2.9 Suppose Y, Bo + B₁t+ X, where (X,) is a zero-mean stationary series with auto- covariance function Y and ẞo and ẞ₁ are constants. (a) Show that (Y) is not stationary but that W, VY, Y,-Y-1 is stationary. (b) In general, show that if Y, μ, + X,, where (X,) is a zero-mean stationary series and u, is a polynomial in 7 of degree d, then V"Y, V(V"-Y) is sta- tionary for m ≥d and nonstationary for 0≤m

Suppose that the stationary time series {x, : t = T} satisfies the equation: x₁ =B₁ X-1 + B₂ x₁-12 - B1 B₂ x₁-13 + U₁ Determine the autocorrelation function of {x, :t € T}. Plot the autocorrelation function with ₂ = 0.6, B₁ = -0.1.

. Suppose {Yt} is a time series such that Yt = et + cet−1 + cet−2 + · · · + ce0 for all t > 0.a) Find the mean and covariance functions for {Yt}, and determine whether or not {Yt} is stationary. Similarly, find the mean and covariance functions for {∇Yt}, and determine whether or not {∇Yt} is stationary.b) Find the values p, d, q so that {Yt} is an ARIMA(p, d, q) model.

Suppose now the time series is linear trend + cycle process Yt = c + Bt + Ct + €t E4 ~ WN(0, o²) (a) Suppose the Cţ is as in MA(1) process. Which process does the first difference follow ? (b) Suppose the Cţ is as in AR(1) process. Which process does the first difference follow ? (c) Extend this reasoning for the case of Ct following ARMA(p,q). (d) What effect does taking differences have on time series dynamics ?

Consider the time series process given by x₁ = Z+0₁ 4-2 where 4 ~WN 0,1 and 6₁ = -0.8. A. Find the autocovariance function for x₁ . √(x₁ + x₂ ) 2 B. Calculate Var .

A time series with a periodic component can be constructed fromxt = U1 sin(2πω0t) + U2 cos(2πω0t),where U1 and U2 are independent random variables with zero means andE(U21 ) = E(U22 ) = σ2. The constant ω0 determines the period or time ittakes the process to make one complete cycle. Show that this series is weaklystationary with autocovariance functionγ(h) = σ2 cos(2πω0h

1 Find a solution of y' = 2y in the form of a power series in x. Find its recurrence relation first!

2. If jk is the rate of simple interest for the k-th period of time, where k = 1, 2,..., n, show that the amount of interest earned from time 0 to time n is j₁ + 2 + +Jn, if an investment of 1 is made at time 0.

Let {e} be a zero mean white noise process. Suppose that the observed process is 0e₁-1, where 0 is either 3 or 1/3. Y₁ = e₁ + et Oet (a) Find the autocorrelation function for {Y} both when 0 = 3 and when 0 = 1/3. (b) You should have discovered that the time series is stationary regardless of the value of 0 and that the autocorrelation functions are the same for 0 = 3 and 0 1/3. For simplicity, suppose that the process mean is known to be zero and the variance of Y, is known to be 1. You observe the series {Y} for t = 1, 2,..., n and suppose that you can produce good estimates of the autocorrelations Pk. Do you think that you could determine which value of 0 is correct (3 or 1/3) based on the estimate of pk? Why or why not? t

Determine two linearly independent series solutions to x2y′′+x(3−x)y′+y =0, x > 0. (11.5.24)