Case GARCH: Modeling Volatility Dynamics
A. Conditional Heteroscedasticity
This lesson introduces you to a recent development in forecasting asset returns.
ARMA forecasting model, which includes a random process as a special case, tracks the
level, defined by the mean at each time period t, of the variable given the information
available at time t. However, asset returns are also subject to variability around the mean,
which is measured by the variance of the variable for t. ARMA models assume
constancy of the variance, i.e., homoscedasticity. However, high frequency asset returns
are not homoscedastic.
Figure 1A below plots 1268 daily returns of CISCO stocks from June 25, 1995 to July
5, 2001. There is no obvious upward or downward trend, i.e., the mean at t appears
constant. However, the variability is not uniform, and shows chunks mixed with
occasional spikes. Let et (residual at time t) = return at time t – sample mean. Then, the
squared residual et2 estimates the variance of the return for t. The plot of the squared
residuals does not appear to follow a random process. Also, similar values of squared
residuals come in chunks. See Figure 1B.
Figure 1A:
The daily return, 6/25/95 to 7/5/01
- CSCO
Figure 1B
The squared daily return, 6/25/95 to 7/5/01
- CSCO
0.06
0.3
0.05
0.2
0.04
0.1
0.03
0.0
0.02
-0.1
-0.2
6/25/96
0.01
5/26/98
0.00
6/25/96
4/25/00
5/26/98
CSCO_R_R
CSCO_R
1
4/25/00
The chunkiness of squared residuals is the result of dependence of the variance of the
return at time t on variances at preceding periods. We can confirm this by computing the
correlogram of squared residuals. See Figure 1C. The standard error of a sample
autocorrelation for 1268 observations from a random process is approximately:
1
 0.028
1268
Comparing ACF values in Figure 1C with the standard error, we conclude that ACF are
highly significant at all lags. Q-stats are also highly significant. This dependence of the
variance at time t on variances of preceding periods is called the conditional
heteroscedasticity. Conditional heteroscedasticity is quite common for high frequency
asset return data.
Figure 1C:
Correlogram of the Squared Daily Returns, 6/25/95 to 7/5/01
- CSCO
Sample: 6/25/1996 7/05/2001
Included observations: 1268
Autocorrelation
Partial Correlation
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1
2
3
4
5
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9
10
AC
PAC
Q-Stat
Prob
0.180
0.175
0.105
0.083
0.057
0.129
0.114
0.185
0.079
0.191
0.180
0.147
0.054
0.036
0.017
0.101
0.069
0.131
-0.003
0.131
41.268
80.204
94.164
103.04
107.20
128.28
144.96
188.82
196.87
243.77
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
We should also note that the spikes in Figure 1A or 1B most likely do not reflect
conditional heteroscedasticity. They could be caused by sudden high variances or shifts
of the mean for the periods.
2
Conditional heteroscedasticity may be present for daily returns, but it may not be so
for the monthly returns. As an example, Figures 2A through 2C show the results for the
monthly return of CISCO stock for the corresponding time range. The correlogram of
squared residuals does not indicate serial correlation.
Figure 2A
The monthly return, 96:7 to 00:6
- CSCO
Figure 2B
The squared monthly returns, 96:7 to 00:6
- CSCO
0.08
0.3
0.2
0.06
0.1
0.04
0.0
-0.1
0.02
-0.2
0.00
96:07 97:01 97:07 98:01 98:07 99:01 99:07 00:01
-0.3
96:07 97:01 97:07 98:01 98:07 99:01 99:07 00:01
CSCO_MR_SQ
CSCO_MR
Figure 2C:
Correlogram of the squared monthly return - CSCO
Sample: 1996:07 2000:06
Included observations: 48
Autocorrelation
Partial Correlation
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2
3
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10
3
AC
PAC
Q-Stat
Prob
-0.030
-0.118
-0.058
-0.104
-0.159
0.118
-0.009
0.103
0.026
-0.142
-0.030
-0.119
-0.066
-0.125
-0.191
0.068
-0.066
0.093
0.002
-0.137
0.0467
0.7718
0.9498
1.5373
2.9436
3.7386
3.7435
4.3818
4.4238
5.6970
0.829
0.680
0.813
0.820
0.709
0.712
0.809
0.821
0.881
0.840
B. Modeling Conditional Heteroscedasticity
Forecasting the conditional standard deviation is important as the volatility affects the
asset price. In 1982, an Economist Engle, developed a statistical model for conditional
heteroscedasticity. In below we introduce the model in the context of forecasting stock
returns. Consider the following formulation for the return of a stock.
rt     t
rt is the return at time, say day, t and t is an independent observation from N (0,  t2 ).
