Analysis Of Arima And Garch Model.docx

Predictive Analysis

Assignment – 1

Analysis of ARIMA and GARCH Models

Submitted by: Vidushi MB18GID283

ARIMA Model for SAIL stock data Log Return Plot

Augmented Dickey-Fuller Test data: stock Dickey-Fuller = -8.1109, Lag order = 9, p-value = 0.01 alternative hypothesis: stationary

Interpretation: p-value is less than 0.05, so the series is stationery. To model a time series with ARIMA model, the series needs to be stationery which is in our case. If the series is non-stationery we need to convert it to stationery using the differencing method. So the value of differencing order d=0

ACF and PACF Plots For AR models, the ACF will dampen exponentially and the PACF will be used to identify the p order of the AR model. For MA models, the PACF will dampen exponentially and the ACF plot will be used to identify the q order of the MA process

Interpretation: The value of p=1, q=1, d=0. So the ARIMA parameters are (1,0,1) To test whether we have chosen the correct order, we do the Ljung-Box test. If pvalue > 0.05 then the order selected is correct.

> Box.test(fit_residual, lag=10, type = "Ljung-Box") Box-Ljung test data: fit_residual X-squared = 3.8172, df = 10, p-value = 0.9552

Since p-value is 0.9552 which is greater than 0.05 so the ARIMA order (1,0,1) is correct.

ARIMA Forecast

Model Information: Call: arima(x = stock_train, order = c(1, 0, 1), include.mean = F ALSE) Coefficients: ar1 -0.0022 s.e. 1.2302

ma1 0.0390 1.2195

sigma^2 estimated as 0.0006478: aic = -3313.48

log likelihood = 1659.74,

Error measures: ME RMSE MAE MPE MAPE MASE ACF1 Training set 0.0002117675 0.02545206 0.01910519 NaN Inf 0. 6908884 0.0001679105 Forecasts: Point Forecast Lo 99 Hi 99 738 0.001041755 -0.0645184 0.06660191 SAIL.NS.Close 2018-12-27 51.5 SAIL.NS.Close 2018-12-28 54.65

Interpretation: From the coefficient we can get the following return equation: Y(t) = -0.0022*Y(t-1) + 0.0390*E(t-1) The forecasted point return is 0.001041755 AIC score is -3313.48. Lower the score, better the model

Forecasted Returns versus Actual Returns

> print(Accuracy_percentage) [1] 56

We can run the model for other possible combinations of (p,d,q) to select the best optimal parameters to run the model.

GARCH Model for SAIL stock data Model 1 Building a GARCH model (sGARCH = Standard GARCH), armaOrder(1,1) = when looking at current stock price

Coefficients: ar1 ma1 0.9879 0.0211 s.e. 0.0068 0.0433

intercept 65.6790 6.2751

sigma^2 estimated as 3.426:

log likelihood = -1006.98,

aic = 2021.96

Model 2 Building a GARCH model (sGARCH = Standard GARCH), armaOrder(0,0) - looking at current stock price

Coefficients: intercept 69.5450 s.e. 0.5143 sigma^2 estimated as 130.7:

log likelihood = -1904.51,

aic = 3813.02

Model 3 Building a GARCH model (sGARCH = Standard GARCH), armaOrder(2,2) - when looking at current stock price

Coefficients: ar1 ar2 0.1074 0.8916 s.e. 0.1353 0.1352

ma1 0.9127 0.1422

sigma^2 estimated as 3.425:

ma2 0.0513 0.0464 log likelihood = -1008.53,

aic = 2027.06

Model 4 Building a GARCH model (sGARCH = Standard GARCH), armaOrder(3,1) - when looking at current stock price

Coefficients: ar1 ar2 0.1615 0.8755 s.e. 0.1635 0.1415

ar3 -0.0595 0.0477

sigma^2 estimated as 3.406:

ma1 0.8537 0.1587

intercept 65.6778 6.1854

log likelihood = -1005.54,

aic = 2023.08

Analysis of GARCH Model We have built here various GARCH models for different order (p, q) ARIMA model. Lower the AIC score, better the model. We found that AIC score of the model is least in the ARIMA model (1,1) i.e. 2021.96 The same results were found when we were doing the ARIMA method.

