Flashcards in Part 2 Deck (13):
News impact curve
- relates past return shocks to current volatility
- measures how new information is incorporated into volatility estimates
- symmetric for GARCH. However, empirical evidence: negative shocks have larger impact
Probability integral transform ut=G(zt)
- cdf has to be uniformly distributed and iid: ut ~ iid U(0,1)
Sandwich estimator in QML estimation
- QML estimation provides "robust" standard errors
- that gives asymptotically valid confidence intervals for the estimators (sandwich estimator)
- obtained as the square root of the diagonal elements of the matrix: omeg = A0^-1*B0*A0^-1 where A0: Hessian matrix and B0: outer product of the gradients
Problem in test of constant correlation?
- we know from cond corr that we should expect different corr depending on the vol
- therefore, a change in corr doesn't necessarily mean that corr indeed changed
Why test the adequacy of a non-normal distr in ML estimation?
- Consistency of QMLE is not guaranteed if incorrect distr used to maximize the likelihood
- Consistent only if either: (1) conditional mean is identically 0; (2) assumed and true error pdfs are symmetric about 0
- If not consistent, the reason is that it fails to capture the effect of the asymmetry distr on the conditional mean
Hill's estimator of the tail index
- Hill's estimator is the MLE of xi for tails drawn from a Pareto distr
- Only for Fréchet distr
- based on the inverse of the assumed cdf
Case of the gev:
- let (y1,...,ytau) be the standardized maxima over N-histories
- and y(1)<=...<=y(tau) the ordered maxima
- plot y(t) to Hxi^-1(t/tau)
Main difficulties of Hill's estimator of tail index
- it depends on the choice of the proportion of the sample used for computing the statistics
- If threshold too much in the tail -> innacurate estimates
- If too many obs -> tail obs are contaminated by obs from the central part
How generate non-linear dependence in multivariate t distr?
- by using normal mixtures
- idea is to introduce randomness into the cov matrix (via a positive mixing var W)
Excess distr fn in EVT?
- Fu(X) = Pr(Xt-u<=x ! Xt>u)
- measures the prob that the excess realization relative to the threshold (Xt-u) is below a certain value, given that u is exceeded
How adapt EVT if time-dependent returns?
- estimate the tail of the cond distr (rather than uncond)
- 2-step strategy:
(1) Fit a cond mean and vol model -> gives approx iid standardized residuals;
(2) Use EVT techniques to model the tail distr
Extrema or tail approach?
- tail approach: requires whole sample iid
- extrema approach: requires subsamples iid
- check for iidness of returns