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STAT 520 HW 1
1. Let {at} ∞ −∞ be a white noise process with E[at ] = 0 and var[at ] = σ 2 a . Define zt = µ + at − .5at−1. Find the mean, autocovariance, and autocorrelation of zt , and verify that {zt} ∞ −∞ is stationary.
2. Let {yt} be a stationary process with mean µy and autocovariance γy(s) = cov[yt , yt−s]. Define zt = yt −yt−1. Obtain the mean and autocovariance of {zt} ∞ −∞ in terms of those of yt and verify that it is stationary.
3. Let {yt} ∞ −∞ and {zt} ∞ −∞ be two stationary processes with means µy and µz and autocovariances γy(s) and γz(s), independent of each other. Find the mean and autocovariance of wt = ayt + bzt , where a and b are constants, and show that {wt} ∞ −∞ is stationary.
4. Let ai , bi be independent r.v.’s with E[ai ] = E[bi ] = 0 and var[ai ] = var[bi ] = σ 2 i . Compute the mean and autocovariance of zt = Pm i=1(ai cos 2πωit + bi sin 2πωit) and show that it is stationary. [Hint: you may want to use the trigonometric identity cos x cos y + sin x sin y = cos(x − y).]
5. Let {at} ∞ 1 be a white noise process with mean 0 and variance σ 2 a . Define z0 = 0, zt = φzt−1 + at , t = 1, 2, . . . , where |φ| < 1.
(a) Express zt explicitly in terms of at .
(b) Calculate the autocovariance cov[zt , zt+s] for s > 0, and show that for large t, zt is approximately stationary.
6. Observing z1, . . . , zN from a stationary process with autocovariance γk and autocorrelation ρk = γk/γ0. It is known that var[¯z] = (γ0/N)[1 + 2PN−1 k=1 (1 − k/N)ρk].
(a) If ρk → 0 as k → ∞, show that var[¯z] → 0 as N → ∞.
(b) Compare var[¯z] with the following autocorrelations: (i) ρk = 0, k 6= 0; (ii) ρ1 = .8, ρ2 = .55, ρk = 0, k > 2.
7. Problem 2.1 in the text (p. 569 in 3rd ed; p.701 in 4th ed.), plus
(d) After inspecting the graphs in (a)-(c), do you think the series is stationary?
(e) Calculate and plot the sample ACF for lags up to 6.
(f) Assume ρk = 0, k > 2. Obtain approximate standard errors for r1, r2, and rk, k > 2. (g) Assume ρk = 0, k > 2. Obtain approximate correlation between r4 and r5. Due January 22, 2024