ARCH and GARCH MODELS
David Leblang
University of Colorado
Leblang
ARCH
Page 1
I.
Motivation: Why ARCH/GARCH Models?
A.
What is ARCH/GARCH?
1)
2)
3)
4)
B.
Generalized—more general than ARCH
Autoregressive—depends on its past
Conditional—variance depends on past info
Heteroscedasticity—non-constant variance.
Econometric—OLS assumes:
1) No Serial Correlation: cov( t ,  t 1 )  0 -- tests
and corrections are standard in the literature.
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ARCH
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2) Homoscedastic Errors:  t ~ NID(  ,  ) --errors
are normally and independently distributed.
Usual for papers to test for heteroscedasticity (i)
in the cross-sectional context but unusual in the
time-series context (t)
3) Consequences: OLS is BLUE and consistent.
HOWEVER, OLS is not efficient (minimum
variance) if we relax the class of estimators to
include nonlinear estimators.
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ARCH
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C.
Empirical Regularities (S&P returns).
1) Volatility Clustering
100*[log(sp(t))-(log(sp(t-1)))]
4.65458
-6.00451
22dec1999
09jul2000
31mar2000
17oct2000
date
Volatility Clustering
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ARCH
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2)
Fat/Heavy Tails (Kurtosis) [k=3]
Fraction
.113636
0
-6.00451
4.65458
100*[log(sp(t))-(log(sp(t-1)))]
Kurtosis
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ARCH
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D.
Theoretical—Variance is of Interest
1) What causes volatility/variance of a series?
Finance literature (risk premium); economics
literature (target zones).
Political science—
political events/information influence variability
of asset prices (e.g., Leblang and Bernhard;
Freeman, Hays and Stix)
2) Are some events/periods/systems conducive
to more/less volatility than others?
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ARCH
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E.
Textbook References
1) Enders, Applied Econometric
Time Series
2) Patterson, An Introduction to
Applied Time Series
3) Franses and van Dijk, NonLinear Time Series Models in
Empirical Finance
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ARCH
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F.
Software (others=PC-GIVE, RATS, TSP)
Software
STATA
www.stata.com
EVIEWS
www.eviews.com
S+ GARCH
www.insightful.com
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ARCH
Advantages
Disadvantages
My favorite in general
Lots of built in models
Choice of algorithm
Easy to program
Only normal dist.
Few built-in
diagnostics
Lots of built in models
Choice of algorithm
Lots of built in diag.
FAST!
Lots of built in models
FIGARCH
MGARCH
t, ged, double exp dist.
Terrific Graphics
Only normal dist.
Difficult to program
Difficult to program
No choice of algorithm
A bit “clunky”
Page 8
II.
Preliminaries: Linear Time Series
A. Variable yt is observed for t=1,2,..,n
B.
The error (t) is a white noise series if
1) E[ t ]  0
2
2
2
2) E[ t ]  E[ t | t 1 ]   t .
The error is
unconditionally and conditionally homoscedastic.
3) E[ t  s ]  0; s  t . Note: this says that the
information set  t 1 does not contain information
to forecast  t .
C. A time series for yt can be thought of as the sum of
a predictable and an unpredictable component:
yt  E[ yt |  t 1 ]   t .
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ARCH
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III. Relax assumption of homoscedasticity
Allow conditional variance of  t to vary over time:
E[ 2t | t 1 ]  ht for some nonnegative function.
A.
B. In general, this is expressed as:  t  zt ht , where zt
is independently and identically distributed normally
with mean zero and unit variance (this can be
relaxed—use student t and ged distributions to allow
for fatter tails).
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ARCH
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C. This means that the distribution of  t conditional
upon the history  t 1 is normal with mean zero and
variance ht. It also means that the unconditional
variance of  t is constant. Using the law of iterated
2
2
2


E
[

]

