# lecture note 12

```MODELING VOLATILITY BY ARCHGARCH MODELS
1
VARIANCE
• A time series is said to be heteroscedastic, if
its variance changes over time, otherwise it is
called homoscedastic.
• When the variance is not constant ( it will
follow mixture normal distribution), we can
expect more outliers than expected from
normal distribution. i.e. when a process is
heteroscedastic, it will follow heavy-tailed or
outlier-prone probability distributions.
2
VARIANCE
• Until the early 80s econometrics had focused almost
solely on modeling the means of series, i.e. their actual
values. Recently however researchers 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
3
VARIANCE
• 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
• This is the true measure of uncertainty
variance
variance
mean
4
VARIANCE
• Stylized Facts of asset returns
i.
ii.
Thick tails: they tend to be leptokurtic
Volatility clustering: Mandelbrot, “large changes tend to be
followed by large changes of either sign”
iii. Leverage Effects: 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. e.g.
stock price volatility on Monday is not three times the volatility on
Tuesday.
v. Forecastable events: volatility is high at regular times such as news
announcements or other expected events, or even at certain times
of day, e.g. less volatile in the early afternoon.
vi. Volatility and serial correlation: There is a suggestion of an inverse
relationship between the two.
vii. Co-movements in volatility: There is considerable evidence that
volatility is positively correlated across assets in a market and even
across markets
5
6
7
ARCH MODEL
• Stock market’s volatility is rarely constant over time.
• In finance, portfolios of financial assets are held as
functions of the expected mean and variance of the
rate of return. Since any shift in asset demand must
be associated with changes in expected mean and
variance of rate of return, ARCH models are the best
suitable models.
• In regression, ARCH models can be used to
approximate the complex models.
8
ARCH MODEL
• Even though the errors may be serially
uncorrelated they are not independent, there
will be volatility clustering and fat tails.
• If there is no serial correlation of the series but
there is of the squared series, then we will say
there is weak dependence. This will lead us to
examine the volatility of the series, since that is
demonstrated by the squared terms.
9
ARCH MODEL
Autocorrelation Function for C1
(with 5% significance limits for the autocorrelations)
1.0
0.8
Autocorrelation
0.6
0.4
0.2
0.0
-0.2
-0.4
-0.6
-0.8
-1.0
1
5
10
15
20
25
30
35
Lag
40
45
50
55
60
Figure 1: Autocorrelation for the log returns for the Intel series
10
Autocorrelation Function for C2
Partial Autocorrelation Function for C2
(with 5% significance limits for the autocorrelations)
(with 5% significance limits for the partial autocorrelations)
1.0
1.0
0.8
0.8
0.6
0.6
Partial Autocorrelation
Autocorrelation
ARCH MODEL
0.4
0.2
0.0
-0.2
-0.4
-0.6
0.4
0.2
0.0
-0.2
-0.4
-0.6
-0.8
-0.8
-1.0
-1.0
1
5
10
15
20
25
30
35
Lag
40
45
50
Figure 2: ACF of the squared returns
55
60
1
5
10
15
20
25
30
35
Lag
40
45
50
55
60
Figure 3 PACF for squared returns
Combining these three plots, it appears that this series is serially
uncorrelated but dependent. Volatility models attempt to capture
such dependence in the return series
11
ARCH MODEL
• Engle(1982) introduced a model in which the
variance at time t is modeled as a linear
combination of past squared residuals and
called it an ARCH (autoregressive conditionally
heteroscedastic) process.
• Bolerslev (1986) introduced a more general
structure in which the variance model looks
more like an ARMA than an AR and called this
a GARCH (generalized ARCH) process.
12
Engle(1982) ARCH Model
Auto-Regressive Conditional Heteroscedasticity
Yt  X t   t
Let  t is the set of all information available at time t.
where  t ~ N ( 0, t )
2
 t  at ht
where at ' s are iid Gaussian0,1 rvs .
2




(
B
)

ht
0
q
t , i  0 ,i  1,2 , , q
an AR(q) model for squared innovations.
13
ARCH(q) MODEL
 t  t 1 ~ N 0,ht 
ht  Var  t  t 1    0  1 t21     q t2q
14
ARCH(q) MODEL
• The equation of ht shows that if t-1 is large,
then the conditional variance of t is also
large, and therefore t tends to be large. This
behavior will spread through out the process
and unusual variability tends to persist, but
not always.
