程序代写代做代考 ECON 3350/7350 Volatility Models – II

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 4 / 7 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