GARCH in finance

1. GARCH
Group 1
Siti Maghfirotul Ulyah M10318801
Uurtsaikhbaatar M10308802
Jean-Raymond Fontin M10318805
Dyah Ayu Kusumaningtyas M10308801
2015/06/08
Homework 5
Stochastic Models for Finance

2.  Daily
 Jan 1, 1999-March 19,
2015
 ARMA(3,3) without MA(1)
term
 Zero mean
-.15
-.10
-.05
.00
.05
.10
.15
.20
99 00 01 02 03 04 05 06 07 08 09 10 11 12 13 14
NASDAQ Daily Log return
𝑟𝑡 = −0.049541𝑟𝑡−1 − 0.408928𝑟𝑡−2 + 0.700079𝑟𝑡−3 − 0.345245𝑎 𝑡−2 + 0.735821𝑎 𝑡−3 + 𝑎 𝑡

3. Diagnostic Checking
There is no linear serial correlation
on residual up to lag 11
There is nonlinear serial correlation
on residual

4. GARCH
Error Distribution-Normal
ARMA(3,3)-GARCH SQ-Residual Coef-Significance AIC BIC
1 ARMA(3,3)-GARCH(1,1) AutoCorr All Significant -5.6147 -5.6023
2 ARMA(3,3)-GARCH(1,2) AutoCorr All Significant -5.6151 -5.6011
3 ARMA(3,3)-GARCH(1,3) No Autocorr GARCH(3) not sign -5.6170 -5.6015
4 ARMA(3,3)-GARCH(1,4) No Autocorr AR(3) MA(3) not sign -5.6158 -5.5972
5 ARMA(3,3)-GARCH(2,1) No Autocorr ARCH(1) not sign -5.6193 -5.6053
6 ARMA(3,3)-GARCH(2,2) No Autocorr ARCH(1) GARCH(2) not sign -5.6183 -5.6028
7 ARMA(3,3)-GARCH(2,3) No Autocorr ARCH(1) GARCH(2,3) not sign -5.6178 -5.6008
8 ARMA(3,3)-GARCH(2,4) No Autocorr ARCH(1) GARCH(2,3,4) not sign -5.6175 -5.5989
9 ARMA(3,3)-GARCH(3,1) No Autocorr ARCH(1,3) not sign -5.6177 -5.6022
10 ARMA(3,3)-GARCH(3,2) No Autocorr ARCH(1,3) GARCH(1) not sign -5.6185 -5.6015
11 ARMA(3,3)-GARCH(3,3) No Autocorr C, ARCH(1) GARCH(3) not sign -5.6209 -5.6023
12 ARMA(3,3)-GARCH(3,4) No Autocorr C, ARCH(1,3) GARCH(1,2,3,4) not sign -5.6170 -5.5969
13 ARMA(3,3)-GARCH(4,1) No Autocorr ARCH(1,3,4) not sign -5.6184 -5.6014
14 ARMA(3,3)-GARCH(4,2) No Autocorr ARCH(1,3,4) GARCH(1) not sign -5.6180 -5.5994
15 ARMA(3,3)-GARCH(4,3) No Autocorr AR(3), MA(3) C, ARCH(1) GARCH(2,3,4) not sign -5.6194 -5.5978
16 ARMA(3,3)-GARCH(4,4) No Autocorr AR(3), MA(3) C, ARCH(1) GARCH(2,3,4) not sign -5.6194 -5.5978

