Modelo de volatilidad del tipo de cambio

En el presente trabajo presentare la elaboración de un modelo Garchs simétricos
y asimetricos, el sotfware eleigido es R y su editor es Rstudio el scrip sera
presentado al final o se podra descargar directo en desde la pagina de Rpub.

Extranción de los datos:

library(quantmod)

Loading required package: xts
Loading required package: zoo

Attaching package: 㤼㸱zoo㤼㸲

The following objects are masked from 㤼㸱package:base㤼㸲:

    as.Date, as.Date.numeric

Loading required package: TTR
Registered S3 method overwritten by 'quantmod':
  method            from
  as.zoo.data.frame zoo 
Version 0.4-0 included new data defaults. See ?getSymbols.

library(PerformanceAnalytics)

Attaching package: 㤼㸱PerformanceAnalytics㤼㸲

The following object is masked from 㤼㸱package:graphics㤼㸲:

    legend

library(tseries)

    㤼㸱tseries㤼㸲 version: 0.10-47

    㤼㸱tseries㤼㸲 is a package for time series analysis and computational finance.

    See 㤼㸱library(help="tseries")㤼㸲 for details.

library(MLmetrics)

Attaching package: 㤼㸱MLmetrics㤼㸲

The following object is masked from 㤼㸱package:base㤼㸲:

    Recall

library(normtest)
library(forecast)
library(tidyverse)

Registered S3 methods overwritten by 'dbplyr':
  method         from
  print.tbl_lazy     
  print.tbl_sql      
-- Attaching packages --------------------------------------- tidyverse 1.3.0 --
v ggplot2 3.3.0     v purrr   0.3.4
v tibble  3.0.3     v dplyr   1.0.0
v tidyr   1.1.0     v stringr 1.4.0
v readr   1.3.1     v forcats 0.5.0
-- Conflicts ------------------------------------------ tidyverse_conflicts() --
x dplyr::filter() masks stats::filter()
x dplyr::first()  masks xts::first()
x dplyr::lag()    masks stats::lag()
x dplyr::last()   masks xts::last()

library(readxl)

fechas<-seq(fecha,by="daily",length=length(data))

Error in seq.Date(fecha, by = "daily", length = length(data)) : 
  invalid string for 'by'

  chartSeries(TC, type = "candlesticks", theme = "white", TA = 'addBBands(); addBBands(draw = "p"); addVo(); addMACD(); addRSI(); addSMA()',
              subset = "last 60 weeks",name = "Tipo de cambio USD/PEN")

requested length is greater than original

retornos<-Return.calculate(TC,method = "log")[-1]
chartSeries(retornos,name="Retorno del tipo de cambio",subset = "last 50 week",theme = "white")

requested length is greater than original

adf.test(retornos)

p-value smaller than printed p-value

    Augmented Dickey-Fuller Test

data:  retornos
Dickey-Fuller = -6.0177, Lag order = 6, p-value = 0.01
alternative hypothesis: stationary

Elegir el mejor modelo ARMA

Por cuestión de tiempo solo utiliare la función ´´´auto.arima()´´´ del paquete
´´´forecast´´´

par(mfrow=c(1,2))
acf(retornos)
pacf(retornos)

layout(1)

Función auto.arima

modelo<-auto.arima(retornos)
summary(modelo)

Series: retornos 
ARIMA(1,0,1) with zero mean 

Coefficients:
          ar1     ma1
      -0.1502  0.4920
s.e.   0.1698  0.1495

sigma^2 estimated as 0.0001242:  log likelihood=791.94
AIC=-1577.88   AICc=-1577.79   BIC=-1567.23

Training set error measures:
                       ME      RMSE        MAE      MPE     MAPE      MASE        ACF1
Training set 0.0001930351 0.0111012 0.00803595 96.56709 126.0553 0.8591846 -0.01743586

El mejor modelo es un ARMA(1,1) para los retornos.

Forecast del tipo de cambio

Lo mismo pero con un LOOP

p<-seq(0,2)
q<-seq(0,2)
datos<-NULL
indice<- NULL
for(i in p){
  for(a in q){
    ata<- summary(arma(retornos,order = c(i,a)))$aic
    datos<- c(datos,ata)
    atita<- paste(i,a)
    indice<- c(indice,atita)
    table<- data.frame("ARMA"=indice,"AIC"=datos)
    print(table)
  }
}

one-dimensional optimization by Nelder-Mead is unreliable:
use "Brent" or optimize() directly

Siendo el mejor modelo escogido por el menor AIC es el modelo ARMA(1,1) para los retornos

Modelo garch simetrico y asimetricos

Por motivo de tiempo solo se procedera hacer los modelos la teoría de tras de ello
no se tocara a fondo.

residuos<-na.omit(modelo$residuals)^2
par(mfrow=c(1,2))
acf(residuos)
pacf(residuos)

layout(1)