E
[
E
[

expectations:
t
t | t 1 ]]  E[ht ] .
D. We now need a model to specify how the
conditional variance of  t evolves over time.
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ARCH
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IV. Autoregressive Conditional Heteroscedasticity
A. Invented by Engle (1982) to explain the volatility
of inflation rates.
B. Basic ARCH (1) model: conditional variance of a
shock at time t is a function of the squares of past
2
h





shocks: t
1 t 1 . (Recall, h is the variance and  is
a “shock,” “news,” or “error”).
C. Since the conditional variance needs to be
nonnegative, the conditions   0;  1  0 have to be
met. If 1 = 0, then the conditional variance is
constant and  t is conditionally homoscedastic.
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ARCH
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V.
Generalized ARCH (GARCH)
A. Because ARCH(p) models are difficult to estimate,
2
2
2
2
and because ( t 2 ,  t 3 ), ( t 3 ,  t 4 ), etc. decay very
slowly, Bollerslev (1986) developed the GARCH
model.
B.
2
h





GARCH (1,1): t
1 t 1   1ht 1 .
C. The variance (ht) is a function of an intercept (), a
shock from the prior period () and the variance from
last period ().
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ARCH
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D.
Higher
order
p
q
j 1
k 1
ht      j  2t  j   k ht2 k
Leblang
ARCH
GARCH
models:
.
Page 14
VI. Linear GARCH Variations.
A.
Integrated GARCH (Engle and Bollerslev 1986).
1) Phenomena is similar to integrated series in
regular (ARMA-type) time-series.
2) Occurs when +=1. When this is the case it
means that there is a unit root in the conditional
variance; past shocks do not dissipate but persist
for very long periods of time.
B. Fractionally Integrated GARCH (Baillie, Bollerslev
and Mikkelsen (1996)).
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ARCH
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C. GARCH in Mean (Engle, Lilien and Robbins
(1987).
1) Idea is that there is a direct relationship
between risk and return of an asset.
2) In the mean equation, include some function of
the conditional variance—usually the standard
deviation.
3) This allows the mean of a series to depend, at
least in part, on the conditional variance of the
series (more later)..
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ARCH
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VII. Non-Linear GARCH Variations (dozens in last 20
years). Linear GARCH models all allow prior shocks
to have a symmetric affect on ht. Non-linear models
allow for asymmetric shocks to volatility. I will focus
on the most common: the Exponentional GARCH
(1,1) (EGARCH) model developed by Nelson (1991).
A.
Conditional variance:
log(ht )     1zt 1   1 (| zt 1 | E[| zt 1 |])   1 log(ht 1 ) , where
zt   t / ht and is the standardized residual.  is the
asymmetric component.
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ARCH
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B. News Impact Curve—differential
positive and negative shocks.
Conditional Variance: GARCH
impact
Conditional Variance: EGARCH
83.8448
.587194
.404425
Conditional Variance: GARCH
Conditional Variance: EGARCH
40.1751
-9.8
9.8
error (t-1)
News Impact Curve: dCPI w/ ARMA(1,1)
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ARCH
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of
VIII. Testing for ARCH
A.
ARCH Tests (Engle 1982).
1) Regress Y on X and obtain some residuals ( t )
2)
2
Regress  t on p lags of
 2t   0   1 2t 1   2 2t 2 .. p 2t  p
 2t ;
that
is,
Assess joint significance of  1   p . If the
coefficients are different from zero then the
null of conditional homoscedasticity can be
rejected.
a.
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ARCH
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b. T*R2 is Engle’s LM test statistic. Under the
null of homoscedasticity it is asymptotically
2

