ARCH GARCH Data Kurs
Library
library("prettydoc")
library("rmdformats")
library(fGarch)
library(aTSA)
library(FinTS)
library(lmtest)
library(forecast)
library(TSA)
library(tseries)
library(xts)
library(readxl)
library(tidyverse)
library("dygraphs")Input Data
Data Kurs adalah nilai tukar rupiah terhadap dolar Amerika Serikat (USD to IDR). Kurs diamati dari 1 Januari 2018 hingga 31 Maret 2020 dalam harian yang ditentukan oleh Bank Indonesia. Sebanyak 548 observasi dicatat. Data berikut diperoleh dari website JISDOR BI dan digunakan Khairunnisa dalam artikel penelitian untuk memodelkan GARCH.
kurs <- read.csv("https://raw.githubusercontent.com/Rifqiaulya/poladatawaktu/main/Informasi%20Kurs%20Jisdor.csv",header=T)
head(kurs) NO Tanggal Kurs
1 1 1/2/2018 12:00:00 AM 13542
2 2 1/3/2018 12:00:00 AM 13498
3 3 1/4/2018 12:00:00 AM 13474
4 4 1/5/2018 12:00:00 AM 13405
5 5 1/8/2018 12:00:00 AM 13397
6 6 1/9/2018 12:00:00 AM 13428tail(kurs) NO Tanggal Kurs
543 543 3/23/2020 12:00:00 AM 16608
544 544 3/24/2020 12:00:00 AM 16486
545 545 3/26/2020 12:00:00 AM 16328
546 546 3/27/2020 12:00:00 AM 16230
547 547 3/30/2020 12:00:00 AM 16336
548 548 3/31/2020 12:00:00 AM 16367Visualisasi
Plot data dan acf memperlihatkan data tidak stasioner pada rataan, namun perlu pengujian secara formal dengan adf test.
Terima H0 dan menerima bahwa data tidak stasioner. Akibatnya, data perlu dilakukan pemebedaan.kurs.ts<-ts(kurs$Kurs)
adf.test(kurs.ts)
Augmented Dickey-Fuller Test
data: kurs.ts
Dickey-Fuller = -1.0975, Lag order = 8, p-value = 0.923
alternative hypothesis: stationaryPembedaan d=1
Plot dan uji ADF menyatakan data pembedaan d=1 telah stasioner.
dif1.kurs<-diff(kurs.ts)
plot(dif1.kurs,ylab="Kurs")
adf.test(dif1.kurs)
Augmented Dickey-Fuller Test
data: dif1.kurs
Dickey-Fuller = -6.216, Lag order = 8, p-value = 0.01
alternative hypothesis: stationaryIdentifikasi Model ARIMA
#ACF menunjukkan kandidat model ARIMA(0,1,2)
acf(dif1.kurs, lag.max = 7)
#PACF menunjukkan kandidat model ARIMA(1,1,0)
pacf(dif1.kurs, lag=6)
#EACF menunjukkan kandidat model ARIMA(0,1,2)
eacf(dif1.kurs)AR/MA
0 1 2 3 4 5 6 7 8 9 10 11 12 13
0 x x o o x o o o o o o o o x
1 x o o o x o o o x o o x o x
2 x o x o o o o o x o o o o x
3 x x x o x o o o o o o o o x
4 x x x o o o o x o o o o o x
[ reached getOption("max.print") -- omitted 3 rows ]#auto ARIMA menunjukkan kandidat model ARIMA(1,1,0)
auto.arima(kurs.ts,trace = T,d = 1)
Fitting models using approximations to speed things up...
ARIMA(2,1,2) with drift : 6091.573
ARIMA(0,1,0) with drift : 6136.04
ARIMA(1,1,0) with drift : 6086.548
ARIMA(0,1,1) with drift : 6094.638
ARIMA(0,1,0) : 6137.337
ARIMA(2,1,0) with drift : 6088.108
ARIMA(1,1,1) with drift : 6087.186
ARIMA(2,1,1) with drift : 6089.748
ARIMA(1,1,0) : 6086.399
ARIMA(2,1,0) : 6087.806
ARIMA(1,1,1) : 6086.846
ARIMA(0,1,1) : 6094.866
ARIMA(2,1,1) : 6089.423
Now re-fitting the best model(s) without approximations...
