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8/18/2019 Arch Uni Slides
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GARCH Models
Eduardo Rossi
University of Pavia
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Empirical regularities
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Empirical regularities
Eduardo Rossi c - Econometria mercati finanziari (avanzato) 3
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Empirical regularities
GARCH models have been developed to account for empirical
regularities in financial data. Many financial time series have a
number of characteristics in common.
• Asset prices are generally non stationary. Returns are usually
stationary. Some financial time series are fractionally integrated.
• Return series usually show no or little autocorrelation.
• Serial independence between the squared values of the series is
often rejected pointing towards the existence of non-linear
relationships between subsequent observations.
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Empirical regularities
• Volatility of the return series appears to be clustered.
• Normality has to be rejected in favor of some thick-tailed
distribution.
• Some series exhibit so-called leverage effect , that is changes in
stock prices tend to be negatively correlated with changes in
volatility. A firm with debt and equity outstanding typicallybecomes more highly leveraged when the value of the firm
falls.This raises equity returns volatility if returns are constant.
Black, however, argued that the response of stock volatility to the
direction of returns is too large to be explained by leverage alone.
• Volatilities of different securities very often move together.
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Why do we need ARCH models?
Wold’s decomposition theorem establishes that any covariance
stationary {yt} may be written as the sum of a linearly deterministic
component and a linearly stochastic with a square-summable,
one-sided moving average representation. We can write,
yt = dt + ut
dt is linearly deterministic and ut is a linearly regular covariancestationary stochastic process, given by
ut = B (L) εt
B (L) =∞i=0
biLi
∞i=0
b2i
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Why do we need ARCH models?
E [εt] = 0
E [εtετ ] =
σ2ε
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Why do we need ARCH models?
Now suppose that yt is a linear covariance stationary process with
i.i.d. innovations as opposed to merely white noise. The
unconditional mean and variance are
E [yt] = 0
E y2t = σ
2ε
∞
i=0
b2i
which are both invariant in time. The conditional mean is time
varying and is given by
E [yt |Φt−1 ] =
∞i=1
biεt−i
where the information set is Φt−1 = {εt−1, εt−2, . . .}.
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Why do we need ARCH models?
This model is unable to capture the conditional variance dynamics.In fact, the conditional variance of yt is constant at
E (yt − E [yt |Φt−1 ])2
|Φt−1 = σ2ε .
This restriction manifests itself in the properties of the k-step-ahead
conditional prediction error variance. The k -step-ahead conditional
prediction is
E [yt+k |Φt ] =∞i=0
bk+iεt−i
and the associated prediction error is
yt+k − E [yt+k |Φt ] =k−1i=0
biεt+k−i
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Why do we need ARCH models?
which has a conditional prediction error variance
E
(yt+k − E [yt+k |Φt ])
2
|Φt
= σ2
ε
k−1i=0
b2
i
E (yt+k − E [yt+k |Φt ])2
|Φt = σ2εk−1
i=0
b2i → σ2ε
∞
i=0
b2i , as k → ∞
For any k, the conditional prediction error variance depends only on
k and not on Φt.
In conclusion, the simple ”i.i.d. innovations model” isunable to take into account the relevant information which
is available at time t.
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The ARCH(q) Model
(Bollerslev, Engle and Nelson, 1994) The process {εt (θ0)} follows an
ARCH (AutoRegressive Conditional Heteroskedasticity) model if
E t−1 [εt (θ0)] = 0 t = 1, 2, . . . (1)
and the conditional variance
σ2t (θ0) ≡ V art−1 [εt (θ0)] = E t−1
ε2t (θ0)
t = 1, 2, . . . (2)
depends non trivially on the σ-field generated by the past
observations: {εt−1 (θ0) , εt−2 (θ0) , . . .} .
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The ARCH(q) Model
Let {yt (θ
0)} denote the stochastic process of interest with
conditional mean
µt (θ0) ≡ E t−1 (yt) t = 1, 2, . . . (3)
µt (θ0) and σ2t (θ0) are measurable with respect to the time t − 1information set. Define the {εt (θ0)} process by
εt (θ0) ≡ yt − µt (θ0) . (4)
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The ARCH(q) Model
It follows from eq.(1) and (2), that the standardized process
zt (θ0) ≡ εt (θ0) σ2t (θ0)
−1/2 t = 1, 2, . . . (5)
with
E t−1 [zt (θ0)] = 0
V art−1 [zt (θ0)] = 1
will have conditional mean zero (E t−1 [zt (θ0)] = 0) and a time
invariant conditional variance of unity.
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The ARCH(q) Model
We can think of εt (θ0) as generated by
εt (θ0) = zt (θ0) σ2t (θ0)
1/2
where ε2t (θ0) is unbiased estimator of σ2t (θ0).
Let’s suppose zt (θ0) ∼ N ID (0, 1) and independent of σ2t (θ0)
E t−1
ε2t
= E t−1
σ2t
E t−1
z2t
= E t−1
σ2t
= σ2t
because z2t |
Φt−
1 ∼ χ2
(1).