• The conditional variance will revert toq
unconditional variance provided 0    i  1
i 1
so that the process will be stationary with
finite variance.
15
ARCH(q) MODEL
• If the ARCH processes have a non-zero mean which
can be expressed as a linear combination of
exogenous and lagged dependent variables, then a
linear regression frame work is appropriate and the
model can be written as,
 t  Yt  X t
 t  t 1 ~ N  X t ,ht 
ht   0   
2
1 t 1
   
2
q t q
 0  0 , i  0 ,i  1, .q
This model is called a “ Linear ARCH(q) Regression “ model.
16
ARCH(q)
• If the regressors include no lagged dependent
variables and can be treated as fixed constants
then the ordinary least square (OLS) estimator
is the best linear unbiased estimator for the
model. However, maximum-likelihood
estimator is nonlinear and is more efficient
than OLS estimator.
17
ESTIMATION OF THE LINEAR ARCH( q )
REGRESSION MODEL
• Let the log likelihood function for the model is
n
l   lt
t 1
1
1  t2
where lt   ln ht 
2
2 ht
• The likelihood function can be maximized with
respect to the unknown parameters  and ’s.
18
ESTIMATION OF THE LINEAR ARCH( q )
• To estimate the parameters, usually we use
Scoring Algorithm.
• Each iteration for parameters  and 
produces the estimates based on the previous
iteration according to,

i 1

i 1
   Î  l / 
i
i  1 ˆ i 
   Î   l / 
i
i  1 ˆ i 
where I and I matrices are information
matrices.
19
STEPS IN ESTIMATION
STEP 1: Estimate  by OLS and obtain residuals.
STEP 2: Compute

where
i 1
   Ẑ Ẑ  ẐẐ f .
1
i 
i 
Ẑ t  1,et 1 , ,et  p  / ht
 i 2
ft  et
i 
i 2
residuals
i 2
i 
 ht
i 
/ h
i 
t
.
Conditional variance
20
STEPS IN ESTIMATION
STEP 3: Using (i+1), compute

i 1

i 
1

  X X  X e
where X and e are matrices with vect ors
X t xt et and et  et st / rt ,
1/ 2

2 2 
rt  1 / ht  2et   j ht  j


j
2 2
2
st  ht 1   j  j ht  j  2et  j  h j t 
21
STEPS IN ESTIMATION
STEP 4: Obtain residuals by using (i+1). Go to
Step 2.
This iterative procedure will be continued until
the convergence of the estimation of .
22
TESTING FOR ARCH DISTURBANCES
• Method 1. The autocorrelation structure of
residuals and the squared residuals can be
examined. An indication of ARCH is that the
residuals will be uncorrelated but the squared
residuals will show autocorrelation.
23
TESTING FOR ARCH DISTURBANCES
• Method 2. A test based on Lagrange Multiplier ( LM ) principle
can be applied. Consider the null hypothesis of no ARCH
errors versus the alternative hypothesis that the conditional
error variance is given by an ARCH(q) process. The test
approach proposed by Engle is to regress the squared
residuals on a constant and q lagged residuals. From the
residuals of this auxiliary regression, a test statistic is
calculated as nR2, where R2 is coming from the auxiliary
regression. The null hypothesis will be rejected if the test
statistic exceeds the critical value from a chi-square
distribution with q degree of freedom.
24
TESTING FOR ARCH DISTURBANCES
•
•
•
•
•
White’s test
Breush-Pagan test
The Goldfeld-Quandt test
Likelihood ratio test
LM tests: the Glejser test; the Harvey-Godfrey
test, and the Park test
25
TESTING FOR
HETEROSCEDASTICITY
• Popular heteroscedasticity LM tests: - Breusch and Pagan
(1979)’s LM test (BP). - White (1980)’s general test.
• Both tests are based on OLS residuals. That is, calculated
under H0: No heteroscedasticity.
• The BP test is an LM test, based on the score of the log
likelihood function, calculated under normality. It is a
general tests designed to detect any linear forms of
heteroskedasticity.
