Lecture 9 Flashcards
What does this large persistence in volatility require for the ARCH model ?
Large p to fit data
How is the forecasting obtained for a GARCH ?
Obtained recursively = similar to ARCH
What does the parameter γ stand for ?
γ = α + β → persitence parameter
What does a GARCH(1,1) imply for ϵ(t)^2 ?
It follows a ARMA(1,1) process
What is the test for homoskedasticity ?
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)
What does it mean if α + β = 0.99 ?
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
What if ω = 0 ?
Then :
• σ^2 → ∞ iff E[log(αz(t)^2 + β)] > 0
• σ^2 → 0 iff E[log(αz(t)^2 + β)] < 0
What if ω > 0 ?
Then :
• σ^2 → ∞ iff E[log(αz(t)^2 + β)] ≥ 0
• σ^2 strictly stationary iff E[log(αz(t)^2 + β)] < 0
What happens if the unconditional variance ϵ(t) is infinite if we have an integrated GARCH model ?
Not CS but SS (unconditional density of r(t) same for all t)
In a IGARCH, what happens if the true model is a regime-switching model for volatility ?
It results in a nearly integrated process
Why is there asymmetric GARCH model ?
Negative returns followed by larger increases in volatility since bad news have stronger effect on volatility
What are the different Asymmetric GARCH models ?
- GJR
- Threshold GARCH
- Exponential GARCH
What is the GJR model ?
Squared volatility depends on size and sign of lagged innovation
When is the GJR model covariance stationary ?
α + γ/β + β < 1
What is the threshold GARCH ?
Closely related to GJR, but not squared
When is the TGARCH covariance stationary ?
β^2 + [α^2 + (α + γ )^2]/2 + 2β(α + γ )/√2 < 1
What is the Exponential GARCH ?
Log of squared volatility depends on sign & size of lagged innovations
What is different regarding the restriction on conditional volatility for the EGARCH ?
No non-negative restriction since conditional volatility always > 0
When is the EGARCH model covariance stationary ?
β < 1
What is the news impact curve ?
It relates past return shocks to current volatility = measure how new information incorporated into volatility estimates
What does the comparison of GARCH and GJR allow ?
Analyze effect of news on conditional volatility
What does the ARCH-in-Mean capture ?
The effect of conditional volatility on conditional mean
What are the several problems with the ARCH-in-Mean ?
- QML estimation still be performed but information matrix not block diagonal
- All parameters of mean & variance equation must be jointly estimated
What is the major change with the SV model ?
Innovation introduced in conditional variance equation
What happened with the first generation models for SV ?
All empirically rejected because volatility requires separate stochastic process
What does the stationarity of volatility process impose in SV ?
β < 1
Why is the SV model more flexible than GARCH ?
- 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
What are the 2 different approaches for the SV model ?
- Regression approach : consistent but not efficient
* ML approach (quasi efficient)
What is the difficulty for the ML estimation of SV models ?
expression f(.) depends on unobservable variable σ(t)
When is the quasi.maximum likelihood estimation efficient ?
Only if volatility is normal
What are the two alternatives based on simulation methods for QML ?
- Indirect inference
* Markov Chain Monte Carlo
