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EXTENSIONS OF GARCH PROCESSES
1. Integrated GARCH Process
Recall from Topic 4 (p. 8) that if tε follows a GARCH(1,1) process, then it can
be shown that 2tε has an ARMA(1,1) representation, namely,
2 20 1 1 1 1 1( )t t t tv vε α α β ε β− −= + + − + (1)
where 2 2t t tv ε σ= − is the difference between the squared innovation and the conditional
variance at time t. In many applications, we find that 1 1α β+ is approximately one.
When 1 1 1α β+ = , equation (1) becomes
2 20 1 1 1t t t tv vε α ε β− −= + − + (2)
so that there is a unit root in the squared residuals 2tε . Equation (2) can be written as:
2 2 2 20 1 1 1t t t t t tv v whereε α β ε ε ε− −∆ = − + ∆ = −
Because the there is a unit root in the squared residuals (they are stationary in first
differences), the model is called an Integrated GARCH(1,1), also known as the
IGARCH(1,1) model.
Recall from Topic 4 notes (p. 8), that the h-step ahead forecast of the conditional
variance from a GARCH(1,1) model is:
0 1 1 1 1 1 1 1 1 1( | ) [1 ( ) ( ) ( ) ] ( )
t h t tE σ α α β α β α β α β σ
+ +Ω = + + + + + + + + +
lim ( | ) var( )
When 1 1 1α β+ = ,
2 20 1( | ) ( 1)t h t tE hσ α σ+ +Ω = − + (4)
so that the forecast of the conditional variance becomes larger and larger as h increases.
In the limit, as h → ∞ , the forecast of the conditional variance becomes infinitely large,
meaning that the unconditional variance of the process is infinite (or undefined) as can be
seen from equation (3) upon substituting 1 1 1α β+ = .
2. Asymmetric GARCH Models
In the GARCH (or ARCH) models that we have discussed so far, a positive or
negative shock last period (that is, 1tε − ) will have the same impact on today’s volatility
because the squared of 1tε − enters the model only. However, negative shocks appear to
contribute more to stock market volatility than do positive shocks. This is called the
leverage effect. A negative shock to aggregate stock prices reduces the aggregate market
value of equity relative to the aggregate market value of corporate debt. Thus the
likelihood of corporate bankruptcy increases as firms are more highly leveraged. This
increases the risk of holding stocks.
The simplest GARCH model allowing for asymmetric response is the threshold
GARCH or the TGARCH model. In this model the GARCH(1,1) conditional variance
function is replaced with:
0 1 1 1 1 1 1
0, 0, 0, 0
σ α α ε γε β σ
The dummy variable tD keeps track of whether the lagged residual is positive or
negative. When 1 0tε − ≥ , the effect of the lagged squared residual on the current
conditional variance 2( )tσ is simply 1α . In contrast, when 1 0tε − < , 1D = so that the
effect of the lagged squared residual on the current conditional variance is 1α γ+ . If
0γ = , the response is symmetric and we have the standard GARCH(1,1) model. If
0γ ≠ , there is an asymmetric response of the conditional variance to “news”, the lagged
residual. If there are leverage effects, 0γ > so that negative shocks have a bigger impact
on the conditional variance than do positive shocks.
Asymmetric response may also be introduced by way of the exponential GARCH
or EGARCH model:
2 21 10 1 1 1
ln( ) ln( )t tt t
σ α α γ β σ
= + + + (6)
There are three important characteristics of the EGARCH model. First, the log of the
conditional variance is being modeled not the conditional variance itself. Regardless of
the magnitude of 2ln( )tσ , the implied value of
tσ can never be negative. Thus, it is
permissible for the coefficients (in equation (6)) to be negative. In other words, the log
specification ensures that the conditional variance is always positive because 2tσ is
obtained by exponentiating 2ln( )tσ . Second, instead of using the value of
1tε − , the
EGARCH model uses the absolute value of the standardized value of 1tε − (that is, 1tε −
divided by it standard error 1tσ − ) as the measure of the size of a shock. Note that the
standardized value of 1tε − is a unit free measure. Third, the EGARCH model allows for
asymmetric response of the log of the conditional variance to “news”. The sign of the
“news” is captured by the term 1 1/t tε σ− − . If 1 1/t tε σ− − is positive, the effect of the
standardized shock on the conditional variance is 1α γ+ . If 1 1/t tε σ− − is negative, the
effect of the standardized shock on the conditional variance is 1α γ− . If 0γ < , the effect
of a negative standardized shock is larger than that of a positive shock so that there is
evidence for a leverage effect.
