Introductory Time Series with R

By Paul S.P. Cowpertwait and Andrew V. Metcalfe · Springer · 2009

Study guide: Exam MAS-II

A concise text that takes the non-specialist, at a fairly quick pace, to where they can confidently apply a range of time series methods using R. It assumes a first-year university statistics course and treats each model the same way: a motivating section, the model defined in mathematical notation, R code that simulates data from it and fits the model to recover its parameters, and a fit to a historical data set with diagnostic plots. Each chapter ends with a summary of the R commands used and exercises; derivations sit in starred sections that can be skipped. Part of Springer's Use R! series.

On the syllabus

• Exam MAS-II — objectives D1–D4; Chapters 1–5 (excluding Sections 3.3 and 3.4), 6, and 7 (Sections 7.1, 7.2 and 7.3).

1 Time Series Data

• 1.1 Purpose
• 1.2 Time series
• 1.3 R language
• 1.4 Plots, trends, and seasonal variation
  • 1.4.1 A flying start: Air passenger bookings
  • 1.4.2 Unemployment: Maine
  • 1.4.3 Multiple time series: Electricity, beer and chocolate data
  • 1.4.4 Quarterly exchange rate: GBP to NZ dollar
  • 1.4.5 Global temperature series
• 1.5 Decomposition of series
  • 1.5.1 Notation
  • 1.5.2 Models
  • 1.5.3 Estimating trends and seasonal effects
  • 1.5.4 Smoothing
  • 1.5.5 Decomposition in R
• 1.6 Summary of commands used in examples
• 1.7 Exercises

2 Correlation

• 2.1 Purpose
• 2.2 Expectation and the ensemble
  • 2.2.1 Expected value
  • 2.2.2 The ensemble and stationarity
  • 2.2.3 Ergodic series*
  • 2.2.4 Variance function
  • 2.2.5 Autocorrelation
• 2.3 The correlogram
  • 2.3.1 General discussion
  • 2.3.2 Example based on air passenger series
  • 2.3.3 Example based on the Font Reservoir series
• 2.4 Covariance of sums of random variables
• 2.5 Summary of commands used in examples
• 2.6 Exercises

3 Forecasting Strategies

• 3.1 Purpose
• 3.2 Leading variables and associated variables
  • 3.2.1 Marine coatings
  • 3.2.2 Building approvals publication
  • 3.2.3 Gas supply
• 3.3 Bass model
  • 3.3.1 Background
  • 3.3.2 Model definition
  • 3.3.3 Interpretation of the Bass model*
  • 3.3.4 Example
• 3.4 Exponential smoothing and the Holt-Winters method
  • 3.4.1 Exponential smoothing
  • 3.4.2 Holt-Winters method
  • 3.4.3 Four-year-ahead forecasts for the air passenger data
• 3.5 Summary of commands used in examples
• 3.6 Exercises

4 Basic Stochastic Models

• 4.1 Purpose
• 4.2 White noise
  • 4.2.1 Introduction
  • 4.2.2 Definition
  • 4.2.3 Simulation in R
  • 4.2.4 Second-order properties and the correlogram
  • 4.2.5 Fitting a white noise model
• 4.3 Random walks
  • 4.3.1 Introduction
  • 4.3.2 Definition
  • 4.3.3 The backward shift operator
  • 4.3.4 Random walk: Second-order properties
  • 4.3.5 Derivation of second-order properties*
  • 4.3.6 The difference operator
  • 4.3.7 Simulation
• 4.4 Fitted models and diagnostic plots
  • 4.4.1 Simulated random walk series
  • 4.4.2 Exchange rate series
  • 4.4.3 Random walk with drift
• 4.5 Autoregressive models
  • 4.5.1 Definition
  • 4.5.2 Stationary and non-stationary AR processes
  • 4.5.3 Second-order properties of an AR(1) model
  • 4.5.4 Derivation of second-order properties for an AR(1) process*
  • 4.5.5 Correlogram of an AR(1) process
  • 4.5.6 Partial autocorrelation
  • 4.5.7 Simulation
• 4.6 Fitted models
  • 4.6.1 Model fitted to simulated series
  • 4.6.2 Exchange rate series: Fitted AR model
  • 4.6.3 Global temperature series: Fitted AR model
• 4.7 Summary of R commands
• 4.8 Exercises

