Modelo ARIMA-GARCH del tipo de Cambio USPEN
Aldair Emerson Chalco Apaza
12/09/2020
Librerias Requeridas
library(readxl)
library(tseries)
library(fGarch)Importamos la base de Datos
La base de datos a importar tiene como fuente el Banco Central De Reserva Del Peru
TCM <- read_excel("C:/Users/aldair/Desktop/TCM.xlsx")
print(TCM)## # A tibble: 344 x 2
## Fecha TCM
## <dttm> <dbl>
## 1 1992-01-01 00:00:00 0.981
## 2 1992-02-01 00:00:00 0.966
## 3 1992-03-01 00:00:00 0.948
## 4 1992-04-01 00:00:00 1.01
## 5 1992-05-01 00:00:00 1.12
## 6 1992-06-01 00:00:00 1.17
## 7 1992-07-01 00:00:00 1.23
## 8 1992-08-01 00:00:00 1.28
## 9 1992-09-01 00:00:00 1.36
## 10 1992-10-01 00:00:00 1.54
## # ... with 334 more rowsGraficamos La Base De Datos
plot(TCM$TCM, x=TCM$Fecha,type= "l", col="blue", xlab="Tiempo", ylab="TCM");grid()
Calculamos los Retornos
RTCM=ts(diff(log(TCM$TCM)))Correlogramas
acf(RTCM,main="Funcion de Autocorrelacion", xlab="Rezagos");grid()
pacf(RTCM,main="Funcion Parcial de Autocorrelacion", xlab="Rezagos");grid()
Estimacion del mejor modelo ARMA
Para estimar un modelo ARMA es necesario conocer los valores que puede tomar \(p\), en la parte Autoregresiva \(AR(p)\), estos valores se pueden visualizar en el correlograma de la funcion de autocorrelacion parcial.
\[p=[1;3;5]\] Y para la parte de media movil \(MA(q)\), sus valores se pueden apreciar en el correlograma de funcion de autocorrelacion. \[ q= [1;2;3]\]
ARMA11=arma(RTCM,order = c(1,1))
print(summary(ARMA11)$aic)## [1] -1960.628ARMA12=arma(RTCM,order = c(1,2))
print(summary(ARMA12)$aic)## [1] -1969.886ARMA13=arma(RTCM,order = c(1,3))
print(summary(ARMA13)$aic)## [1] -2056.494ARMA31=arma(RTCM,order = c(3,1))
print(summary(ARMA31)$aic)## [1] -2062.07ARMA32=arma(RTCM,order = c(3,2))
print(summary(ARMA32)$aic)## [1] -2060.116ARMA33=arma(RTCM,order = c(3,3))
print(summary(ARMA33)$aic)## [1] -2056.937ARMA51=arma(RTCM,order = c(5,1))
print(summary(ARMA51)$aic)## [1] -2054.448ARMA52=arma(RTCM,order = c(5,2))
print(summary(ARMA52)$aic)## [1] -2062.453ARMA55=arma(RTCM,order = c(5,5))
print(summary(ARMA55)$aic)## [1] -2051.857El mejor modelo arma considernado el factor de \(AIC = -2062.453\), es \(MA(5,2)\).
Rescatamos los ERRORES del mejor modelo ARMA
e <- ARMA52$residuals
e2 = e^2e_r=e2[6:343]
acf(e_r)
pacf(e_r)
GARCH11= garchFit(formula = ~garch(1,1),data = e_r,trace = T)##
## Series Initialization:
## ARMA Model: arma
## Formula Mean: ~ arma(0, 0)
## GARCH Model: garch
## Formula Variance: ~ garch(1, 1)
## ARMA Order: 0 0
## Max ARMA Order: 0
## GARCH Order: 1 1
## Max GARCH Order: 1
## Maximum Order: 1
## Conditional Dist: norm
## h.start: 2
## llh.start: 1
## Length of Series: 338
## Recursion Init: mci
## Series Scale: 0.000355476
##
## Parameter Initialization:
## Initial Parameters: $params
## Limits of Transformations: $U, $V
## Which Parameters are Fixed? $includes
## Parameter Matrix:
## U V params includes
## mu -3.83536499 3.835365 0.3835365 TRUE
## omega 0.00000100 100.000000 0.1000000 TRUE
## alpha1 0.00000001 1.000000 0.1000000 TRUE
## gamma1 -0.99999999 1.000000 0.1000000 FALSE
## beta1 0.00000001 1.000000 0.8000000 TRUE
## delta 0.00000000 2.000000 2.0000000 FALSE
## skew 0.10000000 10.000000 1.0000000 FALSE
## shape 1.00000000 10.000000 4.0000000 FALSE
## Index List of Parameters to be Optimized:
## mu omega alpha1 beta1
## 1 2 3 5
## Persistence: 0.9
##
##
## --- START OF TRACE ---
## Selected Algorithm: nlminb
##
## R coded nlminb Solver:
##
## 0: 474.84084: 0.383536 0.100000 0.100000 0.800000
## 