distributed (q )
B.
Graphical Test—Ljung-Box Q Statistic
1) LB (Q) used to diagnose serial correlation in
the residuals
2) LB(Q2) used to diagnose serial correlation in
the squared residuals—heteroscedasticity
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ARCH
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IX. Example—Returns on the S&P 500
. regress dlsp
/returns on the S & P 500 Index
Source |
SS
df
MS
-------------+-----------------------------Model |
0.00
0
.
Residual | 391.285893
219 1.78669358
-------------+-----------------------------Total | 391.285893
219 1.78669358
Number of obs
F( 0,
219)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
220
0.00
.
0.0000
0.0000
1.3367
-----------------------------------------------------------------------------dlsp |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------_cons |
.0096484
.0901184
0.11
0.915
-.167962
.1872588
-----------------------------------------------------------------------------. predict e if e(sample), resid / obtain residuals
. gen e2=e^2
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ARCH
/generate squared residuals
Page 21
. reg e2 l.e2 /regress squared residuals on a lag
Source |
SS
df
MS
-------------+-----------------------------Model |
72.087039
1
72.087039
Residual | 2684.92529
217 12.3729276
-------------+-----------------------------Total | 2757.01233
218 12.6468456
Number of obs
F( 1,
217)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
219
5.83
0.0166
0.0261
0.0217
3.5175
-----------------------------------------------------------------------------e2
|
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------e2
|
L1 |
.1616863
.0669855
2.41
0.017
.0296608
.2937118
_cons
|
1.49788
.2661034
5.63
0.000
.9734018
2.022358
-----------------------------------------------------------------------------. test l1.e2
( 1)
/test H0: homoscedastic residuals
L.e2 = 0.0
F(
1,
217) =
Prob > F =
5.83
0.0166
. display 219*.0261
5.7159
. display chiprob(1, 5.7159) /the value is the p-value to reject H0 of Homoscedasticity
.01681195
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ARCH
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Autocorrelation Function
. ac e2
/autocorrelation function of the squared residuals
Autocorrelations of e2
Bartlett's formula for MA(q) 95% confidence bands
1.00
1.00
0.75
0.75
0.50
0.50
0.25
0.25
0.00
0.00
-0.25
-0.25
-0.50
-0.50
-0.75
-0.75
-1.00
-1.00
0
10
20
Lag
30
40
Correlogram
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ARCH
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. corrgram e2 /correlegram gives the ac and pacs
-1
0
1 -1
0
1
LAG
AC
PAC
Q
Prob>Q [Autocorrelation] [Partial Autocor]
------------------------------------------------------------------------------1
0.1615
0.1617
5.8191 0.0159
||2
0.1511
0.1282
10.937 0.0042
||3
-0.0107 -0.0555
10.963 0.0119
|
|
4
0.0577
0.0505
11.715 0.0196
|
|
5
0.0724
0.0695
12.906 0.0243
|
|
6
0.1087
0.0765
15.603 0.0161
|
|
7
-0.0132 -0.0594
15.643 0.0286
|
|
8
0.0007 -0.0123
15.643 0.0478
|
|
9
-0.0317 -0.0189
15.876 0.0695
|
|
10
0.0070
0.0027
15.887 0.1029
|
|
. wntestq e2, lags(1)
Portmanteau test for white noise
--------------------------------------Portmanteau (Q) statistic =
5.8191
Prob > chi2(1)
=
0.0159
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ARCH
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Remedy: GARCH (1,1) Model
. arch dlsp, arch(1) garch(1) nolog
ARCH family regression
Sample:
4 to 223
Log likelihood = -366.1473
Number of obs
Wald chi2(.)
Prob > chi2
=
=
=
220
.
.
-----------------------------------------------------------------------------|
OPG
dlsp
|
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------dlsp
|
_cons
|
.0232815
.0826522
0.28
0.778
-.1387138
.1852768
-------------+---------------------------------------------------------------ARCH
|
arch
|
L1 |
.1652834
.045527
3.63
0.000
.0760521
.2545146
garch
|
L1 |
.7815966
.0783583
9.97
0.000
.6280172
.935176
_cons
|
.1121176
.0913255
1.23
0.220
-.066877
.2911122
------------------------------------------------------------------------------
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ARCH
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RESIDUAL TESTS
. predict e, resid
. predict v, variance
. gen s=sqrt(v)
. gen se=e/s
. gen se2=se^2
. wntestq se2
Portmanteau test for white noise
--------------------------------------Portmanteau (Q) statistic =
30.0623
Prob > chi2(40)
=
0.8735
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ARCH
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. corrgram se2
