---
title: "MATH 5075 R Project 9"
author: "Your Name Here"
date: "October 20, 2016"
output:
pdf_document:
keep_tex: TRUE
---
*Remember: I expect to see commentary either in the text, in the code with comments created using `#`, or (preferably) both! **Failing to do so may result in lost points!***
*Since this assignment involves simulation, I set the seed to the following in order to get the same results:*
```{r}
set.seed(5292016)
```
## Problem 1
*Use `garchSim()` from the **fGarch** package to simulate 200 observations from the following GARCH(1,1) processes (you may use the default burn-in period), and plot them:*
$$\omega = 0.1, \alpha = 0.1, \beta = 0.1$$
$$\omega = 0.7, \alpha = 0.1, \beta = 0.1$$
$$\omega = 0.001, \alpha = 0.1, \beta = 0.1$$
$$\omega = 0.1, \alpha = 0.7, \beta = 0.1$$
$$\omega = 0.1, \alpha = 0.1, \beta = 0.7$$
$$\omega = 0.1, \alpha = 0.49, \beta = 0.49$$
*In addition to plotting the simulated processes, also plot the ACF and PACF of each simulated process, and compare to what would be seen for* $\text{AR}(p)$ *or* $\text{MA}(q)$ *processes.*
```{r, tidy=TRUE, error=TRUE}
# Your code here
```
## Problem 2
*Use `garchFit()` from the **fGarch** package to fit a $\text{GARCH}(1,1)$ model to the `nyse` data set (**astsa**), using the quasi-maximum likelihood estimator. After fitting a model, simulate a $\text{GARCH}(1,1)$ model with the same parameters as the fitted model, and with 2000 obserivations. Does the simulated process look similar to that of the actual NYSE data?* (Hint: If you save the fitted model in `x`, you can access the coefficients with `x@fit$coef`; the `@` accessor is similar to the `$` accessor, but is used for S4 class R objects.)
```{r, tidy=TRUE, error=TRUE}
# Your code here
```