1: 474.40333: 0.381980 0.0798362 0.0923533 0.793462
## 2: 472.17676: 0.379060 0.0789948 0.0895957 0.814863
## 3: 468.21951: 0.373808 0.0504275 0.0709053 0.842140
## 4: 465.53658: 0.371894 0.0453946 0.0604258 0.856885
## 5: 459.15911: 0.368293 0.0376594 0.0386724 0.886696
## 6: 455.61917: 0.366799 0.0338170 0.0285883 0.898910
## 7: 445.32413: 0.363812 0.0261418 0.00841111 0.923335
## 8: 440.25458: 0.363292 0.0259014 0.00130428 0.929121
## 9: 438.87718: 0.363135 0.0237917 1.00000e-08 0.933161
## 10: 437.39096: 0.366584 0.0247185 1.00000e-08 0.936130
## 11: 436.69509: 0.369938 0.0248800 1.00000e-08 0.939877
## 12: 435.79197: 0.376914 0.0210338 1.00000e-08 0.943950
## 13: 434.49009: 0.391144 0.0195353 1.00000e-08 0.952327
## 14: 433.77874: 0.404581 0.0118008 1.00000e-08 0.965039
## 15: 433.01276: 0.390028 0.0138592 1.00000e-08 0.967234
## 16: 431.92336: 0.375437 0.0126650 1.00000e-08 0.966871
## 17: 431.66360: 0.360851 0.0118755 1.00000e-08 0.968269
## 18: 431.60899: 0.358116 0.0113143 1.00000e-08 0.970067
## 19: 431.58932: 0.355288 0.0111505 1.00000e-08 0.970090
## 20: 431.58541: 0.352463 0.0113312 1.00000e-08 0.969764
## 21: 431.57833: 0.349913 0.0109612 1.00000e-08 0.970708
## 22: 431.57774: 0.349427 0.0109791 1.00000e-08 0.970566
## 23: 431.57756: 0.349819 0.0110264 1.00000e-08 0.970478
## 24: 431.57756: 0.349759 0.0110218 1.00000e-08 0.970489
## 25: 431.57756: 0.349761 0.0110217 1.00000e-08 0.970490
##
## Final Estimate of the Negative LLH:
## LLH: -2252.836 norm LLH: -6.665196
## mu omega alpha1 beta1
## 1.243317e-04 1.392731e-09 1.000000e-08 9.704896e-01
##
## R-optimhess Difference Approximated Hessian Matrix:
## mu omega alpha1 beta1
## mu -4.914942e+09 1.702144e+13 -5.133143e+05 1.198461e+06
## omega 1.702144e+13 -7.094317e+19 -3.864144e+12 -3.326719e+12
## alpha1 -5.133143e+05 -3.864144e+12 -2.758660e+05 -3.007409e+05
## beta1 1.198461e+06 -3.326719e+12 -3.007409e+05 -1.803259e+05
## attr(,"time")
## Time difference of 0.01595807 secs
##
## --- END OF TRACE ---## Warning in sqrt(diag(fit$cvar)): Se han producido NaNs##
## Time to Estimate Parameters:
## Time difference of 0.131649 secssummary(GARCH11)##
## Title:
## GARCH Modelling
##
## Call:
## garchFit(formula = ~garch(1, 1), data = e_r, trace = T)
##
## Mean and Variance Equation:
## data ~ garch(1, 1)
## <environment: 0x0000000018cff058>
## [data = e_r]
##
## Conditional Distribution:
## norm
##
## Coefficient(s):
## mu omega alpha1 beta1
## 1.2433e-04 1.3927e-09 1.0000e-08 9.7049e-01
##
## Std. Errors:
## based on Hessian
##
## Error Analysis:
## Estimate Std. Error t value Pr(>|t|)
## mu 1.243e-04 1.425e-05 8.727 < 2e-16 ***
## omega 1.393e-09 2.127e-10 6.547 5.87e-11 ***
## alpha1 1.000e-08 NA NA NA
## beta1 9.705e-01 NA NA NA
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Log Likelihood:
## 2252.836 normalized: 6.665196
##
## Description:
## Thu Sep 17 18:48:37 2020 by user: ALDAIR EMERSON
##
##
## Standardised Residuals Tests:
## Statistic p-Value
## Jarque-Bera Test R Chi^2 61551.51 0
## Shapiro-Wilk Test R W 0.4555192 0
## Ljung-Box Test R Q(10) 31.11023 0.0005626288
## Ljung-Box Test R Q(15) 32.78485 0.005026209
## Ljung-Box Test R Q(20) 35.23174 0.01890419
## Ljung-Box Test R^2 Q(10) 0.09394783 1
## Ljung-Box Test R^2 Q(15) 0.14117 1
## Ljung-Box Test R^2 Q(20) 0.1630713 1
## LM Arch Test R TR^2 19.47445 0.07770388
##
## Information Criterion Statistics:
## AIC BIC SIC HQIC
## -13.30672 -13.26148 -13.30700 -13.28869Por lo tanto el mejor modelo Arima Grach es \(ARIMA (5,1,2)~ GARCH(1,1)\)