Lecture 9

Question 1

What does this large persistence in volatility require for the ARCH model ?

Answer 1

Large p to fit data

Question 2

How is the forecasting obtained for a GARCH ?

Answer 2

Obtained recursively = similar to ARCH

Question 3

What does the parameter γ stand for ?

Answer 3

γ = α + β → persitence parameter

Question 4

What does a GARCH(1,1) imply for ϵ(t)^2 ?

Answer 4

It follows a ARMA(1,1) process

Question 5

What is the test for homoskedasticity ?

Answer 5

Ho : homoskedastic vs Ha : variance = GARCH(1,1)
→ Ho : α = β = 0 vs Ha : α ≥ 0; β ≥ 0 with at least one strict inequality
→ Similar to test of no ARCH for Ho (GARCH and ARCH locally equilvalent)

Question 6

What does it mean if α + β = 0.99 ?

Answer 6

The result seems stationary. However, it might come from the fact that there is a restriction on α + β and therefore, the process is surely not stationary

Question 7

What if ω = 0 ?

Answer 7

Then :
• σ^2 → ∞ iff E[log(αz(t)^2 + β)] > 0

• σ^2 → 0 iff E[log(αz(t)^2 + β)] < 0

Question 8

What if ω > 0 ?

Answer 8

Then :
• σ^2 → ∞ iff E[log(αz(t)^2 + β)] ≥ 0

• σ^2 strictly stationary iff E[log(αz(t)^2 + β)] < 0

Question 9

What happens if the unconditional variance ϵ(t) is infinite if we have an integrated GARCH model ?

Answer 9

Not CS but SS (unconditional density of r(t) same for all t)

Question 10

In a IGARCH, what happens if the true model is a regime-switching model for volatility ?

Answer 10

It results in a nearly integrated process

Question 11

Why is there asymmetric GARCH model ?

Answer 11

Negative returns followed by larger increases in volatility since bad news have stronger effect on volatility

Question 12

What are the different Asymmetric GARCH models ?

Answer 12

• GJR
• Threshold GARCH
• Exponential GARCH

Question 13

What is the GJR model ?

Answer 13

Squared volatility depends on size and sign of lagged innovation

Question 14

When is the GJR model covariance stationary ?

Answer 14

α + γ/β + β < 1

Question 15

What is the threshold GARCH ?

Answer 15

Closely related to GJR, but not squared

Question 16

When is the TGARCH covariance stationary ?

Answer 16

β^2 + [α^2 + (α + γ )^2]/2 + 2β(α + γ )/√2 < 1

Question 17

What is the Exponential GARCH ?

Answer 17

Log of squared volatility depends on sign & size of lagged innovations

Question 18

What is different regarding the restriction on conditional volatility for the EGARCH ?

Answer 18

No non-negative restriction since conditional volatility always > 0

Question 19

When is the EGARCH model covariance stationary ?

Answer 19

β < 1

Question 20

What is the news impact curve ?

Answer 20

It relates past return shocks to current volatility = measure how new information incorporated into volatility estimates

Question 21

What does the comparison of GARCH and GJR allow ?

Answer 21

Analyze effect of news on conditional volatility

Question 22

What does the ARCH-in-Mean capture ?

Answer 22

The effect of conditional volatility on conditional mean

Question 23

What are the several problems with the ARCH-in-Mean ?

Answer 23

• QML estimation still be performed but information matrix not block diagonal
• All parameters of mean & variance equation must be jointly estimated

Question 24

What is the major change with the SV model ?

Answer 24

Innovation introduced in conditional variance equation

Question 25

What happened with the first generation models for SV ?

Answer 25

All empirically rejected because volatility requires separate stochastic process

Question 26

What does the stationarity of volatility process impose in SV ?

Answer 26

β < 1

Question 27

Why is the SV model more flexible than GARCH ?

Answer 27

• Distribution of returns = fat tails
• Persistence in volatility captured by autoregressive term β
• Correlation between z(t) and v(t) produces volatility asymmetry
• Fits asset returns & residuals closer to standard normal

Question 28

What are the 2 different approaches for the SV model ?

Answer 28

• Regression approach : consistent but not efficient

* ML approach (quasi efficient)

Question 29

What is the difficulty for the ML estimation of SV models ?

Answer 29

expression f(.) depends on unobservable variable σ(t)

Question 30

When is the quasi.maximum likelihood estimation efficient ?

Answer 30

Only if volatility is normal

Question 31

What are the two alternatives based on simulation methods for QML ?

Answer 31

• Indirect inference

* Markov Chain Monte Carlo