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Volatility in Financial Time Series
Autoregressive Conditional
Heteroskedasticity
VECM (Review)
•
Cointegrating Eq:
Error Correction:
D(R1)
CointEq1
-0.029996
(0.01783)
[-1.68255]
0.015287
(0.01140)
[ 1.34155]
D(R1(-1))
0.273219
(0.06803)
[ 4.01619]
-0.028276
(0.04348)
[-0.65026]
D(R1(-2))
-0.087596
(0.06772)
[-1.29358]
0.025434
(0.04328)
[ 0.58761]
D(R10(-1))
0.370337
(0.10747)
[ 3.44593]
0.425735
(0.06869)
[ 6.19757]
D(R10(-2))
-0.263587
(0.10796)
[-2.44152]
-0.266142
(0.06901)
[-3.85675]
C
-0.000459
•
•
•
R1(-1)1.000000
•
•
R10(-1) -0.980444
•
(0.07657)
•
[-12.8046]
•
•
C
0.603495
•
•
•
•
D(R10)
Introduction
• Assess Maximum Likelihood estimation.
• Non linear models
– Three forms
– Testing
• General tests
• Specific tests
• Models for Volatility
– Simple forms
– ARCH
– GARCH
• Constraints and limitations
Maximum Likelihood Estimation
• Since the volatility models used in Financial
Econometrics are non-linear, OLS cannot be used. OLS
requires minimising the residual sum of squares (RSS).
RSS just depends on the coefficients of the mean
equation, β1 , β2 etc. and not on σ2, the conditional
variance.
• So an alternative technique called Maximum Likelihood
Estimation (MLE) must be used.
• Maximum likelihood works by finding the values of the
coefficients that were most likely to have generated our
sample of data. The likelihood function (LF) is formed by
multiplying the formula for the probability of observing a
particular data observation, across all observations (a
joint probability)
ML Estimation
• The Maximum Likelihood approach
assumes a variable x, which follows the
normal distribution, but the mean(μ) and
standard deviation(σ) are unknown. The
probability density function will be:
1
f ( x) 
e
 2
1 / 2(
x

)2
ML Estimation
• To determine the value of the mean, we need to
maximise the likelihood function, or joint
probability density.
• It is easier to use the log-likelihood though, as
the value that maximises the likelihood function
also maximises the log-likelihood function.
• It is then a matter of differentiating with respect
to the mean μ, to find the maximum value of the
log-likelihood function and setting this equal to
zero.
Application to simple regression
model
• Assuming that the error term is normally
distributed with zero mean and standard
deviation σ.
• It is then a process of forming the log-likelihood
function and the values for the constant and
slope parameter that maximise this function are
exactly the same as those obtained by OLS.
• However the estimate of σ is slightly different.
Background and uses
• So far we have considered models which are linear:
y t  0  1x 1t  2x 2t  ...  ut
• Linear models of the behaviour of financial data are
unable to explain:
– Leptokurtosis: the tendency for financial asset returns to have
distributions with fat tails and excess peakedness at the mean
– Volatility clustering or volatility pooling: the tendency for
volatility to be bunched (large returns, +ve or –ve, are expected
to follow other large returns)
– Leverage effects: the tendency for volatility to rise more
following a large price fall than a price rise of the same size
Types of non-linear models:
1)
y t  0  1y t 1y t 2  ut
This is the simplest form of non-linearity – the inclusion of an
interaction term.
2)
y t  f (ut ,ut 1 ,ut 2 ,...)
A general non-linear “data generating process” where the
current value of a series is related non-linearly to current and
past errors
3)
y t  g (ut ,ut 1 ,ut 2 ,...)  ut  2 (ut 1 ,ut 2 ,...)
A more specific definition of 2) where in addition the current
value of series is non-linearly related to the variance of the
error, σ2.
ARCH models
• It is this last type that we will concentrate on, in particular
models which are “linear in mean” { g(.) } but “non-linear
in variance” { σ2(.) }.
• Such models are known as Autoregressive Conditional
Heteroscedasticity (ARCH) models.
• This can be interpreted as autocorrelation of the
heteroskedasticity.
Models for volatility
In Finance volatility is often used as a proxy for the underlying
riskiness of an asset, so is particularly important in finance.
Volatility can be measured by the standard deviation or
variance of the returns to a financial asset (a crude measure
of total risk)
The simplest measure is historical volatility – the variance or
std. dev. over some historical period
Volatility often appears in financial theory – the Black-Scholes
formula for an option price includes the variance of stock
market prices. One might derive the implied volatility of a
stock from the option price by manipulation of the BlackScholes formula.
Autoregressive Volatility
Autoregressive volatility
• Volatility is defined by the trading range over a particular
day. An AR(p) model is estimated for the volatility
measure, to see if current volatility relates to past.
 hight 