-1
0
1 -1
0
1
LAG
AC
PAC
Q
Prob>Q [Autocorrelation] [Partial Autocor]
------------------------------------------------------------------------------1
-0.0100 -0.0100
.02243 0.8810
|
|
2
0.0873
0.0875
1.7295 0.4212
|
|
3
-0.0914 -0.0911
3.6084 0.3070
|
|
4
0.0091
0.0009
3.6269 0.4588
|
|
5
-0.0114
0.0044
3.6562 0.5999
|
|
6
0.0214
0.0127
3.7612 0.7090
|
|
7
-0.0549 -0.0547
4.4529 0.7264
|
|
8
-0.0243 -0.0290
4.5894 0.8004
|
|
9
-0.0238 -0.0117
4.7205 0.8580
|
|
10
0.0067
0.0017
4.7311 0.9084
|
|
No Remaining ARCH…BUT, what about normality??
Recall: Normal distribution has skewness of 0 and
kurtosis of 3 and we know that financial series tend
to be fat tailed.
Leblang
ARCH
Page 27
.graph se, norm bin(50)
Fraction
.1
0
-4.20146
2.73526
se
. sktest se
Skewness/Kurtosis tests for Normality
------- joint -----Variable | Pr(Skewness)
Pr(Kurtosis) adj chi2(2)
Prob>chi2
-------------+------------------------------------------------------se |
0.067
0.012
8.77
0.0125
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ARCH
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Solution: Use Robust Standard Errors—robust to
departures from normality (Bollerslev & Wooldridge
1982)
. arch dlsp, arch(1) garch(1) nolog robust
ARCH family regression
Sample:
4 to 223
Log likelihood = -366.1473
Number of obs
Wald chi2(.)
Prob > chi2
=
=
=
220
.
.
-----------------------------------------------------------------------------|
Semi-robust
dlsp
|
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------dlsp
|
_cons
|
.0232815
.0786518
0.30
0.767
-.1308732
.1774362
-------------+---------------------------------------------------------------ARCH
|
arch
|
L1 |
.1652834
.2083251
0.79
0.428
-.2430264
.5735931
garch
|
L1 |
.7815966
.3140995
2.49
0.013
.165973
1.39722
_cons
|
.1121176
.2578869
0.43
0.664
-.3933314
.6175666
------------------------------------------------------------------------------
Leblang
ARCH
Page 29
Inclusion of Exogenous Variables
. arch dlsp, arch(1) garch(1) nolog robust het(gore) bhhh
ARCH family regression -- multiplicative heteroskedasticity
Sample:
4 to 223
Log likelihood = -365.4092
Number of obs
Wald chi2(.)
Prob > chi2
=
=
=
220
.
.
-----------------------------------------------------------------------------|
Semi-robust
dlsp
|
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------dlsp
|
_cons
|
.0135455
.080307
0.17
0.866
-.1438533
.1709443
-------------+---------------------------------------------------------------HET
|
gore
| -.1355259
.0615727
-2.20
0.028
-.2562061
-.0148457
_cons
|
5.006925
3.286556
1.52
0.128
-1.434607
11.44846
-------------+---------------------------------------------------------------ARCH
|
arch
|
L1 |
.1945511
.0973455
2.00
0.046
.0037575
.3853447
garch
|
L1 |
.6837859
.1219819
5.61
0.000
.4447057
.922866
------------------------------------------------------------------------------
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ARCH
Page 30
. wntestq se2
Portmanteau test for white noise
--------------------------------------Portmanteau (Q) statistic =
29.1930
Prob > chi2(40)
=
0.8965
. sktest se2
Skewness/Kurtosis tests for Normality
------- joint -----Variable | Pr(Skewness)
Pr(Kurtosis) adj chi2(2)
Prob>chi2
-------------+------------------------------------------------------se2 |
0.000
0.000
.
0.0000
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ARCH
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ARCH IN MEAN
. arch dlsp, arch(1) garch(1) nolog robust het(gore) archm archmexp(sqrt(X))
ARCH family regression -- multiplicative heteroskedasticity
Sample:
4 to 223
Log likelihood = -362.6718
Number of obs
Wald chi2(1)
Prob > chi2
=
=
=
220
4.84
0.0278
-----------------------------------------------------------------------------|
Semi-robust
dlsp
|
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------dlsp
|
_cons
|
-.80408
.3986817
-2.02
0.044
-1.585482
-.0226782
-------------+---------------------------------------------------------------ARCHM
|
sigma2ex
|
.7068768
.3213948
2.20
0.028
.0769545
1.336799
-------------+---------------------------------------------------------------HET
|
gore
| -.1067959
.016462
-6.49
0.000
-.1390609
-.0745308
_cons
|
3.790934
.9837613
3.85
0.000
1.862797
5.71907
-------------+---------------------------------------------------------------ARCH
|
arch
|
L1 |
.1835399
.0981365
1.87
0.061
-.0088041
.3758838
garch
|
L1 |
.6634369
.1133509
5.85
0.000
.4412733
.8856006
------------------------------------------------------------------------------
Leblang
ARCH
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