exercises103ts

# exercises103ts - Stochastic Modelling Exercises on Time...

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Stochastic Modelling Exercises on Time Series * Dr. Iqbal Owadally March 3, 2003 Elementary Problems Q1. Rewrite the following time series models using the backward shift notation. Classify each of them as an ARIMA( p , d , q ) process (that is, determine p , d and q in each case). State whether each is (i) stationary, (ii) invertible. (i) X t = 0 . 5 X t - 1 + Z t (ii) X t - 0 . 5 X t - 1 = Z t - 1 . 3 Z t - 1 + 0 . 4 Z t - 2 (iii) X t - 1 . 5 X t - 1 + 0 . 6 X t - 2 = Z t (iv) ( X t - 0 . 2) - 1 . 2( X t - 1 - 0 . 2) + 0 . 2( X t - 2 - 0 . 2) = Z t - 0 . 5 Z t - 1 Q2. The following autoregressive processes are stationary. Calculate ρ 1 , ρ 2 and ρ 3 . (i) X t + 0 . 5 X t - 1 - 0 . 1 X t - 2 = Z t ; (ii) X t = - 0 . 6 X t - 2 + Z t ; (iii) (1 - 1 . 1 B + 0 . 18 B 2 ) X t = Z t ; (iv) X t = - αX t - 1 - α 2 X t - 2 - α 3 X t - 3 + Z t . Q3. Calculate ρ 1 and ρ 2 for the following MA processes: (i) Y t = Z t - βZ t - 1 (ii) Y t = (1 + 2 . 4 B + 0 . 8 B 2 ) Z t Q4. (i) Describe the key diﬀerence between the correlograms of a stationary AR process and an MA process of the same order. (ii) Derive the autocorrelation function for the stationary ARMA(1, 1) process: ( X t - μ ) - α ( X t - 1 - μ ) = Z t - βZ t - 1 . (iii) Comment on the correlogram of the ARMA(1, 1) process above. * Elementary problems should be attempted ﬁrst. Past exam questions are included for exam practice and could be attempted later. They are adapted from papers set by the Exam Board of the Institute and Faculty of Actuaries. Papers set by the Exam Board, Faculty of Actuarial Science and Statistics, Cass Business School, City University, are separately available. Contact details: Cass Building Room 5071, extension 8478, [email protected] 1

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Q5. The AR(1) process X t - 0 . 4 X t - 1 = Z t is started arbitrarily at t = 0 with initial condition X 0 = x 0 R . The sequence { Z t } is a set of independent and identically distributed random variables with zero mean and variance
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exercises103ts - Stochastic Modelling Exercises on Time...

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