# Daily sales figures for a store were found to be summarized well by the ARIMA(2,1,0)…

4. Daily sales figures for a store were found to be summarized well by the ARIMA(2,1,0) model t – 2.4t-1 + 2.1t-2 – 0.7t-3 = At, where ˆvA = 58, 000. Sales figures this week (5 days) were, in order, 560, 580, 640, 770, and 800. Compute forecasts for the next three days. 5. A stationary Gaussian process {t} has correlations satisfying |?±1| = 1 2 and ?` = 0 if |`| = 2. Show that the process has an MA(1) representation. [Hint: Define At = P8 j=0 ? j t-j for a value of ? which you must determine, and verify the conditions which define an MA(1) process.] 6. This is a simple stochastic volatility model. You will have to chase up a reference for even-order moments of the standard normal law, or compute them using its moment generating function. Let At = w 2 t-1wt where the wt’s comprise an independent sequence of Gaussian N(0, v) innovations. (a) Show that E(At) = 0, V ar(At) = 3v 3 and that the At’s are uncorrelated. (b) Compute V ar(A 2 t ) and show that the sequence (A 2 t ) has MA(1)-type correlations, i.e., they are zero for lags |`| = 2, and non-zero for |`| = 1. Compute this correlation. (c) It follows from Question 5 that there is an MA(1) representation Yt := A 2 t = Bt – ?Bt-1, where ? is a constant which you should specify, and (Bt) is a white noise process. Explain why it cannot be Gaussian white noise, and why the Bt’s cannot be independent. 7. Let I0, I1, . . . be independent and all with a Bern( 1 2 ) distribution. Define a random variable

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The University of Western Australia School of Mathematics & Statistics STAT7450: Time Series Methods and Applications Assignment 3 (2012) Submit solutions to all except #5 by the end of semester. It’s solution is given below, and it indicates that you may have to dig deeply to answer some of the other questions. 1. (a) Consider the AR(1) model t = 3t-1 + At. Decide whether it is stationary, giving reasons. (b) Show that the process defined by t = -Pj1 3-jAt+j is stationary and solves the AR(1) difference equation in (a). (c) In what way is {t} unsatisfactory for data modelling? 2. Let Xt = 0.9Xt-1 + 0.09Xt-2 + At and Zt = Zt-1 + At – 0.1At-1. Identify each model as a specific ARIMA model. In which ways are they similar/different? You will need to think about their AR(1) and MA(1) representations. 3. The following table gives sample correlations ` for a series {t : 1 t 100}, and correlations ` for its differenced version rt: ` 1 2 3 4 5 6 ` 0.97 0.97 0.93 0.85 0.8 0.71 ` -0.42 0.18 -0.02 0.07 -0.01 -0.09 Determine an ARIMA model which could have these correlations. 4. Daily sales figures for a store were found to be summarized well by the ARIMA(2,1,0) model t – 2.4t-1 + 2.1t-2 – 0.7t-3 = At, where ˆvA = 58, 000. Sales figures this week (5 days) were, in order, 560, 580, 640, 770, and 800. Compute forecasts for the next three days. 5. A stationary Gaussian process {t} has correlations satisfying |±1| 12 and ` = 0 if |`| 2. Show that the process has an MA(1) representation. [Hint: Define At = P1j=0 jt-j for a value of which you must determine, and verify the conditions which define an MA(1) process.] 6. This is a simple stochastic volatility model. You will have to chase up a reference for even-order moments of the standard normal law, or compute them using its moment generating function. Let At = w2t-1wt where the wt’s comprise an independent sequence of Gaussian N(0, v) innovations. (a) Show that E(At) 0, V ar(At) 3v3 and that the At’s are uncorrelated. (b) Compute V…

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