CHAPTER 15
ASSET PRICE VOLATILITY:
THE ARCH AND GARCH MODELS
Damodar Gujarati
Econometrics by Example
VOLATILITY CLUSTERING
 Volatility Clustering: Periods of turbulence in which
prices show wide swings and periods of tranquility in
which there is relative calm.
 Financial time series often exhibit the phenomenon of
volatility clustering.
 This results in correlation in error variance over time.
 Use autoregressive conditional heteroscedasticity
(ARCH) models to take into account such correlation or
time-varying volatility.
Damodar Gujarati
Econometrics by Example
THE ARCH MODEL
 This model shows that conditional on the information available up to time
(t-1), the value of the random variable Y is a function of the variable X:
Yt I t 1     X t  ut
 We assume that given the information available up to time (t – 1), the
error term is independently and identically normally distributed with mean
value of 0 and variance of σt2 (heteroscedastic variance):
ut It 1  iid N (0, t2 )
 Assume that the error variance at time t is equal to some constant plus a
constant multiplied by the squared error term in the previous time period:
 t2
= 0  1ut 1 ,
where 0 ≤ λ1 < 1
Damodar Gujarati
Econometrics by Example
2
THE ARCH MODEL (CONT.)
 The ARCH(1) model includes only one lagged squared value of the
error term.
 An ARCH(p) model has p lagged squared error terms, as follows:
  0  1 u
2
t
2
t 1
 2 u
2
2
t 2
 ...  p ut  p
 If there is an ARCH effect, it can be tested by the statistical
significance of the estimated coefficients.
 If they are significantly different from zero, we can conclude that
there is an ARCH effect.
Damodar Gujarati
Econometrics by Example
ESTIMATION OF THE ARCH MODEL
 The Least-squares Approach
 Once we obtain the squared error term from the chosen model, we
can estimate the ARCH model by the usual least squares method.
 The Akaike or Schwarz information criterion can determine the
number of lagged terms to include.
 Choose the model that gives the lowest value on the basis of these criteria
 The Maximum-likelihood Approach
 An advantage of the ML method is that we can estimate the mean
and variance functions simultaneously.
 The mathematical details of the ML method are somewhat involved, but
statistical packages, such as STATA and EVIEWS, have built-in routines to
estimate the ARCH models.
Damodar Gujarati
Econometrics by Example
DRAWBACKS OF THE ARCH MODEL
 1. The ARCH model requires estimation of the coefficients of p
autoregressive terms, which can consume several degrees of
freedom.
 2. It is often difficult to interpret all the coefficients, especially if
some of them are negative.
 3. The OLS estimating procedure does not lend itself to estimate
the mean and variance functions simultaneously.
 Therefore, the literature suggests that an ARCH model higher than
ARCH (3) is better estimated by the Generalized Autoregressive
Conditional Heteroscedasticity (GARCH) model.
Damodar Gujarati
Econometrics by Example
THE GARCH MODEL
 In its simplest form, the variance equation in the GARCH model is
modified as follows:
  0   u  2
2
t
2
1 t 1
2
t 1
 This is known as the GARCH (1,1) model.
 The ARCH (p) model is equivalent to GARCH (1,1) as p increases.
 Note that in the ARCH (p) we have to estimate (p+1) coefficients,
whereas in the GARCH (1,1) model given in we have to estimate only
three coefficients.
 The GARCH (1,1) model can be generalized to the GARCH (p, q) model
with p lagged squared error terms and q lagged conditional variance terms,
but in practice GARCH (1,1) has proved useful to model returns on
financial assets.
Damodar Gujarati
Econometrics by Example
FURTHER EXTENSIONS OF THE ARCH MODEL
 GARCH-M Model
 Explicitly introduce a risk factor, the conditional variance, in the
original regression:
Yt     X t    ut
2
t
 This is called the GARCH-M (1,1) model.
 Further Extensions of ARCH and GARCH Models
 AARCH, SAARCH, TARCH, NARCH, NARCHK, EARCH, are
all variants of the ARCH and GARCH models.
Damodar Gujarati
Econometrics by Example