Garch-m
• The process or return is dependent on
the volatility
r (t )    c  a (t )
2
t
, c are
constants
a (t )   t  (t )
   0   a  1
2
t
2
1 t 1
2
t 1
C is the “risk premium parameter”; c>0 indicates the return
is positively related to its volatility.
Spring 2004
K. Ensor, STAT 421
1
-0.2
0.2
S&P 500
0
200
400
600
800
Time
Histogram
ACF
Series: x
AR ( 21 ) Spectrum using yule-walker
0.0
S&P 500
Spring 2004
0.2
0.4
0
5
10
15
20
25
-24
spectrum
-26
-28
-0.5
-1.0
-1.0
0
-0.2
0.0
ACF
0.0
-0.5
100
ACF
200
0.5
0.5
-22
300
1.0
1.0
PACF
0
5
10
Lag
K. Ensor, STAT 421
15
Lag
20
25
0.0
0.1
0.2
0.3
0.4
0.5
frequency
2
Output from Splus m-garch fit
garch(x~1+var.in.mean,~garch(1,1))
Estimated Coefficients:
-------------------------------------------------------------Value
Std.Error
t value Pr(>|t|)
0.00548675
0.00226173
2.426
7.747e-003
ARCH-IN-MEAN 1.08783589
0.81822755
1.330
9.203e-002
A
0.00008764
0.00002507
3.496
2.494e-004
ARCH(1)
0.12268468
0.02047268
5.993
1.571e-009
GARCH(1)
0.84939373
0.01957565
43.390 0.000e+000
C
--------------------------------------------------------------
Differs from Tsay’s fit slightly.
Spring 2004
K. Ensor, STAT 421
3
Series and Conditional SD
0.20
Original Series
0.05
Conditional SD
0.4
Values
0.10
0.15
Square root
Of volatility
-0.2
0.0
0.2
S&P500 Index
0
Spring 2004
200
400
K. Ensor, STAT 421
600
800
4
Summary Graphs
ACF of Observations
ACF of Squared Observations
ACF
ACF
1.0
1.0
0.8
0.8
0.6
0.6
GARCH Standardized Residuals
residuals
0.4
0.4
2
0.2
Standardized Residuals
0.2
0.0
0.0
0
5
10
15
20
25
30
0
5
10
15
Lags
20
25
30
Lags
0
-2
ACF of Std. Residuals
-4
ACF
1.0
0
200
400
600
800
0.8
ACF of Squared Std. Residuals
0.6
ACF
1.0
0.4
QQ-Plot of Standardized Residuals
0.8
0.2
QQ-Plot
0.6
0.0
2
0
5
10
15
Lags
20
25
Standardized Residuals
0.4
30
0.2
0.0
0
5
10
15
20
25
0
-2
30
Lags
147
-4
173
742
Spring 2004
K. Ensor, STAT 421
-3
-2
-1
0
1
2
Quantiles of gaussian distribution
3
5
Hong Kong stock market index return
(bottom graph) and estimated volatility.