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The ARCH(q) Model
If the conditional distribution of zt is time invariant with a finite
fourth moment, the fourth moment of εt is
E
ε4t
= E
z4t
E
σ4t
≥ E
z4t
E
σ2t2
= E
z4t
E
ε2t2
where the last equality follows from
E [σ2t ] = E [E t−1(ε2t )] = E [ε
2t ]
E
ε4t
≥ E
z4t
E
ε2t
2
by Jensen’s inequality. The equality holds true for a constantconditional variance only.
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The ARCH(q) Model
If zt ∼ N ID (0, 1), then E
z4t
= 3, the unconditional distribution forεt is therefore leptokurtic
E ε4t ≥ 3E ε
2t
2
E
ε4t
/E
ε2t2 ≥ 3
The kurtosis can be expressed as a function of the variability of the
conditional variance.
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The ARCH(q) Model
In fact, if εt |Φt−1 ∼ N
0, σ2t
E t−1
ε4t
= 3E t−1
ε2t
2
E
ε4t
= 3E
E t−1
ε2t2
≥ 3
E
E t−1
ε2t2
= 3
E
ε2t2
E
ε4t
− 3
E
ε2t
2
= 3E
E t−1
ε2t
2
− 3
E
E t−1
ε2t
2
E
ε4t
= 3
E
ε2t2
+ 3E
E t−1
ε2t2
− 3
E
E t−1
ε2t2
k = E
ε4t
[E (ε2t )]2 = 3 + 3E
E t−1
ε2t2
−
E
E t−1
ε2t2
[E (ε2t )]2
= 3 + 3V ar
E t−1
ε2t
[E (ε2t )]2 = 3 + 3
V ar
σ2t
[E (ε2t )]2 .
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The ARCH(q) Model
Another important property of the ARCH process is that the process
is conditionally serially uncorrelated. Given that
E t−1 [εt] = 0
we have that with the Law of Iterated Expectations:
E t−h [εt] = E t−h [E t−1 (εt)] = E t−h [0] = 0.
This orthogonality property implies that the {εt} process is
conditionally uncorrelated:
Covt−h [εt, εt+k] = E t−h [εtεt+k] − E t−h [εt] E t−h [εt+k] =
= E t−h [εtεt+k] = E t−h [E t+k−1 (εtεt+k)] =
= E t−h [εtE t+k−1 [εt+k]] = 0
The ARCH model has showed to be particularly useful in modeling
the temporal dependencies in asset returns.
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The ARCH(q) Model
The ARCH (q) model introduced by Engle (Engle (1982)) is a linearfunction of past squared disturbances:
σ2t = ω +
q
i=1
αiε2t−i (6)
In this model to assure a positive conditional variance the parameters
have to satisfy the following constraints: ω > 0 e
α1 ≥ 0, α2 ≥ 0, . . . , αq ≥ 0. Defining
σ2t ≡ ε2t − vt
where E t−1 (vt) = 0 we can write (6) as an AR(q ) in ε2t :
ε2t = ω + α (L) ε2t + vt
where α (L) = α1L + α2L2 + . . . + αqL
q.
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The ARCH(q) Model
(1 − α(L))ε2t = ω + vt
The process is weakly stationary if and only if q
i=1
αi
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The ARCH(q) Model
The process is characterised by leptokurtosis in excess with respect
to the normal distribution. In the case, for example, of ARCH(1)
with εt |Φt−1 ∼ N
0, σ2t
, the kurtosis is equal to:
E
ε4t
/E
ε2t
2
= 3
1 − α21
/
1 − 3α21
(8)
with 3α21
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The ARCH(q) Model
The result is readily obtained:
E
ε4t
= 3E
σ4t
E
ε4t
= 3
ω2 + α21E
ε4t−1
+ 2ωα1E
ε2t−1
E
ε4t
= 3
ω2 + 2ωα1E
ε2t−1
(1 − 3α21)
= 3
ω2 + 2ωα1σ
2
(1 − 3α
2
1)
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The ARCH(q) Model
substituting σ2
= ω/ (1 − α1):
E
ε4t
= 3
ω2 (1 − α1) + 2ω2α1
(1 − 3α21) (1 − α1)
= 3ω2 (1 + α1)
(1 − 3α21) (1 − α1)
finally
E
ε4t
/E
ε2t2
= 3ω2 (1 + α1) (1 − α1)
2
(1 − 3α21) (1 − α1) ω2
= 3
1 − α21
1 − 3α21. (9)
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The ARCH Regression Model
We have an ARCH regression model when the disturbances in a
linear regression model follow an ARCH process:
yt = x′
tb + εt
E t−1 (εt) = 0
εt |Φt−1 ∼ N
0, σ2t
E t−1 ε2t ≡ σ2t = ω + α (L) ε2t
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The GARCH(p,q) Model
In order to model in a parsimonious way the conditional
heteroskedasticity, Bollerslev (1986) proposed the Generalised ARCH
model, i.e GARCH(p,q):
σ2t = ω + α (L) ε2t + β (L) σ
2t . (10)
where α (L) = α1L + . . . + αqLq
, β (L) = β 1L + . . . + β pL p
.The GARCH(1,1) is the most popular model in the empirical
literature:
σ2t = ω + α1ε2t−1 + β 1σ2t−1. (11)
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The GARCH(p,q) Model
To ensure that the conditional variance is well defined in a
GARCH (p,q ) model all the coefficients in the corresponding linear
ARCH (∞) should be positive:
σ2t =
1 −
p
i=1β iLi
−1
ω +
q
j=1αjε
2t−j
= ω∗ +∞k=0
φkε2t−k−1 (12)
σ
2
t ≥ 0 if ω
∗
≥ 0 and all φk ≥ 0.The non-negativity of ω∗ and φk is also a necessary condition for the
non negativity of σ2t .
