5]    2.2830 1.440 1.667  0.4120
ARCH Lag[7]    3.4629 2.315 1.543  0.4307

Nyblom stability test
------------------------------------
Joint Statistic:  1.9785
Individual Statistics:              
mu     0.11717
ar1    0.32459
ar2    0.08229
ar3    0.39508
ma1    0.32534
omega  0.20236
alpha1 0.15393
beta1  0.18699

Asymptotic Critical Values (10% 5% 1%)
Joint Statistic:         1.89 2.11 2.59
Individual Statistic:    0.35 0.47 0.75

Sign Bias Test
------------------------------------
                   t-value    prob sig
Sign Bias            2.531 0.01186  **
Negative Sign Bias   1.463 0.14437    
Positive Sign Bias   1.355 0.17641    
Joint Effect         6.408 0.09337   *


Adjusted Pearson Goodness-of-Fit Test:
------------------------------------
  group statistic p-value(g-1)
1    20     16.85       0.5997
2    30     24.20       0.7192
3    40     33.83       0.7042
4    50     41.36       0.7728


Elapsed time : 0.121119 

Modelo GARCH (3,2)

ugarch32 = ugarchspec(mean.model = list(armaOrder = c(3,2)))
ugfit32 = ugarchfit(spec = ugarch32, data = D_Serie_plomo_boxCox)
ugfit32

*---------------------------------*
*          GARCH Model Fit        *
*---------------------------------*

Conditional Variance Dynamics   
-----------------------------------
GARCH Model : sGARCH(1,1)
Mean Model  : ARFIMA(3,0,2)
Distribution    : norm 

Optimal Parameters
------------------------------------
        Estimate  Std. Error   t value Pr(>|t|)
mu      0.008950    0.006973    1.2834 0.199362
ar1     0.372878    0.055930    6.6668 0.000000
ar2     0.116392    0.063477    1.8336 0.066713
ar3     0.080350    0.063736    1.2607 0.207432
ma1    -1.373027    0.009566 -143.5268 0.000000
ma2     0.457797    0.006917   66.1821 0.000000
omega   0.006893    0.007456    0.9245 0.355226
alpha1  0.022881    0.016000    1.4301 0.152702
beta1   0.960092    0.023406   41.0184 0.000000

Robust Standard Errors:
        Estimate  Std. Error   t value Pr(>|t|)
mu      0.008950    0.006730    1.3298 0.183579
ar1     0.372878    0.059105    6.3087 0.000000
ar2     0.116392    0.064855    1.7946 0.072711
ar3     0.080350    0.059511    1.3502 0.176959
ma1    -1.373027    0.006561 -209.2797 0.000000
ma2     0.457797    0.004852   94.3527 0.000000
omega   0.006893    0.005235    1.3167 0.187941
alpha1  0.022881    0.015322    1.4933 0.135356
beta1   0.960092    0.016563   57.9661 0.000000

LogLikelihood : -316.2401 

Information Criteria
------------------------------------
                   
Akaike       1.9652
Bayes        2.0686
Shibata      1.9638
Hannan-Quinn 2.0064

Weighted Ljung-Box Test on Standardized Residuals
------------------------------------
                         statistic p-value
Lag[1]                   0.0005441  0.9814
Lag[2*(p+q)+(p+q)-1][14] 3.1748314  1.0000
Lag[4*(p+q)+(p+q)-1][24] 8.4258622  0.9529
d.o.f=5
H0 : No serial correlation

Weighted Ljung-Box Test on Standardized Squared Residuals
------------------------------------
                        statistic p-value
Lag[1]                     0.1799  0.6715
Lag[2*(p+q)+(p+q)-1][5]    1.3048  0.7877
Lag[4*(p+q)+(p+q)-1][9]    2.8247  0.7875
d.o.f=2

