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T05 GARCH_3 example.pptx

T05 GARCH_3 example

pptx

School

Georgia Institute Of Technology *

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Course

6402

Subject

Industrial Engineering

Date

Dec 6, 2023

Type

pptx

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23

Uploaded by ConstableGazelle2633

I. Exchange Rates Prediction: ARCH Modeling II. Exchange Rates Prediction: ARMA+GARCH Modeling III. Exchange Rates Prediction: Comparing Predictions Time Series Analysis: Modeling Heteroskedasticity (3): Examples ARCH ARMA GARCH compare

# What order? pacf(resids^2, ,main="Squared Residuals") # Fit ARCH model of order 8 library(tseries) garch.fit = garch(resids, order = c(0,8),trace=F) summary(garch.fit) #Evaluate goodness of fit resids.fgarch = residuals(garch.fit)[-c(1:7)] resids.fgarch=resids.fgarch[!is.na(resids.fgarch)] par(mfcol=c(2,1)) acf(resids.fgarch,main="ACF of ARCH Residuals (USD/EUR)") acf(resids.fgarch^2,main="ACF of Squared ARCH Residuals (USD/EUR)") Box.test(resids.fgarch,lag=9,type='Ljung',fitdf=8) Box.test(resids.fgarch^2,lag=9,type='Ljung',fitdf=8) ARCH Fit ARCH ARMA GARCH compare

ARCH Fit: Summary (USD/EUR) Order: p=8 All coefficients are statistically significant Reject the null of uncorrelated residuals but do not reject for squared residuals ARCH ARMA GARCH compare

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ARCH Fit: Summary (USD/BRL) Order: p=6 All coefficients are statistically significant Reject the null of uncorrelated residuals but do not reject for squared residuals ARCH ARMA GARCH compare

ARCH Fit: Residual Analysis ACF of White Noise ACF of White Noise ARCH ARMA GARCH compare

I. Exchange Rates Prediction: ARCH Modeling II. Exchange Rates Prediction: ARMA+GARCH Modeling III. Exchange Rates Prediction: Comparing Predictions Time Series Analysis: Modeling Heteroskedasticity: Examples ARCH ARMA GARCH compare

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#GARCH Order Selection library(rugarch) #Select model with smallest BIC final.bic = Inf final.order = c(0,0) for (m in 0:3) for (n in 0:3){ spec = ugarchspec(variance.model=list(garchOrder=c(m,n)), mean.model=list(armaOrder=c(6, 3), include.mean=T), distribution.model="std") fit = ugarchfit(spec, data.train, solver = 'hybrid') current.bic = infocriteria(fit)[2] if (current.bic < final.bic){ final.bic = current.bic final.order = c(m, n) }} GARCH Order Selection Selected Orders for modeling the conditional variance : m=1, n=2 USD/EUR case ARCH ARMA GARCH compare

#Refine the ARMA order final.bic = Inf final.order.arma = c(0,0) for (p in 0:6) for (q in 0:6){ spec = ugarchspec(variance.model=list(garchOrder=c(1,2), mean.model=list(armaOrder=c(p, q), include.mean=T), distribution.model="std") fit = ugarchfit(spec, data.train, solver = 'hybrid') current.bic = infocriteria(fit)[2] if (current.bic < final.bic){ final.bic = current.bic final.order.arma = c(p, q) } } ARMA Order Selection (refined) Selected Orders for modeling the conditional mean : AR=0, MA=0 USD/EUR case ARCH ARMA GARCH compare

#Refine the GARCH order final.bic = Inf final.order.garch = c(0,0) for (m in 0:3) for (n in 0:3){ spec = ugarchspec(variance.model=list(garchOrder=c(m,n)), mean.model=list(armaOrder=c(final.order.arma[1], final.order.arma[2]), include.mean=T), distribution.model="std") fit = ugarchfit(spec, data.train, solver = 'hybrid') current.bic = infocriteria(fit)[2] if (current.bic < final.bic){ final.bic = current.bic final.order.garch = c(m, n) } } GARCH Order Selection (refined) USD/EUR case Selected Orders for modeling the conditional variance : m=1, n=2 ARCH ARMA GARCH compare

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Order Selection: Comparison ARMA(6,3)+GARCH(2,1) ARMA(0,0)+GARCH(2,1) ARMA(0,0)+GARCH(2,1) USD/EUR currency USD/BRL currency USD/CYN currency ARMA(9,8)+GARCH(1,1) ARMA(0,0)+GARCH(1,1) ARMA(0,0)+GARCH(1,1) ARMA(9,1)+GARCH(3,3) ARMA(1,0)+GARCH(3,3) ARMA(1,0)+GARCH(3,3) ARCH ARMA GARCH compare

