Financial Time Series Analysis
(MFM3113)
ARCH/GARCH Models
Dr.K.D.Prasangika
Department of Mathematics
University of Ruhuna
Introduction
Autoregressive integrated moving average (ARIMA) models that al-
low modeling of volatility are unable to deal with volatility over time.
Monetary and financial series are characterized by volatility cluster-
ing means periods of high volatility alternate with periods of low
volatility. This phenomenon is called conditional heterocedents that
is particularly common in stock market data.
In order to highlight the persistence of volatility, Engle (1982) pro-
posed a new model called Heteroscedastic Conditional Autoregres-
sive (ARCH) able to capture the behavior of volatility as a function
of time.
ARCH/GARCH Models
• An ARCH (autoregressive conditionally heteroscedastic)
model is a model for the variance of a time series.
• ARCH models are used to describe a changing, possibly
volatile variance. Although an ARCH model could possibly be
used to describe a gradually increasing variance over time,
most often it is used in situations in which there may be short
periods of increased variation.
• ARCH models were created in the context of econometric and
finance problems having to do with the amount that
investments or stocks increase (or decrease) per time period,
so there’s a tendency to describe them as models for that type
of variable.
• An ARCH model could be used for any series that has periods
of increased or decreased variance.
The ARCH(1) Variance Model
Suppose that we are modeling the variance of a series yt . The
ARCH(1) model for the variance of model yt is that conditional on
yt−1 , the variance at time t is
2
Var (yt |yt−1 ) = σt2 = α0 + α1 yt−1
We impose the constraints α0 ≥ 0 and α1 ≥ 0 to avoid negative
variance.
If we assume that the series has mean = 0 (this can always be done
by centering), the ARCH model could be written as
yt = σt ϵt
q
σt = 2
α0 + α1 yt−1
ϵt ∼ iid(µ = 0, σ 2 )
ARCH(q) Model
Let Yt the return be explained by some variables Xt which can be
an ARMA model (p, q).
Yt = Xt β + et
where et /It1 ∼ N(0, σt2 ). It−1 refers to all the information available
in the returns data up to the time t − 1.
The ARCH process allows you to set the conditional variance σt2 .
σt2 = α0 + α1 et−1
2 2
+ α2 et−2 2
+ . . . + αq et−q
• et = σt zt , zt ∼ N(0, 1) where:
• σt is the conditional variance.
• zt is the standard residue
• et is conditional errors of the asset at time t
Example
The following plot is a time series plot of a simulated series (n =
300) for the ARCH model
Var (yt |yt−1 ) = σt2 = 5 + 0.5yt−1
2
No correlations are significant, so the series looks to be white noise.
The PACF of the squared values has a single spike at lag 1 suggesting
an AR(1) model for the squared series.
Generalization
An ARCH(m) process is one for which the variance at time is condi-
tional on observations at the previous m times, and the relationship
is
Var (yt |yt−1 , . . . , yt−m ) = σt2 = α0 + α1 yt−1
2 2
+ . . . + αm yt−m
With certain constraints imposed on the coefficients, the yt series
squared will theoretically be AR(m).
GARCH model
A GARCH (generalized autoregressive conditionally heteroscedastic)
model uses values of the past squared observations and past vari-
ances to model the variance at time t. As an example, a GARCH(1, 1)
is
σt2 = α0 + α1 yt−1
2 2
+ β1 σt−1
In the GARCH notation, the first subscript refers to the order of the
y 2 terms on the right side, and the second subscript refers to the
order of the σ 2 terms.
Identifying an ARCH/GARCH Model
• The best identification tool may be a time series plot of the
series. It’s usually easy to spot periods of increased variation
sprinkled through the series.
• It can be fruitful to look at the ACF and PACF of both yt and
yt2 . For instance, if yt appears to be white noise and yt2
appears to be AR(1), then an ARCH(1) model for the
variance is suggested.
• GARCH models may be suggested by an ARMA type look to
the ACF and PACF of yt2 .
• You might have to experiment with various ARCH and
GARCH structures.
Example
The following plot is a time series plot of a simulated series, x, (n
= 300) for the GARCH(1,1) model
Var (xt |xt−1 ) = σt2 = 5 + 0.5xt−1
2 2
+ 0.5σt−1
The ACF of the series below shows that the series looks to be white
noise.
The ACF of the squared series follows an ARMA pattern because of
both the ACF and PACF taper. This suggest a GARCH(1,1) model.
Let’s use the fGarch package to fit GARCH(1,1) model to x where
we center the series to work with a mean of 0.
Here is a part of output.
This suggest the following model for yt = xt − 0.5423 :
The fGARCH summary provides the Jarque Bera Test for the null
hypothesis that the residuals are normally distributed and the famil-
iar Ljung Bos tests.
Diagnostics all look okay.
Thank you.
