Ch.2 Univariate Modelling

Question 1

What is the simplest model of volatility, and what is the formula for conditional volatility.

Answer 1

Moving Average, with

Question 2

What are two drawbacks of Moving Averages?

Answer 2

1. Very sensitive to size of estimation window.
2. Weights all history equally.

Question 3

What are the weights given to each history in an EWMA? How does one ensure that the sum is equal to one?

Answer 3

To ensure sum of 1, think about the sum of a power series.

Question 4

Using the Sum equation of the EWMA, derive an equation without sum, containing only a lagged y and a lagged conditional volatility.

Answer 4

Question 5

What is the usual value of lambda in EWMA?

Answer 5

0.94

Question 6

What is one characteristic of unconditional volatility in EWMA?

Answer 6

It doesn’t exist.

Question 7

What is the unconditional volatility of an ARCH(1) model. How did you derive it?

Answer 7

See slides 33-37 in Lecture 2.

Question 8

What moment is usually used to assess tail-fatness?

Answer 8

Kurtosis: Fourth moment divided by the squared second moment.

Kurtosis higher than 3 means fatter tails.

Question 9

What are two parameter restrictions on the ARCH(1) model? Do we always impose them?

Answer 9

1. Positive Parameters - always assumed.
2. Stationarity, i.e. alpha is between 0 and 1. - Not always assumed.

Question 10

Given an ARCH(1) model, what happens to unconditional volatility if we do not assume stationarity?

Answer 10

It doesn’t exist.

Question 11

What is the main difference between a GARCH and ARCH model?

Answer 11

ARCH only uses passed square returns while GARCH use past values of the dependent variable.

Question 12

Calculate the unconditional volatility of an ARCH(1) and GARCH(1,1) model.

Answer 12

Question 13

Why is stationarity often not imposed for GARCH(1,1) model.

Answer 13

1. Misspecification
2. Multiple Solutions to objective function which we maximize

Question 14

How do we transform a GARCH model into an EWMA model. What is its unconditional variance?

Answer 14

Question 15

Give a brief explanation about the meaning of Alpha, Beta and their sum in the GARCH model.

Answer 15

Alpha: how vol. reacts to new information.
Beta: How much vol. remembers the past.
Sum: Predictability

Question 16

GARCH Half-Life??

Answer 16

60-62…. (Just the formula.)

Question 17

What are two downsides of tGARCH?

Answer 17

1. Needs much more observations than normal GARCH.
2. Degrees of freedom estimates are highly imprecise.

Question 18

Maximum likelihood?

Answer 18

Not sure it’ll be on the exam.

Question 19

Give one test to compare different nested models.

Answer 19

Likelihood Ratio Test. Higher values suggest better performance.H

Question 20

How would you test for model specification?

Answer 20

1. QQ plot of residuals to check for normality.
2. Formal tests of residual autocorrelation like LB tests.
3. If different models are nested, one can use Likelihood Ratio Test.

Question 21

What is an implied model of volatility.

Answer 21

Models volatility using Black-Scholes Model. It is based on options pricing.

Question 22

What is the formula of GARCH’s half life?

Answer 22

Question 23

What is the minimum number of data points needed to estimate a GARCH model?

Answer 23

500

Question 24

What is the minimum number of data points needed to estimate a t-GARCH model?

Answer 24

3000