Por teoria el GARCH(1,1) es el mejor asi que solo hare:

• GARCh(1,1)
• EGARCH(1,1)
• TGARCH(1,1)

library(rugarch)

Loading required package: parallel

Attaching package: 㤼㸱rugarch㤼㸲

The following object is masked from 㤼㸱package:purrr㤼㸲:

    reduce

The following object is masked from 㤼㸱package:stats㤼㸲:

    sigma

Formulación de los modelos

ata=ugarchfit(data = residuos,spec = ugarchspec())
ata

*---------------------------------*
*          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      0.000113    0.000019  6.018828 0.000000
ar1     0.308770    0.222481  1.387848 0.165183
ma1     0.008990    0.217228  0.041387 0.966988
omega   0.000000    0.000000  0.005223 0.995833
alpha1  0.056306    0.012983  4.336888 0.000014
beta1   0.901117    0.021822 41.293699 0.000000

Robust Standard Errors:
        Estimate  Std. Error  t value Pr(>|t|)
mu      0.000113    0.003430 0.032801  0.97383
ar1     0.308770  882.720544 0.000350  0.99972
ma1     0.008990  945.645293 0.000010  0.99999
omega   0.000000    0.002016 0.000001  1.00000
alpha1  0.056306  351.463825 0.000160  0.99987
beta1   0.901117  310.203534 0.002905  0.99768

LogLikelihood : 1794.972 

Information Criteria
------------------------------------
                    
Akaike       -13.922
Bayes        -13.839
Shibata      -13.923
Hannan-Quinn -13.889

Weighted Ljung-Box Test on Standardized Residuals
------------------------------------
                        statistic p-value
Lag[1]                    0.07418  0.7853
Lag[2*(p+q)+(p+q)-1][5]   1.52283  0.9976
Lag[4*(p+q)+(p+q)-1][9]   3.37917  0.8252
d.o.f=2
H0 : No serial correlation

Weighted Ljung-Box Test on Standardized Squared Residuals
------------------------------------
                        statistic p-value
Lag[1]                   0.006731  0.9346
Lag[2*(p+q)+(p+q)-1][5]  0.213784  0.9915
Lag[4*(p+q)+(p+q)-1][9]  0.399407  0.9993
d.o.f=2

Weighted ARCH LM Tests
------------------------------------
            Statistic Shape Scale P-Value
ARCH Lag[3]    0.1677 0.500 2.000  0.6821
ARCH Lag[5]    0.2393 1.440 1.667  0.9560
ARCH Lag[7]    0.2955 2.315 1.543  0.9930

Nyblom stability test
------------------------------------
Joint Statistic:  74.1615
Individual Statistics:              
mu     0.07828
ar1    0.02974
ma1    0.05880
omega  7.69870
alpha1 0.23736
beta1  0.09644

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
------------------------------------

Adjusted Pearson Goodness-of-Fit Test:
------------------------------------
  group statistic p-value(g-1)
1    20     327.7    4.134e-58
2    30     367.9    2.300e-60
3    40     380.7    1.294e-57
4    50     398.1    3.477e-56


Elapsed time : 0.1089871 

specificacion= ugarchspec(
  variance.model=list(model="fGARCH", submodel="TGARCH",garchOrder=c(1,1)), 
  mean.model=list(armaOrder=c(2,1))
)

ata2=ugarchfit(data = residuos,spec = specificacion)
curva2=newsimpact(ata2)
plot(curva2$zx,curva2$zy,type="l",main="Curva de impacto")

Proyección

pre %>% plot()

Make a plot selection (or 0 to exit): 

1:   Time Series Prediction (unconditional)
2:   Time Series Prediction (rolling)
3:   Sigma Prediction (unconditional)
4:   Sigma Prediction (rolling)

1

Make a plot selection (or 0 to exit): 

1:   Time Series Prediction (unconditional)
2:   Time Series Prediction (rolling)
3:   Sigma Prediction (unconditional)
4:   Sigma Prediction (rolling)

3

Make a plot selection (or 0 to exit): 

1:   Time Series Prediction (unconditional)
2:   Time Series Prediction (rolling)
3:   Sigma Prediction (unconditional)
4:   Sigma Prediction (rolling)

0

Grafico del VaR

m=mean(retornos)
s=sd(retornos)
par.2<-VaR(retornos,p=0.99,method = "gaussian")
ggplot(retornos,aes(TC))+geom_histogram(aes(
  y=..density..
),binwidth = 0.01,fill="blue",color="black",alpha=0.4)+geom_vline(xintercept = par.2,lty=2)+
  stat_function(fun = dnorm,args =
    list(mean=m,sd=s),col="red")+labs(title="VaR de los retornos")

par(mfrow=c(1,1))
plot(density(retornos), main = "Retorno de Distribución Empírica"); curve(dnorm(x, mean = m, sd = s), from = -0.1, to = 0.1, add = TRUE, col = 'blue')