ams316_hw2

# ams316_hw2 - AMS316 HW2 Due Oct 19, 2009 1. Show that the...

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Unformatted text preview: AMS316 HW2 Due Oct 19, 2009 1. Show that the acf of teh mth-order MA process given by m Xt = k =0 Zt−k /(m + 1) is ρ(k ) = (m + 1 − k )/(m + 1) k = 0, 1, . . . , m 0 k>m 2. Show that the inﬁnite-order MA process {Xt } deﬁned by Xt = Zt + C (Zt−1 + Zt−2 + . . . ) where C is a constant, is non-stationary. Also show that the series of ﬁrst diﬀerences {Yt } deﬁned by Yt = Xt − Xt−1 is a ﬁrst-order MA process and is stationary. Find the ACF of {Yt }. 3. Find the values of λ1 and λ2 such that the 2nd-order AR process deﬁned by Xt = λ1 Xt−1 + λ2 Xt−2 + Zt is stationary. If λ1 = 1/3, λ2 = 2/9, show that the ACF of Xt is given by ρ(k ) = 16 2 27 3 |k | + 1 5 − 21 3 |k | , k = 0, ±1, ±2, . . . 4. Show that the ACF of the stationary 2nd-order AR process Xt = is given by ρ(k ) = 1 1 Xt−1 + Xt−2 + Zt 12 12 1 32 − 77 4 |k | 45 1 77 3 |k | + , k = 0, ±1, ±2, . . . ...
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