5. 𝑟𝑡 = −0.038572𝑟𝑡−1 − 0.633548𝑟𝑡−2 + 0.556353𝑟𝑡−3 − 0.611625𝑎 𝑡−2 + 0.584794𝑎 𝑡−3 + 𝑎 𝑡
𝑎 𝑡 = 𝜎𝑡 𝜀𝑡
𝜎𝑡
2
= 5.84 × 10−7
+ 0.035835𝛼 𝑡−1
2
+ 1.948578𝜎𝑡−1
2
− 1.445541𝜎𝑡−2
2
+ 0.459692𝜎𝑡−3
2
𝜀𝑡~𝑡 𝑣
ARMA(3,3)-GARCH(1,3)
with Student-t
error distribution
Error distribution-Student-t
ARMA(3,3)-GARCH SQ-Residual Coef-Significance AIC BIC
17 ARMA(3,3)-GARCH(1,1) AutoCorr All Significant -5.6214 -5.6075
18 ARMA(3,3)-GARCH(1,2) AutoCorr All Significant -5.6231 -5.6076
19 ARMA(3,3)-GARCH(1,3) No Autocorr All Significant -5.6237 -5.6067
20 ARMA(3,3)-GARCH(1,4) No Autocorr AR(2,3), MA(2,3) not sign -5.6222 -5.6037
21 ARMA(3,3)-GARCH(2,1) No Autocorr AR(2,3), MA(2,3), C, ARCH(1) not sign -5.6240 -5.6085
22 ARMA(3,3)-GARCH(2,2) No Autocorr AR(2,3), MA(2,3), ARCH(1) GARCH(2) not sign -5.6236 -5.6066
23 ARMA(3,3)-GARCH(2,3) No Autocorr AR(2,3), MA(2,3), ARCH(1) GARCH(2,3) not sign -5.6231 -5.6045
24 ARMA(3,3)-GARCH(2,4) No Autocorr ARCH(1), GARCH(2,3,4) not sign -5.6175 -5.5989
25 ARMA(3,3)-GARCH(3,1) No Autocorr AR(2,3), MA(2,3) ARCH(1,3) not sign -5.6236 -5.6066
26 ARMA(3,3)-GARCH(3,2) No Autocorr ARCH(1,3), GARCH(1) not sign -5.6185 -5.6015
27 ARMA(3,3)-GARCH(3,3) No Autocorr C, ARCH(1), GARCH(3) not sign -5.6282 -5.6081
28 ARMA(3,3)-GARCH(3,4) No Autocorr
AR(2,3), MA(2,3), C, ARCH(1,3) GARCH(1,2,3,4) n
ot sign -5.6225 -5.6008
29 ARMA(3,3)-GARCH(4,1) No Autocorr AR(2,3), MA(2,3), ARCH(1,3,4) not sign -5.6232 -5.6046
30 ARMA(3,3)-GARCH(4,2) No Autocorr
AR(2,3), MA(2,3), ARCH(1,3,4),GARCH(1,2) not si
gn -5.6227 -5.6025
31 ARMA(3,3)-GARCH(4,3) No Autocorr AR(2,3), MA(2,3), ARCH(1) not sign -5.6245 -5.6028
32 ARMA(3,3)-GARCH(4,4) No Autocorr ARCH(1), GARCH(3,4) not sign -5.6252 -5.6020

6. TGARCH

7. EGARCH

8. GARCH-M
ARMA(3,3)-GARCH(1,1)-M zero mean and without MA(1) term
Criteria σt log(σt ) σt
2
error
distribution
Normal Student-t Normal Student-t Normal Student-t
significance of p
arameters
yes,
after dropping
AR(3) and MA(3)
terms
yes,
after dropping
AR(3) and MA(3)
terms
yes,
after dropping
AR(3) and MA(3)
terms
yes,
after dropping
AR(3) and MA(3)
terms
yes yes
residual is
uncorrelated
yes yes yes yes yes yes
squared residual
is uncorrelated
no no no no no no
No adequate model

9. GARCH-M
Criteria σt log(σt ) σt
2
error distribution Normal Student-t Normal Student-t Normal Student-t
significance of
parameters
yes
arch(1) is in
significant
α0, arch(1),
garch(2) and (3)
are insignificant
arch(1) is
insignificant
α0 and arch(1)
are insignificant
AR(2), MA(2),
and ARCH(1) are
insignificant
residual is
uncorrelated
yes
squared residual
is uncorrelated
yes
AIC -5.623055
BIC -5.606018
𝑟𝑡 = −0.043532 𝑟𝑡−1 + 0.533481 𝑟𝑡−2 + 0.572861 𝑎 𝑡−1 + 0.059908 𝜎𝑡 + 𝑎 𝑡 where 𝑎 𝑡 = 𝜎𝑡 𝜖 𝑡
𝜎𝑡
2
= 2.56 × 10−8 + 0.011724 𝑎 𝑡−1
2
+ 0.104195 𝑎 𝑡−2
2
− 0.11416 𝑎 𝑡−3
2
+ 1.547368 𝜎𝑡−1
2
− 0.290231 𝜎𝑡−2
2
− 0.25907𝜎𝑡−3
2
with 𝜖 𝑡~𝑁(0,1)
ARMA(2,2)-GARCH(3,3)-M zero mean and without MA(1) term

10. COMPARISON
Model AIC BIC
GARCH -5.6237 -5.6067
TGARCH
EGARCH
GARCH-M -5.623055 -5.606018

11. THANK YOU