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The GARCH(p,q) Model
In order to make ω∗ e {φk}∞
k=0 well defined, assume that :
i. the roots of the polynomial β (x) = 1 lie outside the unit circle,
and that ω ≥ 0, this is a condition for ω∗
to be finite and positive.
ii. α (x) e 1 − β (x) have no common roots.
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The GARCH(p,q) Model
These conditions are establishing nor that σ2t ≤ ∞ neither thatσ2t∞
t=−∞is strictly stationary. For the simple GARCH(1,1) almost
sure positivity of σ2t requires, with the conditions (i) and (ii), that
(Nelson and Cao, 1992),
ω ≥ 0
β 1 ≥ 0
α1 ≥ 0 (13)
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The GARCH(p,q) Model
For the GARCH(1,q) and GARCH(2,q) models these constraints can
be relaxed, e.g. in the GARCH(1,2) model the necessary and
sufficient conditions become:
ω ≥ 0
0 ≤ β 1
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The GARCH(p,q) Model
For the GARCH(2,1) model the conditions are:
ω ≥ 0
α1 ≥ 0
β 1 ≥ 0
β 1 + β 2 < 1
β 21 + 4β 2 ≥ 0 (15)
These constraints are less stringent than those proposed by Bollerslev
(1986):
ω ≥ 0
β i ≥ 0 i = 1, . . . , p
αj ≥ 0 j = 1, . . . , q (16)
Eduardo Rossi c - Econometria mercati finanziari (avanzato) 30
GA C ( )
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The GARCH(p,q) Model
These results cannot be adopted in the multivariate case, where the
requirement of positivity for σ2t means the positive definiteness forthe conditional variance-covariance matrix.
Eduardo Rossi c - Econometria mercati finanziari (avanzato) 31
T Y W
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The Yule-Walker equations for the squared process
A GARCH(p,q) can be represented as an ARMA process, given thatε2t = σ
2t + υt, where E t−1 [υt] = 0, υt ∈
−σ2t , ∞
:
ε2
t
= ω +
max( p,q)
j=1
(αj + β j) ε2
t−
j
+ υt − p
i=1
β iυt−iε2t ∼ARMA(m,p) with m = max( p, q ). We can apply the classical
results of ARMA model.
Eduardo Rossi c - Econometria mercati finanziari (avanzato) 32
T Y W
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The Yule-Walker equations for the squared process
Example: GARCH(1,1)
σ2t = ω + α1ε2t−1 + β 1σ
2t−1
replacing σ2t with ε2t − υt, we obtain
ε2t − υt = ω + α1ε
2t−1 + β 1(ε
2t−1 − υt−1)
hence
ε2t = ω + (α1 + β 1)ε2t−1 + υt − β 1υt−1
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The Yule Walker equations for the squared process
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The Yule-Walker equations for the squared process
We can study the autocovariance function, that is:
γ 2 (k) = cov
ε2t , ε
2t−k
γ 2 (k) = cov
ω + mj=1
(αj + β j) ε2t−j +
υt −
pi=1
β iυt−i
, ǫ2t−k
γ 2 (k) =
mj=1
(αj + β j) cov
ε2t−j, ε2t−k
+cov
υt − pi=1
β iυt−i, ε2t−k
(17)
When k is big enough, the last term on the right of expression (17) is
null.
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The Yule Walker equations for the squared process
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The Yule-Walker equations for the squared process
The sequence of autocovariances satisfy a linear difference equationof order max ( p, q ), for k ≥ p + 1
γ 2 (k) = m
j=1
(αj + β j) γ 2 (k − j)
This system can be used to identify the lag order m and p, that is the
p and q order if q ≥ p, the order p if q < p.