Weighted ARCH LM Tests
------------------------------------
            Statistic Shape Scale P-Value
ARCH Lag[3]    0.1063 0.500 2.000  0.7444
ARCH Lag[5]    2.2247 1.440 1.667  0.4237
ARCH Lag[7]    3.3163 2.315 1.543  0.4559

Nyblom stability test
------------------------------------
Joint Statistic:  1.8652
Individual Statistics:             
mu     0.1209
ar1    0.5124
ar2    0.1089
ar3    0.1084
ma1    0.1028
ma2    0.1088
omega  0.1944
alpha1 0.1506
beta1  0.1811

Asymptotic Critical Values (10% 5% 1%)
Joint Statistic:         2.1 2.32 2.82
Individual Statistic:    0.35 0.47 0.75

Sign Bias Test
------------------------------------
                   t-value   prob sig
Sign Bias            2.164 0.0312  **
Negative Sign Bias   1.210 0.2273    
Positive Sign Bias   1.240 0.2158    
Joint Effect         4.686 0.1963    


Adjusted Pearson Goodness-of-Fit Test:
------------------------------------
  group statistic p-value(g-1)
1    20     16.85       0.5997
2    30     27.64       0.5372
3    40     37.46       0.5402
4    50     51.33       0.3827


Elapsed time : 0.1440229 

Modelo GARCH (3,3)

ugarch33 = ugarchspec(mean.model = list(armaOrder = c(3,3)))
ugfit33 = ugarchfit(spec = ugarch33, data = D_Serie_plomo_boxCox)

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

ugfit33

*---------------------------------*
*          GARCH Model Fit        *
*---------------------------------*

Conditional Variance Dynamics   
-----------------------------------
GARCH Model : sGARCH(1,1)
Mean Model  : ARFIMA(3,0,3)
Distribution    : norm 

Optimal Parameters
------------------------------------
        Estimate  Std. Error     t value Pr(>|t|)
mu      0.007646    0.000140    54.62758  0.00000
ar1     0.170414    0.000119  1435.98395  0.00000
ar2    -0.888152    0.000092 -9692.21943  0.00000
ar3    -0.147493    0.000090 -1636.69761  0.00000
ma1    -1.215910    0.000182 -6693.14455  0.00000
ma2     1.395152    0.000122 11423.98714  0.00000
ma3    -0.773084    0.000092 -8359.15899  0.00000
omega   0.007814    0.009716     0.80421  0.42128
alpha1  0.016873    0.011432     1.47598  0.13995
beta1   0.961803    0.013974    68.82895  0.00000

Robust Standard Errors:
        Estimate  Std. Error     t value Pr(>|t|)
mu      0.007646    0.000226    33.84649  0.00000
ar1     0.170414    0.001263   134.87946  0.00000
ar2    -0.888152    0.001166  -761.73752  0.00000
ar3    -0.147493    0.000932  -158.21625  0.00000
ma1    -1.215910    0.002157  -563.58784  0.00000
ma2     1.395152    0.001805   772.97047  0.00000
ma3    -0.773084    0.000433 -1785.67791  0.00000
omega   0.007814    0.034929     0.22370  0.82299
alpha1  0.016873    0.070900     0.23798  0.81189
beta1   0.961803    0.161358     5.96069  0.00000

LogLikelihood : -303.3641 

Information Criteria
------------------------------------
                   
Akaike       1.8934
Bayes        2.0083
Shibata      1.8917
Hannan-Quinn 1.9393

Weighted Ljung-Box Test on Standardized Residuals
------------------------------------
                         statistic p-value
Lag[1]                      0.2031  0.6522
Lag[2*(p+q)+(p+q)-1][17]    6.3861  1.0000
Lag[4*(p+q)+(p+q)-1][29]   13.9990  0.6141
d.o.f=6
H0 : No serial correlation

Weighted Ljung-Box Test on Standardized Squared Residuals
------------------------------------
                        statistic p-value
Lag[1]                   0.000421  0.9836
Lag[2*(p+q)+(p+q)-1][5]  1.877322  0.6477
Lag[4*(p+q)+(p+q)-1][9]  3.524780  0.6701
d.o.f=2

