Short questions 2008-2016

Question 1

1b. What is the risk of using a given non-normal distribution in the context of Maximum-Likelihood estimation?

Answer 1

ML approach cannot be directly used since if the innovations do not follow the non-normal distribution, MLE will be inconsistent.

Question 2

1c. To estimate the tail index, how would you decide to follow the extrema approach or the tail approach?

Answer 2

The tail approach requires that the whole sample of returns is iid, while the extrema approach only requires that the subsamples are iid. Therefore, we should check for iid-ness of returns before deciding which approach to follow.

Question 3

1j. Give the definition of the Value-at-Risk and expected shortfall and explain why you should use the expected shortfall as a measure of risk instead of the Value-at- Risk.

Answer 3

The VaR of a portfolio is the minimum potential loss that the portfolio can suffer in the q% worst cases, over a given time horizon. It does not satisfy subadditivity and disregards the form of the pdf beyond the confidence level. The ES is the expected value of the loss of the portfolio in the q% worst cases over a given time horizon

Question 4

1c. What are the main advantages of a copula over a joint distribution?

Answer 4

For joint distribution:
Marginal distributions are not Gaussian = impossible to define a joint distribution = two variables have different marginal distributions

Marginal distributions do not have a multivariate extension

In these contexts: use copula since:
Relate two marginal distributions instead of two series directly.
Not require any additional assumption on the non-linear dependence.

Question 5

1d. How would you define concordance?

Answer 5

Concordance is an association, which says that 2 variables are concordant when small values of one are likely to be associated with small values of the other, and large values of one are likely to be associated with large values of the other.

Question 6

1a. What is the dynamics of squared returns in an ARCH(p) process?

Answer 6

ε_t^2=σ_t^2 z_t^2=σ_t^2+σ_t^2 (z_t^2-1)=σ_t^2+v_t=E[ε_t^2 ]+v_t
ε_t^2=σ_t^2+v_t=ω+〖∑α〗i ε(t-i)^2+v_t
Therefore, ε_t^2 is an AR(p) process.

Question 7

1b. What are the main differences between a (G)ARCH model and a stochastic volatility model?

Answer 7

• Innovation is introduced in the conditional variance equation in SV

• While in the GARCH model, σ_t only depends on information known at date t −1, in the SV model σ_t^2 depends on the innovation v_t, so that σ_t^2is not measurable on I_(t-1)
.
• While GARCH models are written in discrete time, SV models come from continuous time literature.

Question 8

1c. Give the main characteristics of an Integrated GARCH model and explain why it fits many financial time series very well.

Answer 8

Model :σ_t^2=ω+(1-β_1 ) ϵ_(t-1)^2+β_1 σ_(t-1)^2
Unconditional variance of ϵ_t = not CS but still strictly stationary
IGARCH process may reflect other dynamics for volatility. For instance, if the true model is a regime-switching model for volatility, estimating a GARCH model will generally result in a nearly integrated volatility process.

Question 9

1d. Why are GARCH models expected to take non-normality of innovations into account?

Answer 9

An attractive feature of GARCH models is that even when the conditional distribution of innovations is normal, the unconditional distribution of the error term ε_t has fatter tails than the normal distribution. It is then possible to capture a high unconditional skewness using a conditional distribution with a small skewness. Even in the case where innovations are assumed to be normal, a GARCH model yields to an unconditional distribution with fatter tails than the normal distribution.

Question 10

1e. What are the advantages and drawbacks of the Garm-Charlier distribution?

Answer 10

Drawbacks:
• For moments (m3,m4) distant from normality (0,3), g(z |η) may be negative for some z.
• For other pairs, the pdf g(z |η) may be multimodal.
• The domain of definition, for which the distribution is well defined, is small

Advantages
• Additional flexibility over a normal distribution
• Two additional parameters m3 and m4 are directly the third and fourth moments.

Question 11

1f. How is defined the sandwich estimator in QML estimation?

