Exercises

Question 1

What are the steps to derive the unconditional variance?

Usually, e.g. when deriving from a random walk

Answer 1

Usually you want to rewrite the model to sum e.g. an AR or AR-MA model. Then you can derive it.

Question 2

Derive the ARCH(inf) of a GARCH(1,1)

Answer 2

Question 3

Solve

Answer 3

Question 4

How to find the derivation of a function of sigma with respect to theta?

Answer 4

Find the derivative of sigma with respect to each variable in theta.

Question 5

What is the Newton-Rapson algorithm?

Answer 5

x^(k+1) = x^(k) - f’(x^(k))/f’‘(x^(k))

Question 6

What is the normal density?

Answer 6

Question 7

How to test a parameter from a model, while knowing Omega hat?

Answer 7

Fill this value into a normal cdf (obv.)

Question 8

How to draw a News Impact Curve?

Answer 8

y_t = 0 is omega,
y_t = +/-1 is omega + alpha,
y_t = +/- 2 is omega + alpha^2
etc.

Question 9

Do you want higher or lower AIC/BIC?

Answer 9

Lower is better

Question 10

Which of AIC/BIC should always be lower?

Answer 10

BIC since it “punishes” parameters more

Question 11

What is z_0.05? And z_0.025?

Answer 11

-1.64, -1.96

Question 12

When is the approximation for large h of sigma(h) better?

Answer 12

When alpha + beta is highest

Question 13

What is the cond. expectation E[X | Y] and cond. variance Var[X | Y] of a bivariate model with

Answer 13

Question 14

What is a requirement for a valid covariance matrix of a DVECH model (and other multivariate GARCH models)?

Answer 14

It needs to be symmetric (diagonally)

Question 15

What is the uncond. variance of a VECH model

Answer 15

(I - A_1 - A_2 -..)W

Question 16

What is corr rho?

Answer 16

rho_12 = sigma_12/(sigma_1 sigma_2)

Question 17

What are important facts you need to know when deriving the kurtosis?

Answer 17

• The kurtosis of a normal distribution is three
• sigma^4 = sigma^2*2 = exp(2mu +2sigma)

Question 18

A LogN + LogN distribution is?

Answer 18

LogN

Question 19

Cov(a, b) = ?

In terms of the expectation

Answer 19

E[ab] - E[a]E[b]

Question 20

How do you show that an indirect estimator is efficient?

Answer 20

Question 21

If you have to check whether the auxiliary statistics of a estimation make sense, what do you need to look at?

Answer 21

1. Are there enough auxiliary statistics
2. Are the statistics relevant (e.g., no skewness for a symmetric distribution)

Question 22

Which type of models make sense to check an SV model with using aux. statistics?

Answer 22

Models that check y^2_t and the autocov.

Question 23

What are the steps of deriving the loglikelyhood?

Answer 23

1. Find the mean and variance of the observation equation
2. Fill these values into a normal pdf (1/sqrt(2pi sigma^2))exp((y - mu)^2/2sigma^2)
3. Sum these into a log likelyhood, you may remove constants
4. Start the sum at a t for which you have values

Question 24

Answer 24