1. (a) What stylised features of financial data cannot be explained using linear time series models?

(b) Which of these features could be modelled using a GARCH(1,1) process?

(c) Why, in recent empirical research, have researchers preferred GARCH(1,1) models to pure ARCH(p)?

(d) Describe two extensions to the original GARCH model. What additional characteristics of financial data might they be able to capture?

(e) Consider the following GARCH(1,1) model yt = x + ut, ut - N(0,at2) (8.110)

If yt is a daily stock return series, what range of values are likely for the coefficients x, a0, ai and p?

(f) Suppose that a researcher wanted to test the null hypothesis that ai + p = 1 in the equation for part (e). Explain how this might be achieved within the maximum likelihood framework.

(g) Suppose now that the researcher had estimated the above GARCH model for a series of returns on a stock index and obtained the following parameter estimates: fx = 0.0023, a0 = 0.0172, pi = 0.9811, ai = 0.1251. If the researcher has data available up to and including time T, write down a set of equations in at2 and u2 their lagged values, which could be employed to produce one-, two-, and three-step-ahead forecasts for the conditional variance of yt. (h) Suppose now that the coefficient estimate of p for this model is

0.98 instead. By re-considering the forecast expressions you derived in part (g), explain what would happen to the forecasts in this case.

2. (a) Discuss briefly the principles behind maximum likelihood.

(b) Describe briefly the three hypothesis testing procedures that are available under maximum likelihood estimation. Which is likely to be the easiest to calculate in practice, and why?

(c) OLS and maximum likelihood are used to estimate the parameters of a standard linear regression model. Will they give the same estimates? Explain your answer.

3. (a) Distinguish between the terms 'conditional variance' and

'unconditional variance'. Which of the two is more likely to be relevant for producing:

1. 1-step-ahead volatility forecasts ii. 20-step-ahead volatility forecasts.

(a) If ut follows a GARCH(1,1) process, what would be the likely result if a regression of the form (8.110) were estimated using OLS and assuming a constant conditional variance?

(b) Compare and contrast the following models for volatility, noting their strengths and weaknesses:

i. Historical volatility ii. EWMA

iv. Implied volatility.

4. Suppose that a researcher is interested in modelling the correlation between the returns of the NYSE and LSE markets.

(a) Write down a simple diagonal VECH model for this problem. Discuss the values for the coefficient estimates that you would expect.

(b) Suppose that weekly correlation forecasts for two weeks ahead are required. Describe a procedure for constructing such forecasts from a set of daily returns data for the two market indices.

(c) What other approaches to correlation modelling are available?

(d) What are the strengths and weaknesses of multivariate GARCH models relative to the alternatives that you propose in part (c)?

5. (a) What is a news impact curve? Using a spreadsheet or otherwise, construct the news impact curve for the following estimated EGARCH and GARCH models, setting the lagged conditional variance to the value of the unconditional variance (estimated from the sample data rather than the mode parameter estimates), which is 0.096

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