Answer 11

Ω=A_0^(-1) B_0 A_0^(-1)
A_0=-1/T∑E[(∂^2 logl(θ_0 ))/(∂θ∂θ’ )]
B_0=1/T∑E[∂logl(θ_0 )/∂θ ∂logl(θ_0 )’/∂θ]

Question 12

1g. How do you define the excess distribution function in extreme value theory?

Answer 12

Definition: Let u be a fixed real number, the threshold, in the support of Xt . The excess distribution of the r.v. Xt over the threshold u is defined as
F_u (x)=Pr⁡[X_t-u≤x│X_t>u]=(F_X (x+u)-F_X (u))/(1-F_X (u) )
The excess distribution measures the probability that the excess realization relative to the threshold (X_t-u) is below a certain value, given that u is exceeded.

Question 13

1j. How can we use the method of moments for estimating copula parameters?

Answer 13

Estimator of the parameter is obtained by equalizing the theoretical and empirical quantities using usin Kendall’s tau or Spearson’s rho

Question 14

1a. What is the dynamics of the squared return for a standard GARCH(1,1) model?

Answer 14

ε_t^2=σ_t^2+v_t=ω+αε_(t-1)^2+βσ_(t-1)^2+v_t=ω+γε_(t-1)^2+v_t+βv_(t-1)
Therefore, ε_t^2 is an ARMA(1,1) process.

Question 15

1c. What is the main difference between the data generating process of GARCH models and stochastic volatility models?

Answer 15

Innovation is introduced in the conditional variance equation.

Question 16

1e. What are the main difficulties in the use of the Hill’s estimator of the tail index?

Answer 16

A problem with the Hill estimator is that it depends on the choice of the portion q/T of the sample used for computing the statistics.

• If we choose a threshold too much in the tail, one obtains very inaccurate estimates, because just too few observations are used in the estimation.
• If we use too many observations, tail observations are contaminated by observations from the central part.

Question 17

1c. What kind of information about the volatility is provided by the news impact curve?

Answer 17

The news impact curve relates past return shocks (news) to current volatility. It measures how new information is incorporated into volatility estimates.

Question 18

1d. When does the Gram-Charlier distribution fail to fit financial returns?

Answer 18

When Kurtosis and Skewness fall out of domain of definition.

Question 19

1e. How do you describe the Hill’s estimator of the tail index?

Answer 19

The Hill estimator of the tail index is obtained by calculating:
ξ ̂q^H=1/(q -1) ∑(j=1)^(q-1)〖log⁡〖(x_( j))〗-log⁡〖(x_q)〗 〗for the minimum
ξ ̂q^H=1/(q -1) ∑(j=1)^(q-1)〖log⁡〖(x_( T-j+1))〗-log⁡〖(x_(T-q+1))〗 〗 for the maximum
The Hill’s index is asymptotically normal, with √q (ξ ̂_q^H-ξ)~N(0,ξ^2)
If q grows such that q / T → 0 as T →∞ and Xt is iid, then ξ ̂_q^H is a strongly consistent estimator of ξ . For non-iid but stationary sequences, ξ ̂_q^H remains weakly consistent, provided the dependency is not too strong

Question 20

1f. How would you adapt the extreme value theory in the case of time-dependent returns?

Answer 20

1. Fit a conditional mean and volatility model

2. Use EVT techniques to model the tail distribution of the standardized data.

Question 21

1g. How do you construct a QQ-plot?

Answer 21

Case of the gev:
Let {y1,…, yτ} be the standardized maxima over N-histories y_τ=(m_τ-μ_T)/φ_T and y_1≤⋯≤y_τ the ordered maxima, then the plot {y_t,H_ξ^(-1) (t/τ)} is named the quantile plot (with p=t/τ ).
yt → empirical quantiles (t/τ)
H_ξ^(-1) (t/τ) → theoretical quantiles consistent with the distribution Hξ

Question 22

1h. What are the main advantages of using a DCC model instead of a standard multivariate GARCH model?

