Flashcards in Part 2 Deck (13):

1

## News impact curve

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- 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

2

## Probability integral transform ut=G(zt)

### - cdf has to be uniformly distributed and iid: ut ~ iid U(0,1)

3

## Sandwich estimator in QML estimation

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- 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

4

## Problem in test of constant correlation?

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- 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

5

## Why test the adequacy of a non-normal distr in ML estimation?

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- 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

6

## Hill's estimator of the tail index

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- Hill's estimator is the MLE of xi for tails drawn from a Pareto distr

- Only for Fréchet distr

7

## QQ-plot

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- 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)

8

## Main difficulties of Hill's estimator of tail index

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- 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

9

## How generate non-linear dependence in multivariate t distr?

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- by using normal mixtures

- idea is to introduce randomness into the cov matrix (via a positive mixing var W)

10

## Excess distr fn in EVT?

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- 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

11

## How adapt EVT if time-dependent returns?

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- 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

12

## Extrema or tail approach?

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- tail approach: requires whole sample iid

- extrema approach: requires subsamples iid

- check for iidness of returns

13