ARIMA Model R Code library(quantmod);library(tseries); library(timeSeries);library(forecast);library(xts);

# Step 1: Testing for stationarity getSymbols('SAIL.NS', from='2016-01-01', to='2019-01-01') stock_prices = SAIL.NS[,4] stock = diff(log(stock_prices),lag=1) stock = stock[!is.na(stock)] plot(stock,type='l', main='log returns plot') print(adf.test(stock))

# Step 2: Find values of p and q breakpoint = floor(nrow(stock)*(2.9/3)) par(mfrow = c(1,1)) acf.stock = acf(stock[c(1:breakpoint),], main='ACF Plot', lag.max=100) pacf.stock = pacf(stock[c(1:breakpoint),], main='PACF Plot', lag.max=100) auto.arima(stock) fit=arima(stock, order = c(1,0,1)) fit_residual=residuals(fit) Box.test(fit_residual, lag=10, type = "Ljung-Box")

# Step 3: Estimation and Forecasting Actual_series = xts(0,as.Date("2014-11-25","%Y-%m-%d")) forecasted_series = data.frame(Forecasted = numeric())] for (b in breakpoint:(nrow(stock)-1)) { stock_train = stock[1:b, ] stock_test = stock[(b+1):nrow(stock), ] fit = arima(stock_train, order = c(1, 0, 1),include.mean=FALSE) summary(fit) acf(fit$residuals,main="Residuals plot")

arima.forecast = forecast(fit, h = 1,level=99) summary(arima.forecast) par(mfrow=c(1,1)) plot(arima.forecast, main = "ARIMA Forecast") forecasted_series = rbind(forecasted_series,arima.forecast$mean[1]) colnames(forecasted_series) = c("Forecasted") Actual_return = stock[(b+1),] Actual_series = c(Actual_series,xts(Actual_return)) rm(Actual_return) print(stock_prices[(b+1),]) print(stock_prices[(b+2),])

}

# Step 4: Check the accuracy of the ARIMA model by comparing the forecasted returns versus the actual returns. Actual_series = Actual_series[-1] forecasted_series = xts(forecasted_series,index(Actual_series)) plot(Actual_series,type='l',main='Actual Returns Vs Forecasted Returns') lines(forecasted_series,lwd=1.5,col='red') legend('bottomright',c("Actual","Forecasted"),lty=c(1,1),lwd=c(1.5,1.5),col=c('black','red')) comparsion = merge(Actual_series,forecasted_series) comparsion$Accuracy = sign(comparsion$Actual_series)==sign(comparsion$Forecasted) print(comparsion) Accuracy_percentage = sum(comparsion$Accuracy == 1)*100/length(comparsion$Accuracy) print(Accuracy_percentage)

# The accuracy percentage of the model is 56%

GARCH Model R Code library(quantmod); library(rugarch); library(forecast);

s<-getSymbols('SAIL.NS', from='2017-01-01', to='2019-01-01') chartSeries(s) stock= SAIL.NS[,4] stock = stock[!is.na(stock)] head(stock) #Using Arima model to Know AIC & BIC Value for same order as GARCH fit1 <- Arima(stock, order=c(1,0,1)) summary(fit1) #Building a GARCH model (sGARCH = Standard GARCH), armaOrder(1,1) = when looking at current stock price s1 <- ugarchspec(variance.model = list(model = "sGARCH", garchOrder = c(1,1)), mean.model = list(armaOrder=c(1,1)), distribution.model = "std") sGarch1 <- ugarchfit(spec=s1, data = stock) sPredict1<- ugarchboot(sGarch1, n.ahead=10, method =c("Partial", "Full")[1]) plot(sPredict1, which = 2) #Building a GARCH model (sGARCH = Standard GARCH), armaOrder(0,0) = when looking at current stock price fit2 <- Arima(stock, order=c(0,0,0)) summary(fit2) s2 <- ugarchspec(variance.model = list(model = "sGARCH", garchOrder = c(1,1)), mean.model = list(armaOrder=c(0,0)), distribution.model = "std") sGarch2 <- ugarchfit(spec=s2, data = stock) sPredict2<- ugarchboot(sGarch2, n.ahead=10, method =c("Partial", "Full")[1]) plot(sPredict2, which = 2) #Building a GARCH model (sGARCH = Standard GARCH), armaOrder(2,2) = when looking at current stock price fit3 <- Arima(stock, order=c(2,0,2),include.mean = FALSE) summary(fit3)

s3 <- ugarchspec(variance.model = list(model = "sGARCH", garchOrder = c(1,1)), mean.model = list(armaOrder=c(2,2)), distribution.model = "std") sGarch3 <- ugarchfit(spec=s3, data = stock) sPredict3<- ugarchboot(sGarch3, n.ahead=10, method =c("Partial", "Full")[1]) plot(sPredict3, which = 2) #Building a GARCH model (sGARCH = Standard GARCH), armaOrder(3,1) = when looking at current stock price fit4<- Arima(stock, order=c(3,0,1)) summary(fit4) s4 <- ugarchspec(variance.model = list(model = "sGARCH", garchOrder = c(1,1)), mean.model = list(armaOrder=c(3,1)), distribution.model = "std") sGarch4 <- ugarchfit(spec=s4, data = stock) sPredict4<- ugarchboot(sGarch4, n.ahead=10, method =c("Partial", "Full")[1]) plot(sPredict4, which = 2)