FMBF-Seminar1-Answers.doc

Financial Modelling and Business Forecasting
Seminar 1

Question 1 Write down the algebraic form for the following time-series models:

i. A stationary AR (1) model with a constant and iid errors with mean zero and variance 1.

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ii. A stationary AR (2) model with a constant, trend and normal iid errors with mean zero and variance 2.

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iii. An MA (2) with a constant and iid errors with mean zero and variance 1. Is it stationary?

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It is stationary because

,

and

iv. A stationary ARMA (2, 1) with a constant and iid errors with mean zero and variance 1.

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v. A random walk with iid errors with mean zero and variance 1.

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Question 2 Derive the mean, the variance and covariances of the AR (1) process in Question1 i.

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Note that in determining the variance and covariances of AR processes, we assume that the intercept parameter , so that the mean of the time-series is Setting is equivalent to measuring the series in terms of deviations about its mean, or (). This adjustment in the mean does not affect the variances or covariances of the time series is demonstrated as follows:

,
We have shown above that hence . Thus we can write:

When deriving autocovariances and autocorrelations, this last equation is easier to work with because the model has no intercept term and has a zero mean. To simplify the notation we do not add the star in the derivations.

Question 3 Derive the mean, variance of the random walk in Question 1 iii. Why is the random walk not stationary? How can you transform a random walk into a stationary process?

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By iterative substitution we obtain:

The random walk is nonstationary because its variance varies with time.

By differencing, we obtain a stationary process: .

Question 4 Derive the mean and variance of a random walk with a constant (drift). Why is the random walk with drift not stationary?

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It is clearly nonstatinary because the mean and the variance vary with time.

Question 5 Derive the mean and variance of a constant plus a time trend with iid errors with mean zero and variance 1. Is the process stationary.

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It is obviously nonstationary as the mean varies with time.

Question 6 Explain Box-Jenkins methodology for univariate time series modelling.

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See lecture note 2.

Relevant reading: lecture notes 1 and 2, HS Ch. 1 and 2, Brooks Ch. 5 up to page 258.