Time Series Analysis: Forecasting and Control, Fourth Edition

PART SIX

Exercises and Problems

This part of the book is a collection of exercises and problems for the separate chapters. We hope that these will further enhance the value of the book when used as a course text and also assist private study. A number of examples point to extensions of the ideas and act as a first introduction to additional methods.

CHAPTER 2

2.1 The following are temperature measurements z_t made every minute on a chemical reactor:

200, 202, 208, 204, 204, 207, 207, 204, 202, 199, 201, 198, 200, 202, 203, 205, 207, 211, 204, 206, 203, 203, 201, 198, 200, 206, 207, 206, 200, 203, 203, 200, 200, 195, 202, 204.

• (a) Plot the time series.
• (b) Plot z_t+1 versus z_t.
• (c) Plot z_t+2 versus z_t.

After inspecting the graphs, do you think that the series is autocorrelated?

2.2. State whether or not a stationary stochastic process can have the following values of autocorrelations:

• (a) ρ₁ = 0.80, ρ₂ = 0.55, ρ_k = 0 for k > 2
• (b) ρ₁ = 0.80, ρ₂ = 0.28, ρ_k = 0 for k > 2

2.3. Two stationary stochastic processes z_1t and z_2t have the following autocovariance functions:

Calculate the autocovariance function of the process z_3t = z_1t + 2z_2t and verify that it is a valid stationary process.

2.4. Calculate c₀, c₁ c₂, c₃, r₁, r₂, r₃ for the series given in Exercise 2.1. Make a graph of r_k, k = 0, 1, 2, 3.

2.5. On the supposition that ρ_j = 0 for j > 2,

(a) Obtain approximate standard ...