ARMA+GARCH: Goodness of Fit #Goodness of Fit spec.1 = ugarchspec(variance.model=list(garchOrder=c(1,2)), mean.model=list(armaOrder=c(6,3), include.mean=T), distribution.model="std") final.model.1 = ugarchfit(spec.1, data.train, solver = 'hybrid') spec.2 = ugarchspec(variance.model=list(garchOrder=c(1,2)), mean.model=list(armaOrder=c(0,0), include.mean=T), distribution.model="std") final.model.2 = ugarchfit(spec.2, data.train, solver = 'hybrid') #Compare Information Criteria infocriteria(final.model.1) infocriteria(final.model.2) ARCH ARMA GARCH compare

ARMA+GARCH: Goodness of Fit (cont’d) Both models perform similarly: choose least complex model USD/EUR USD/BRL USD/CNY ARCH ARMA GARCH compare

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I. Exchange Rates Prediction: ARCH Modeling II. Exchange Rates Prediction: ARMA+GARCH Modeling III. Exchange Rates Prediction: Comparing Predictions Time Series Analysis: Modeling Heteroskedasticity: Examples ARCH ARMA GARCH compare

#Prediction of the return time series and the volatility sigma nfore = length(data.test) fore.series.1 = NULL fore.sigma.1 = NULL for(f in 1: nfore){ data = data.train if(f>2) data = c(data.train,data.test[1:(f-1)]) final.model.1 = ugarchfit(spec.1, data, solver = 'hybrid') fore = ugarchforecast(final.model.1, n.ahead=1) fore.series.1 = c(fore.series.1, fore@forecast$seriesFor) fore.sigma.1 = c(fore.sigma.1, fore@forecast$sigmaFor) } Computing Prediction ARCH ARMA GARCH compare

Prediction Accuracy (USD/EUR) ARMA-GARCH Model 2 performs best for all measures ARCH ARMA GARCH compare

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Prediction Accuracy (USD/BRL) ARMA+GARCH Model 2 performs best across all measures ARCH ARMA GARCH compare

Prediction Accuracy (USD/CYN) ARMA+GARCH Model 2 performs best across all but MAPE measures ARCH ARMA GARCH compare

#Mean Prediction Comparison Plots ymin = min(c(as.vector(data.test),fore.series.1,fore.series.2)) ymax = max(c(as.vector(data.test),fore.series.1,fore.series.2)) data.plot = data.test names(data.plot)="Fore" plot(data[c(n-53):n],type="l", ylim=c(ymin,ymax), xlab=" ", ylab="USD/EUR Exchange Rate",main="Mean Prediction Comparison (USD/EUR)") data.plot$Fore=fore.series.1 points(data.plot,lwd= 2, col="blue") data.plot$Fore=fore.series.2 points(data.plot,lwd= 2, col="brown") Mean Prediction Comparison ARCH ARMA GARCH compare

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Mean Prediction Comparison ARCH ARMA GARCH compare

#Compare squared observed time series with variance forecasts ymin = min(c(as.vector(data.test^2),fore.sigma.1^2,fore.sigma.2^2) ymax = max(c(as.vector(data.test^2),fore.sigma.1^2,fore.sigma.2^2)) plot(data.test^2,type="l", ylim=c(ymin,ymax), xlab=" ", ylab="USD/EUR Exchange Rate") data.plot$Fore=fore.sigma.1^2 points(data.plot,lwd= 2, col="blue") data.plot$Fore=fore.sigma.2^2 points(data.plot,lwd= 2, col="brown") Variance Prediction Comparison ARCH ARMA GARCH compare

Variance Prediction Comparison ARCH ARMA GARCH compare

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Other Models: R Implementation #GARCH spec.1 = ugarchspec(variance.model=list(garchOrder=c(1,2)), mean.model= list(armaOrder=c(0,0), include.mean=T), distribution.model="std") #GJR-GARCH spec.2 = ugarchspec(variance.model=list(model = "gjrGARCH",garchOrder=c(1,2)), mean.model=list(armaOrder=c(0,0), include.mean=T), distribution.model="std") #EGARCH spec.3 = ugarchspec(variance.model=list(model ="eGARCH",garchOrder=c(1,2)), mean.model=list(armaOrder=c(0,0),include.mean=T), distribution.model="std") #APARCH spec.4 = ugarchspec(variance.model=list(model = "apARCH",garchOrder=c(1,2)), mean.model=list(armaOrder=c(0,0), include.mean=T), distribution.model="std") #IGARCH spec.5 = ugarchspec(variance.model=list(model = "iGARCH",garchOrder=c(1,2)), mean.model=list(armaOrder=c(0,0), include.mean=T), distribution.model="std") ARCH ARMA GARCH compare

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Variance Prediction Comparison ARCH ARMA GARCH compare

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