Answer 22

• Describe the dynamics of covariances but not of correlations, which we are mostly interested in.
• The conditional correlation matrix (Γt) is guaranteed to be positive definite after imposing a few restrictions
• Conditional variances (θV) and the conditional correlations (θC) can be estimated separately.

Question 23

1i. What is the relation between the cdf of a joint distribution and the corresponding copula, as stated by Sklar’s theorem?

Answer 23

Let H be a joint distribution function of X and Y with marginal distributions F and G, respectively. Then,
if C is a copula and F and G are univariate distribution functions, then H(x,y)=C(F(x),G(x)) is a joint distribution function with marginal distributions F and G.

Question 24

1j. What are the advantages and drawbacks of the Elliptical copulas relative to the Archimedean copulas?

Answer 24

Disadvantage of Elliptical vs Archimedean: most Archimedean copulas have closed form expressions.
Advantage of Elliptical vs Archimedean: multivariate extensions of Archimedean copulas are somewhat difficult to establish.

Question 25

How would you explain the success of the Integrated GARCH model?

Answer 25

In many cases, estimates of the GARCH(1,1) model on returns yield α +β ≈ 1, so that the conditional variance is nearly integrated. Hence, many models of volatility have assumed Integrated GARCH processes for its simplicity..

Question 26

1d. Why is it really necessary to test the adequacy of a given non-normal distribution in case of Maximum-Likelihood estimation?

Answer 26

The consistency of the QMLE is not guaranteed if we use an incorrect distribution to maximize the likelihood. They show that consistency of a non-normal QMLE is achieved only if either
(1) the conditional mean is identically zero; or
(2) the assumed (theoretical) and true (empirical) error pdfs are symmetric about zero.
When these two conditions are not satisfied, the QMLE may be inconsistent. The reason is that we fail to capture the effect of the asymmetry distribution on the conditional mean. Thus, a crucial issue when a non-normal likelihood is used for the QML estimation is whether adequacy tests confirm that the assumed distribution correctly fits the data.

Question 27

1g. Are first-generation multivariate GARCH models well designed to test for a constant correlation?

Answer 27

First-generation multivariate GARCH models describe the dynamics of covariances but not of correlations, which we are mostly interested in. Thus, they are not well designed to test for a constant correlation.

Question 28

1j. In case of non-normal returns, when do you expect that the mean-variance criterion performs well?

Answer 28

If the investor only cares about mean and variance of his/her portfolio, nothing has to be changed in the case of non-normality.

Question 29

How would you estimate a GARCH model if you do not know (and do not careabout) the distribution of the innovation process?

Answer 29

If we do not want to acknowledge that returns are normal, we estimate the GARCH model using the QML estimation technique. The parameter estimates are unchanged, but the standard errors are now estimated without assuming normality. Under normality, the asymptotic covariance matrix Σ is the inverse of the information matrix A0. When normality is not assumed, we obtain the robust covariance matrix Ω=A_0^(-1) B_0 A_0^(-1). The main effect of estimating QML standard errors is to increase the uncertainty about the parameter estimates.

Question 30

1d. What should be the distribution and dynamics of the probability integral transform ut =G(zt) if the estimated distribution G is consistent with the true distribution?

Answer 30

Cdf has to be uniformly distributed and iid required.

Question 31

1b. Give the main characteristics of a GARCH-in-Mean model and explain the rationale behind it and why it is difficult to estimate.

Answer 31

r_t =ϕ⋅σ^2 +ε_t. Several specifications are possible and the ARCH/GARCH specification is as usual. The mean of the return equation is a risk premium for the risk an investor is taking. We expect assets with higher variance/risk to have a higher remuneration for that risk. Estimation requires estimating both equations simultaneously instead of a two-step procedure.

Question 32

1e. What is the main problem in the test of constant correlation between two-time series, when it is based on the equality of the correlations over two subsamples?

Answer 32

We know from conditional correlations that we should expect higher or lower correlations depending on the volatility. Therefore, a change in correlations does not necessarily mean that correlations indeed changed.