Weighted ARCH LM Tests
------------------------------------
            Statistic Shape Scale P-Value
ARCH Lag[3]     0.132 0.500 2.000  0.7164
ARCH Lag[5]     3.495 1.440 1.667  0.2256
ARCH Lag[7]     3.999 2.315 1.543  0.3468

Nyblom stability test
------------------------------------
Joint Statistic:  2.1823
Individual Statistics:               
mu     0.009463
ar1    0.084749
ar2    0.040388
ar3    0.127400
ma1    0.112359
ma2    0.025122
ma3    0.127944
omega  0.240478
alpha1 0.156217
beta1  0.235842

Asymptotic Critical Values (10% 5% 1%)
Joint Statistic:         2.29 2.54 3.05
Individual Statistic:    0.35 0.47 0.75

Sign Bias Test
------------------------------------
                   t-value   prob sig
Sign Bias          0.79369 0.4280    
Negative Sign Bias 0.07393 0.9411    
Positive Sign Bias 0.45202 0.6516    
Joint Effect       1.06756 0.7849    


Adjusted Pearson Goodness-of-Fit Test:
------------------------------------
  group statistic p-value(g-1)
1    20     25.31       0.1505
2    30     21.48       0.8412
3    40     40.36       0.4100
4    50     43.47       0.6960


Elapsed time : 0.4752219 

Modelo GARCH (3,4)

ugarch34 = ugarchspec(mean.model = list(armaOrder = c(3,4)))
ugfit34 = ugarchfit(spec = ugarch34, data = D_Serie_plomo_boxCox)

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

ugfit34

*---------------------------------*
*          GARCH Model Fit        *
*---------------------------------*

Conditional Variance Dynamics   
-----------------------------------
GARCH Model : sGARCH(1,1)
Mean Model  : ARFIMA(3,0,4)
Distribution    : norm 

Optimal Parameters
------------------------------------
        Estimate  Std. Error     t value Pr(>|t|)
mu      0.008015    0.000079   101.50629  0.00000
ar1    -0.058561    0.000042 -1402.70621  0.00000
ar2    -0.808669    0.000095 -8471.40062  0.00000
ar3    -0.385453    0.000094 -4092.49264  0.00000
ma1    -0.998260    0.000157 -6367.08244  0.00000
ma2     1.097369    0.000143  7655.44604  0.00000
ma3    -0.456657    0.000090 -5064.26237  0.00000
ma4    -0.207323    0.000063 -3280.09611  0.00000
omega   0.006964    0.007469     0.93243  0.35112
alpha1  0.020134    0.018875     1.06671  0.28610
beta1   0.960539    0.018883    50.86845  0.00000

Robust Standard Errors:
        Estimate  Std. Error     t value Pr(>|t|)
mu      0.008015    0.000034   238.29797 0.000000
ar1    -0.058561    0.000303  -193.51751 0.000000
ar2    -0.808669    0.001139  -710.11115 0.000000
ar3    -0.385453    0.000165 -2336.75373 0.000000
ma1    -0.998260    0.001491  -669.52397 0.000000
ma2     1.097369    0.002044   536.92994 0.000000
ma3    -0.456657    0.000276 -1655.80388 0.000000
ma4    -0.207323    0.000516  -402.04624 0.000000
omega   0.006964    0.027471     0.25352 0.799865
alpha1  0.020134    0.183959     0.10945 0.912845
beta1   0.960539    0.253914     3.78293 0.000155

LogLikelihood : -300.8067 

Information Criteria
------------------------------------
                   
Akaike       1.8840
Bayes        2.0104
Shibata      1.8819
Hannan-Quinn 1.9344

Weighted Ljung-Box Test on Standardized Residuals
------------------------------------
                         statistic p-value
Lag[1]                      0.3109  0.5771
Lag[2*(p+q)+(p+q)-1][20]    6.7537  1.0000
Lag[4*(p+q)+(p+q)-1][34]   16.3545  0.6244
d.o.f=7
H0 : No serial correlation

