TianyuDu/AnnEconForecast

---
title: "set_11_Volatility2"
output:
  html_notebook: default
  pdf_document: default
  html_document:
    df_print: paged
---

Set data directory path
```{r setup, include=FALSE, cache = FALSE}
if (!require("knitr")) install.packages("knitr")
library(knitr)
opts_knit$set(root.dir = "C:/Teaching/Undergrad/data ECO374")
```
 
Install and load required packages
```{r}
chooseCRANmirror(graphics=FALSE, ind=33)
if (!require("fGarch")) install.packages("fGarch")

library(aTSA)
library(xts)
library(anytime)
library(fGarch)
```

Dowload data on S&P500 index, daily frequency
Source: https://finance.yahoo.com/quote/%5EGSPC/history?p=%5EGSPC

Load, extract Close price, convert to xts format, and plot
```{r}
data_SP500_full <- read.csv(file="SP500_daily.csv", header=TRUE, sep=",")

xtsdata_SP500 <- xts(x=data_SP500_full[c("Close")], order.by=anytime(data_SP500_full$Date))
 
plot.xts(xtsdata_SP500, plot.type = c("single"), main ="S&P 500 index, close price" , legend.loc = "topleft", grid.col="lightgray")
```

Plot of returns
```{r}
xtsdata_SP500_returns <- diff(log(xtsdata_SP500))
xtsdata_SP500_returns <- na.omit(xtsdata_SP500_returns)

plot.xts(xtsdata_SP500_returns, plot.type = c("single"), main ="S&P 500 index, returns" , legend.loc = "topleft", grid.col="lightgray", col="blue")
```

Plot of returns SQUARED (variance)
```{r}
xtsdata_SP500_returns_SQUARED <- xtsdata_SP500_returns^2

plot.xts(xtsdata_SP500_returns_SQUARED, plot.type = c("single"), main ="S&P 500 index, returns SQUARED (variance)" , legend.loc = "topleft", grid.col="lightgray", col="red")
```

ACF and PACF plots of returns SQUARED (variance)
```{r}
maxlag <- 30
ACF <- acf(as.matrix(xtsdata_SP500_returns_SQUARED$Close), main=NULL, lag.max=maxlag, plot=FALSE)
PACF <- pacf(as.matrix(xtsdata_SP500_returns_SQUARED$Close), main=NULL, lag.max=maxlag, plot=FALSE)

par(mfcol=c(1,2), mai=c(0.4, 0.4, 0.7, 0.1), cex.main=0.9)
plot(ACF[1:maxlag],ylim=c(-0.05,0.25), main="ACF of returns variance")
plot(PACF[1:maxlag],ylim=c(-0.05,0.25), main="PACF of returns variance")
```

ARCH model estimation
```{r}
ARCH <- garchFit(~garch(12,0), data = xtsdata_SP500_returns, trace = FALSE)
summary(ARCH)
```

GARCH model estimation
Note that GARCH(1,1) with 4 parameters has lower AIC and higher Log likelihood than ARCH(12) with 14 parameters
```{r}
GARCH <- garchFit(~garch(1,1), data = xtsdata_SP500_returns, trace = FALSE)
summary(GARCH)
```

Forecast (plot along with last 100 observations)
Note that we have not included any model for the mean here
```{r}
forecast <- predict(GARCH, plot=TRUE, nx=100)
```