ARCH and GARCH. Modeling Volatility Dynamics

1. ARCH and GARCH

Modeling Volatility Dynamics

2. Modeling Unequal Variability

• Equal Variability: Homoscedasticity
• Unequal Variability: Heteroscedasticity
– Means any variability (around the mean)
that is not homoscedasticity
– Models must be developed for specific
cases

3. What These Acronym Mean?

• ARCH
– Autoregressive Conditional
Heteroscedasticity
• GARCH
– Generalized ARCH

4. Information in e2

• Let et have the mean 0 and the variance st.
• Let et be the residual of a model fitted.
• Then:
– et estimates et
– et2 estimates the variance st2.

5. ARCH Modeling of st2.

• ARCH(1)
s e
2
t
• ARCH as AR(1) on
e s t
e e
2
t
2
( t 1)
2
t
2
( t 1)
2
t
t

6. GARCH

• GARCH(1)
s e
2
t
2
( t 1)
s
2
( t 1)
2
2
e
s
• GARCH (1) as ARMA(1,1) on
t
t t
e e
2
t
2
( t 1)
t t 1

7. Asymmetry in GARCH - TARCH

et 0
Asymmetry in GARCH - TARCH
• TARCH(1,1)
s e
2
t
2
t 1
de
2
t 1s
d = 1 if et < 0, and = 0 if et > 0
s
2
t 1

8. Asymmetry in GARCH - EGARCH

• EGARCH(1,1)
log s
s t2 0
2
t
log s
2
t 1
e t 1
e t 1
s t 1
s t 1
0 for asymmetric effect

9. Eviews Command

ARCH(p, q) series_name c