TianyuDu/AnnEconForecast

---
title: "Set_11_Garch"
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(MASS)
library(moments)
library(fGarch)
```

Simulate GARCH(1,1) process with low and high persistence
```{r}
spec = garchSpec(model = list(omega = 2, alpha = 0.4, beta = 0.4))
GARCH1 <- garchSim(spec, n = 1000, extended = TRUE)
spec = garchSpec(model = list(omega = 2, alpha = 0.1, beta = 0.88))
GARCH2 <- garchSim(spec, n = 1000, extended = TRUE)
```

Plot the simulated series
```{r}
par(mfcol=c(1,2), mai=c(0.4, 0.4, 0.7, 0.1), cex.main=0.9)
plot(GARCH1$garch, type="l", ylim=c(-40,40), main = expression(paste("GARCH(1,1): ", alpha, " = 0.4,  ", beta, " = 0.4;","  persistence = 0.67")), cex.main=0.8)
plot(GARCH2$garch, type="l", ylim=c(-40,40), main = expression(paste("GARCH(1,1): ", alpha, " = 0.1,  ", beta, " = 0.88;", "  persistence = 0.83")), cex.main=0.8)
```

Plot the conditional standard deviations of the simulated series
```{r}
par(mfcol=c(1,2), mai=c(0.4, 0.4, 0.7, 0.1), cex.main=0.9)
plot(GARCH1$sigma, type="l", ylim=c(0,20), main = expression(paste("GARCH(1,1): std. dev.;","  persistence = 0.67")), cex.main=0.8)
plot(GARCH2$sigma, type="l", ylim=c(0,20), main = expression(paste("GARCH(1,1): std. dev.;","  persistence = 0.83")), cex.main=0.8)
```

Unconditional moments: persistence = 0.67
```{r}
all.moments(GARCH1$garch, central=TRUE, order.max=4)
```

Unconditional moments: persistence = 0.83
```{r}
all.moments(GARCH2$garch, central=TRUE, order.max=4)
```

Distributions of GARCH draws (persistence 0.67)
Note that the distribution is not Normal in a similar way to a typical financial time series.
```{r}
r_t1 <- GARCH1$garch
fit1 <- fitdistr(r_t1, "normal")
para1 <- fit1$estimate

r_t2 <- GARCH2$garch
fit2 <- fitdistr(r_t2, "normal")
para2 <- fit2$estimate

par(mfcol=c(1,2), mai=c(0.4, 0.4, 0.7, 0.1), cex.main=0.9)
hist(r_t1, prob = TRUE, breaks=50, xlim=c(-15,15), col="lightgrey", main="Histogram of GARCH draws, persistence = 0.67")
curve(dnorm(x, para1[1], para1[2]), col = 2, add = TRUE)
hist(r_t2, prob = TRUE, breaks=50, xlim=c(-45,45), col="lightgrey", main="Histogram of GARCH draws, persistence = 0.83")
curve(dnorm(x, para2[1], para2[2]), col = 2, add = TRUE)
```

ACF and PACF of the GARCH draws square (variance)
```{r}
maxlag <- 20
ACF1 <- acf(r_t1^2, main=NULL, lag.max=maxlag, plot=FALSE)
PACF1 <- pacf(r_t1^2, main=NULL, lag.max=maxlag, plot=FALSE)

ACF2 <- acf(r_t2^2, main=NULL, lag.max=maxlag, plot=FALSE)
PACF2 <- pacf(r_t2^2, main=NULL, lag.max=maxlag, plot=FALSE)

par(mfcol=c(2,2), mai=c(0.4, 0.4, 0.7, 0.1), cex.main=0.9)
plot(ACF1[1:maxlag], ylim=c(-0.1,0.3), main="ACF of variance, persistence = 0.67")
plot(PACF1[1:maxlag], ylim=c(-0.1,0.3),main="PACF of variance, persistence = 0.67")
plot(ACF2[1:maxlag], ylim=c(-0.1,0.3), main="ACF of variance, persistence = 0.83")
plot(PACF2[1:maxlag], ylim=c(-0.1,0.3),main="PACF of variance, persistence = 0.83")
```

ACF and PACF of the normalized GARCH draws 
Note that all the information of the process is contained in the conditional variance.
Once we normalize with it, there is no more information left.
```{r}
maxlag <- 20
r_t_normalized1 = GARCH1$garch / GARCH1$sigma
ACF1 <- acf(r_t_normalized1, main=NULL, lag.max=maxlag, plot=FALSE)
PACF1 <- pacf(r_t_normalized1, main=NULL, lag.max=maxlag, plot=FALSE)

r_t_S_normalized2 = GARCH2$garch / GARCH2$sigma
ACF2 <- acf(r_t_S_normalized2, main=NULL, lag.max=maxlag, plot=FALSE)
PACF2 <- pacf(r_t_S_normalized2, main=NULL, lag.max=maxlag, plot=FALSE)

par(mfcol=c(2,2), mai=c(0.4, 0.4, 0.7, 0.1), cex.main=0.9)
plot(ACF1[1:maxlag], main="ACF of normalized r_t, persistence = 0.67")
plot(PACF1[1:maxlag], main="PACF of normalized r_t, persistence = 0.67")

plot(ACF2[1:maxlag], main="ACF of normalized r_t, persistence = 0.83")
plot(PACF2[1:maxlag], main="PACF of normalized r_t, persistence = 0.83")
```