Econ 623 Econometrics II Topic 7: ARCH Models 1 Forms and Properties of ARCH#type Models

21 Pages·2012·0.17 MB·English

Econ 623 Econometrics II
Topic 7: ARCH Models
1 Forms and Properties of ARCH-type Models
1.1 ARCH(q) model (Engle, 1982)
It models the conditional variance as a deterministic function of past returns. Suppose
y is the return of a (cid:133)nancial asset.
t
y = (cid:22)+(cid:27) "
t t t :
(cid:27)2 = (cid:11)+(cid:12) y2 + +(cid:12) y2
(cid:26) t 1 t 1 (cid:1)(cid:1)(cid:1) q t q
(cid:0) (cid:0)
A special ARCH(q) model that we will consider here is,
iid
y = (cid:27) " ; " N(0;1)
t t t t :
(cid:24)
(cid:27)2 = (cid:11)+(cid:12) y2 + +(cid:12) y2
( t 1 t 1 (cid:1)(cid:1)(cid:1) q t q
(cid:0) (cid:0)
1
Properties:
E(y I ) = 0
t t 1
(cid:15) j (cid:0)
E(y2 I ) = (cid:27)2 = (cid:11)+(cid:12) y2 + +(cid:12) y2
(cid:15) tj t(cid:0)1 t 1 t(cid:0)1 (cid:1)(cid:1)(cid:1) q t(cid:0)q
y2 AR(q)
(cid:15) t (cid:24)
Since (cid:27)2 is time variant, the model is called the Autoregressive Conditional Het-
(cid:15) t
eroskedasticity (ARCH) model.
According to the model, the conditional variance is completely determined by
(cid:15)
the lagged squared returns.
2
The kurtosis of ARCH is bigger than 3. In particular, when q = 1, the kurtosis
(cid:15)
of y is given by 3(1 (cid:12)2)=(1 3(cid:12)2) > 1 as long as (cid:12) = 0. So ARCH models
t (cid:0) 1 (cid:0) 1 1 6
have fatter tails than the normal distribution.
If (cid:12) ; ;(cid:12) ;(cid:11) > 0, and (cid:12) + + (cid:12) < 1; the ARCH(q) model is covariance
1 q 1 q
(cid:15) (cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1)
stationary. If so, E(y2) = (cid:11)=(1 (cid:12) (cid:12) )
t (cid:0) 1 (cid:0)(cid:1)(cid:1)(cid:1)(cid:0) q
E(y y ) = 0 for any t = s:
t s
(cid:15) 6
E(y2 y2) = 0 for some t = s:
(cid:15) t s 6 6
In most empirical studies, we need many lags in the variance equation (ie q must
(cid:15)
be large).
3
1.2 GARCH(p,q) (Bollerslev, 1986)
It models the conditional variance as a deterministic function of past returns and
conditional variance
y = (cid:27) "
t t t ;
(cid:27)2 = (cid:11) + q (cid:12) y2 + p (cid:11) (cid:27)2
(cid:26) t 0 j=1 j t(cid:0)j j=1 j t(cid:0)j
P P
Properties:
E(y I ) = 0, E(y2 I ) = (cid:27)2
(cid:15) tj t(cid:0)1 tj t(cid:0)1 t
y2 = (cid:11) + ((cid:11) + (cid:12) )y2 + + ((cid:11) + (cid:12) )y2 + w (cid:11) w (cid:11) w ;
(cid:15) t 0 1 1 t(cid:0)1 (cid:1)(cid:1)(cid:1) m m t(cid:0)m t (cid:0) 1 t(cid:0)1 (cid:0) (cid:1)(cid:1)(cid:1) (cid:0) p t(cid:0)p
where m = max p;q ; w = y2 (cid:27)2
f g t t (cid:0) t
y2 ARMA(max p;q ;p)
(cid:15) t (cid:24) f g
According to the model, the conditional variance is the weighted average of
(cid:15)
the lagged squared returns (ARCH term) and the lagged conditional variance
(GARCH term).
4
If (cid:11) ; ;(cid:11) > 0; (cid:12) ; ;(cid:12) > 0; (cid:11) > 0 and ((cid:11) +(cid:12) )+ +((cid:11) +(cid:12) ) < 1; the
1 p 1 q 1 1 m m
(cid:15) (cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1)
GARCH(p,q) model is covariance stationary. E(y2) = (cid:11)=(1 (cid:11) (cid:11)
t (cid:0) 1 (cid:0) (cid:1)(cid:1)(cid:1) (cid:0) m (cid:0)
(cid:12) (cid:12) )
1 m
(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)
In the GARCH (1,1) model, E(y2 I ) = (cid:11) + ((cid:11) + (cid:12) )y2 + w (cid:11) w ; so
(cid:15) tj t 0 1 1 t(cid:0)1 t (cid:0) 1 t(cid:0)1
((cid:11) +(cid:12) ) governs the persistence of volatilities.
