\documentclass[11pt]{article}

\usepackage{../Math545}

\usepackage{amsmath,amssymb}

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\def\myGt{\mathbf{G}_t}
\def\myFt{\mathbf{F}_t}
\def\myHonet{\mathbf{H}_{1t}}
\def\myHtwot{\mathbf{H}_{2t}}
\def\myZ{\mathbf{Z}}
\def\evec{\mathbf{e}}
\def\bP{\mathbf{P}}
\def\bM{\mathbf{M}}
\def\bK{\mathbf{K}}
\def\bL{\mathbf{L}}
\def\bv{\mathbf{v}}
\def\ba{\mathbf{a}}
\def\bOmega{{\boldsymbol{\Omega}}}
\def\bSigma{{\boldsymbol{\Sigma}}}
\def\beps{{\boldsymbol{\epsilon}}}


\begin{document}

<<setup, cache=FALSE, echo=FALSE>>=
library(knitr)
# global chunk options
opts_chunk$set(cache=TRUE, autodep=TRUE)
options(scipen=999)
options(repos=c(CRAN="https://cloud.r-project.org/"))
@

\begin{center}
\textsclarge{MATH 545: Introduction to Time Series Analysis}

\smallskip
\textsc{Conditional Heteroscedastic Models}
\end{center}

In a \textit{conditional heteroscedastic model}, we have an observed data set with \textit{time-varying variance}.  Typically, these models can be considered as special forms of state space models.

\bigskip
\textbf{Stochastic Volatility:} The \textit{stochastic volatility} model uses a stochastic state space form that allows the variance of the observed process to vary stochastically with time.   The simplest form specifies that
\[
Y_t = \exp\{X_t/2\} Z_t \qquad \{Z_t\} \sim \WN(0,\sigma_Z^2)
\]
and then that
\[
X_{t+1} = \phi X_t + V_{t+1}  \qquad \{Z_t\} \sim \WN(0,\sigma_V^2)
\]
that is, given $X_t = x_t$, $Y_t$ has mean zero and variance $\exp\{x_t\}$, and $\{X_t\} \sim \AR(1)$.


<<CH1,comment='+', fig.width=7.25, fig.height=5.0, fig.align='center'>>=
set.seed(24)
n<-1000;phi<-0.98;sigZ<-0.01;sigV<-0.05
X<-arima.sim(n,model=list(ar=phi),rand.gen = function(n) sqrt(sigV) * rnorm(n))
Z<-rnorm(n)*sigZ
Y<-exp(X/2)*Z
par(mar=c(4,3,2,0));plot(Y,type='l',ylim=range(Y));
points(Y,pch=19,cex=0.3)
@

This model allows for \textit{volatility clustering}, that is, bursts of high or low variance.

\pagebreak

\textbf{ARCH/GARCH MODEL:} The \textit{ARCH} and \textit{GARCH} models use \textit{deterministic} state space forms that allows the variance of the observed process to vary with time.   The ARCH(1) model specifies
\[
Y_t = \sqrt{X_t} Z_t \qquad \{Z_t\} \sim \WN(0,1)
\]
and then that
\[
X_{t+1} = \alpha_0 + \alpha_1 Y_t^2  
\]
whereas the GARCH(1,1) model specifies
\[
X_{t+1} = \alpha_0 + \alpha_1 Y_t^2 + \beta_1 X_t
\]
where, in both cases, given $X_t = x_t$, $Y_t$ has mean zero and variance $\sqrt{x_t}$.


<<CH2,comment='+', fig.width=7.25, fig.height=5.25, fig.align='center'>>=
set.seed(345)
n<-1000;al0<-0.01;al1<-0.95
X<-Y<-rep(0,n);X[1]<-al0;Y[1]<-rnorm(1)*sqrt(X[1])
for(j in 2:n){
    X[j]<-al0+al1*Y[j-1]^2
    Y[j]<-rnorm(1)*sqrt(X[j])
}
par(mar=c(3,3,2,0));plot(Y,type='l',ylim=range(Y));title('ARCH(1)')
points(Y,pch=19,cex=0.3)
al0<-0.1;al1<-0.08;be1<-0.90
for(j in 2:n){
    X[j]<-al0+al1*Y[j-1]^2+be1*X[j-1]
    Y[j]<-rnorm(1)*sqrt(X[j])
}
par(mar=c(4,3,2,0));plot(Y,type='l',ylim=range(Y));title('GARCH(1,1)')
points(Y,pch=19,cex=0.3)
@

These models also allow for volatility clustering, but only have one source of randomness.

\end{document}