ARIMA(1,1,0) : 6094.328
Best model: ARIMA(1,1,0) Series: kurs.ts
ARIMA(1,1,0)
Coefficients:
ar1
0.3049
s.e. 0.0407
sigma^2 estimated as 4015: log likelihood=-3045.15
AIC=6094.31 AICc=6094.33 BIC=6102.92Pemilihan Model
modelarima012<-arima(kurs.ts, order=c(0,1,2), method="ML")
modelarima110<-arima(kurs.ts, order=c(1,1,0), method="ML")
(modelarima012)
Call:
arima(x = kurs.ts, order = c(0, 1, 2), method = "ML")
Coefficients:
ma1 ma2
0.2933 0.1466
s.e. 0.0423 0.0432
sigma^2 estimated as 3990: log likelihood = -3043.96, aic = 6091.92(modelarima110)
Call:
arima(x = kurs.ts, order = c(1, 1, 0), method = "ML")
Coefficients:
ar1
0.3049
s.e. 0.0407
sigma^2 estimated as 4008: log likelihood = -3045.15, aic = 6092.31#Model Terbaik pada AIC terkecil model ARIMA(0,1,2)Signifikansi Koefisien
coeftest(modelarima012)
z test of coefficients:
Estimate Std. Error z value Pr(>|z|)
ma1 0.293312 0.042262 6.9404 3.911e-12 ***
ma2 0.146573 0.043222 3.3911 0.000696 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1coeftest(modelarima110)
z test of coefficients:
Estimate Std. Error z value Pr(>|z|)
ar1 0.304912 0.040683 7.4948 6.638e-14 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1#Semua Koefisien Model ARIMA(0,1,2) siginfikanPlot data dan Fitted Value
fit<-kurs$Kurs-modelarima012$residuals
plot(kurs$Kurs,type="l")
lines(fit,col="Red")
Analisis Residual
Sisaan terlihat acak dan bebas.
tsdiag(modelarima012)
checkresiduals(modelarima012$residuals)
Tolak H0, akibatnya sisaan tidak menyebar normal
jarque.bera.test(residuals(modelarima012))
Jarque Bera Test
data: residuals(modelarima012)
X-squared = 1774.7, df = 2, p-value < 2.2e-16Terlihat ada korelasi pada sisaan kuadrat yang menandakan terdapat heteroskedastis. LM test dan PQ test semua lag p value observasinya dibawah garis maka ada indikasi heteroskedastisitas. Berdasarkan plot sisaan kuadrat terdapat titik titik yang melebihi nilai kebanyakannya sehingga terjadi heteroskedastisitas.
acf(modelarima012$residuals^2,main="ACF Squared Residual")
#atau
arch.test(modelarima012)ARCH heteroscedasticity test for residuals
alternative: heteroscedastic
Portmanteau-Q test:
order PQ p.value
[1,] 4 263 0
[2,] 8 323 0
[3,] 12 341 0
[4,] 16 365 0
[5,] 20 365 0
[6,] 24 365 0
Lagrange-Multiplier test:
order LM p.value
[1,] 4 254.1 0.00e+00
[2,] 8 111.5 0.00e+00
[3,] 12 67.4 3.83e-10
[4,] 16 29.0 1.62e-02
[5,] 20 20.8 3.51e-01
[6,] 24 16.7 8.22e-01
Pemeriksaan efek ARCH
Tolak H0 untuk uji LM test dari lag 1 hingga 5. Akibatnya terdapat efek ARCH jangka panjang atau GARCH.