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The GARCH Regression Model
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The GARCH Regression Model
Let wt = 1, ε2t−1, · · · , ε2t−q, σ2t−1, · · · , σ2t− p′
,
γ = (ω, α1, · · · , αq, β 1, · · · , β p) and θ ∈ Θ, where θ = (b′, γ ′) and Θ isa compact subspace of a Euclidean space such that εt possesses finite
second moments. We may write the GARCH regression model as:
εt = yt − x′
tb
εt |Φt−1 ∼ N
0, σ2t
σ2t = w′
tγ
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Stationarity
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Stationarity
The process {εt} which follows a GARCH(p,q) model is a martingale
difference sequence. In order to study second-order stationarity it’s
sufficient to consider that:
V ar [εt] = V ar [E t−1 (εt)] + E [V art−1 (εt)] = E
σ2t
and show that is asymptotically constant in time (it does not depend
upon time).A process {εt} which satisfies a GARCH(p,q) model with positive
coefficient ω ≥ 0, αi ≥ 0 i = 1, . . . , q , β i ≥ 0 i = 1, . . . , p is covariance
stationary if and only if:
α (1) + β (1)
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Stationarity
This is a sufficient but non necessary conditions for strict
stationarity. Because ARCH processes are thick tailed, the conditionsfor covariance stationarity are often more stringent than the
conditions for strict stationarity.
A GARCH(1,1) model can be written as
σ2t = ω
1 +
∞k=1
ki=1
β 1 + α1z2t−i
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Stationarity
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Stationarity
In fact,
σ2t = ω + σ2t−1
α1z
2t−1 + β 1
σ2t−1 = ω + σ2t−2
α1z2t−2 + β 1
σ2t−2 = ω + σ
2t−3
α1z
2t−3 + β 1
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Stationarity
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Stationarity
σ2t = ω + ω + σ2t−2 α1z2t−2 + β 1 α1z2t−1 + β 1= ω + ω
α1z2t−1 + β 1
+ σ2t−2
α1z2t−2 + β 1
α1z2t−1 + β 1
= ω + ω
α1z
2t−1 + β 1
+ ω
α1z
2t−1 + β 1
α1z
2t−2 + β 1
+σ2t−3 α1z2t−3 + β 1 α1z2t−2 + β 1 α1z2t−1 + β 1
finally,
σ2t = ω 1 +∞
k=1
k
i=1
α1z2t−i + β 1
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Stationarity
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Nelson shows that when ω > 0, σ2t
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Forecasting volatility
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Forecasting with a GARCH(p,q) (Engle and Bollerslev 1986):
σ2t+k = ω +
qi=1
αiε2t+k−i +
pi=1
β iσ2t+k−i
we can write the process in two parts, before and after time t:
σ2t+k = ω +n
i=1
αiε
2t+k−i + β iσ
2t+k−i
+
mi=k
αiε
2t+k−i + β iσ
2t+k−i
where n = min {m, k − 1} and by definition summation from 1 to 0
and from k > m to m both are equal to zero.
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Forecasting volatility
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Thus
E t σ2t+k
= ω+n
i=1
(αi + β i) E t σ2t+k
−
i+
m
i=k
αiε2t+k
−
i
+ β iσ2
t+k−
i .
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Forecasting volatility
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In particular for a GARCH(1,1) and k > 2:
E t
σ
2
t+k
=
k−2
i=0 (α1 + β 1)
i
ω + (α1 + β 1)
k−1
σ
2
t+1
= ω
1 − (α1 + β 1)
k−1
[1 − (α1 + β 1)] + (α1 + β 1)
k−1 σ2t+1
= σ2
1 − (α1 + β 1)k−1
+ (α1 + β 1)
k−1σ2t+1
= σ2 + (α1 + β 1)k−1
σ2t+1 − σ
2
When the process is covariance stationary, it follows that E t
σ2t+k
converges to σ2 as k → ∞.
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The IGARCH(p,q) model
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The GARCH(p,q) process characterized by the first two conditional
moments:
E t−1 [εt] = 0
σ2t ≡ E t−1 ε2t
= ω +
q
i=1αiε
2t−i +
p
i=1β iσ
2t−i
where ω ≥ 0, αi ≥ 0 and β i ≥ 0 for all i and the polynomial
1 − α (x) − β (x) = 0
has d > 0 unit root(s) and max { p, q } − d root(s) outside the unitcircle is said to be:
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The IGARCH(p,q) model
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• Integrated in variance of order d if ω = 0
• Integrated in variance of order d with trend if ω > 0.
The Integrated GARCH(p,q) models, both with or without trend, are
therefore part of a wider class of models with a property called”persistent variance” in which the current information remains
important for the forecasts of the conditional variances for all horizon.
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The IGARCH(p,q) model
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So we have the Integrated GARCH(p,q) model when (necessarycondition)
α (1) + β (1) = 1
To illustrate consider the IGARCH(1,1) which is characterised by
α1 + β 1 = 1
σ2t = ω + α1ε2t−1 + (1 − α1) σ2t−1
σ2t = ω + σ2t−1 + α1
ε2t−1 − σ
2t−1
0 < α1 ≤ 1
For this particular model the conditional variance k steps in the
future is:
E t
σ2t+k
= (k − 1) ω + σ2t+1
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Persistence
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• In many studies of the time series behavior of asset volatility the
question has been how long shocks to conditional variance persist.
• If volatility shocks persist indefinitely, they may move the whole
term structure of risk premia.
• There are many notions of convergence in the probability theory(almost sure, in probability, in L p), so whether a shock is
transitory or persistent may depend on the definition of
convergence.