Under a random walk hypothesis:
 t2   2
for all t, i.e., the variance of  t is constant for all t. As noted earlier, we say that the return
rt is homoscedastic. Figure 1A through 1C, the variance of the daily return of CISCO
stocks do not follow homoscedasticity, but show a dependence on the variances of
preceding periods. One way to model this serial correlation of the variance is:
 t2     t21
(1)
This formulation is called an Autoregressive Conditional Heteroscedastic (ARCH)
model. The reason why it is called so is that  t2 is an unbiased estimate of  t2 , i.e.,
E   t2    t2 , so that defining the estimation error  t   t2   t2 , we can re-write (1) for
 t2 as follows:
 t2   t2   t     t21   t
(2)
(2) is an AR(1) specification for the squared residual. Unlike the AR(1) model for the
mean, it is the error term  t is not constant. We could thus say that (2) is a
heteroscedastic AR(1).
One can generalize (2) to include squared residuals with lags 2, 3 and so on as
explanatory variables. Following the standard notation for AR models, it is convenient to
denote (2) as ARCH(1). Let (L) be a q-th order polynomial in lag operator L, i.e.,
  L   1L   p Lp
4
Then, ARCH (p) is:
 t2      L  t2
As we have learned in ARMA modeling for the mean, it is not wise to keep
increasing lagged squared residual terms for explaining the serial dependence of the
variance. Instead, a useful parsimonious generalization of (1) is to include  t21 term as an
explanatory variable, i.e:
 t2   0    t21    t21
(3)
The generalization is proposed by Bollerslev in 1986 and is called GARCH (Generalized
ARCH). It can be shown that (3) is an heteroscedastic ARMA (1,1) specification for the
squared residual. See below. From (3),
 t2   t2   t     t21   t21  
    t21     t21   (t 1)   
t
t
    t21   t21  1 (t 1)  t
        t21  t   (t 1)
It is heteroscedastic, because  t is not constant. Following ARCH(1), (3) is denoted as
GARCH(1,1).
Let   L  be a q-th order polynomial in L, i.e.,
  L   1L 
q Lq
Then GARCH(p, q) is:
 t2      L  t2    L  t2
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C. Model Fitting Using Eviews
EViews offer routines that are especially suited for testing for conditional
heteroscedasticity in the residual and proceeding to fit ARCH or a GARCH models. We
will illustrate these features in below. We use the daily return of CSCO stocks as our
data. We begin with fitting a homoscedastic model:
csco_r     t
Select Quick/Estimate Equation and open the panel as shown below. Specify the
model for the mean in the upper space. Note that the Method window indicates LS –
Least Squares, which is the best estimation method for homoscedastic data. Click OK
and the output is generated.
Select View/Residual Tests and open the menu. See the panel below. A variety of tests
for goodness of random process appears. A useful test for the presence of conditional
heteroscedasticity in the residual is Correlogram – Squared Residuals which we have
seen before in Figure 1C. Another useful test is ARCH – LM test. LM stands for
Lagrange multiplier.
6
Select the test, the following window opens:
The LM test uses autoregression of the squared residuals. If ARCH(1) is suspected, we
enter 1 in the window. Here is the result of the test.
7
ARCH Test:
F-statistic
Obs*R-squared
45.18800
43.69846
Probability
Probability
0.000000
0.000000
Test Equation:
Dependent Variable: RESID^2
Method: Least Squares
Sample(adjusted): 6/27/1996 5/04/2001
Included observations: 1267 after adjusting endpoints
Variable
Coefficient
Std. Error
t-Statistic
Prob.
C
RESID^2(-1)
0.001039
0.186028
8.70E-05
0.027674
11.95071
6.722202
0.0000
0.0000
R-squared
Adjusted R-squared
S.E. of regression
Sum squared resid
Log likelihood
Durbin-Watson stat
0.034490
0.033726
0.002831
0.010138
5636.875
2.053261
Mean dependent var
S.D. dependent var
Akaike info criterion
Schwarz criterion
F-statistic
Prob(F-statistic)
0.001275
0.002880
-8.894830
-8.886710
45.18800
0.000000
The test statistic is the number of observations, T times R2 of the autoregression, i.e.,
T  R 2 . In the absence of ARCH(1), the coefficient of et21 is zero, the null hypothesis,
and T  R 2 should follow a  2 distribution with the number of degrees of freedom 1,
which predicts that the T  R 2 exceeds 3.84 (from a table of  2 distribution) only 5% of
the time. The value of the test statistic for this example is so high that we will not
hesitate rejecting the null hypothesis.
We now proceed to fit an ARCH(1) model for the data. Going back to the beginning
of this section, this time, after selecting Quick/Estimate Equation, open the pull down
menu for Method and select ARCH – Autoregressive Conditional Heteroscedasticity.
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The panel as shown below opens and indicates that it is ready to fit a GARCH(1, 1).
For fitting ARCH(1), replace 1 in GARCH window by 0 and click OK. Here are the
results for ARCH(1).
Table 1: Fitting ARCH(1) for daily returns of CSCO stocks
Dependent Variable: CSCO_R
Method: ML - ARCH
Sample(adjusted): 6/26/1996 5/04/2001
Included observations: 1268 after adjusting endpoints
Convergence achieved after 13 iterations
C
Coefficient
Std. Error
z-Statistic
Prob.