Weighted Ljung-Box Test on Standardized Squared Residuals
------------------------------------
                        statistic p-value
Lag[1]                  0.0001802  0.9893
Lag[2*(p+q)+(p+q)-1][5] 1.8362757  0.6576
Lag[4*(p+q)+(p+q)-1][9] 3.6498050  0.6486
d.o.f=2

Weighted ARCH LM Tests
------------------------------------
            Statistic Shape Scale P-Value
ARCH Lag[3]     0.132 0.500 2.000  0.7164
ARCH Lag[5]     3.523 1.440 1.667  0.2224
ARCH Lag[7]     4.240 2.315 1.543  0.3133

Nyblom stability test
------------------------------------
Joint Statistic:  2.6224
Individual Statistics:               
mu     0.030452
ar1    0.098253
ar2    0.009115
ar3    0.078315
ma1    0.096031
ma2    0.010425
ma3    0.074367
ma4    0.053526
omega  0.239634
alpha1 0.160443
beta1  0.232196

Asymptotic Critical Values (10% 5% 1%)
Joint Statistic:         2.49 2.75 3.27
Individual Statistic:    0.35 0.47 0.75

Sign Bias Test
------------------------------------
                   t-value   prob sig
Sign Bias           0.5411 0.5888    
Negative Sign Bias  0.1697 0.8654    
Positive Sign Bias  0.3600 0.7191    
Joint Effect        0.6439 0.8863    


Adjusted Pearson Goodness-of-Fit Test:
------------------------------------
  group statistic p-value(g-1)
1    20     17.34      0.56696
2    30     33.80      0.24652
3    40     42.05      0.34018
4    50     66.13      0.05181


Elapsed time : 0.6751161 

Comparamos el criterio de Akaike de estos modelos y elegimos el de menor valor.

Modelo GARCH (0,1): 1.9844 Modelo GARCH (0,2): 1.9519 Modelo GARCH (0,3): 1.9581 Modelo GARCH (0,4): 1.9603 Modelo GARCH (1,1): 1.9542 Modelo GARCH (1,2): 1.9579 Modelo GARCH (1,3): 1.9624 Modelo GARCH (1,4): 1.9665 Modelo GARCH (2,1): 1.9618 Modelo GARCH (2,3): 1.9687 Modelo GARCH (2,4): 1.9615 Modelo GARCH (3,1): 1.9618 Modelo GARCH (3,2): 1.9652 Modelo GARCH (3,3): 1.8934 Modelo GARCH (3,4): 1.8840

El mejor modelo es un GARCH(3,4) con un Akaike igual a 1.8840.

Validación

Validaciones al modelo garch (3,3)

Los coeficientes del modelo final son:

ugfit34@fit$coef

          mu          ar1          ar2          ar3          ma1          ma2 
 0.008014971 -0.058561355 -0.808668686 -0.385453033 -0.998260358  1.097368548 
         ma3          ma4        omega       alpha1        beta1 
-0.456657328 -0.207322932  0.006964399  0.020134402  0.960538562 

La varianza del modelo final es:

ug_var = ugfit33@fit$var
ug_var

  [1] 0.3676427 0.3676427 0.3676427 0.3635047 0.3676412 0.3643084 0.3641986
  [8] 0.3605524 0.3578195 0.3545035 0.3551585 0.3603292 0.3562924 0.3535610
 [15] 0.3656628 0.3598516 0.3634407 0.3714937 0.3787383 0.3727789 0.3663640
 [22] 0.3625566 0.3676219 0.3649026 0.3719243 0.3756022 0.3701214 0.3835421
 [29] 0.3793661 0.3835659 0.3768116 0.3756050 0.3792004 0.3727211 0.3665795
 [36] 0.3604386 0.3564278 0.3543751 0.3489187 0.3478997 0.3429434 0.3376586
 [43] 0.3338787 0.3308601 0.3263995 0.3305577 0.3259034 0.3253628 0.3597818
 [50] 0.3561672 0.3530243 0.3488234 0.3433265 0.3394126 0.3364778 0.3314542
 [57] 0.3315254 0.3279098 0.3255426 0.3337938 0.3543764 0.3599517 0.3554383
 [64] 0.3512984 0.3461774 0.3432691 0.3407153 0.3483043 0.3428229 0.3516880
 [71] 0.3462461 0.3411836 0.3362341 0.3355030 0.3351284 0.3301905 0.3320840
 [78] 0.3368319 0.3341193 0.3310481 0.3444940 0.3511874 0.3456718 0.3403181
 [85] 0.3525657 0.3484420 0.3429641 0.3504364 0.3450051 0.3396449 0.3348298
 [92] 0.3332607 0.3291442 0.3245010 0.3614046 0.3660978 0.3606919 0.3547287
 [99] 0.3490683 0.3439290 0.3449397 0.3408462 0.3371949 0.3504073 0.3450712
[106] 0.3483571 0.3443451 0.3433967 0.3469587 0.3666649 0.3605770 0.3597076
[113] 0.3599361 0.3636130 0.3585532 0.3740919 0.3677597 0.3659722 0.3753538
[120] 0.3797288 0.3874556 0.3894430 0.3951508 0.4148949 0.4080853 0.4189850
[127] 0.4127174 0.4063488 0.3991383 0.3972144 0.3899037 0.3831109 0.3783441
[134] 0.3845384 0.4086850 0.4028583 0.3972913 0.3906799 0.3880680 0.3831394
[141] 0.3769815 0.3705196 0.3642495 0.3730915 0.3693801 0.3642504 0.3583246
[148] 0.3768345 0.3703778 0.3710731 0.3714779 0.3651269 0.3718615 0.3937805
[155] 0.3865541 0.4431137 0.4345062 0.4268681 0.4240458 0.4160138 0.4128634
[162] 0.4056088 0.4115819 0.4087113 0.4055277 0.4043563 0.4060058 0.4130330
[169] 0.4211892 0.4184386 0.4105295 0.4031245 0.3958031 0.3929729 0.3876112
[176] 0.3877931 0.3820380 0.3752784 0.3864641 0.3908362 0.4325803 0.4238707
[183] 0.4206829 0.4145628 0.4121620 0.4088403 0.4079033 0.4014975 0.4043225
[190] 0.3978421 0.4102651 0.4430776 0.4349378 0.4460114 0.4398472 0.4308698
[197] 0.4224470 0.4149481 0.4110652 0.4031816 0.4028566 0.4123143 0.4201028
[204] 0.4134400 0.4054701 0.3979381 0.3906376 0.3896997 0.4135905 0.4056074
[211] 0.4080397 0.4006441 0.3974537 0.3983423 0.4128970 0.4068070 0.4036318
[218] 0.4214909 0.4135183 0.4104644 0.4069037 0.4022878 0.3979527 0.3935152
[225] 0.3871278 0.3807593 0.3742687 0.3730768 0.4002948 0.4287017 0.4403818
[232] 0.4333734 0.4347464 0.4260434 0.4242166 0.4308502 0.4339136 0.4253104
[239] 0.4256298 0.4173787 0.4128477 0.4685204 0.4592514 0.4522762 0.4430160
[246] 0.4364100 0.4558083 0.4501901 0.4523588 0.4461748 0.4405606 0.4327910
[253] 0.4243938 0.4181216 0.4102655 0.4035172 0.3989222 0.3940532 0.4031022
[260] 0.3992396 0.3920569 0.3879719 0.3811714 0.3799643 0.4000282 0.3977637
[267] 0.3918083 0.3905832 0.3835676 0.3778886 0.3767524 0.3709634 0.3685223
[274] 0.3627078 0.3647968 0.3599055 0.3685241 0.3626341 0.3741147 0.3684995
[281] 0.3632180 0.3588244 0.3529899 0.3498717 0.3516472 0.3481079 0.3426521
[288] 0.3375084 0.3353278 0.3306603 0.3279421 0.3251698 0.3235166 0.3218239
[295] 0.3243003 0.3255028 0.3219054 0.3191025 0.3273402 0.3226510 0.3228142
[302] 0.3191898 0.3152298 0.3133238 0.3115172 0.3077247 0.3045823 0.3008819
[309] 0.2972287 0.2960532 0.2927948 0.2955150 0.2920478 0.2887254 0.2895646
[316] 0.3019606 0.3064515 0.3082782 0.3043450 0.3177068 0.3167485 0.3127004
[323] 0.3086047 0.3046948 0.3021918 0.2992841 0.2974641 0.2944741 0.3107706
[330] 0.3069691 0.3079925