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Persistence
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• In linear models it typically makes no difference which of the
standard definitions we use, since the definitions usually agree.
• In GARCH models the situation is more complicated.In the IGARCH(1,1):
σ2t = ω + α1ε2t−1 + β 1σ
2t−1
where α1 + β 1 = 1. Given that ε2t = z
2t σ
2t , we can rewrite the
IGARCH(1,1) process as
σ2
t = ω + σ2
t−
1(1 − α1) + α1z2
t−
1 0 < α1 ≤ 1.
When ω = 0, σ2t is a martingale.
Eduardo Rossi c - Econometria mercati finanziari (avanzato) 50
Persistence
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Based on the nature of persistence in linear models, it seems that
IGARCH(1,1) with ω > 0 and ω = 0 are analogous to random walks
with and without drift, respectively, and are therefore natural modelsof ”persistent” shocks.
This turns out to be misleading, however:
• in IGARCH(1,1) with ω = 0, σ2t collapses to zero almost
surely
• in IGARCH(1,1) with ω > 0, σ2t is strictly stationary and ergodic
and therefore does not behave like a random walk, since randomwalks diverge almost surely.
Eduardo Rossi c - Econometria mercati finanziari (avanzato) 51
Persistence
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Two notions of persistence.
1. Suppose σ2t is strictly stationary and ergodic. Let F σ2t be theunconditional cdf for σ2t , and F s
σ2t
the conditional cdf for σ2t ,
given information at time s < t. For any s F
σ2t
− F s
σ2t
→ 0
at all continuity points as t → ∞. There is no persistence when
σ
2
t
is stationary and ergodic.2. Persistence is defined in terms of forecast moments. For some
η > 0, the shocks to σ2t fail to persist if and only if for every s,
E sσ2η
t converges, as t → ∞, to a finite limit independent of
time s information set.
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Persistence
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Whether or not shocks to
σ2t
”persist” depends very much on
which definition is adopted. The conditional moment may diverge to
infinity for some η, but converge to a well-behaved limit independent
of initial conditions for other η, even when the
σ2t
is stationary and
ergodic.
Eduardo Rossi c - Econometria mercati finanziari (avanzato) 53
Persistence
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GARCH(1,1):
σ2t+1 = ω + α1ǫ2t + β 1σ
2t = ω + σ
2t
α1z
2t + β 1
σ2t = ω
1 +
∞k=1
ki=1
α1z2t−i + β 1
σ2t = ω + σ
2t−1
α1z
2t−1 + β 1
σ2t−1 = ω + σ
2t−2
α1z
2t−2 + β 1
σ2t = ω +
ω + σ2t−2
α1z
2t−2 + β 1
α1z
2t−1 + β 1
= ω + ω α1z2t−1 + β 1 + σ2t−2 α1z2t−2 + β 1 α1z2t−1 + β 1= ω + ω
α1z
2t−1 + β 1
+ ω
α1z
2t−1 + β 1
α1z
2t−2 + β 1
+σ2t−3
α1z
2t−3 + β 1
α1z
2t−2 + β 1
α1z
2t−1 + β 1
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Persistence
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E t−3
σ2t
= ω
t−(t−3)−1
k=0
(α1 + β 1)k
+σ2t−3 (α1 + β 1) (α1 + β 1) (α1 + β 1)
E s σ2t = ω
t−s−1
k=0
(α1 + β 1)k
+ σ2t−s (α1 + β 1)
t−s
E s
σ2t
converges to the unconditional variance of ω/ (1 − α1 − β 1)
as t → ∞ if and only if α1 + β 1 0 and α1 + β 1 = 1
E s
σ2t
→ ∞ a.s. as t → ∞. Nevertheless, IGARCH models are
strictly stationary and ergodic.
Eduardo Rossi c - Econometria mercati finanziari (avanzato) 55
The EGARCH(p,q) Model
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• GARCH models assume that only the magnitude and not the
positivity or negativity of unanticipated excess returnsdetermines feature σ2t .
• There exists a negative correlation between stock returns and
changes in returns volatility, i.e. volatility tends to rise inresponse to ”bad news”, (excess returns lower than expected)
and to fall in response to ”good news” (excess returns higher
than expected).
Eduardo Rossi c - Econometria mercati finanziari (avanzato) 56
The EGARCH(p,q) Model
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If we write σ2t as a function of lagged σ2t and lagged z
2t , where
ε2t = z2t σ
2t
σ2t = ω +
qj=1
αjz2t−jσ
2t−j +
pi=1
β iσ2t−i
it is evident that the conditional variance is invariant to changes in
sign of the z′
t
s. Moreover, the innovations z2
t−
j
σ2
t−
j
are not i.i.d .
• The nonnegativity constraints on ω∗ and φk, which are imposed
to ensure that σ2t remains nonnegative for all t with probability
one. These constraints imply that increasing z2t in any period
increases σ2t+m for all m ≥ 1, ruling out random oscillatory
behavior in the σ2t process.