0.001883
0.000879
2.141806
0.0322
22.65828
7.835731
0.0000
0.0000
Variance Equation
C
ARCH(1)
R-squared
Adjusted R-squared
S.E. of regression
Sum squared resid
Log likelihood
0.000884
0.331873
3.90E-05
0.042354
-0.000129
-0.001710
0.035747
1.616493
2475.264
Mean dependent var
S.D. dependent var
Akaike info criterion
Schwarz criterion
Durbin-Watson stat
9
0.001478
0.035717
-3.899471
-3.887298
2.104299
The following extracts the main results for the variable.
csco _ rt  0.003038   t
t
N  0,  t2 
 t2  0.000884  0.33187 t21
(4)
This time, we fit GARCH(1,1) for the same data.
Sample(adjusted): 6/26/1996 5/04/2001
Included observations: 1268 after adjusting endpoints
Convergence achieved after 28 iterations
C
Coefficient
Std. Error
z-Statistic
Prob.
0.003038
0.000821
3.700891
0.0002
2.972380
6.651751
63.36457
0.0030
0.0000
0.0000
Variance Equation
C
ARCH(1)
GARCH(1)
R-squared
Adjusted R-squared
S.E. of regression
Sum squared resid
Log likelihood
1.69E-05
0.083822
0.904858
-0.001909
-0.004287
0.035793
1.619371
2565.158
5.69E-06
0.012601
0.014280
Mean dependent var
S.D. dependent var
Akaike info criterion
Schwarz criterion
Durbin-Watson stat
0.001478
0.035717
-4.039682
-4.023451
2.100559
The estimated model for the daily return of CSCO stocks is:
r  0.003038   t with  t
N  0,  t2 
(5)
 t2  0.0000169  0.083822  t21  0.904858 t21
Which model – ARCH(1) or GARCH(1,1) fitts the data best? First of all, the
coefficient of GARCH(1) term, i.e.,  t21 is highly significant. Second, the estimation
method uses the method of maximum likelihood. Comparing the value of Log
likelihood, GARCH has a higher value, which means that the data are more likely from
from GARCH process than ARCH. Third, GARCH uses one more parameter than
ARCH. Both Akaike information criterion and Schwarz criterion adjust the
likelihood for the number of parameters. The lower the values of these criteria, the better
the fit. GARCH has lower values than ARCH for both criteria. From these tests, we
would opt for GARCH over ARCH.
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Is there rooms for improvement? To answer the question, we select View/Rresidual
Tests/Correlogram – Squared Residuals in the GARCH output. Eviews computes the
correlogram for the standardized residual:
zt 
et
t
Here is the result:
We can also select View/Rresidual Tests/ARCH LM test and get the following result
(only a part of the output is shown).
ARCH Test:
F-statistic
Obs*R-squared
4.324190
4.316272
Probability
Probability
0.037775
0.037750
Test Equation:
Dependent Variable: STD_RESID^2
Method: Least Squares
Sample(adjusted): 6/27/1996 5/04/2001
Included observations: 1267 after adjusting endpoints
Variable
Coefficient
Std. Error
t-Statistic
Prob.
C
STD_RESID^2(-1)
0.942968
0.058457
0.055291
0.028111
17.05459
2.079469
0.0000
0.0378
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These tests indicate that the conditional heteroscedasticity is mostly eliminated for the
standardized residual. (AC are still significant but their values are small.) Therefore, we
conclude that the daily stock return of CISCO stocks follow GARCH(1,1) as summarized
in (4). It is instructive to compute the correlogram for the standardized squared residual.
The autocorrelations are highly significant after lag 1, which indicates there is still
significant conditional heteroscedasticity left in the standardized residual.
How does the estimated model track the heteroscedasticity in the data? In the
GARCH(1,1) estimation output, open View/Conditional SD menu:
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Then, the graph of the standard deviation as computed by equation (4) appears. See
below.
0.10
0.08
0.06
0.04
0.02
0.00
6/26/96
5/27/98
4/26/00
D. Forecasting Using GARCH
Here is the GARCH(1,1) estimated for the daily return of CSCO stocks shown earlier.
r  0.003038   t with  t
N  0,  t2 
(5)
 t2  0.0000169  0.083822  t21  0.904858 t21
How can we use this model for forecasting purposes? First, for constructing an interval
forecast of the return, we can account for changing standard deviation by applying the
second equation in (5) for  t2 . Second, forecasting the future value of the variance itself
is important for assessing the risk of investing in CSCO stocks. These forecast
advantages of the model are for short run. For a long run forecast of the variance, the
MA effect will quickly disappear and the AR effect converges to the following value:

0.0000169

 0.001493
1      1   0.083822  0.904858 
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which is the sample variance of the data. That is, for a long run forecast, the sample
mean and the sample variance are the best forecast of the expected return and the risk.
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