Damodar Gujarati

1
CHAPTER 15

• ASSET PRICE VOLATILITY
• THE ARCH AND GARCH MODELS

2
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.

3
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
• 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 st2 (heteroscedastic
  variance)
• 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
• where 0 ?1 lt 1

4
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
• 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.

5
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.

6
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.

7
THE GARCH MODEL

• In its simplest form, the variance equation in
  the GARCH model is modified as follows
• 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
  (p1) 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.

8
FURTHER EXTENSIONS OF THE ARCH MODEL

• GARCH-M Model
• Explicitly introduce a risk factor, the
  conditional variance, in the original regression
• 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.