3. Tests for Leverage Effects
First estimate the mean equation with, say, a GARCH(1,1) specification for the
variance equation, by maximum likelihood methods and form the standardized residuals
To test for leverage effects, one could estimate a regression of the form
η− − −= + + + + +
0 1 1 2 2t t t n t k ts a a s a s a s (7)
where tη is the regression disturbance. If there are no leverage effects, the squared
standardized residuals should be uncorrelated with the levels of the standardized
residuals. If the regression slope coefficients were negative and statistically significant,
that would indicate negative shocks are associated with large values of the conditional
variance and, thus, there are leverage effects.
Engle and Ng (1993) developed a second way to determine whether positive and
negative shocks have different effects on the conditional variance. Let
The Sign Bias test uses the regression equation of the form
2 0 1 1t t ts a a D η−= + + (8)
where tη is the regression disturbance. If a t-test indicates that 1a is statistically different
from zero, the sign of the current period shock is helpful in predicting volatility. In
particular, if 1a is positive and statistically different from zero, negative shocks tend to
increase the conditional variance. To generalize the test, one could estimate the
regression:
2 0 1 1 2 1 1 3 1 1(1 )t t t t t t ts a a D a D s a D s η− − − − −= + + + − +
Note that 1(1 )tD −− assigns a value of one to positive or zero shocks. The presence of
1 1t tD s− − and 1 1(1 )t tD s− −− is designed to determine whether the effects of positive and
negative shocks on the conditional variance depend on their size. Statistical significance
of 2a and 3a would suggest the presence of size bias, where not only the sign (indicated
by the statistical significance of 1a ) but also the magnitude or size of the shock is
important for predicting the conditional variance.
4. Leverage Effects in the Composite NYSE Index
Recall from Topic 4 notes that we estimated an MA(1)-GARCH(1,1) model for
the percentage daily logarithmic change in the NYSE index, denoted tsr , over the period
January 3, 1995 to August 30, 2002, a total of 1,931 observations. Having done this, we
now save the standardized residuals from this model (denoted ts ) and estimate the
regression given by equation (7) for three lags. The results are shown in Table 1.
Table 1: Estimation of Regression Equation for Leverage Effects
Dependent Variable: S2
Method: Least Squares
Sample (adjusted): 5 1931
Included observations: 1927 after adjustments
Variable Coefficient Std. Error t-Statistic Prob.
C 0.974715 0.046069 21.15789 0
S(-1) -0.15996 0.045941 -3.48179 0.0005
S(-2) -0.25772 0.045936 -5.6104 0
S(-3) -0.0882 0.045937 -1.92 0.055
R-squared 0.024002 Mean dependent var 1.000512
Adjusted R-squared 0.022479 S.D. dependent var 2.037487
S.E. of regression 2.014457 Akaike info criterion 4.24065
Sum squared resid 7803.602 Schwarz criterion 4.252199
Log likelihood -4081.87 F-statistic 15.76358
Durbin-Watson stat 2.075408 Prob(F-statistic) 0
The coefficients on 1ts − , 2ts − and 3ts − are negative and statistically significant. Thus,
negative shocks are associated with large values of the conditional variance, suggesting
the presence of leverage effects. Table 2 reports the results of the sign bias test given by
equation (8).
Table 2: Results of the Sign Bias Test
Dependent Variable: S2
Method: Least Squares
Sample (adjusted): 3 1931
Included observations: 1929 after adjustments
Variable Coefficient Std. Error t-Statistic Prob.
C 0.637753 0.126028 5.060401 0
D(-1) 0.418212 0.135489 3.086678 0.0021
R-squared 0.00492 Mean dependent var 0.999597
Adjusted R-squared 0.004404 S.D. dependent var 2.036632
S.E. of regression 2.032143 Akaike info criterion 4.257095
Sum squared resid 7957.747 Schwarz criterion 4.262864
Log likelihood -4103.97 F-statistic 9.527584
Durbin-Watson stat 1.967774 Prob(F-statistic) 0.002053
Since the coefficient on ( 1)D − is positive and significant, we again conclude that
negative shocks tend to increase the conditional variance of tsr .
In view of these findings, we estimated the MA(1)-TGARCH(1,1) model. The
results are reported in table 3. The coefficient on the asymmetric term is 0.1948. It is
positive and statistically significant. Thus, there is evidence for leverage effects in the
returns to the NYSE Composite Index.