Eduardo Rossi c - Econometria mercati finanziari (avanzato) 57
The EGARCH(p,q) Model
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• The GARCH models are not able to explain the observed
covariance between ε2t and εt−j . This is possible only if the
conditional variance is expressed as an asymmetric function of εt−j .
• In GARCH(1,1) models, shocks may persist in one norm and die
out in another, so the conditional moments of GARCH(1,1) may
explode even when the process is strictly stationary and ergodic.
• GARCH models essentially specify the behavior of the square of
the data. In this case a few large observations can dominate the
sample.
Eduardo Rossi c - Econometria mercati finanziari (avanzato) 58
The EGARCH(p,q) Model
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In the EGARCH(p,q) model (Exponential GARCH(p,q)) put forwardby Nelson the σ2t depends on both size and the sign of lagged
residuals. This is the first example of asymmetric model:
ln
σ2t
= ω +
pi=1
β i ln
σ2t−i
+
q
i=1
αi [φzt−i + ψ (|zt−i| − E |zt−i|)]
α1 ≡ 1, E |zt| = (2/π)1/2given that zt ∼ N ID(0, 1), where the
parameters ω, β i, αi are not restricted to be nonnegative. Let define
g (zt) ≡ φzt + ψ [|zt| − E |zt|]
by construction {g (zt)}∞
t=−∞ is a zero-mean, i.i.d. random sequence.
Eduardo Rossi c - Econometria mercati finanziari (avanzato) 59
The EGARCH(p,q) Model
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• The components of g (zt) are φzt and ψ [|zt| − E |zt|], each with
mean zero.
• If the distribution of zt is symmetric, the components are
orthogonal, but not independent.
• Over the range 0 < zt
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• If ψ > 0 and φ = 0, the innovation in ln
σ2t+1
is positive
(negative) when the magnitude of zt is larger (smaller) than its
expected value.
• If ψ = 0 and φ 0 and φ
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In the EGARCH model ln
σ2t+1
is homoskedastic conditional on σ2t ,
and the partial correlation between zt and ln
σ2t+1
is constant
conditional on σ2t
.
An alternative possible specification of the news impact curve is the
following (Bollerslev, Engle, Nelson (1994))
g(zt, σ2t ) = σ−2θ0t θ1zt1 + θ2 |zt|+σ−2γ 0t
γ 1 |zt|
ρ
1 + γ 2 |zt|ρ − E t
γ 1 |zt|
ρ
1 + γ 2 |zt|ρ
The parameters γ 0 and θ0 parameters allow both the conditional
variance of ln
σ2
t+1
and its conditional correlation with zt to varywith the level of σ2t .
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The EGARCH(p,q) Model
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If θ1 0, the
model downweights large |zt|’s.
Nelson assumes that zt has a GED distribution (exponential power
family). The density of a GED random variable normalized is:
f (z; υ) = υ exp − 1
2 |z/λ|υ
λ2(1+1/υ)Γ (1/υ) − ∞ < z
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where Γ (·) is the gamma function, and
λ ≡
2(−2/υ)Γ (1/υ) /Γ (3/υ)1/2
υ is a tail thickness parameter.
z’s distribution
υ = 2 standard normal distribution
υ 2 thinner tails than the normal
υ = ∞ uniformly distributed on
−31/2
, 31/2
With this density, E |zt| = λ21/υΓ (2/υ)
Γ (1/υ) .
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The EGARCH(p,q) Model
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The Generalized t Distribution takes the form:
f
εtσ−1
t ; η, ζ
=
η
2σtbζ 1/ηB (1/η,ζ ) [1 + |εt|η / (ζbησηt )]
ζ +1/η
where B (1/η,ζ ) ≡ Γ (1/η) Γ (ζ ) Γ ( 1/η + ζ ) denotes the beta function,
b ≡ [Γ (ζ ) Γ ( 1/η) /Γ (3/η) Γ (ζ − 2/η)]1/2
and ζη > 2, η > 0 and ζ > 0. The factor b makes V ar
εtσ−1t
= 1.
The Generalized t nests both the Student’s t distribution and the
GED.
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The EGARCH(p,q) Model
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The Student’s t-distribution sets η = 2 and ζ = 12 times the degree of freedom
The GED is obtained for ζ = ∞. The GED has only one shapeparameter η, which is apparently insufficient to fit both the central
part and the tails of the conditional distribution.
Eduardo Rossi c - Econometria mercati finanziari (avanzato) 66
Stationarity of EGARCH(p,q)
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In order to simply state the stationarity conditions, we write the
EGARCH(p,q) model as:
1 −
pi=1
β iLi
ln
σ2t
= ω +
qi=1
αiLi [φzt + ψ (|zt| − E |zt|)]
ln
σ2t
=
1 −
pi=1
β i
−1
ω +
1 −
pi=1
β iLi
−1 qi=1
αiLi
g (zt)
ln
σ2t
= ω∗
+
∞i=1
ϕig (zt−i)
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Stationarity of EGARCH(p,q)
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Given φ = 0 or ψ = 0, thenln
σ2t
− ω∗
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The first two moments of ln
σ2t
− ω∗ are finite and time invariant :
E ln σ2t − ω∗ = 0 for all tV ar
ln
σ2t
− ω∗
= V ar (g (zt))∞i=1
ϕ2i
since V ar (g (zt)) is finite and the distribution of
ln
σ2t
− ω∗
isindependent of t, the first two moments of
ln
σ2t
− ω∗
are finite
and time invariant, so
ln
σ2t
− ω∗
is covariance stationary if
∞i=1ϕ
2
i
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Since ln
σ2t
is written in ARMA(p,q) form, when
1 −
p
i=1β ix
i
and
qi=1
αixi
have no common roots, conditions for strict stationarity of exp(−ω∗) σ2t
and {exp(−ω∗/2) εt} are equivalent to all the roots
of 1 − p
i=1
β ixi lying outside the unit circle.