1 1
To describe the real (cid:133)nancial time series, a parsimonious GARCH model is often
(cid:15)
doing the job as well as an ARCH model with a long lag length. In the empirical
study, the GARCH(1,1) speci(cid:133)cation has been found to be adequate in most
application. In (cid:133)nancial data, often ((cid:11) +(cid:12) ) is very found to be close to 1 in the
1 1
GARCH(1,1) model. Therefore, the volatility process is persistent, suggesting
volatility is predictable.
5
1.3 IGARCH (Engle and Bollerslev, 1986)
Volatility is as persistent as a unit root
y = (cid:27) "
t t t
(cid:27)2 = (cid:11) + q (cid:12) y2 + p (cid:11) (cid:27)2 ;
8 tq (cid:11)0 + jp=1 (cid:12)j =t(cid:0)j1 j=1 j t(cid:0)j
< j=1 j Pj=1 j P
Properties: : P P
A shock to the conditional variance remains forever.
(cid:15)
Unconditional expected value of y2 is in(cid:133)nite. Hence, the IGARCH model is not
(cid:15) t
covariance stationary.
It is parsimonious compared with GARCH(p,q).
(cid:15)
6
1.4 GJR-GARCH (Glosten, Jaganathan and Runkle, 1993)
Conditional variance responds to good news in di⁄erent ways to bad news.
y = (cid:27) "
t t t
(cid:27)2 = (cid:11) + q (cid:12) y2 +(cid:13)y2 d + p (cid:11) (cid:27)2 ;
8 dt =01 if yi=1 <i 0t(cid:0);id =t(cid:0)01 itf(cid:0)1y i=01 i t(cid:0)i
< t 1 t 1 t 1 t 1
(cid:0) P(cid:0) (cid:0) (cid:0)P(cid:21)
Properties: :
d is a decision rule variable. The threshold is y = 0:
t t 1
(cid:15) (cid:0)
The model can lead to the leverage e⁄ect. When (cid:13) > 0; bad news (y < 0) has
t 1
(cid:15) (cid:0)
larger impact on volatility than good news (y 0):
t 1
(cid:0) (cid:21)
It is also known as the threshold ARCH (TARCH) model.
(cid:15)
7
1.5 EGARCH (Nelson, 1991)
Model the log-conditional volatility to ensure volatility is positive.
y = (cid:27) "
t t t
;
ln((cid:27)2) = (cid:11) + p (cid:11) ln((cid:27)2 )+ q ((cid:12) yt i +(cid:13) yt i)
( t 0 j=1 j t(cid:0)j i=1 i (cid:27)t(cid:0)(cid:0)i i(cid:27)t(cid:0)(cid:0)i
(cid:12) (cid:12)
Properties: P P (cid:12) (cid:12)
(cid:12) (cid:12)
Conditional variance depends on both the size and the sign of lagged residuals.
(cid:15)
(cid:12) captures the asymmetric e⁄ect. When (cid:12) < 0; bad news (y < 0) has larger
i i t i
(cid:15) (cid:0)
impact on volatility than good news (y 0):
t i
(cid:0) (cid:21)
In the volatility equation, ln((cid:27)2) but (cid:27)2 is modeled such that no restriction on
(cid:15) t t
the parameters is needed.
8
1.6 Absolute Value ARCH (Schwert, 1989)
p
(cid:27) = (cid:11) + (cid:12) y + (cid:11) (cid:27)
t 0 i t i i t i
j (cid:0) j (cid:0)
i=1;q i=1
X X
1.7 NARCH (non-linear ARCH) (Higgin and Bera, 1992)
q
(cid:27)(cid:13) = (cid:11) + (cid:12) y (cid:13) + (cid:11) (cid:27) (cid:13)
t 0 i t i i t i
j (cid:0) j (cid:0)
i=1 i=1;p
X X
Nest the standard GARCH model. If (cid:13) = 2, it is a GARCH; if (cid:13) = 1, it is an
(cid:15)
absolute value ARCH.
1.8 Quadratic ARCH (Sentana, 1991)
(cid:27)2 = (cid:11) +(cid:12)y2 +(cid:14)y +(cid:11)h
t 0 t 1 t 1 t 1
(cid:0) (cid:0) (cid:0)
(cid:14) < 0 means positive returns increase volatility less than negative returns.
(cid:15)
1.9 Switching ARCH (Cai, 1994; Hamilton and Susmel, 1994)
y = (cid:27) "
t t t
(cid:27)2 = (cid:13)(S )+ (cid:12) y2
8 t t i=1;q i t i ;
(cid:0)
(cid:13)(S ) = (cid:13) +(cid:13) S
> t 0 1 t
>< P(S = i S =Pj) = p
t t ij
j
>
>
There are two ARCH: models and the economy switches from one to another
(cid:15)
following a Markov chain.
9
1.10 ARCH in Mean (Engle, Lilien and Robin(1987))
Allow inter-temporal risk-return trade-o⁄s
y = (cid:14)(cid:27)2 +(cid:27) "
t t t t ;
(cid:27)2 = (cid:11)+(cid:12) y2 + +(cid:12) y2
(cid:26) t 1 t 1 (cid:1)(cid:1)(cid:1) q t q
(cid:0) (cid:0)
This model is the extension of the Capital Asset Pricing Model (CAPM).
(cid:15)
10