PSTAT 174/274, Fall 2022: Homework # 2.
Note: {Zt} ∼ W N(0, σ2
Z
) denotes white noise.
1. Bellow, you are given the following graphs of autocorrelation functions for three separate data sets, each
with n observations. The dotted lines in each graph correspond to 95% confidence intervals. Determine
which of the above data sets exhibit statistically significant autocorrelations. Explain how you came to
this conclusion.
A. I only; B. II only; C. III only; D. I, II and III; E. The answer is not given by (A), (B), (C), or (D).
2. For each of the two time series models, check stationarity and invertibility. Fully justify your answer.
(2.a) Xt = Zt −
2
3
Zt−1 −
1
3
Zt−2.
(2.b) Xt =
2
3Xt−1 +
1
3Xt−2 + Zt
.
3. (3.a) For a MA(3) process with coefficients θ1 = 2, θ2 = 0.5, and θ3 = −0.1, (i) write the mathematical
equation for MA(3) model with these coefficients, and (ii) calculate the autocorrelation function at lags 1,
2, 3, 4: ρ(1), ρ(2), ρ(3) and ρ(4).
(3.b) For an AR(1) process with coefficient ϕ1 = −0.5, (i) write the mathematical equation for AR(1))
model with these coefficients, and (ii) calculate the autocorrelation function at lags 1, 2, 3, 4: ρ(1), ρ(2), ρ(3)
and ρ(4).
4. You are given the following process: Xt = 3 + Y + Zt
, where Y is a mean zero random variable with
variance σ
2
Y
, independent of the white noise {Zt}. Determine whether the process X is stationary and find
its autocovariance and autocorrelation functions.
5. Let Xt = Zt + 2Zt−1 − 8Zt−2.
(i) Identify the model as the model as MA(q) or AR(p), specify q or p respectively.
(ii) Is the model stationary and invertible? Explain fully and show calculations where needed.
(Hint: review 4 from homework 1!)
(iii) Find ρX(2). Use R to simulate 300 values of {Xt} and use your simulated values to plot sample acf.
Compare your sample estimate of ρX(2) to its true value found by calculations. Redo this part using 10,000
simulated values of Xt
.
The following problems are for students enrolled in PSTAT 274 ONLY
G1 Let {Zt} ∼ W N(0, 1) and {Xt} be given by Xt = Zt + θZt−2.
(a) Find the autocovariance and autocorrelation function for this process when θ = 0.8.
(b) Compute the variance of the sample mean (X1 + X2 + X3 + X4)/4 when θ = 0.8.
(c) Repeat (b) when θ = −0.8 and compare your answer with the result obtained in (b).
G2 Provide at least two examples of AR(2) models with autocovariance functions exhibiting very different
behavior pattern. Include plots of corresponding theoretical acfs and the corresponding R code.
G3 Let Xt = Zt + θZt−1, t = 1, 2, . . ., where Zt ∼ IID(0, σ2
Z
). Show that Xt
is both weakly and strictly
stationary.
(Hint: for the last part express the joint moment generating function E exp(Pn
i=1 λiXi) in terms of function
m(λ) = E exp(λZi).)