It is interesting to compare the value of the likelihood function from the MA(1)-
TGARCH model, which is 2475.02− with that from the MA(1)-GARCH(1,1) model,
which is 2516.63− . It is valid to make such a comparison since the MA(1)-
TGARCH(1,1) model nests the MA(1)-GARCH(1,1). In other words, the MA(1)-
GARCH(1,1) model can be viewed as a restricted model with respect to the MA(1)-
TGARCH(1,1) model since it is obtained from the latter when the coefficient on the
asymmetric term is restricted to be zero. Clearly the maximized value of the likelihood
function from the MA(1)-TGARCH(1,1) model is larger than that from the MA(1)-
GARCH(1,1) model. We would expect this since the coefficient on the asymmetric term
in the TGARCH model is highly statistically significant. Nevertheless, we could perform
a likelihood ratio test of the restriction that the coefficient on the asymmetric term is zero
as follows:
2( 2516.63 ( 2475.02))
R ULR LL LL= − −
The LR statistic is distributed as a 2(1)χ since there is only one restriction here. Since
0.0583.2 (1) 3.841χ> = , we reject the null that there is no asymmetric response and
conclude that the MA(1)-TGARCH(1,1) model is better than the MA(1)-GARCH(1,1)
Table 3: Results of Estimation of MA(1)-TGARCH(1,1) Model
Dependent Variable: SR
Method: ML – ARCH (Marquardt) – Normal distribution
Sample (adjusted): 2 1931
Included observations: 1930 after adjustments
Convergence achieved after 19 iterations
MA backcast: 1, Variance backcast: ON
GARCH = C(3) + C(4)*RESID(-1)^2 + C(5)*RESID(-1)^2*(RESID(-1)<0)
+ C(6)*GARCH(-1)
Mean Equation
Coefficient Std. Error z-Statistic Prob.
C 0.041635 0.020227 2.05835 0.0396
MA(1) 0.114205 0.023915 4.775395 0
Variance Equation
C 0.018143 0.002799 6.482478 0
RESID(-1)^2 -0.00644 0.009726 -0.66206 0.5079
RESID(-1)^2*(RESID(-1)<0) 0.194754 0.015796 12.32918 0
GARCH(-1) 0.893656 0.009227 96.85522 0
R-squared 0.003704 Mean dependent var 0.0353
Adjusted R-squared 0.001115 S.D. dependent var 1.006207
S.E. of regression 1.005646 Akaike info criterion 2.571005
Sum squared resid 1945.788 Schwarz criterion 2.588306
Log likelihood -2475.02 F-statistic 1.430784
Durbin-Watson stat 2.081274 Prob(F-statistic) 0.209991
The results of estimating the MA(1)-EGARCH(1,1) model are shown in table 4.
The coefficient on the asymmetric term (shown in the table as C(5)) is 0.15524− . Since
this coefficient is negative and statistically significant, there is evidence for a leverage
effect, that is negative shocks have a bigger impact on the log of the conditional variance
than do positive shocks. We cannot compare the maximized log-likelihood value from
the MA(1)-EGARCH(1,1) model with that from the MA(1)-GARCH(1,1) model since
the models are not nested: the GARCH model cannot be viewed as a restricted EGARCH
model since in the EGARCH the log of the conditional variance is being modeled
whereas in the GARCH, the level of the conditional variance is being modeled.
Table 4: Results of Estimating MA(1)-EGARCH(1,1) Model
Dependent Variable: SR
Method: ML - ARCH (Marquardt) - Normal distribution
Sample (adjusted): 2 1931
Included observations: 1930 after adjustments
Convergence achieved after 16 iterations
MA backcast: 1, Variance backcast: ON
LOG(GARCH) = C(3) + +
+ C(6)*LOG(GARCH(-1))
Mean Equation
Coefficient Std. Error z-Statistic Prob.
C 0.027938 0.019166 1.457706 0.1449
MA(1) 0.11858 0.023734 4.996317 0
Variance Equation
C(3) -0.10572 0.013741 -7.69377 0
C(4) 0.126307 0.016369 7.716313 0
C(5) -0.15524 0.011328 -13.7042 0
C(6) 0.96458 0.004179 230.8339 0
R-squared 0.003355 Mean dependent var 0.0353
Adjusted R-squared 0.000765 S.D. dependent var 1.006207
S.E. of regression 1.005822 Akaike info criterion 2.555701
Sum squared resid 1946.47 Schwarz criterion 2.573003
Log likelihood -2460.25 F-statistic 1.295357
Durbin-Watson stat 2.089712 Prob(F-statistic) 0.263051
5. Exogenous Variables in the GARCH Specification
Sometimes it is useful to include an exogenous variable in the variance equation. For
example, financial market volume often helps to explain financial market volatility. In
this case, the standard GARCH(1,1) model would be augmented in the following way
2 2 20 1 1 1 1t t t txσ α α ε β σ γ− −= + + +
where γ is a parameter and tx is a positive exogenous variable, for example, the volume
of trades on the NYSE today.
