ECON 3350/7350 Volatility Models – II
Eric Eisenstat
The University of Queensland
Tutorial 8
Eric Eisenstat
(School of Economics)
ECON3350/7350 Week 8
1 / 7
The EGARCH Model
EGARCH was suggested by Nelson (1991). The variance equation is given by
ln ht = α0 + β1 ln ht−1 + λ
t−1
Advantage: Since we model lnht, then even if the parameters are negative, ht will be positive.
The effect of a shock in εt on ln ht−1 If εt−1 is positive, the effect is (α1 + λ).
h0.5 εt−1
If t−1 is negative, the effect is (α1 − λ). h0.5
t−1
Leverage effect exists as long as λ1 ̸= 0 (i.e., λ1 < 0).
εt−1 h0.5
εt−1 + α1 .
h0.5 t−1
Eric Eisenstat (School of Economics) ECON3350/7350 Week 8 2 / 7
Representing The Leverage Effect
Source: Enders (2010)
When λ < 0, positive shocks generate less volatility than negative shocks (“bad news”).
The standardised εt−1 is unit h0.5
t−1 free and permits a more
natural interpretation of the size and persistence of the shocks.
Eric Eisenstat (School of Economics) ECON3350/7350 Week 8 3 / 7
The EGARCH model (cont.)
Note that we can rewrite as:
lnht =α0+β1lnht−1+(α1+λ)εt−1 ifεt−1 >0
h0.5 t−1
lnht =α0+β1lnht−1+(α1−λ)εt−1 ifεt−1 <0 h0.5
t−1
The logarithmic transformation guarantees that variances are
non-negative.
Weexpectλ+α1 >0whileλ<0.
Eric Eisenstat (School of Economics) ECON3350/7350 Week 8
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The TGARCH model
A form of GARCH model:
ht = α0 + α1ε2t−1 + λdt−1ε2t−1 + β1ht−1, where dt−1 = 1 if εt−1 < 0 and dt−1 = 0 otherwise.
We require λ + α1 0 and α1 < 0 for non-negativity. Leverage effect exists when λ > 0.
Eric Eisenstat (School of Economics) ECON3350/7350 Week 8 5 / 7
Testing for Leverage Effects
Estimate the TARCH or EGARCH model, i.e.
ht = α0 + α1ε2t−1 + λdt−1ε2t−1 + β1ht−1
or
ln ht = α0 + β1 ln ht−1 + λ Test: H0 : λ = 0 using a t-test.
εt−1
h0.5 t−1
εt−1 + α1 .
h0.5 t−1
Eric Eisenstat (School of Economics) ECON3350/7350
Week 8
6 / 7
The (G)ARCH-M model
The mean model together with the variance model can be written as:
r
y t = x ′t β + γ g ( h t ) + ε t + θ j ε t − j ,
j=1 qp
h t = α 0 + α j ε 2t − j + β i h t − i . j=1 i=1
Usually, g(ht) is set to lnht, √ht, or hmt .
Eric Eisenstat (School of Economics) ECON3350/7350 Week 8 7 / 7