Series and Conditional SD
15
Conditional SD
-10
-5
0
5
10
Values
2
4
6
8
Original Series
0
Spring 2004
100
200
K. Ensor, STAT 421
300
400
500
6
garchfit<-garch(HK~1+arma(1,0),~garch(1,1),cond.dist="t",dist.est=T)
Estimated Coefficients:
-------------------------------------------------------------Value Std.Error t value Pr(>|t|)
AR(1) 0.0450
0.04578
0.983 0.163052
A 0.1688
0.08404
2.009 0.022568
ARCH(1) 0.1700
0.05835
2.913 0.001871
GARCH(1) 0.7732
0.06454
11.980 0.000000
--------------------------------------------------------------
Spring 2004
K. Ensor, STAT 421
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HK - Garch fit +/- 2SD
Series with 2 Conditional SD Superimposed
garchfit
Values
10
0
-10
0
Spring 2004
100
200
K. Ensor, STAT 421
300
400
500
8
ACF of Observations
ACF of Squared Observations
ACF
ACF
1.0
1.0
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0.0
0.0
-0.2
0
5
10
15
20
25
0ACF of Std. Residuals
5
Lags
10
15
20
25
Lags
ACF
1.0
0.8
GARCH Standardized Residuals
ACF of Squared Std. Residuals
ACF
0.6
1.0
residuals
0.4
0.8
2
0.2
0.6
0.0
0.4
QQ-Plot
0
6
0.2
0
5
10
15
20
25
4
Lags
0.0
-2
0
5
10
15
Lags
-4
Standardized Residuals
Standardized Residuals
QQ-Plot of Standardized Residuals
2
20
0
25
-2
2 3
-4
-6
-6
1
-5
0
100
Spring 2004
200
300
400
500
K. Ensor, STAT 421
0
5
Quantiles of t distribution
9
garchfit<-garch(HK~1+arma(1,0),~garch(1,1),cond.dist="gaussian",dist.est=T)
--------------------------------------------------------------
Estimated Coefficients:
-------------------------------------------------------------Value Std.Error t value
Pr(>|t|)
AR(1) 0.1199
0.05709
2.100 1.811e-002
A 0.1424
0.04834
2.946 1.687e-003
ARCH(1) 0.1782
0.03693
4.827 9.287e-007
GARCH(1) 0.7592
0.04913
15.452 0.000e+000
--------------------------------------------------------------
Spring 2004
K. Ensor, STAT 421
10
Series and Conditional SD
15
Conditional SD
-10
-5
0
5
10
Values
2
4
6
8
10
Original Series
0
Spring 2004
100
200
300
K. Ensor, STAT 421
400
500
11
Series with 2 Conditional SD Superimposed
garchfit
20
Values
10
0
-10
-20
0
Spring 2004
100
200
300
K. Ensor, STAT 421
400
500
12
GARCH Standardized Residuals
residuals
0
-2
-4
QQ-Plot of Standardized Residuals
-6
QQ-Plot
0
100
200
300
400
500
2
Standardized Residuals
Standardized Residuals
2
0
-2
471
434
-4
-6
Spring 2004
K. Ensor, STAT 421
51
-3
-2
-1
0
1
Quantiles of gaussian distribution
2
3
13
Japanese stock market index and volatility based on
Gaussian GARCH(1,1) model
Series and Conditional SD
15
Conditional SD
-10
-5
0
5
10
Values
2
4
6
8
Original Series
0
Spring 2004
100
200
300
K. Ensor, STAT 421
400
500
14
garchfit<-garch(JI~-
1,~garch(1,1),cond.dist="gaussian",dist.est=
T)
--------------------------------------------------------------
Estimated Coefficients:
-------------------------------------------------------------Value Std.Error t value
Pr(>|t|)
A 0.1352
0.04517
2.993 1.452e-003
ARCH(1) 0.1713
0.03409
5.024 3.552e-007
GARCH(1) 0.7708
0.04609
16.722 0.000e+000
--------------------------------------------------------------
Spring 2004
K. Ensor, STAT 421
15
JI
Series with 2 Conditional SD Superimposed
garchfit
20
Values
10
0
-10
-20
0
Spring 2004
100
200
300
K. Ensor, STAT 421
400
500
16
ACF of Observations
ACF of Squared Observations
ACF
ACF
1.0
1.0
0.8
0.8
JI
0.6
0.6
0.4
0.4
0.2
0.2
ACF of Std. Residuals
0.0
0.0
ACF
-0.2
1.0
0
5
10
15
20
0
25
5
Lags
10
15
0.8
20
25
Lags
0.6
ACF of Squared Std. Residuals
GARCH Standardized Residuals
0.4
ACF
residuals
1.0
0.2
2
0.8
0.6
0
0
5
10
15
Lags0.4
20
QQ-Plot of Standardized Residuals
25
QQ-Plot
-2
0.2
-4
2
0.0
0
-6
5
10
15
Lags
0
100
200
300
400
500
Standardized Residuals
Standardized Residuals
0.0
0
20-2
471
434
-4
-6
Spring 2004
K. Ensor, STAT 421
25
51
-3
-2
-1
0
1
2
Quantiles of gaussian distribution
17
3
Let’s trying looking at the multivariate GARCH.
