Lecture 10 Flashcards
What are the distinction between condition vs unconditional distribution ?
- Time varying volatility → fat tails in unconditional distribution but not enough to fit actual data
- Introduce non-normal distribution to adjust data
How to extend normal distribution into these two directions of time varying volatility and non-normal distribution
- Natural extensions to normal distribution allowing fat tails (student)
- But distribution not designed to capture asymmetry
If r(t) is a time series of realization of log-returns, how to break down its dynamics in 3 ?
- Conditional mean = location parameter
- Conditional variance = scale parameters
- Conditional distribution = shape parameters
In r(t) = μ(θ) + ϵ(t), what does θ include ?
All parameters associated with conditional mean and variance equations
What are the several issues related to modeling of non-normal returns ?
• 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 ?
What is one of the attractive feature of Garch model ?
Even if conditional distribution of innovations z(t) = normal, unconditional distribution of ϵ(t) has fatter tails than normal.
What does symmetric conditional distribution imply ?
Garch effects do not imply asymmetric distribution
What does an asymmetric conditional distribution imply ?
|Su| > |Sc|
What happens if conditional distribution g not normal ?
ML approach not directly used since based on normality assumption
What if the first and second moments are correctly specified ?
Consitent estimate θ by maximizing normal likelihood under conditional normality assumption even if true distribution not normal
What is the difference between MLE and QMLE ?
- 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
What is the asymptotic of θ(QMLE) ?
√T (θ(QML) - θo) ̴ N(0,Ω)
What does QMLE provide ?
robust standard errors → asymptotically valid confidence intervals for estimators
How is the sandwich estimator obtained ?
As square roots of diagonal elements of matrix
- Ω = A^-1BA^-1
- A = hessian
- B = outer product of gradients
What is different between the sandwich estimator for finite sample ?
It is evaluated at θ(QML)
By what is the asymptotic covariance matrix of ML estimator given ?
By inverse of Hessian since B = A under normality :
√T (θ(ML) - θo) ̴ N(0,A^-1)
What happens for normal likelihood when using QMLE ?
QMLE = consistent since robust w.r. to true distribution of model
What happens when the true distribution is not normal while using the QMLE ?
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
What is the common practice for inefficiency ?
Using non-normal distribution + robust covariance matrix
When is the consistency of non normal QMLE achieved ?
- 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
How can we handle this problem of unconsistency ?
By introducing additional location parameter making QMLE robust to asymmetry
What is a crucial issue when non normal likelihood used for QMLE ?
Adequacy tests confirm assumed distribtuion fits data
What is the moment problem ?
Sequence of moments {μ(j)} → necessary and sufficient conditions for existence and expression for positive pdf.
What is the Hamburger moment problem ?
nsc = moment matrices = positive definite for alll n
→ det ||μ|| ≥ 0 where ||μ|| = Henkel matrix
What are the maximal values of skewness and kurtosis ?
μ(3) ≤ μ(4) - 1 with μ(4) ≥ 1
What are the several ways of dealing with asymmetry or fat tails in distribution ?
- 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
What does it mean when we have a semi-parametric ARCH model ?
- 1st and 2nd moments given by ARMA process and ARCH model
* Condional density approximated by non parametric and ARCH model
What is the procedure to maximize the log likelihood function with semi parametric estimation ?
- 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
What does semi-paramteric method avoid ?
Problems of distribution mis-specification, since using non normal distribution may lead to inconsistent parameter estimates if distribution incorrect
What if we believe that the true pdf of the random variable Z is close to normal ?
Use approximation of pdf around normal density : g(z|η) = φ(z)p(n)(z|η)
What does φ(z) represent ?
standard normal density with mean 0 and unit variance
How is p(n)(z|η) chosen ?
s.t. g(z|η) same firest moments as pdf of z
What is the special case fore series expansion ?
Gram-Charlier type A expansion
→ describe deviations from normality of innovations in GARCH framework
What are the properties of the Gram- Charlier distribution ?
• 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)
What are the drawback of GC expansions ?
- For moments (m3,m4) distant from normality (0,3) → negative g(.) for some z
- Pdf may be multimodal
- Domain of definition = small
What are the characteristics of the student t distribution ?
- Capture fat tails of financial returns
- Symmetric distribution
- Normalized
What are the characteristics of the skewed student t distribution ?
- 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
What is the shape parameter ?
η = (v,λ)’ with
• v = degree of freedom parameter
• λ = asymmetry parameter
o = 0 → traditional student
o = 0 and v = ∞ → normal distribution
When does the desity and the various moments exist for all parameters ?
- Only for v > 2 and -1 < λ < 1
- Skewness exists if v > 3
- Kurtosis exists if v > 4
What are the constraints of the domain of definition of distribution ?
(v,λ) ϵ ]2,∞[x[-1,1[
On what are the adequacy tests based ?
Distance between empirical distribution and assumed distribution
On which steps is the adequacy test based ?
• 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
On what are based the simple estimate of standard error of estimated number observations ?
Binaomial distribution
What happens to Fn under Ho ?
Fn ̴ drawn from binomial distribution B(T,p) where T = number of observations and p = 1/N = probability to fall in cell n
What is the expectation of Fn and its variance ?
- E[Fn] = Tp = T/N
* V[Fn] = Tp(1-p)
What are the confidence band for Fn ?
E[Fn] ± 2√V[Fn]
How is the binomial distibution in Pearson’s test statistic ?
≅ N(Tp,Tp)
