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\textit{EC821:\ Time Series Econometrics, Spring 2003}

\bigskip \textit{Notes Section 11}

\textbf{ARCH:\ Modelling volatility}

Consider the $AR(p)$ model:
\[
y_{t}=c+\sum_{i=1}^{p}\phi _{i}y_{t-i}+u_{t}
\]
with $u$ i.i.d. Covariance stationarity requires that the $AR\;$polynomial
has roots outside the unit circle. The optimal linear forecast of the level
of $y_{t},$ given covariance stationarity, is:
\[
E\left( \hat{y}_{t}|y_{t-1},y_{t-2},...\right) =c+\sum \hat{\phi}_{i}y_{t-i}
\]
So that the conditional mean is changing as the process evolves. However,
given covariance stationarity, the unconditional mean is constant:
\[
Ey_{t}=c\left( 1-\phi _{1}-\phi _{2}-...-\phi _{p}\right) ^{-1}
\]
What if we wanted to forecast the variance of the series, rather than its
mean? We consider $u_{t}$ as a process with a fixed unconditional variance, $%
\sigma ^{2}.$ But the conditional variance of $y_{t}$ could change over
time. If it followed a systematic pattern, we might have something like:
\[
u_{t}^{2}=\eta +\sum_{i=1}^{m}\alpha _{i}u_{t-i}^{2}+\omega _{t}
\]
where $\omega $ is an i.i.d. error process. This law of motion implies that:
\[
E\left( u_{t}^{2}|u_{t-1}^{2},u_{t-2}^{2},...\right) =\eta
+\sum_{i=1}^{m}\alpha _{i}u_{t-i}^{2}
\]
which is then the $m^{th}-$order model of Autoregressive Conditional
Heteroskedasticity ($ARCH),$ as proposed by Engle (1982). This conditional
expectation must be non--negative for all realizations of the $u_{t}$
process; a necessary condition is that $\alpha _{i}>0$ for all $i.$ For $%
u_{t}^{2}$ itself to be covariance stationary, we further require that all
the roots of the $\alpha $ polynomial lie outside the unit circle. If the $%
\alpha _{i}$ are all non--negative, this condition may be written as $\sum
\alpha _{i}<1.$

An alternative way to write such a model is in the form:
\[
u_{t}=\sqrt{h_{t}}v_{t}
\]
where $v_{t}$ is distributed with mean zero and variance of unity. If we
then write:
\[
h_{t}=\eta +\sum_{i=1}^{m}\alpha _{i}u_{t-i}^{2}
\]
then the conditional expectation gives us the same expression in terms of $%
\eta $ and the $\alpha _{i}$ terms. 

This $ARCH$ model may then be used to augment a regression equation, as a
way of modelling the conditional variance of that equation's errors. The
presence of $ARCH\;$effects (detected, for instance, by the Lagrange
Multiplier test for $ARCH:$\texttt{archlm} in Stata) does not invalidate the
use of OLS\ to estimate the equation, but if there are systematic movements
in the conditional variance, we might want to be able to model them jointly
with the level of the series. The ``mean equation'' and the $ARCH\;$equation
for the conditional variance may be jointly estimated in a maximum
likelihood context. Various solutions have been proposed to deal with the
non--negativity constraints, which are quite difficult to impose in a ML\
estimation procedure.

Non--Gaussian distributions may also be used: to cope with the stylized
facts of excess kurtosis in asset returns, it may be desirable to allow for
this in the model. $ARCH$ models have often been fit using a $t$
distribution, where an addditional parameter: the degrees of freedom---is
estimated in the process. An even more general solution relies upon the
Generalized Error Distribution (GED), which encompasses both the Normal and $%
t$ distributions as special cases, allowing for both excess and
less--than--normal kurtosis. 

\textbf{The GARCH\ model of Bollerslev}

The primary extension of the $ARCH(m)\;$methodology of Engle is the
Generalized $ARCH,$ or $GARCH(r,m)$ model of Bollerslev (1986). The
extension from $ARCH\;$to $GARCH\;$considers:
\[
h_{t}=\eta +\Pi (L)u_{t}^{2}
\]
where $\Pi (L)$ is an infinite--order lag polynomial. Under appropriate
conditions, we can rewrite this as a rational lag in two finite--order
polynomials in the lag operator:
\[
\Pi (L)=\alpha (L)\left[ 1-\delta (L)\right] ^{-1}
\]
where the roots of $\delta (L)$ are outside the unit circle. This gives rise
to the $GARCH(r,m)$ model:
\[
h_{t}=\kappa +\sum_{i=1}^{r}\delta _{i}h_{t-i}+\sum_{j=1}^{m}\alpha
_{j}u_{t-j}^{2}
\]
where $\kappa =(1-\sum \delta _{i})\eta .$ This is an $ARMA(p,r)$ process
for the squared errors, where the $j^{th}$ AR coefficient is $\left( \delta
_{j}+\alpha _{j}\right) $and the  $j^{th}$ MA\ coefficient is $-\delta _{j},$
with $p=\max (r,m).$ The non--negativity requirement is that all parameters
in this process are non--negative, with $\kappa >0.$ The process is CS\ if $%
\left( \sum \delta _{i}+\sum \alpha _{i}\right) <1.$ Just as a low--order $%
ARMA(p,q)$ process will often work as well as a high--order $AR(p),$ a
low--order $GARCH(r,m)$ will often suffice to capture the dynamics of the
conditional variance as well as an $ARCH(m)$ for large $m.$ The ability to
specify a more parsimonious model, especially given the non--negativity
constraints on the maximum likelihood problem, is attractive.