Original Observations
-4
15
Series 1
-10
-5
0
5
10
Values
-2
0
2
4
6
8
Series 2
0
Spring 2004
100
200
300
K. Ensor, STAT 421
400
500
18
Series 1: Hong Kong Stock Index
Series 2: Japanese Stock Index
ACF of Observations
Series 1 and Series 2
-0.2
-0.1
0.0
0.0
0.2
0.1
ACF
0.4
0.6
0.2
0.8
0.3
1.0
Series 1
0
5
10
15
20
25
0
5
10
20
25
20
25
Series 2
-0.1
0.0
0.0
0.2
0.4
ACF
0.1
0.6
0.2
0.8
0.3
1.0
Series 2 and Series 1
15
-25
Spring 2004
-20
-15
Lag
-10
-5
0
0
5
10
K. Ensor, STAT 421
Lag
15
19
Series 1: Hong Kong Stock Index Squared
Series 2: Japanese Stock Index Squared
Multivariate Series : X
Series 1 and Series 2
0.0
-0.05
0.2
0.0
ACF
0.4
0.05
0.6
0.10
0.8
0.15
1.0
Series 1
0
5
10
15
20
0
5
15
20
15
20
Series 2
0.0
0.0
0.2
0.2
0.4
ACF
0.6
0.4
0.8
0.6
1.0
Series 2 and Series 1
10
-20
Spring 2004
-15
Lag
-10
-5
0
0
5
K. Ensor, STAT 421
10
Lag
20
-------------------------------------------------------------mgarchfit=mgarch(X~-
1+arma(1,0),~garch(1,
1))
Estimated Coefficients:
-------------------------------------------------------------Value Std.Error t value
Pr(>|t|)
AR(1; 1, 1) 0.124329
0.058850
2.1126 1.757e-002
AR(1; 2, 2) 0.017088
0.047872
0.3569 3.606e-001
A(1, 1) 0.144756
0.050129
2.8877 2.027e-003
A(2, 2) 0.003265
0.006921
0.4718 3.187e-001
ARCH(1; 1, 1) 0.186976
0.039732
4.7059 1.649e-006
ARCH(1; 2, 2) 0.069114
0.016141
4.2818 1.117e-005
GARCH(1; 1, 1) 0.755876
0.050284 15.0320 0.000e+000
GARCH(1; 2, 2) 0.937297
0.017199 54.4981 0.000e+000
-------------------------------------------------------------Page 367 of text
Spring 2004
K. Ensor, STAT 421
21
MGARCH Residuals
-4
15
Series 1
-10
-5
0
5
10
Residuals
-2
0
2
4
6
8
Series 2
0
Spring 2004
100
200
300
K. Ensor, STAT 421
400
500
22
MGARCH Volatility
1.0
10 0.6
Series 1
2
4
6
8
Conditional SD
1.4
1.8
Series 2
0
Spring 2004
100
200
300
K. Ensor, STAT 421
400
500
23
ACF of Standardized Residuals
ACF of Squared Std. Residuals
Series 1
Series 1 and Series 2
0.20
0.10
0.15
0.8
0.0
0.05
ACF
0.4
-0.05
0.2
-0.1
-0.10
0.0
0.0
0.0
0.2
ACF
0.4
0.1
0.6
0.6
0.2
0.8
1.0
Series 1 and Series 2
1.0
Series 1
0
5
10
15
20
25
0
5
10
20
25
0
5
10
20
25
5
10
Series 2 and Series 1
15
20
25
20
25
Series 2
0.8
0.05
0.8
0.4
0.2
-0.1
0.0
0.0
-0.05
0.2
0.4
ACF
0.0
0.6
0.6
0.2
ACF
0.1
0.0
0
1.0
Series 2
15
1.0
Series 2 and Series 1
15
-25
-20
-15
Lag
-10
-5
0
0
5
10
Lag
15
20
25
-25
-20
-15
Lag
-10
-5
0
0
5
10
Lag
15
Standardized Residuals
4
Series 2
2
QQ-Plot of Standardized Residuals
-3
-2
-1
Series 1
Standardized Residuals
0
-2
2
0
1
2
3
491
352
2
0
-2
37
471
434
-2
-4
-6
51
-4
-3
-2
-1
0
1
2
3
Quantiles of gaussian distribution
-6
Standardized Residuals
Series 1
0
Series 2
4
0
100
Spring 2004
200
300
400
500
K. Ensor, STAT 421
24