for (i in 1:5) {
ArchTest <- ArchTest(modelarima012$residuals, lags=i, demean=TRUE)
cat("P Value LM Test lag ke", i,"adalah" , ArchTest$p.value, "\n") }P Value LM Test lag ke 1 adalah 9.601294e-28
P Value LM Test lag ke 2 adalah 1.218508e-29
P Value LM Test lag ke 3 adalah 2.931241e-29
P Value LM Test lag ke 4 adalah 2.312342e-28
P Value LM Test lag ke 5 adalah 1.739022e-29 Pemodelan GARCH
#alpha1 unsur ARCH 1 nya di lag residual kuadrat
#LM Archtestnya p value siginifikan, yg mau kita tuju adalah tidak signfikan, terdapat efek masih
arch1<-garchFit(~arma(0,2)+garch(1,0),data = dif1.kurs, trace = F)
summary(arch1)
Title:
GARCH Modelling
Call:
garchFit(formula = ~arma(0, 2) + garch(1, 0), data = dif1.kurs,
trace = F)
Mean and Variance Equation:
data ~ arma(0, 2) + garch(1, 0)
<environment: 0x0000000025ac1748>
[data = dif1.kurs]
Conditional Distribution:
norm
Coefficient(s):
mu ma1 ma2 omega alpha1
1.21476 0.11216 0.12915 1837.09768 0.52789
Std. Errors:
based on Hessian
Error Analysis:
Estimate Std. Error t value Pr(>|t|)
mu 1.215e+00 2.514e+00 0.483 0.62892
ma1 1.122e-01 4.250e-02 2.639 0.00832 **
ma2 1.291e-01 3.242e-02 3.984 6.78e-05 ***
omega 1.837e+03 1.613e+02 11.391 < 2e-16 ***
alpha1 5.279e-01 8.914e-02 5.922 3.18e-09 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Log Likelihood:
-2957.233 normalized: -5.406275
Description:
Fri May 07 22:51:34 2021 by user: DK0158AU
Standardised Residuals Tests:
Statistic p-Value
Jarque-Bera Test R Chi^2 85.48614 0
Shapiro-Wilk Test R W 0.9778342 2.273594e-07
Ljung-Box Test R Q(10) 16.1978 0.09410852
Ljung-Box Test R Q(15) 31.59991 0.007295258
Ljung-Box Test R Q(20) 36.92794 0.01193701
Ljung-Box Test R^2 Q(10) 54.29062 4.283436e-08
Ljung-Box Test R^2 Q(15) 78.84286 1.136172e-10
Ljung-Box Test R^2 Q(20) 87.39625 2.107072e-10
LM Arch Test R TR^2 56.64841 9.147279e-08
Information Criterion Statistics:
AIC BIC SIC HQIC
10.83083 10.87018 10.83067 10.84621 #beta 1 garch atau lag variansnya sendiri, p-value LM besar, AIC lebih kecil, tidak terdapat efek
garch11<-garchFit(~arma(0,2)+garch(1,1),data = dif1.kurs, trace = F)
summary(garch11)
Title:
GARCH Modelling
Call:
garchFit(formula = ~arma(0, 2) + garch(1, 1), data = dif1.kurs,
trace = F)
Mean and Variance Equation:
data ~ arma(0, 2) + garch(1, 1)
<environment: 0x0000000020b61ff0>
[data = dif1.kurs]
Conditional Distribution:
norm
Coefficient(s):
mu ma1 ma2 omega alpha1 beta1
1.678450 0.096288 0.047817 165.468900 0.191526 0.772454
Std. Errors:
based on Hessian
Error Analysis:
Estimate Std. Error t value Pr(>|t|)
mu 1.67845 2.21700 0.757 0.449
ma1 0.09629 0.04803 2.005 0.045 *
ma2 0.04782 0.04591 1.042 0.298
omega 165.46890 81.35917 2.034 0.042 *
alpha1 0.19153 0.04621 4.145 3.4e-05 ***
beta1 0.77245 0.05995 12.885 < 2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Log Likelihood:
-2928.937 normalized: -5.354548
Description:
Fri May 07 22:51:34 2021 by user: DK0158AU
Standardised Residuals Tests:
Statistic p-Value
Jarque-Bera Test R Chi^2 35.09304 2.396868e-08
Shapiro-Wilk Test R W 0.9857588 3.493584e-05
Ljung-Box Test R Q(10) 16.4892 0.08645869
Ljung-Box Test R Q(15) 26.13412 0.03663633
Ljung-Box Test R Q(20) 31.9502 0.04383181
Ljung-Box Test R^2 Q(10) 6.446059 0.7764995
Ljung-Box Test R^2 Q(15) 15.74557 0.3991598
Ljung-Box Test R^2 Q(20) 22.77047 0.3002051
LM Arch Test R TR^2 11.07645 0.5223783
Information Criterion Statistics:
AIC BIC SIC HQIC
10.73103 10.77825 10.73080 10.74949 garch12<-garchFit(~arma(0,2)+garch(1,2),data = dif1.kurs, trace = F)
summary(garch12)
Title:
GARCH Modelling
Call:
garchFit(formula = ~arma(0, 2) + garch(1, 2), data = dif1.kurs,
trace = F)
Mean and Variance Equation:
data ~ arma(0, 2) + garch(1, 2)
<environment: 0x000000002695bfe8>
[data = dif1.kurs]
Conditional Distribution:
norm
Coefficient(s):
mu ma1 ma2 omega alpha1 beta1
1.570774 0.090214 0.051667 188.588790 0.223673 0.547426
beta2
0.188204
Std. Errors:
based on Hessian
Error Analysis:
Estimate Std. Error t value Pr(>|t|)
mu 1.57077 2.21547 0.709 0.47832
ma1 0.09021 0.04882 1.848 0.06464 .
ma2 0.05167 0.04540 1.138 0.25513
omega 188.58879 90.67619 2.080 0.03754 *
alpha1 0.22367 0.05632 3.972 7.14e-05 ***
beta1 0.54743 0.19176 2.855 0.00431 **
beta2 0.18820 0.16114 1.168 0.24283
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Log Likelihood:
-2928.631 normalized: -5.353987
Description:
Fri May 07 22:51:35 2021 by user: DK0158AU
Standardised Residuals Tests:
Statistic p-Value
Jarque-Bera Test R Chi^2 35.85958 1.633775e-08
Shapiro-Wilk Test R W 0.9857627 3.503186e-05
Ljung-Box Test R Q(10) 16.57499 0.08431343
Ljung-Box Test R Q(15) 26.45063 0.03354461
Ljung-Box Test R Q(20) 32.2248 0.0409611
Ljung-Box Test R^2 Q(10) 5.831729 0.8291969
Ljung-Box Test R^2 Q(15) 15.13517 0.4417249
Ljung-Box Test R^2 Q(20) 22.49684 0.3141704
LM Arch Test R TR^2 10.64375 0.5596704
Information Criterion Statistics:
AIC BIC SIC HQIC
10.73357 10.78865 10.73325 10.75510 garch21<-garchFit(~arma(0,2)+garch(2,1),data = dif1.kurs, trace = F)
summary(garch21)
Title:
GARCH Modelling
Call:
garchFit(formula = ~arma(0, 2) + garch(2, 1), data = dif1.kurs,
trace = F)
Mean and Variance Equation:
data ~ arma(0, 2) + garch(2, 1)
<environment: 0x0000000020e4ee08>
[data = dif1.kurs]
Conditional Distribution:
norm
Coefficient(s):
mu ma1 ma2 omega alpha1 alpha2
1.7173e+00 9.6406e-02 4.8124e-02 1.6890e+02 1.9297e-01 1.0000e-08
beta1
7.6987e-01
Std. Errors:
based on Hessian
Error Analysis:
Estimate Std. Error t value Pr(>|t|)
mu 1.717e+00 2.226e+00 0.771 0.44049
ma1 9.641e-02 4.874e-02 1.978 0.04795 *
ma2 4.812e-02 4.630e-02 1.039 0.29867
omega 1.689e+02 9.492e+01 1.779 0.07517 .