The strict stationarity of
exp(−ω∗) σ2t
, {exp(−ω∗/2) εt} need
not imply covariance stationarity, since
exp(−ω∗) σ2t
,
{exp(−ω∗/2) εt} may fail to have finite unconditional means and
variances.
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Stationarity of EGARCH(p,q)
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For some distribution of {zt} (e.g., the Student t with finite degrees
of freedom),
exp(−ω∗
) σ2
t
and {exp(−ω∗
/2) εt} typically have nofinite unconditional moments.
If the distribution of zt is GED and is thinner-tailed than the double
exponential (ν > 1), and if
∞i=1ϕ
2
i
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The Non linear ARCH(p,q) model (Engle - Bollerslev 1986):
σγ t = ω +
qi=1
αi |εt−i|γ +
pi=1
β iσγ t−i
σγ
t
= ω +
q
i=1
αi |εt−i − k|γ
+
p
i=1
β iσγ
t−
i
for k = 0, the innovations in σγ t will depend on the size as well as the
sign of lagged residuals, thereby allowing for the leverage effect in
stock return volatility.
Eduardo Rossi c - Econometria mercati finanziari (avanzato) 72
Other Asymmetric Models
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The Glosten - Jagannathan - Runkle model (1993):
σ2t = ω + pi=1
β iσ2t−i +
qi=1
αiε
2t−1 + γ iS
−
t−iε2t−i
where
S −t =
1 if εt
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The Asymmetric GARCH(p,q) model (Engle, 1990):
σ2t = ω +
q
i=1αi (εt−i + γ )2
+
p
i=1β iσ2t−i
The QGARCH by Sentana (1995):
σ
2
t = σ
2
+ Ψ
′
xt−
q + x
′
t−qAxt−
q +
p
i=1β iσ
2
t−i
when xt−q = (εt−1, . . . , εt−q)′
. The linear term (Ψ′xt−q) allows for
asymmetry. The off-diagonal elements of A accounts for interaction
effects of lagged values of xt on the conditional variance. The
QGARCH nests several asymmetric models.
Eduardo Rossi c - Econometria mercati finanziari (avanzato) 74
The GARCH-in-mean Model
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The GARCH-in-mean (GARCH-M) proposed by Engle, Lilien and
Robins (1987) consists of the system:
yt = γ 0 + γ 1xt + γ 2g
σ2t
+ ǫt
σ2t = β 0 +
q
i=1αiǫ
2t−1 +
p
i=1β iσ
2t−1
ǫt | Φt−1 ∼ N (0, σ2t )
When yt ≡ (rt − rf ), where (rt − rf ) is the risk premium on holding
the asset, then the GARCH-M represents a simple way to model the
relation between risk premium and its conditional variance.
Eduardo Rossi c - Econometria mercati finanziari (avanzato) 75
The GARCH-in-mean Model
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This model characterizes the evolution of the mean and the variance
of a time series simultaneously.
The GARCH-M model therefore allows to analize the possibility of
time-varying risk premium.
It turns out that:
yt | Φt−1 ∼ N (γ 0 + γ 1xt + γ 2g
σ2t
, σ2t )
In applications, g σ2t = σ
2t , g σ
2t = ln σ
2t and g σ
2t = σ
2t have
been used.
Eduardo Rossi c - Econometria mercati finanziari (avanzato) 76
The News Impact Curve
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• The news have asymmetric effects on volatility.• In the asymmetric volatility models good news and bad news
have different predictability for future volatility.
• The news impact curve characterizes the impact of past returnshocks on the return volatility which is implicit in a volatility
model.
Eduardo Rossi c - Econometria mercati finanziari (avanzato) 77
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The News Impact Curve
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For the GARCH model the News Impact Curve (NIC) is centered on
ǫt−1 = 0.
GARCH(1,1):
σ2t = ω + αǫ2t−1 + βσ
2t−1
The news impact curve has the following expression:
σ2t = A + αǫ2t−1
A ≡ ω + βσ2
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The News Impact Curve
In the case of EGARCH model the curve has its minimum at
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ǫt−1 = 0 and is exponentially increasing in both directions but withdifferent paramters.