6. GARCH-in-Mean Models
The GARCH(1,1)-in-Mean model (which is written in abbreviated form as
GARCH(1,1)-M) is:
σ α α ε β σ
Let ty be the return on a financial asset or portfolio. Then 1 0 1( | )t t tE y a a σ−Ω = + . Thus,
the conditional mean return depends on the conditional standard deviation. Since the
conditional standard deviation can be viewed as a measure of the risk associated with the
asset or portfolio, the specification for the mean equation captures the notion in finance
of a trade-off between mean return and risk. The mean return is time-varying since tσ is
time-varying. Only in the case of where 1 0a = is the mean return constant, although
there is time-varying volatility in the model given by the GARCH(1,1) specification.
Note that in some empirical applications the conditional variance rather than the
conditional standard deviation appears in the mean equation.
As a practical matter, if there appears to be a shift in the conditional mean of ty in
response to changing volatility, then that is indicative of a GARCH-M process.
Table 5 presents a GARCH-in-Mean model for the term premium between the
three and six month U.S. zero coupon bonds.
Table 5: GARCH(1,1)-in-Mean Model for the Term Premium
Dependent Variable: TERM
Method: ML - ARCH (Marquardt) - Normal distribution
Sample (adjusted): 1947M01 1987M02
Included observations: 482 after adjustments
Convergence achieved after 32 iterations
Variance backcast: OFF
GARCH = C(4) + C(5)*RESID(-1)^2 + C(6)*GARCH(-1)
Mean Equation
Coefficient Std. Error z-Statistic Prob.
STDDEV 0.380607 0.076123 4.999884 0
C 0.003369 0.001901 1.772456 0.0763
TERM(-1) 0.712542 0.039053 18.24564 0
Variance Equation
C 2.75E-06 6.21E-06 0.443379 0.6575
RESID(-1)^2 0.385388 0.038328 10.05507 0
GARCH(-1) 0.756158 0.015696 48.17537 0
R-squared 0.484561 Mean dependent var 0.223071
Adjusted R-squared 0.479147 S.D. dependent var 0.220262
S.E. of regression 0.158963 Akaike info criterion -1.72331
Sum squared resid 12.0282 Schwarz criterion -1.6713
Log likelihood 421.3169 F-statistic 89.49705
Durbin-Watson stat 2.176771 Prob(F-statistic) 0
The term premium is defined as the yield to maturity on six month bills less the
yield to maturity on three month bills. The data cover the period December 1946 to
February 1987. The coefficient on the conditional standard deviation is positive and
statistically significant as expected since the higher the risk, the higher the term premium
required on the long bond relative to the short bond. Also included in the mean equation
is the lagged term premium to account for serial correlation. It is apparent that the term
premium is quite persistent. Finally, 1 1 1.15α β+ = , which is quite a bit larger than one,
violating the sign restrictions on the model.
7. Maximum Likelihood Estimation of the ARMA-GARCH Models
Consider the ARMA(1,1)-GARCH(1,1) model:
The only observed series we have is { }
y . Thus we will have to reconstruct { }
σ from observed { }
y . We do so iteratively and need to assume values for t=0:
.σ Usually, we set
0ε = and 2 2
σ σ= , where 2σ is the (unconditional) sample
variance. Then, for given values of parameters , , , , ,cγ φ θ α β and given
0ε = , 2 2
y we can compute
All subsequent values of { }
ε and 2{ }
σ are reconstructed in a similar way:
Next step is to specify the likelihood, that is the joint probability to observe specific
values of { }
y for given , , , , ,cγ φ ϕ α β and
0ε = , 2 2
σ σ= and maximize it for given
0ε = , 2 2
σ σ= with respect to the parameters , , , , ,cγ φ ϕ α β .
In order to specify the likelihood we need to know the joint (unconditional) distribution
y . However what we are given, instead, is the conditional distributions of
Ω . Moreover { }
y are not independent.
There are two way around this problem both of which lead to the same solution.
One way it to consider maximizing the joint likelihood of the standardized innovations
= . By assumption { }
ξ are iid standard normal random
variables and their joint pdf is
( , , , | , , , , , , , ) ( | , )
f c fξ ξ ξ γ φ θ α β ε σ ξ ε σ
σ are computed iteratively as in Eq. (9).
The other (I would say more proper) way is to use the following decomposition (applying
the Bayes formula for conditional probability):
1 1 0 1 1 0 1 1 0
1 1 0 1 2 1 0 2 1 0
1 1 2 1 0 0
( , ,..., , ) ( | ,,..., , ) ( ,..., , )
( | ,,..., , ) ( | ,..., , ) ( ,..., , )
( | ) ( | ) ( | ) ( )
f y y y y f y y y y f y y y
f y y y y f y y y y f y y y
f y f y f y f
= Ω Ω Ω Ω
and specify pdf conditional on parameters as
1 1 0 0 0 1 0
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