Question 33

1g. Why is the Kendall’s tau a better measure of dependence than the usual Pearson’s correlation coefficient?

Answer 33

Pearson’s correlation is a valid measure for linear dependence within the elliptical family of distributions. Kendall’s tau also captures non-linear associations.

Question 34

1h. What is the main motivation of using copula models from an estimation point of view compared to standard joint distributions?

Answer 34

Estimation is possible in two steps. We do not need to refer to multivariate distributions, which often do not exist, and we can assume different margins for the different assets

Question 35

1i. What are the main drawbacks of historical simulations to estimate the Value-at-Risk?

Answer 35

It assumes iidness of returns and it creates rare but sharp changes in the level of VaR (ghost features)

Question 36

1e. Why do factor models help in dealing with estimation error?

Answer 36

Reduce the number of unknown parameters

Question 37

1g. How can we avoid the ghost effect in the estimate of the VaR based on the historical simulation approach?

Answer 37

Evaluate with EWMA and put more weights on new information

Question 38

1a. What are the advantages and drawbacks of the semi-parametric GARCH estimation?

Answer 38

Ads:
• avoids some problems of distribution mis-specification

• capture the non-normality directly

Question 39

1c. How are related the probability that the maximum over subsamples of size N reaches a given level and the probability that the return reaches this level? Under what assumption?

Answer 39

probability that the level x is reached for the maximum/minimum of the distribution.
θ=Pr⁡[m≤x]=Pr⁡[r_(t,1)≤x,…,r_(t,n)≤x]=(〖Pr⁡〖[r_t≤x])〗〗^N=(θ^* )^N
obtain that the probability that the level x is reached by a daily return is
θ^*=θ^(1/N)
Under the assumption that subsamples are iid

Question 40

1f. Why is VaR not a coherent measure of risk?

Answer 40

Because time dependency and non-normality effect affect its computation, since it is based on the unconditional distribution of returns.

Question 41

1d. What should be the distribution and dynamics of the probability integral transform ut =G(zt) if the estimated distribution G is consistent with the true distribution?

Answer 41

Cdf has to be uniformly distributed and iid required.

Question 42

1g. Why should not we use the usual Pearson’s correlation coefficient to measure concordance?

Answer 42

= Measure of association but not necessarily of concordance. Measure of concordance under normality only. ρ[X,Y] has a set of undesirable properties:

1. ρ[X,Y] not invariant not under non-linear increasing transformations.
2. ρ[X,Y] is bounded
3. ρ[X,Y] for comonotonic (or counter-monotonic) variables can be different from 1 (or –1).
4. ρ[X,Y] = 0 does not imply independence

Question 43

1h. Why is the two-step estimation approach consistent in the estimation of a copula model?

Answer 43

The log of copulas can be decomposed in two main terms. First as marginal and second to dependence distribution.

Question 44

What are the main drawbacks of the RiskMetrics approach?

Answer 44

Basic assumption: log-returns are conditionnally normal. 0.94 may be close to the actual value for some assets, but far from it for some other assets

Question 45

What happen when we assume no correlation between asset when estimating the VaR?

Answer 45

We dramatically under-estimate the VaR of the whole position. In other words, in case of crash, we are likely to have big problems.

Question 46

What are the main advantages and drawbacks of the diagonal BEKK model?

Answer 46

The number of parameters is considerably reduced for the diagonal BEKK model.
If we have a large number of series, the nbr of parameters to be estimated with the diagonal BEKK model is larger than the number of parameters of a DCC model. Since the correlation is not modeled directly, it is difficult to perform specification tests about the correlation.

Question 47

What are the main advantages and drawbacks of the DCC model?

Answer 47

One can estimate the correlation parameters and the variance parameters separately, thus the estimation is easier to perform. Also, the DCC model explicitly describes the correlation matrix and it is also more intuitive in that the dynamics in Qt, are modeled as a GARCH process as well