• The White test is an asymptotic Wald-type test,
normality is not needed. It allows for nonlinearities by
using squares and crossproducts of all the x’s in the
auxiliary regression.
26
TESTING FOR
HETEROSCEDASTICITY
• Drawbacks of the Breusch-Pagan test: - It has been
shown to be sensitive to violations of the normality
assumption. - Three other popular LM tests: the Glejser
test; the Harvey-Godfrey test, and the Park test, are also
sensitive to such violations.
• Drawbacks of the White test - If a model has several
regressors, the test can consume a lot of df’s. - In cases
where the White test statistic is statistically significant,
heteroscedasticity may not necessarily be the cause, but
model specification errors. - It is general, but does not
give us a clue about how to model heteroscedasticity to
do FGLS. The BP test points us in a direction.
27
PROBLEMS IN ARCH MODELING
• In most of the applications of the ARCH model a
relatively long lag in the conditional variance is often
called for, and this leads to the problem of negative
variance and non-stationarity. To avoid this problem,
generally a fixed lag structure is typically imposed. So
it is necessary to extent the ARCH models to a new
class of models allowing for a both long memory and
much more flexible lag structure.
• Bollerslev introduced a Generalized ARCH (GARCH)
models which allows long memory and flexible lag
structure.
28
GARCH (Bollerslev,1986)
• In empirical work with ARCH models high q
is often required, a more parsimonious
representation is the Generalized ARCH model
q
ht   0    
i 1
2
i t i
p
   j ht  j
j 1
 0  0, i  0, j  0,i  1,,q , j  1, p
which is an ARMA(max(p,q),p) model for the squared
innovations.
• GARCH (p, q) process allows lagged
conditional variances to enter as well.
29
GARCH MODEL
• The GARCH (p, q) process is stationary iff
q
p
i 1
j 1
  i    j  1.
• The simplest but often very useful GARCH
process is the GARCH (1,1) process given by
 t  t 1 ~ N 0,ht 
ht   0   
2
1 t 1
  1ht 1
30
TESTING FOR GARCH
DISTURBANCES
• METHOD 1: Use the previous LM test for
ARCH. If the null hypothesis is rejected for
long disturbances, GARCH model is
appropriate.
• METHOD 2: A test based on Lagrange
Multiplier (LM) principle can be applied.
Consider the null hypothesis of ARCH (q) for
errors versus the alternative hypothesis that
the errors are given by a GARCH (p, q) process.
31
INTEGRATED GARCH
• When
q
p
i 1
j 1
  i    j  1,
the process has a unit root.
Use Integrated GARCH or IGARCH process
32
SYMMETRYCITY OF GARCH
MODELS
• In ARCH, GARCH, IGARCH processes, the effect
of errors on the conditional variance is
symmetric, i.e., positive error has the same
effect as a negative error.
• However, in finance, good and bad news have
different effects on the volatility.
• Positive shock has a smaller effect than the
negative shock of the same magnitude.
33
EGARCH
• For the asymmetric relation between many
financial variables and their volatility changes
and to relax the restriction on the coefficients
in the model, Nelson (1991) proposed
EGARCH process.

log ht    0    j g at 1 j ,at   t / ht
j 0
Usually, g function is chosen as
g at   at    at  E at .
34
THRESHOLD GARCH (TGARCH) OR
GJR-GARCH
• Glosten, Jaganathan and Runkle (1994)
proposed TGARCH process for asymmetric
volatility structure.
• Large events have an effect but small events
not.
2
2
ht   0  1 t 1  1dt 1 t 1   1ht 1
TARCH(1,1)
where
1, t 1  0
dt 1  
0 , t 1  0
35
TGARCH
• When 1&gt;0, the negative shock will have
larger effect on the volatility.
• THE LEVERAGE EFFECT: The tendency for
volatility to decline when returns rise and to
rise when returns falls.
• TEST FOR LEVERAGE EFFECT: Estimate TGARCH
or EGARCH and test whether 1=0 or =0.
36
NONLINEAR ARCH (NARCH) MODEL
• This then makes the variance depend on both
the size and the sign of the variance which
helps to capture leverage type effects.


