## Practice Questions final exam

### Question 1

Consider the constant mean ARCH(1) model:

a) Show that the unconditional variance of at is given by:

b) Write down the conditions required to ensure that both

i.the conditional variance, σt 2 , is positive, and

ii.the unconditional variance exists and is positive?

c) The fourth moment of the at series is given by

What additional conditions are required to ensure positivity of this fourth moment?

d) The unconditional kurtosis of the at series is defined as

Find an expression for the unconditional kurtosis, κ, in terms of the model parameters.

e) Discuss when the unconditional kurtosis found in d) will be greater than 3 and the implications for modelling stock returns.

f) The table below shows ML and LS estimates for the ARCH(1) models for CBA and NWS

Write down the estimated models for CBA and NWS using maximum likelihood estimators;

g) What is the average volatility for CBA and NWS when you use ML estimates?

### Question 2  加拿大统计Final exam代考

a) Discuss one advantage and one disadvantage of each of the following measures of risk:

a.Volatility

b.VaR

c.ES

b) Discuss how you could use Historical Simulation to obtain estimates of each of these measures.

c) Can Historical Simulation allow for heteroskedasticity in returns? If so how?

d) Discuss one advantage and one disadvantage of Historical Simulation for estimating market risk

e) If the change in value distribution over the next h periods is in Gaussian ∆𝑉|𝐹~𝑁(𝜇, 𝜎2),derive the VaR and ES at probability level p. You can use the following result to help you:

Here 𝜙(. )and Φ(. ) represent the probability density function and the cumulative distribution function, respectively, for a standard normal variable.

f) Provide formulae for VaR and ES under the assumptions of both the Risk Metrics and the Independent Normal models?

g) Give an example of each of the following model types for log returns:

a.Parametric

b.Non-parametric

c.Semi-parametric

h) Describe the steps involved in Monte Carlo simulation of VaR and ES for 10-day returns using an AR(1)-GARCH(1,1) model?

### Question 3  加拿大统计Final exam代考

Consider the AR(1)-GJR-GARCH(1,1) model with Gaussian errors:

a) Derive the conditional mean of rt, stating any required assumptions. Under what conditions will the mean be stationary?

b) Derive the unconditional variance of rt. Under what conditions will the variance be stationary?

c) Write down conditions to ensure the conditional variance is always positive.

d) Write down the full likelihood function for this model.

e) Write down the conditional likelihood function for this model.  加拿大统计Final exam代考

f) Write down the conditional log-likelihood function for this model.

g) Derive expressions for the average variance in each regime, i.e., 𝑉ar(𝑎t|𝑎t-1 < 0) and𝑉ar(𝑎t|𝑎t-1 ≥ 0).

h) What is the persistence in each regime?

i) Discuss ways in which we can assess the level of asymmetry for an estimated AR (1)-GJR-GARCH(1,1) model.

j) When plotting NIC curves for a GJR-GARCH model discuss the advantages and disadvantages of using standardised and non-standardised residuals?

k) Write down the conditional volatility equation for a GJR-ARCH(2) model.

### Question 4

Consider the constant mean GARCH (1,1) model:

a) If , where 𝜂t is the innovation of the squared process, show that

b) Use this expression to derive the unconditional variance of returns for the GARCH(1,1) model.

c) The excess kurtosis of 𝑎𝑎𝑡𝑡is given by:

where 𝐾𝐾𝜀𝜀 is the excess kurtosis for the specific distribution assumed for 𝜀𝜀𝑡𝑡.

d) Find an expression for 𝐾𝐾𝑎𝑎 𝜙𝜙 the excess kurtosis of 𝑎𝑎𝑡𝑡for a Gaussian GARCH(1,1) model.

e) Use the espresso in in d) to find conditions for the excess kurtosis to exist for a Gaussian GARCH(1,1) model.

f) What is the excess kurtosis if 𝛼𝛼1 = 0?

g) What is the excess kurtosis if 𝛽𝛽1 = 0? (i.e. for an ARCH(1,1) model)?

l) Write down the full quasi-likelihood function for this model.

m) Write down the conditional quasi-likelihood function for this model.

n) Write down the conditional quasi-log-likelihood function for this model.

### Question 5  加拿大统计Final exam代考

a) How can we assess the accuracy of volatility forecasts?

b) Describe, compare and contrast the 4 volatility proxies used in this unit.

Consider a constant mean ARCH(m) model with volatility equation given by:

c) Find the one step-ahead volatility forecast from origin t, 𝜎𝜎𝑡𝑡 2(1) = 𝑉𝑉𝑉𝑉𝑉𝑉(𝑟𝑟𝑡𝑡+1|𝐹𝐹𝑡𝑡).

d) Find the 2-step ahead volatility forecast from origin t, 𝜎𝜎𝑡𝑡 2(2) in terms of 𝜎𝜎𝑡𝑡 2(1).

e) For a constant mean ARCH(1) model, find the one and two-step-ahead volatility forecasts, 𝜎2t(1) and 𝜎2t(2).

f) Find an expression for the h-step ahead volatility forecast for a constant mean ARCH(1) model,𝜎2t(ℎ).

g) Describe the behaviour of 𝜎2t(ℎ) as 𝜎2t(ℎ) → ∞.

h) Repeat part e) – g) for a GARCH(1,1) model with constant mean.