library(ggplot2)

Warning: package 'ggplot2' was built under R version 4.2.3

autoplot(ts(ug_var)) + 
  geom_line(color = "blue") + labs(title = "Varianza de nuestro modelo GARCH (3,4)")

Análisis de los cuadrados de los residuales de la nueva serie

ug_resid = (ugfit34@fit$residuals)^2
autoplot(ts(ug_resid)) + autolayer(ts(ug_resid), series = "Residuales al cuadrado") + autolayer(ts(ug_var), series = "Varianza") + labs(title = "Residuales al cuadrado y Varianza de nuestro modelo GARCH (3,4)") + ylab("") + xlab("Tiempo")

Esta serie temporal presenta doble heterocedasticidad.

Correlogramas de los residuales al cuadrado de la nueva serie

par(mfrow = c(2,1))
acf(ug_resid, lag.max=34, main = "Función de autocorrelación")
pacf(ug_resid, lag.max=34, main = "Función de autocorrelación parcial")

par(mfrow = c(1,1))

Esta nueva serie de errores al cuadrado sigue siendo heterocedástica.

checkresiduals(ugfit34@fit$residuals)

    Ljung-Box test

data:  Residuals
Q* = 6.7484, df = 10, p-value = 0.749

Model df: 0.   Total lags used: 10

Pronóstico

prediccion <- forecast(modelo_auto, h=4)
autoplot(prediccion)

Modelo estimado

Para obtener la ecuación estimada de una serie temporal con un modelo ARIMA y un modelo GARCH, necesitamos combinar los componentes de cada modelo.

El modelo ARIMA(2,1,1)(2,0,0)[12] puede expresarse como:

\[[ (1 - \phi_1 B - \phi_2 B^2) (1 - B)^1 (y_t - \mu) = (1 + \theta_1 B) \varepsilon_t ]\]

Donde: \(- ( \phi_1 ) y ( \phi_2 )\) son los coeficientes autorregresivos. \(- ( \theta_1 )\) es el coeficiente de la parte de media móvil. \(- ( B )\) es el operador de rezago. \(- ( \mu )\) es la media del proceso. \(- ( \varepsilon_t )\) es el término de error.

Y el modelo GARCH(3,4) puede expresarse como:

\[[ \sigma_t^2 = \omega + \alpha*1* \varepsilon{t-1}^2 + \beta*1* \sigma{t-1}^2 + \alpha*2* \varepsilon{t-2}^2 + \beta*2* \sigma{t-2}^2 + \alpha*3* \varepsilon{t-3}^2 + \beta*3* \sigma{t-3}^2 + \alpha*4* \varepsilon{t-4}^2 ]\]

Donde: \(- ( \omega )\) es la constante. \(- ( \alpha\_i )\) son los coeficientes de la parte ARCH. \(- ( \beta\_i )\) son los coeficientes de la parte GARCH.