q
t
p
     i |  t i   |   j
i 1


t j
j 1
37
ARCH in MEAN (G)ARCH-M
Many classic areas of finance suggest that the
mean of a relationship will be affected by the
volatility or uncertainty of a series. Engle Lilien
and Robins(1987) allow for this explicitly using
an ARCH framework.
yt  xt   t   t
2
q
 t     i 
2
i 1
p
2
t i
   j
2
t j
j 1
typically either the variance or the standard
deviation are included in the mean relationship.
38
NORMALITY ASSUMPTION
• While the basic GARCH model allows a certain
amount of leptokurtic behaviour, this is often
insufficient to explain real world data. Some
authors therefore assume a range of
distributions other than normality which help
to allow for the fat tails in the distribution.
t Distribution
The t distribution has a degrees of freedom
parameter which allows greater kurtosis.
39
THE GARCH ZOO
• TARCH = threshold ARCH
• STARCH = structural ARCH
• SWARCH = switching ARCH
• QTARCH = quantitative threshold ARCH
• vector ARCH
• diagonal ARCH
• factor ARCH
40
S&amp;P COMPOSITE STOCK MARKET
RETURNS
• Monthly data on the S&amp;P Composite index returns
over the period 1954:1–2001:9. Lags of the inflation
rate and the change in the three-month Treasury bill
(T-bill) rate are used as regressors, in addition to lags
of the returns. We begin by modeling the returns
series as a function of a constant, one lag of returns
(Ret_l), one lag of the inflation rate (Inf_l) and one
lag of the first-difference of the three-month T-bill
rate (DT-bill_l).
41
S&amp;P COMPOSITE STOCK MARKET
RETURNS
Ret
20
10
0
-10
-20
01JAN1950
01JAN1955
01JAN1960
01JAN1965
01JAN1970
01JAN1975
01JAN1980
01JAN1985
01JAN1990
01JAN1995
01JAN2000
01JAN2005
date
42
S&amp;P COMPOSITE STOCK MARKET
RETURNS
proc autoreg data=returns maxit=50;
model ret = ret_1 inf_1 dt_bill_1/ archtest;
Ordinary Least Squares Estimates
SSE
MSE
SBC
MAE
MAPE
Durbin-Watson
6023.21652
10.62296
2991.08622
2.44852035
258.49597
1.9457
Variable
DF
Estimate
Intercept
ret_1
Inf_1
DT_bill_1
1
1
1
1
1.1771
0.2080
-1.1795
-1.2501
DFE
Root MSE
AIC
AICC
HQC
Regress R-Square
Total R-Square
Parameter Estimates
Standard
Error
t Value
0.2100
0.0410
0.4480
0.2914
5.60
5.07
-2.63
-4.29
567
3.25929
2973.69666
2973.76733
2980.48101
0.1011
0.1011
Approx
Pr &gt; |t|
&lt;.0001
&lt;.0001
0.0087
&lt;.0001
Variable
Label
Inf_1
43
DT-bill_1
S&amp;P COMPOSITE STOCK MARKET
RETURNS
Tests for ARCH Disturbances Based on OLS Residuals
Order
1
2
3
4
5
6
7
8
9
10
11
12
Q
Pr &gt; Q
LM
Pr &gt; LM
6.3810
6.5739
8.5749
8.7780
10.8979
17.7423
18.4102
18.5711
18.7501
18.7956
19.2071
19.8498
0.0115
0.0374
0.0355
0.0669
0.0534
0.0069
0.0103
0.0173
0.0274
0.0429
0.0575
0.0700
6.9874
7.0149
9.0576
9.0917
11.2837
16.7401
16.9397
16.9607
17.0048
17.0058
17.1694
17.5120
0.0082
0.0300
0.0285
0.0588
0.0460
0.0103
0.0178
0.0305
0.0486
0.0742
0.1030
0.1313
44
ARCH(9)
proc autoreg data=returns maxit=50;
model ret = ret_1 inf_1 dt_bill_1/ garch=(q=9);run;
GARCH Estimates
