Question

QUESTION4 (a) Let e be a zero-mean, unit-variance white noise process. Consider a process that begins at time t = 0 and is defined recursively as follows. Let Y0 = ceo and Y1-CgY0-ei. Then let Y,-φ1Yt-it wt-1-et for t > ï as in an AR(2) process. Show that the process mean, E(Y.), is zero. (b) Suppose that (a is generated according to }.-10 e,-tet-+扣-1 with e,-N(0.) 0 Find the mean and covariance functions for (Y). Is (Y) stationary? Justify your answer. (i) Determine p, and p2 (iii) Using (ii) or otherwise, determine and φ22. QUESTION5 (a) From a series Y of length 100, the sample autocorrelations at lags 1-3 are 0.8, 0.5 and 0.4, respectively. Furthermore, the respective sample mean and sample variance of the series are Ỹ-2 and s-. Suppose that the appropriate model for the series is the AR(2) model where o,, o2 and θί are model parameters, and 6,, are independent and identically distributed random variables with mean 0 and variance ơ2 Find the method of moments estimates of θο-0,1-Og and σ2. (b) For the ARMA(1,2) model Y 0.8Y-et 0.7e-1+0.6et-2, show that: (i) p,-0.8A-i for k > 2. (ii) ρ,-0.8p, +0.6/70.

Answer 1

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