True, False, Maybe 2006-2007

Question 1

The squared return is an unbiased estimator of the conditional volatility.

Answer 1

Solution: True.

The squared return is generally not a precise estimator, but it is unbiased.

Question 2

1b. When α +β = 1 in a GARCH(1,1) model (σ_t^2=ω+αε_(t-1)^2+βσ_(t-1)^2) the return process is not strictly stationary.

Answer 2

Solution: False.

It is not covariance stationary, but it is still strictly stationary.

Question 3

1c. A GARCH-in-Mean model requires a joint estimation of the conditional mean and variance equations.

Answer 3

Solution: True.

A joint estimation is required, because the Hessian matrix is not bloc-diagonal.

Question 4

When the true distribution is known to be non-normal, it is more efficient to estimate the model by QMLE using a non-normal distribution.

Answer 4

Solution: False.
If we correctly approximate the true distribution, it is more efficient to use an estimation of this distribution, rather than the normal distribution.

Question 5

Under non-normality, a semi-parametric estimation of the distribution avoids the mis-specification issue.

Answer 5

Solution: True.
The semi-parametric estimation allows to approximate the true distribution, but avoids any mis-specification of the true distribution.

Question 6

The skewed Student t distribution is able to capture all the possible levels of skewness and kurtosis.

Answer 6

Solution: False.

The skewed t is not able to capture the complete range of skewness and kurtosis.

Question 7

In the extreme value theory, the extrema approach and the tail approach give the same estimate of the tail index.

Answer 7

Solution: False.
The tail index is the same for the two approaches, but the estimates are different, since they are based on a different information.

Question 8

When returns are time dependent, the tail approach can be applied to standardized
returns to remove time dependency.

Answer 8

Solution: True.
In case of time dependency, the use of standardized returns (or residuals) to the tail approach is recommended (McNeil and Frey, 2002).

Question 9

A test for a correlation constant through time can be easily performed using first-generation
multivariate GARCH models.

Answer 9

Solution: False.
The first-generation multivariate GARCH models focus on the covariance matrix, so that it is extremely difficult to perform any test regarding the correlation. For this purpose, a Dynamic Conditional Correlation model is required.

Question 10

Under normality of returns, the Constant Relative Risk Aversion utility reduces to the mean-variance criterion.

Answer 10

Solution: True.
Under normality, the CRRA utility can be approximated as a mean-variance criterion (while the exponential utility is exactly equivalent to the mean-variance criterion)

Question 11

The News Impact Curve measures the effect of past innovation on the current volatility.

Answer 11

True. This curve measures how the effect of past innovation is incorporated into volatility estimates. Under a standard GARCH model, the News Impact Curve is symmetric, meaning that good news and bad news have the same effect on volatility.

Question 12

The characteristics of the unconditional distribution of returns combine the dynamics of volatility and the shape of the conditional distribution of innovations.

Answer 12

True. The unconditional distribution of returns combines the dynamics of volatility (typically GARCH model) and the shape of the conditional distribution of innovations, since returns can be written as r_t =μ +σ_t z_t

Question 13

Provided the conditional mean and variance are correctly specified, the Quasi-Maximum-Likelihood estimator is a consistent estimator under non-normality

Answer 13

True. Quasi-Maximum-Likelihood estimation does not assume normality, although it is obtained using the normal log-likelihood. As a consequence, it is consistent even under non-normal distribution. Of course, the parameters of the first and second moment equations will be consistent only if these equations are correctly specified.

Question 14

The skewed Student t distribution is able to reach the maximum boundary for skewness and kurtosis.

Answer 14

False. The parameters of the skewed student t are restricted to the domain (ν ,λ )∈]2,∞[× ]−1,1[ . The range of skewness and kurtosis is restricted to certain domain, which is smaller than the maximum domain possible given by μ_3^2

Question 15

The test developed by Diebold, Gunther, and Tay (1996) tests whether returns are iid normally distributed.

Answer 15

False. It tests if returns adjust a particular marginal distribution, but it is not restricted to the case of the normal distribution.

Question 16

Extremes of returns drawn from the normal and Student t distributions have the same asymptotic distribution.

Answer 16

False. Extremes of a normal distribution converge to a Gumbel distribution (characterizing distributions with thin tails), while extremes of a student t distribution converge to a Fréchet distribution (characterizing distributions with fat tails).

Question 17

When returns are time dependent, the estimation of the tail parameter based on the maxima/minima over subsamples still applies.

Answer 17

Maybe. The extremes approach only imposes the iid-ness of subsamples over which extremes àre computed. Therefore using maxima/minima over subsamples is more likely to provide consistent estimators although this approach still has to satisfy that subsamples are iid.

Question 18

To incorporate some dynamics in the extreme value theory, one way is to model the tail of the standardized residuals of a GARCH process.

Answer 18

True. Modeling the tail of the standardized residual is a way to incorporate some dynamics in the extreme value theory. Once the time-invariant characteristics of the tails of standardized residuals are known, one still has to incorporate the effect of the time-varying volatility to obtain the characteristics of the tails of returns. These characteristics of the tail of returns are therefore themselves time varying.

Question 19

The main advantage of the expected shortfall relative to the Value at Risk is that it is a consistent measure of risk.

Answer 19

True. VaR does not satisfy the sub-additivity property, so that diversification does not necessarily result in a reduction of risk, as measured by VaR

Question 20

A test for a correlation constant through time can be easily performed using a DCC (Dynamic Conditional Correlation) model.

Answer 20

True. The DCC model has been designed for such a test of constant correlation. The test is based on the nullity of the parameter of the cross-product of lagged error terms.