Lecture 10

Question 1

What are the distinction between condition vs unconditional distribution ?

Answer 1

• Time varying volatility → fat tails in unconditional distribution but not enough to fit actual data
• Introduce non-normal distribution to adjust data

Question 2

How to extend normal distribution into these two directions of time varying volatility and non-normal distribution

Answer 2

• Natural extensions to normal distribution allowing fat tails (student)
• But distribution not designed to capture asymmetry

Question 3

If r(t) is a time series of realization of log-returns, how to break down its dynamics in 3 ?

Answer 3

• Conditional mean = location parameter
• Conditional variance = scale parameters
• Conditional distribution = shape parameters

Question 4

In r(t) = μ(θ) + ϵ(t), what does θ include ?

Answer 4

All parameters associated with conditional mean and variance equations

Question 5

What are the several issues related to modeling of non-normal returns ?

Answer 5

• Unconditional distribution = non-normal → conditional distrribution g(.) also non-normal ?
Whatever you do, you generate non-normal returns so the answer is no. Even if you filter for the volatility
• Conditional distribution non.normal → model it explicietly ?
Yes we can, we can have a model where we don’t have the conditional distribution but we can estimate parameters.

• Model conditional explicitly → asymmetry and fat-tailedness ?

Question 6

What is one of the attractive feature of Garch model ?

Answer 6

Even if conditional distribution of innovations z(t) = normal, unconditional distribution of ϵ(t) has fatter tails than normal.

Question 7

What does symmetric conditional distribution imply ?

Answer 7

Garch effects do not imply asymmetric distribution

Question 8

What does an asymmetric conditional distribution imply ?

Answer 8

|Su| > |Sc|

Question 9

What happens if conditional distribution g not normal ?

Answer 9

ML approach not directly used since based on normality assumption

Question 10

What if the first and second moments are correctly specified ?

Answer 10

Consitent estimate θ by maximizing normal likelihood under conditional normality assumption even if true distribution not normal

Question 11

What is the difference between MLE and QMLE ?

Answer 11

• MLE : maximize assuming true conditional distribution of errors = normal
• QMLE : maximize normal likelihood even when true distribution not normal

→ same estimates of θ since same maximization problem

→ covariance matrices of estimator differs → QMLE computed without assuming conditional normality

Question 12

What is the asymptotic of θ(QMLE) ?

Answer 12

√T (θ(QML) - θo) ̴ N(0,Ω)

Question 13

What does QMLE provide ?

Answer 13

robust standard errors → asymptotically valid confidence intervals for estimators

Question 14

How is the sandwich estimator obtained ?

Answer 14

As square roots of diagonal elements of matrix

• Ω = A^-1BA^-1
• A = hessian
• B = outer product of gradients

Question 15

What is different between the sandwich estimator for finite sample ?

Answer 15

It is evaluated at θ(QML)

Question 16

By what is the asymptotic covariance matrix of ML estimator given ?

Answer 16

By inverse of Hessian since B = A under normality :

√T (θ(ML) - θo) ̴ N(0,A^-1)

Question 17

What happens for normal likelihood when using QMLE ?

Answer 17

QMLE = consistent since robust w.r. to true distribution of model

Question 18

What happens when the true distribution is not normal while using the QMLE ?

Answer 18

It is inefficient because:
• Degree of inefficiency increases with degree of departure from normality
o Simulation evidence 84% loss under normality instead of MLE
o Loss of efficiency of 59% if t with 5 degrees of freedom

→ Use correct distribution of Zt = improve efficiency of estimator

Question 19

What is the common practice for inefficiency ?

Answer 19

Using non-normal distribution + robust covariance matrix

Question 20

When is the consistency of non normal QMLE achieved ?

Answer 20

• Assumed & true errors pdfs = unimodal and symmetric around 0
• Conditional mean of return process = identically 0

→ if 2 conditions not satisfied = inconsistent QMLE since fails to capture effect of asymmetry of distribution on conditional mean

Question 21

How can we handle this problem of unconsistency ?

Answer 21

By introducing additional location parameter making QMLE robust to asymmetry

Question 22

What is a crucial issue when non normal likelihood used for QMLE ?

Answer 22

Adequacy tests confirm assumed distribtuion fits data

Question 23

What is the moment problem ?

Answer 23

Sequence of moments {μ(j)} → necessary and sufficient conditions for existence and expression for positive pdf.

Question 24

What is the Hamburger moment problem ?

