Transcript IDENTIFICATION - Vaibhav Gupta

ARCH AND GARCH
VAIBHAV GUPTA
MIB, DOC, DSE, DU
REFS
RISK AND VOLATILITY: ECONOMETRIC MODELS
AND FINANCIAL PRACTICE
Nobel Lecture, December 8, 2003
Robert F. Engle III
INTRODUCTION
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Advantage of knowing about risks: You can change
your behaviour to avoid it.
To avoid all risks would be impossible: no flying, no
driving, no walking, eating, drinking, no to sunshine.
To some who are obsessed with a early morning bath,
NO BATH as well.
There are some risks we choose to take because the
benefits exceed the costs.
Optimal behaviour takes risks that are worthwhile.
This is central paradigm to finance.
Thus we optimize our behaviour, in particular our
portfolio, to maximize rewards and minimize risks.
SIMPLE CONCEPT OF RISK CAN MEAN A
LOT OF NOBEL CITATIONS
Markowitz (1952) and Tobin (1958) associated
risk with the variance in the value of a portfolio:
From the avoidance of risk they derived
optimizing portfolio and banking behaviour.
(Nobel Prize 1981)
 Sharpe (1964) developed the implications when
all investors follow the same objectives with the
same information. This theory is called the
Capital Asset Pricing Model or CAPM, CAPM,
and shows that there is a natural relation
between expected returns and variance. (Nobel
Prize 1990)
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Black and Scholes (1972) and Merton (1973)
developed a model to evaluate the pricing of
options. (1997 Nobel Prize)
Typically the square root of the variance, called the
volatility, was reported. They immediately recognized
that the volatilities were changing over time.
 A simple approach, sometimes called historical
volatility, was and widely used. (sample standard
deviations over a short period)
 What is the right period:
 Too long: Not relevant for today
 Too short: Very noisy
 Furthermore, it is volatility over a future period
that should be considered as risk, forecast also
needed in the measure of today.
 Theory of dynamic volatility is needed: ARCH
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Until the early 80s econometrics had focused almost
solely on modelling the means of series, i.e. their
actual values. Recently however we have focused
increasingly on the importance of volatility, its
determinates and its effects on mean values.
A key distinction is between the conditional and
unconditional variance.
the unconditional variance is just the standard
measure of the variance
var(x) =E(x -E(x))2
the conditional variance is the measure of our uncertainty
about a variable given a model and an information set.
cond var(x) =E(x-E(x|  ))2
Conditional
variance
this is the true measure of uncertainty
variance
mean
Stylised Facts of asset returns
i) Thick tails, they tend to be leptokurtic
ii)Volatility clustering, Mandelbrot, ‘large changes tend to be
followed by large changes of either sign’
iii)Leverage Effects, refers to the tendency for changes in stock
prices to be negatively correlated with changes in volatility.
iv)Non-trading period effects. when a market is closed information
seems to accumulate at a different rate to when it is open. eg stock
price volatility on Monday is not three times the volatility on
Tuesday.
v) Forcastable events, volatility is high at regular times such
as news announcements or other expected events, or even at
certain times of day, eg less volatile in the early afternoon.
vi)
Co-movements in volatility. There is considerable evidence
that volatility is positively correlated across assets in a market and
even across markets
Engle(1982) ARCH Model
Auto-Regressive Conditional Heteroscedasticity
Yt  X t   t

2
t

t
q
     i 2 t i     ( L) 2
define
2
 t ~ N (0,  t 2 )
i 1
 t   t  t
2
2
    ( L)   t
2
an AR(q) model for squared innovations.
note as we are dealing with a variance
  0 i  0 all i
even though the errors may be serially uncorrelated they are not
independent, there will be volatility clustering and fat tails.
if the standardised residuals
zt   t /  t
are normal then the fourth moment for an ARCH(1) is
E( t ) / E( t )  3(1   ) /(1  3 ) if 3  1
4
2
2
2
2
2
VOLATILITY
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Volatility – conditional variance of the process
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Don’t observe this quantity directly (only one
observation at each time point)
Common features
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Serially uncorrelated but a depended process
Stationary
Clusters of low and high volatility
Tends to evolve over time with jumps being rare
Asymmetric as a function of market increases or market
decreases
THE BASIC MODELS
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Consider a process r(t) where
r (t )   (t )  a(t )
 (t )  E (r (t ) | F (t  1))
Conditional mean
evolves as an ARMA
process
p
q
j 1
k 1
 (t )  0   j r (t  j )   k a(t  k )
 (t )  Var(r (t ) | F (t 1))
2
How does the conditional variance
evolve?
MODELING THE VOLATILITY
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Evolution of the conditional variance follows to
basic sets of models
The evolution is set by a fixed equation (ARCH,
GARCH,…)
 The evolution is driven by a stochastic equation
(stochastic volatility models).
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Notation:
a(t)=shock or mean-corrected return;
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is the positive square root of the volatility
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 (t )
ARCH MODEL
We have the general format as before
 The equation defining the evolution of the
volatility (conditional variance) is an AR(m)
process.
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Why would this
model yield
“volatility
clustering”?
a(t )   (t ) (t )
 (t )  0  1a (t 1)  ma (t  m)
2
2
2
BASIC PROPERTIES ARCH(1)
Unconditional mean is 0.
E[a(t )]  E[ E (a(t ) | F (t  1))]
 E[ E ( (t ) (t ) | F (t  1))]
 E[ (t ) E ( (t )))
0
BASIC PROPERTIES, ARCH(1)
Unconditional variance
Var[a(t )]  Var[ E (a(t ) | F (t  1))] E[Var (a(t ) | F (t  1)]
 0  E[ 2 (t )]
 E[ 0  1a 2 (t  1)]
  0  1 E[a 2 (t  1)]
  0  1Var[a(t  1)]
  0  1Var[a(t )]
Var[a(t )]   0 /(1  1 )
What
constraint
does this
put on 1?
BASIC PROPERTIES OF ARCH
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01<1
Higher order moments lead to additional constraints on the
parameters
 Finite positive (always the case) fourth moments
requires
0 12<1/3
Moment conditions get more difficult as the order increases
– see general framework of equation 3.6
Note – in general the kurtosis for a(t) is greater than 3
even if the ARCH model is built from normal random
variates.
Thus the tails are heavier and you expect more “outliers”
than “normal”.
ARCH ESTIMATION, MODEL FITTING AND
FORECASTING
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MLE for normal and t-dist ’s is given on pages 88 and 89.
The full likelihood is very difficult and thus the conditional
likelihood is most generally used.
The conditional likelihood ignores the component of the
likelihood that involves unobserved values (in other words,
obs 1 through m)
MLE for joint estimation of parameters and degree of the tdistribution is given.
Model selection
 Fit ARMA model to mean structure
 Review PACF to identify order of ARCH
 Check the standardized residuals – should be WN
Forecasting – identical to AR forecasting but we forecast
volatility first and then forecast the process.
GARCH MODEL
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Generalize the ARCH model by including an MA
component in the model for the volatility or the
conditional variance.
a(t )   (t ) (t )
m
s
j 1
k 1
 2 (t )   0   j a 2 (t  j )    k 2 (t  k )
a(t )   (t ) (t )
m
s
 (t )   0   j a (t  j )    k (t  k )
2
2
j 1
2
k 1
Proceed as before – using all you learned from ARMA models.