alpha1 1.930e-01 5.927e-02 3.256 0.00113 **
alpha2 1.000e-08 7.635e-02 0.000 1.00000
beta1 7.699e-01 7.796e-02 9.875 < 2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Log Likelihood:
-2929.205 normalized: -5.355037
Description:
Fri May 07 22:51:35 2021 by user: DK0158AU
Standardised Residuals Tests:
Statistic p-Value
Jarque-Bera Test R Chi^2 35.60396 1.856515e-08
Shapiro-Wilk Test R W 0.9856454 3.222841e-05
Ljung-Box Test R Q(10) 16.49147 0.08640154
Ljung-Box Test R Q(15) 26.12109 0.03676898
Ljung-Box Test R Q(20) 32.03505 0.0429263
Ljung-Box Test R^2 Q(10) 6.467754 0.7745541
Ljung-Box Test R^2 Q(15) 15.7345 0.3999132
Ljung-Box Test R^2 Q(20) 22.85614 0.2959133
LM Arch Test R TR^2 11.01519 0.5276174
Information Criterion Statistics:
AIC BIC SIC HQIC
10.73567 10.79075 10.73535 10.75720 #kandidat model lain ketika ARIMA(0,1,1) GARCH(1,1) karena ada koefisien tak siginfikan di garch11
garch11k<-garchFit(~arma(0,1)+garch(1,1),data = dif1.kurs, trace = F)
summary(garch11k)
Title:
GARCH Modelling
Call:
garchFit(formula = ~arma(0, 1) + garch(1, 1), data = dif1.kurs,
trace = F)
Mean and Variance Equation:
data ~ arma(0, 1) + garch(1, 1)
<environment: 0x0000000025931370>
[data = dif1.kurs]
Conditional Distribution:
norm
Coefficient(s):
mu ma1 omega alpha1 beta1
1.734972 0.090323 167.654611 0.192608 0.771104
Std. Errors:
based on Hessian
Error Analysis:
Estimate Std. Error t value Pr(>|t|)
mu 1.73497 2.12681 0.816 0.4146
ma1 0.09032 0.04561 1.980 0.0477 *
omega 167.65461 82.09439 2.042 0.0411 *
alpha1 0.19261 0.04618 4.171 3.04e-05 ***
beta1 0.77110 0.05998 12.857 < 2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Log Likelihood:
-2929.521 normalized: -5.355615
Description:
Fri May 07 22:51:35 2021 by user: DK0158AU
Standardised Residuals Tests:
Statistic p-Value
Jarque-Bera Test R Chi^2 35.99119 1.529723e-08
Shapiro-Wilk Test R W 0.9855314 2.972663e-05
Ljung-Box Test R Q(10) 18.53264 0.04661644
Ljung-Box Test R Q(15) 27.96422 0.02179408
Ljung-Box Test R Q(20) 34.30147 0.02415621
Ljung-Box Test R^2 Q(10) 5.940365 0.8202483
Ljung-Box Test R^2 Q(15) 15.56604 0.4114636
Ljung-Box Test R^2 Q(20) 23.55279 0.2624729
LM Arch Test R TR^2 10.83636 0.5429924
Information Criterion Statistics:
AIC BIC SIC HQIC
10.72951 10.76886 10.72935 10.74489 #AIC terkecil pada ARIMA(0,1,1) GARCH(1,1)Plot Simpangan Baku
#plot simpangan baku
sb<-garch11k@sigma.t
ragam<-garch11k@sigma.t^2
#sequense kurg 1 obs sehingga harus dimodifikasi
dates2<-kurs$Tanggal[-1]
stdv=xts(sb,order.by = dates2)
vragam=xts(ragam,order.by = dates2)
plot(stdv, main="Simpangan Baku")
plot(vragam, main="Ragam")
Sumber :
Website JISDOR BI : https://www.bi.go.id/id/statistik/informasi-kurs/jisdor/default.aspx
Khairunnisa, dan Siti Sunendiari.2020.Perbandingan Model Exponential GARCH dan Glosten Jaganathan Runkle GARCH dalam Meramalkan Nilai Tukar Rupiah terhadap Dolar Amerika Serikat.Prosiding Statistika Universitas Islam Bandung.Vol 6 No2 Hal 96-104