 low t 
 t2  log
2
p
 t   0    j  t2 j  t
j 1
Autoregressive Conditional Heteroscedasticity (ARCH) Models
Recall that the assumption of a constant error variance
[var(ut)=σ2] is called homoscedasticity. If the variance is not
constant it is heteroscedastic.
Residuals and Two Std Err. Bands (BT return
regressed on UK market return)
0.15
0.10
0.05
0.00
-0.05
-0.10
-0.15
-0.20
2
82
162
242
322
402
482
562
642
Days
( Heteroscedasticity test statistic significant at 4.4%)
722
802
882
Volatility occurs in “bursts” – it could be “autocorrelated”.
The ARCH model parameterises this.
Definition of a conditional variance:
t2  var(ut | ut 1 ,ut 2 ,...)  E [ut2 | ut 1 ,ut 2...]
(Because E(ut)=0, variance is just E(ut2))
The conditional variance of a random variable with a zero
mean and normal distribution is equal to the conditional
expected value of the square of ut . The conditional variance
changes as more data becomes available.
“Autocorrelation in volatility” is modelled by allowing conditional
variance to depend on the immediately previous value of the
squared error:
 t2   0  1u t21
ARCH(1) process
Nothing has been said yet about the variable whose variance
we are modelling – the modelling of the conditional mean. So
a full ARCH(1) model might look like:
y t  1  2 x 2t  3x 3t   4 x 4t  ut
ut ~ N (0,  t2 )
 t2   0  1ut21
This can be extended to a general case where the error
variance depends on q lags of the squared errors:
t2  0  1ut21  2ut22  ...  q ut2q
Sometimes σt2 is called ht
y t  1  2x 2t  3x 3t   4 x 4t  ut
ht   0  1ut21   2ut22  ...  q ut2q
ut ~ N (0, ht )
Constraints and Limitations
• As ht is a measure of variance it has to be a positive
number. This means, given that the squared residuals on
the RHS of the last equation must also be positive, that all
the coefficients α0 … αq must be positive.
• This is an important constraint when an ARCH model is
estimated. We need to remodel the relationship if negative
coefficients appear.
Testing for ARCH
• Run the regression model that you wish to investigate and save the
residuals.
• Compute the square of the residuals and regress them on q lags of
themselves:
• Note the R2 from this regression.
• Compute the test statistic T.R2 (where T is the number of
observations in the sample. This has a Chi2 (q) distribution. The null
hypothesis is no “ARCH effects”, so if T.R2 is less than the Chi2
critical value we accept the null hypothesis of no ARCH effects in the
model.
Introduction to the GARCH model
• The GARCH model or the Generalised
Autoregressive Conditional
Heteroskedasticity model introduces a
lagged dependent variable into the model,
i.e. lagged variance term:
  0  
2
t
2
u
1 t 1
 
2
2 t 1
GARCH Model
• The GARCH model has been shown to
have a better performance than the ARCH
model using a smaller number of terms,
i.e. it is more parsimonious
• Often a GARCH (1,1) model is sufficient to
explain the volatility in a financial time
series.
GARCH Models
• The GARCH model is really an ARMA type
of process.
• To show this the squared residual at time t is
equal to the conditional variance and a
constant term, to give:
 t2  ut2   t
GARCH model
• By substituting this into our original GARCH
model, we can produce the following ARMA
type process:
2
ut
 0  
2
1ut 1
  2 t 1   t
ARCH/GARCH Example
• These models have the same interpretation as the
usual models in terms of the coefficients and tstatistics:
yt  0.50
(0.20)
 t2  0.12  0.16ut21  0.56 t21
(0.10) (0.04)
R 2  0.64, DW  1.98
(0.16)
ARCH Example
• The first step involves estimating the mean,
often this involves regressing the variable of
interest against a constant, the resulting error
term (u) is used to produce the conditional
variance term in the resulting GARCH model.
• However the mean equation can include other
terms, often ARIMA models are used.
• In the previous slide the GARCH model has
significant variables, which are correctly signed
and the diagnostic tests are passed
ARCH Example
• The use of GARCH models is not only
required to ensure the model is well
specified, it can also be used for
forecasting.
• GARCH models have proven to be
particularly useful at forecasting returns on
risky assets.
Conclusion
• Volatility is important in finance as it represents
risk and is a particular feature of financial data.
• Many assets exhibit volatility clustering, such
that there is autocorrelation in the
heteroskedasticity.
• The ARCH model is an important way of
modelling this phenomenon
• The GARCH model has better time series
properties than the ARCH model.