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Homework 1

STATS 4A03

Due on Crowdmark by Friday, January 26, 11:59pm

Guidelines: Unless otherwise specified, you are required to justify and prove all your answers.

You are welcome to collaborate with other students on homework assignments, and you should feel free to discuss the problems with each other. However, you are expected to write all your solutions independently of any collaborators and you should not share written solutions with other students before the deadline. If you collaborate with other students, you must cite any collaborators that you had on any given problem.

You may use the textbook and lecture slides/notes. You are discouraged from using outside resources (online, Math stack, etc.), but if you do, you must cite all your sources. If your solution is too similar to the cited one, you may lose credit on the problem.

Your homework grade will be based on completeness plus the correctness of a random subset of four (4) problems.

Exercise 1. Suppose that {Yt} is stationary with autocovariance function γk. Show that for any fixed positive integer n and any constants c1, c2, . . . , cn, the process {Wt} defined by Wt = c1Yt + c2Yt−1 + · · · + cnYt−n+1 is stationary.

Exercise 2. Let {Xt} be a zero-mean, unit-variance stationary process with autocorrelation func-tion ρk. Suppose that µt is a nonconstant function and that σt is a positive-valued nonconstant function. The observed series is formed as Yt = µt + σtXt .

a) Find the mean and covariance function for the {Yt} process.

b) Show that the autocorrelation function for the {Yt} process depends only on the time lag. Is the {Yt} process stationary?

c) Is it possible to have a time series with a constant mean and with corr(Yt , Yt−k) free of t but with {Yt} not stationary?

Exercise 3. Evaluate the mean and covariance function for each of the following processes. In each case, determine whether or not the process is stationary.

a) Yt = θ0 + tet .

b) Wt = ∇Yt , where Yt is as given in part (a).

c) Yt = etet−1, where {et} is white noise.

Exercise 4. Let Y1 = θ0 + e1, where θ0 is a constant, and for t > 1 define Yt recursively by Yt = θ0 + Yt−1 + et . The process {Yt} is random walk with drift.

a) Show that Yt can be written as Yt = tθ0 + et + et−1 + · · · + e1.

b) Find the mean function for Yt .

c) Find the autocovariance function for Yt .

Exercise 5. Suppose that

Yt = R cos(2π(f t + Φ)) for t = 0, ±1, ±2, . . .

where 0 < f < 1/2 is a fixed frequency and R and Φ are uncorrelated random variables where Φ is uniformly distributed on the interval (0, 1).

a) Show that E(Yt) = 0 for all t.

b) Show that the process is stationary with γk = 2/1E(R2) cos(2πfk).

Exercise 6. Suppose Yt = µ + et + et−1. Find Var(Y¯). Compare your answer to what would have been obtained if Yt = µ + et . Describe the effect that the autocorrelation in {Yt} has on Var(Y¯).