SSE
MSE
Log Likelihood
SBC
MAE
MAPE
6048.16958
10.59224
-1466.4833
3021.82996
2.4465657
258.891059
Observations
Uncond Var
Total R-Square
AIC
AICC
HQC
Normality Test
Pr &gt; ChiSq
571
12.094028
0.0973
2960.96652
2961.72191
2984.71174
33.6859
&lt;.0001
45
ARCH(9)
Parameter Estimates
Variable
Intercept
ret_1
Inf_1
DT_bill_1
ARCH0
ARCH1
ARCH2
ARCH3
ARCH4
ARCH5
ARCH6
ARCH7
ARCH8
ARCH9
DF
Estimate
Standard
Error
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1.1960
0.2146
-0.8773
-0.9640
4.7466
0.1207
0.005981
0.1341
1.021E-19
0.1266
0.1499
0.0179
-5.49E-21
0.0524
0.1897
0.0443
0.3807
0.2491
0.9261
0.0477
0.0356
0.0548
0
0.0553
0.0510
0.0402
0
0.0486
t Value
Approx
Pr &gt; |t|
6.30
4.84
-2.30
-3.87
5.13
2.53
0.17
2.45
Infty
2.29
2.94
0.45
-Infty
1.08
&lt;.0001
&lt;.0001
0.0212
0.0001
&lt;.0001
0.0114
0.8667
0.0144
&lt;.0001
0.0220
0.0033
0.6563
&lt;.0001
0.2812
Variable
Label
Inf_1
DT-bill_1
46
ARCH(9)
47
ARCH(8)
GARCH Estimates
SSE
MSE
Log Likelihood
SBC
MAE
MAPE
6053.67303
10.60188
-1133.4449
2355.75315
2.44715477
259.470044
Observations
Uncond Var
Total R-Square
AIC
AICC
HQC
Normality Test
Pr &gt; ChiSq
571
11.1188142
0.0965
2294.8897
2295.6451
2318.63492
50.4365
&lt;.0001
48
ARCH(8)
Parameter Estimates
Variable
Intercept
ret_1
Inf_1
DT_bill_1
ARCH0
ARCH1
ARCH2
ARCH3
ARCH4
ARCH5
ARCH6
ARCH7
ARCH8
TDFI
DF
Estimate
Standard
Error
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1.2017
0.1964
-0.7688
-1.0448
6.1876
0.0793
0.001457
0.0588
0.0145
0.1349
0.1230
0.0316
2.88E-20
0.1365
0.1926
0.0443
0.3936
0.2853
1.3140
0.0533
0.0401
0.0542
0.0580
0.0715
0.0614
0.0521
0
0.0502
t Value
Approx
Pr &gt; |t|
6.24
4.43
-1.95
-3.66
4.71
1.49
0.04
1.08
0.25
1.89
2.01
0.61
Infty
2.72
&lt;.0001
&lt;.0001
0.0508
0.0003
&lt;.0001
0.1372
0.9710
0.2787
0.8030
0.0591
0.0449
0.5441
&lt;.0001
0.0065
Variable Label
Inf_1
DT-bill_1
Inverse of t DF
49
ARCH(8)
50
GARCH(1,1)
proc autoreg data=returns maxit=100;
model ret = ret_1 inf_1 dt_bill_1/ garch=( q=1, p=1 ) dist = t ;run;
GARCH Estimates
SSE
MSE
Log Likelihood
SBC
MAE
MAPE
6038.45467
10.57523
-1136.1108
2323.00074
2.44725923
260.336638
Observations
Uncond Var
Total R-Square
AIC
AICC
HQC
Normality Test
Pr &gt; ChiSq
571
10.9965281
0.0988
2288.22162
2288.47785
2301.79032
54.3846
&lt;.0001
51
GARCH(1,1)
Parameter Estimates
Variable
Intercept
ret_1
Inf_1
DT_bill_1
ARCH0
ARCH1
GARCH1
TDFI
DF
1
1
1
1
1
1
1
1
Estimate
Standard
Error
t Value
Approx
Pr &gt; |t|
1.2560
0.1858
-1.0253
-1.1092
1.2423
0.0841
0.8029
0.1508
0.1917
0.0446
0.3834
0.2768
0.8819
0.0416
0.1085
0.0499
6.55
4.16
-2.67
-4.01
1.41
2.02
7.40
3.02
&lt;.0001
&lt;.0001
0.0075
&lt;.0001
0.1589
0.0432
&lt;.0001
0.0025
Variable Label
Inf_1
DT-bill_1
Inverse of t DF
52
GARCH(1,1)
53
EGARCH(1,1)
Exponential GARCH Estimates
SSE
MSE
Log Likelihood
SBC
MAE
MAPE
6033.42807
10.56642
-1461.7473
2974.27381
2.44961323
246.96188
Observations
Uncond Var
Total R-Square
AIC
AICC
HQC
Normality Test
Pr &gt; ChiSq
571
.
