(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 = μ + ut , ut ∼ N



0, σ2 t



(8.110)

σ2 t

= α0 + α1u2t

−1

+ βσ2 t−1 (8.111)

If yt is a daily stock return series, what range of values are likely for the coefficients μ, α0, α1 and β?

(f) Suppose that a researcher wanted to test the null hypothesis that

α1 + β = 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: μˆ = 0.0023, αˆ 0 = 0.0172,

ˆβ

= 0.9811, ˆα1 = 0.1251. If the researcher has data available up to and including time T , write down a set of equations in σ2 t and u2t 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 ˆβ 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.

AppendixLO1