Answer 24

nsc = moment matrices = positive definite for alll n

→ det ||μ|| ≥ 0 where ||μ|| = Henkel matrix

Question 25

What are the maximal values of skewness and kurtosis ?

Answer 25

μ(3) ≤ μ(4) - 1 with μ(4) ≥ 1

Question 26

What are the several ways of dealing with asymmetry or fat tails in distribution ?

Answer 26

* Non or semi parametric estimation → capture non normality directly * Asymmetry introduced using expansion about symmetric distribution → can be applied to any symmetric distribution * Exists distributions with asymmetry or fat tails → skewed student t distribution

Question 27

What does it mean when we have a semi-parametric ARCH model ?

Answer 27

* 1st and 2nd moments given by ARMA process and ARCH model | * Condional density approximated by non parametric and ARCH model

Question 28

What is the procedure to maximize the log likelihood function with semi parametric estimation ?

Answer 28

* Initial estimate of θ given by θ' (obtained by QMLE) * Fitted resituals and fitted variances used to compute standardized residuals with mean zero and unit variance * Density g(z(θ)) estimated using non parametric method to get density g(t) * Compute log-likelihood → maximize with g(t) fixed and iterate steps 2-4 until convergence

Question 29

What does semi-paramteric method avoid ?

Answer 29

Problems of distribution mis-specification, since using non normal distribution may lead to inconsistent parameter estimates if distribution incorrect

Question 30

What if we believe that the true pdf of the random variable Z is close to normal ?

Answer 30

Use approximation of pdf around normal density : g(z|η) = φ(z)p(n)(z|η)

Question 31

What does φ(z) represent ?

Answer 31

standard normal density with mean 0 and unit variance

Question 32

How is p(n)(z|η) chosen ?

Answer 32

s.t. g(z|η) same firest moments as pdf of z

Question 33

What is the special case fore series expansion ?

Answer 33

Gram-Charlier type A expansion | → describe deviations from normality of innovations in GARCH framework

Question 34

What are the properties of the Gram- Charlier distribution ?

Answer 34

• GC expansions allow for additional flexibility over normal distribution since introduce skewness and kurtosis of distribution as unknown parameters ``` • First four moments of Z o E(Z) = 0 o V(Z) = 1 o Sk(Z) = m(3) o Ku(Z) = m(4) ```

Question 35

What are the drawback of GC expansions ?

Answer 35

* For moments (m3,m4) distant from normality (0,3) → negative g(.) for some z * Pdf may be multimodal * Domain of definition = small

Question 36

What are the characteristics of the student t distribution ?

Answer 36

* Capture fat tails of financial returns * Symmetric distribution * Normalized

Question 37

What are the characteristics of the skewed student t distribution ?

Answer 37

* Capture fat tails and asymmetry of distribution * Introduces generalized student-t distribution with asymmetries and zero mean and unit variance * Parameters depend on past realizations + subsequently higher moments may be time varying

Question 38

What is the shape parameter ?

Answer 38

η = (v,λ)' with • v = degree of freedom parameter • λ = asymmetry parameter o = 0 → traditional student o = 0 and v = ∞ → normal distribution

Question 39

When does the desity and the various moments exist for all parameters ?

Answer 39

* Only for v > 2 and -1 < λ < 1 * Skewness exists if v > 3 * Kurtosis exists if v > 4

Question 40

What are the constraints of the domain of definition of distribution ?

Answer 40

(v,λ) ϵ ]2,∞[x[-1,1[

Question 41

On what are the adequacy tests based ?

Answer 41

Distance between empirical distribution and assumed distribution

Question 42

On which steps is the adequacy test based ?

Answer 42

• Test whether ut is serially correlated using standard LM test o Regress [u(t)- û]^I on k lags of variable o LM statistic • Test Ho: ut is U(0,1) o Cut empirical and theoretical distribution into N cells o Test if 2 distributions same using Pearson’s test statistic

Question 43

On what are based the simple estimate of standard error of estimated number observations ?

Answer 43

Binaomial distribution

Question 44

What happens to Fn under Ho ?

Answer 44

Fn ̴ drawn from binomial distribution B(T,p) where T = number of observations and p = 1/N = probability to fall in cell n

Question 45

What is the expectation of Fn and its variance ?

Answer 45

* E[Fn] = Tp = T/N | * V[Fn] = Tp(1-p)

Question 46

What are the confidence band for Fn ?

Answer 46

E[Fn] ± 2√V[Fn]

Question 47

How is the binomial distibution in Pearson's test statistic ?

Answer 47

≅ N(Tp,Tp)