EGARCH(1,1):
σ2t = ω + β ln
σ2t−1
+ φzt−1 + ψ (|zt−1| − E |zt−1|)
where zt = ǫt/σt. The news impact curve is
σ2t =
A exp
φ + ψσ ǫt−1
f or ǫt−1 > 0
A exp
φ − ψ
σ ǫt−1
f or ǫt−1 0
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The News Impact Curve
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• The EGARCH allows good news and bad news to have differentimpact on volatility, while the standard GARCH does not.
• The EGARCH model allows big news to have a greater impact
on volatility than GARCH model. EGARCH would have higher
variances in both directions because the exponential curve
eventually dominates the quadrature.
Eduardo Rossi c - Econometria mercati finanziari (avanzato) 81
The News Impact Curve
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The Asymmetric GARCH(1,1) (Engle, 1990)
σ2t = ω + α (ǫt−1 + γ )2 + βσ2t−1
the NIC is
σ2t = A + α (ǫt−1 + γ )2
A ≡ ω + βσ2
ω > 0, 0 ≤ β 0, 0 ≤ α
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σ2t = ω + αǫ2t + βσ
2t−1 + γS
−
t−1ǫ2t−1
S −
t−1 = 1 if ǫt−1 0
A + (α + γ ) ǫ2t−1 if ǫt−1 0, 0 ≤ β 0, 0 ≤ α
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The GARCH-in-mean (GARCH-M) proposed by Engle, Lilien and
Robins (1987) consists of the system:
yt = γ 0 + γ 1xt + γ 2g
σ2t
+ ǫt
σ2t = β 0 +
q
i=1αiǫ
2t−1 +
p
i=1β iσ
2t−1
ǫt | Φt−1 ∼ N (0, σ2t )
When yt ≡ (rt − rf ), where (rt − rf ) is the risk premium on holding
the asset, then the GARCH-M represents a simple way to model the
relation between risk premium and its conditional variance.
Eduardo Rossi c - Econometria mercati finanziari (avanzato) 84
The GARCH-in-mean Model
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This model characterizes the evolution of the mean and the variance
of a time series simultaneously.
The GARCH-M model therefore allows to analyze the possibility of
time-varying risk premium.
It turns out that:
yt | Φt−1 ∼ N (γ 0 + γ 1xt + γ 2g
σ2t
, σ2t )
In applications, g σ2t = σ
2t , g σ
2t = ln σ
2t and g σ
2t = σ
2t have
been used.
Eduardo Rossi c - Econometria mercati finanziari (avanzato) 85
The News Impact Curve
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• The news have asymmetric effects on volatility.
• In the asymmetric volatility models good news and bad news
have different predictability for future volatility.
• The news impact curve characterizes the impact of past returnshocks on the return volatility which is implicit in a volatility
model.
Eduardo Rossi c - Econometria mercati finanziari (avanzato) 86
The News Impact Curve
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Holding constant the information dated t − 2 and earlier, we can
examine the implied relation between εt−1 and σ2t , with σ2t−i = σ2
i = 1, . . . , p .
This impact curve relates past return shocks (news) to current
volatility.This curve measures how new information is incorporated into
volatility estimates.
Eduardo Rossi c - Econometria mercati finanziari (avanzato) 87
The News Impact Curve
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For the GARCH model the News Impact Curve (NIC) is centered on
ǫt−1 = 0.
GARCH(1,1):
σ2t = ω + αǫ2t−1 + βσ
2t−1
The news impact curve has the following expression:
σ2t = A + αǫ2t−1
A ≡ ω + βσ2
Eduardo Rossi c - Econometria mercati finanziari (avanzato) 88
The News Impact Curve
In the case of EGARCH model the curve has its minimum at
t 1 0 d i ti ll i i i b th di ti b t ith
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ǫt−1 = 0 and is exponentially increasing in both directions but withdifferent paramters.
EGARCH(1,1):
σ2t = ω + β ln
σ2t−1
+ φzt−1 + ψ (|zt−1| − E |zt−1|)
where zt = ǫt/σt. The news impact curve is
σ2t =
A expφ + ψ
σ ǫt−1
f or ǫt−1 > 0
A exp
φ − ψ
σ ǫt−1
f or ǫt−1 0
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The News Impact Curve
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• The EGARCH allows good news and bad news to have differentimpact on volatility, while the standard GARCH does not.
• The EGARCH model allows big news to have a greater impact
on volatility than GARCH model. EGARCH would have higher
variances in both directions because the exponential curve
eventually dominates the quadrature.
Eduardo Rossi c - Econometria mercati finanziari (avanzato) 90
The News Impact Curve
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The Asymmetric GARCH(1,1) (Engle, 1990)
σ2t = ω + α (ǫt−1 + γ )2 + βσ2t−1
the NIC is
σ2t = A + α (ǫt−1 + γ )2
A ≡ ω + βσ2
ω > 0, 0 ≤ β 0, 0 ≤ α
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σ2t = ω + αǫ2t + βσ
2t−1 + γS
−
t−1ǫ2t−1
S −
t−1 = 1 if ǫt−1 0A + (α + γ ) ǫ2t−1 if ǫt−1 0, 0 ≤ β 0, 0 ≤ α