TianyuDu/AnnEconForecast

---
title: "Set_11_Archp"
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 ARCH(p)
```{r}
spec = garchSpec(model = list(omega = 2, alpha = c(0.3, 0.1, 0.4), beta = 0))
ARCHp <- garchSim(spec, n = 1000, extended = TRUE)

plot(ARCHp$garch, type="l", main = expression(paste("ARCH(3): ", alpha[1], " = 0.3,  ", alpha[2], " = 0.1,  ", alpha[3], " = 0.4")), cex.main=0.9)
```

Distributions of ARCH draws
Note that the distribution is not Normal in a similar way to a typical financial time series.
```{r}
r_t <- ARCHp$garch
fit <- fitdistr(r_t, "normal")
para <- fit$estimate
hist(r_t, prob = TRUE, breaks=100, xlim=c(-11,11), col="lightgrey", main="Histogram of ARCH draws", cex.main=0.9)
curve(dnorm(x, para[1], para[2]), col = 2, add = TRUE)
```

Moments of the ARCH draws (alpha=0.6)
```{r}
all.moments(r_t, central=TRUE, order.max=4)
```

ACF and PACF of the ARCH draws and their square (variance)
```{r}
maxlag <- 30
ACF <- acf(r_t, main=NULL, lag.max=maxlag, plot=FALSE)
PACF <- pacf(r_t, main=NULL, lag.max=maxlag, plot=FALSE)

ACF_S <- acf(r_t^2, main=NULL, lag.max=maxlag, plot=FALSE)
PACF_S <- pacf(r_t^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(ACF[1:maxlag], ylim=c(-0.1,0.5), main="ACF of ARCH draws")
plot(PACF[1:maxlag], ylim=c(-0.1,0.5),main="PACF of ARCH draws")
plot(ACF_S[1:maxlag], ylim=c(-0.1,0.5), main="ACF of ARCH draws SQUARED")
plot(PACF_S[1:maxlag], ylim=c(-0.1,0.5),main="PACF of ARCH draws SQUARED")
```

ACF and PACF of the normalized ARCH draws (variance) 
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_normalized = ARCHp$garch / ARCHp$sigma
ACF <- acf(r_t_normalized, main=NULL, lag.max=maxlag, plot=FALSE)
PACF <- pacf(r_t_normalized, main=NULL, lag.max=maxlag, plot=FALSE)

r_t_S_normalized = ARCHp$garch^2 / ARCHp$sigma^2
ACF_S <- acf(r_t_S_normalized, main=NULL, lag.max=maxlag, plot=FALSE)
PACF_S <- pacf(r_t_S_normalized, 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(ACF[1:maxlag], main="ACF of ARCH draws normalized")
plot(PACF[1:maxlag], main="PACF of ARCH draws normalized")

plot(ACF_S[1:maxlag], main="ACF of ARCH draws SQUARED normalized")
plot(PACF_S[1:maxlag], main="PACF of ARCH draws SQUARED normalized")
```