An interesting special case is that of $IGARCH,$ or integrated $GARCH:$
where  $\left( \sum \delta _{i}+\sum \alpha _{i}\right) =1$ (or cannot be
distinguished from 1). This causes the unconditional variance of $u_{t}$ to
be infiinite, so that neither $u_{t}$ nor $u_{t}^{2}$ is CS. The issue is
essentially that of a unit root in the $ARMA\;$process for $u_{t}^{2},$ and
is often encountered in practice.

\textbf{The GARCH--in--mean model}

A very useful variation on the $GARCH$ model is $GARCH-in-mean:$ a
specification where the conditional variance itself enters the mean
equation. For assets, we might expect higher return and higher risk to be
positively correlated, and thus a positive $ARCH-in-mean$ term would be
expected. A similar rationale would apply if we confront the stylized fact
that countries with higher levels of inflation often are observed to have
higher variances of the inflation process. An example of this model is
provided by Engle et al. (1987).

\textbf{Alternative GARCH\ specifications}

A huge literature on alternative $GARCH\;$specifications exists; many of
these models are preprogrammed in Stata's \texttt{arch} command, and
references for their analytical derivation are given in the Stata manual.
One of particular interest is Nelson's (1991) exponential $GARCH,$ or $%
EGARCH.$ He proposed:
\[
\log h_{t}=\eta +\sum_{j=1}^{\infty }\pi _{j}\left( \left| \nu _{t-j}\right|
-E\left| \nu _{t-j}\right| +\theta \nu _{t-j}\right) 
\]
which is then parameterized as a rational lag of two finite--order
polynomials, just as in Bollerslev's $GARCH.$ Advantages of the $EGARCH\;$%
specification include the positive nature of $h_{t}$ irregardless of the
estimated parameters, and the asymmetric nature of the impact of
innovations: with $\theta \neq 0,$ a positive shock will have a different
effect on volatility than will a negative shock, mirroring findings in
equity market research about the impact of ``bad news'' and ``good news'' on
market volatility. Nelson's model is only one of several extensions of $%
GARCH\;$that allow for asymmetry, or consider nonlinearities in the process
generating the conditional variance:\ for instance, the threshold $ARCH$
model of Zakoian (1990) and the Glosten et al. model (1993). 

The $ARCH\;$and $GARCH\;$models have also been extended in a multivariate
context (although considering more than two variables is quite difficult, as
the number of parameters to be estimated grows very rapidly). 

Useful surveys of the literature (although now somewhat dated) are provided
by Bollerslev et al. (1992, 1994). 

{\small References}

\begin{thebibliography}{9}
\bibitem{}  {\small Bollerslev, T., 1986. Generalized autoregressive
conditional heteroskedasticity. Journal of Econometrics 31, 307--327.}

\bibitem{bck}  {\small Bollerslev, T., Chou, R. and K. Kroner, 1992. ARCH\
modeling in finance. Journal of Econometrics 52, 5--59.}

\bibitem{ben}  {\small Bollerslev, T., Engle, R. and D. Nelson, 1994. ARCH\
models. In Handbook of Econometrics, Vol. IV, ed. R. Engle and D. McFadden.
Elsevier.}

\bibitem{eng82}  {\small Engle, R., 1982. Autoregressive conditional
heteroskedasticity with estimates of the variance of U.K. inflation.
Econometrica, 50, 987--1008.}

\bibitem{elr}  {\small Engle, R., Lilien, D., and R. Robins, 1987.
Estimating time varying risk premia in the term structure: the ARCH--M
model. Econometrica, 55, 391--407.}

\bibitem{gjr}  {\small Glosten, L. Jagannathan, R., and D. Runkle, 1993. On
the relation between the expected value and the volatility of the nominal
excess return on stocks. Journal of Finance, 48, 1779--1801.}

\bibitem{zak}  {\small Zakoian, J., 1990. Threshold heteroskedastic models.
Unpublished manuscript, CREST, INSEE. }
\end{thebibliography}

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