0.0995
2939.49469
2939.75092
2953.06339
21.0253
&lt;.0001
54
EGARCH(1,1)
Parameter Estimates
Variable
Intercept
ret_1
Inf_1
DT_bill_1
EARCH0
EARCH1
EGARCH1
THETA
DF
Estimate
Standard
Error
1
1
1
1
1
1
1
1
1.1517
0.1903
-1.0770
-1.0312
0.3394
0.2360
0.8553
-0.6143
0.1978
0.0469
0.3677
0.2466
0.1176
0.0598
0.0504
0.2099
t Value
Approx
Pr &gt; |t|
5.82
4.06
-2.93
-4.18
2.88
3.94
16.95
-2.93
&lt;.0001
&lt;.0001
0.0034
&lt;.0001
0.0039
&lt;.0001
&lt;.0001
0.0034
Variable
Label
Inf_1
DT-bill_1
55
EGARCH(1,1)
56
IGARCH(1,1)
Integrated GARCH Estimates
SSE
MSE
Log Likelihood
SBC
MAE
MAPE
6034.97631
10.56914
-1140.055
2324.54181
2.44746877
259.845571
Observations
Uncond Var
Total R-Square
AIC
AICC
HQC
Normality Test
Pr &gt; ChiSq
571
0.0993
2294.11008
2294.30902
2305.98269
49.0458
&lt;.0001
57
IGARCH(1,1)
Parameter Estimates
Variable
Intercept
ret_1
Inf_1
DT_bill_1
ARCH0
ARCH1
GARCH1
TDFI
DF
Estimate
Standard
Error
1
1
1
1
1
1
1
1
1.2475
0.1851
-1.0481
-1.1441
0.3691
0.1357
0.8643
0.1865
0.1879
0.0465
0.3879
0.2988
0.2422
0.0450
0.0450
0.0559
t Value
Approx
Pr &gt; |t|
6.64
3.98
-2.70
-3.83
1.52
3.01
19.20
3.34
&lt;.0001
&lt;.0001
0.0069
0.0001
0.1275
0.0026
&lt;.0001
0.0008
Variable Label
Inf_1
DT-bill_1
Inverse of t DF
58
IGARCH(1,1)
59
S&amp;P COMPOSITE RETURNS VS FITTED DATA
Ret
20
10
0
-10
-20
01JAN1950
01JAN1955
01JAN1960
01JAN1965
01JAN1970
01JAN1975
01JAN1980
date
01JAN1985
01JAN1990
01JAN1995
01JAN2000
01JAN2005
60
ESTIMATED CONDITIONAL VARIANCE OF
S&amp;P COMPOSITE RETURNS FROM
IGARCH(1,1) MODEL
v
50
40
30
20
10
0
01JAN1950
01JAN1955
01JAN1960
01JAN1965
01JAN1970
01JAN1975
01JAN1980
date
01JAN1985
01JAN1990
01JAN1995
01JAN2000
01JAN2005
61
REFERENCES
• Bollerslev, Tim. 1986. “Generalized Autoregressive
Conditional Heteroskedasticity.” Journal of Econometrics.
April, 31:3, pp. 307–27.
• Engle, Robert F. 1982. “Autoregressive Conditional
Heteroskedasticity with Estimates of the Variance of United
Kingdom Inflation.” Econometrica. 50:4, pp. 987–1007.
• Glosten, Lawrence R., Ravi Jagannathan and David E. Runkle.
1993. “On the Relation Between the Expected Value and the
Volatility of the Nominal Excess Returns on Stocks.” Journal of
Finance. 48:5, pp. 1779–801.
• Nelson, Daniel B. 1991. “Conditional Heteroscedasticity in
Asset Returns: A New Approach.” Econometrica. 59:2, pp.
347–70.
62
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