[2019] STAT 153 Introduction to Time Series - Midterm Exam - Q2 ACF
2. Consider the stationary, zero-mean AR(1) model Xt = 0.5Xt−1 + Zt and the MA(1) model Wt = 0.5Zt−1 + Zt, where Zt is some white noise with variance σ .
(a) For each of Zt,Wt, and Xt give the ACVF and ACF function.
i. For Zt: s (1 Points)
ii. For Wt: s (2 Points)
iii. For Xt: s (2 Points)
(b) For each of Zt,Wt, and Xt give the approximate mean and variance of its sample ACF at lag 2 for n = 100 observations.
Hint: Recall Bartlett’s formula Wij =
Sum ∞m=1 (ρ(m + i) + ρ(m − i) − 2ρ(i)ρ(m)) (ρ(m + j) + ρ(m − j) − 2ρ(j)ρ(m))
i. For Zt: s (2 Points)
ii. For Wt: s (4 Points)
iii. For Xt: s (4 Points)
Figure 1: Sample ACFs of different time series data.
(c) Figure 1 shows sample ACFs for each of the three models for n = 100 observations. Which figure corresponds to which process? Explain. (3 Points)
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