5 Regression

• 5.1 Purpose
• 5.2 Linear models
  • 5.2.1 Definition
  • 5.2.2 Stationarity
  • 5.2.3 Simulation
• 5.3 Fitted models
  • 5.3.1 Model fitted to simulated data
  • 5.3.2 Model fitted to the temperature series (1970–2005)
  • 5.3.3 Autocorrelation and the estimation of sample statistics*
• 5.4 Generalised least squares
  • 5.4.1 GLS fit to simulated series
  • 5.4.2 Confidence interval for the trend in the temperature series
• 5.5 Linear models with seasonal variables
  • 5.5.1 Introduction
  • 5.5.2 Additive seasonal indicator variables
  • 5.5.3 Example: Seasonal model for the temperature series
• 5.6 Harmonic seasonal models
  • 5.6.1 Simulation
  • 5.6.2 Fit to simulated series
  • 5.6.3 Harmonic model fitted to temperature series (1970–2005)
• 5.7 Logarithmic transformations
  • 5.7.1 Introduction
  • 5.7.2 Example using the air passenger series
• 5.8 Non-linear models
  • 5.8.1 Introduction
  • 5.8.2 Example of a simulated and fitted non-linear series
• 5.9 Forecasting from regression
  • 5.9.1 Introduction
  • 5.9.2 Prediction in R
• 5.10 Inverse transform and bias correction
  • 5.10.1 Log-normal residual errors
  • 5.10.2 Empirical correction factor for forecasting means
  • 5.10.3 Example using the air passenger data
• 5.11 Summary of R commands
• 5.12 Exercises

6 Stationary Models

• 6.1 Purpose
• 6.2 Strictly stationary series
• 6.3 Moving average models
  • 6.3.1 MA(q) process: Definition and properties
  • 6.3.2 R examples: Correlogram and simulation
• 6.4 Fitted MA models
  • 6.4.1 Model fitted to simulated series
  • 6.4.2 Exchange rate series: Fitted MA model
• 6.5 Mixed models: The ARMA process
  • 6.5.1 Definition
  • 6.5.2 Derivation of second-order properties*
• 6.6 ARMA models: Empirical analysis
  • 6.6.1 Simulation and fitting
  • 6.6.2 Exchange rate series
  • 6.6.3 Electricity production series
  • 6.6.4 Wave tank data
• 6.7 Summary of R commands
• 6.8 Exercises

7 Non-stationary Models

• 7.1 Purpose
• 7.2 Non-seasonal ARIMA models
  • 7.2.1 Differencing and the electricity series
  • 7.2.2 Integrated model
  • 7.2.3 Definition and examples
  • 7.2.4 Simulation and fitting
  • 7.2.5 IMA(1, 1) model fitted to the beer production series
• 7.3 Seasonal ARIMA models
  • 7.3.1 Definition
  • 7.3.2 Fitting procedure
• 7.4 ARCH models
  • 7.4.1 S&P500 series
  • 7.4.2 Modelling volatility: Definition of the ARCH model
  • 7.4.3 Extensions and GARCH models
  • 7.4.4 Simulation and fitted GARCH model
  • 7.4.5 Fit to S&P500 series
  • 7.4.6 Volatility in climate series
  • 7.4.7 GARCH in forecasts and simulations
• 7.5 Summary of R commands
• 7.6 Exercises

8 Long-Memory Processes

• 8.1 Purpose
• 8.2 Fractional differencing
• 8.3 Fitting to simulated data
• 8.4 Assessing evidence of long-term dependence
  • 8.4.1 Nile minima
  • 8.4.2 Bellcore Ethernet data
  • 8.4.3 Bank loan rate
• 8.5 Simulation
• 8.6 Summary of additional commands used
• 8.7 Exercises

9 Spectral Analysis

• 9.1 Purpose
• 9.2 Periodic signals
  • 9.2.1 Sine waves
  • 9.2.2 Unit of measurement of frequency
• 9.3 Spectrum
  • 9.3.1 Fitting sine waves
  • 9.3.2 Sample spectrum
• 9.4 Spectra of simulated series
  • 9.4.1 White noise
  • 9.4.2 AR(1): Positive coefficient
  • 9.4.3 AR(1): Negative coefficient
  • 9.4.4 AR(2)
• 9.5 Sampling interval and record length
  • 9.5.1 Nyquist frequency
  • 9.5.2 Record length
• 9.6 Applications
  • 9.6.1 Wave tank data
  • 9.6.2 Fault detection on electric motors
  • 9.6.3 Measurement of vibration dose
  • 9.6.4 Climatic indices
  • 9.6.5 Bank loan rate
• 9.7 Discrete Fourier transform (DFT)*
• 9.8 The spectrum of a random process*
  • 9.8.1 Discrete white noise
  • 9.8.2 AR
  • 9.8.3 Derivation of spectrum
• 9.9 Autoregressive spectrum estimation
• 9.10 Finer details
  • 9.10.1 Leakage
  • 9.10.2 Confidence intervals
  • 9.10.3 Daniell windows
  • 9.10.4 Padding
  • 9.10.5 Tapering
  • 9.10.6 Spectral analysis compared with wavelets
• 9.11 Summary of additional commands used
• 9.12 Exercises

10 System Identification

• 10.1 Purpose
• 10.2 Identifying the gain of a linear system
  • 10.2.1 Linear system
  • 10.2.2 Natural frequencies
  • 10.2.3 Estimator of the gain function
• 10.3 Spectrum of an AR(p) process
• 10.4 Simulated single mode of vibration system
• 10.5 Ocean-going tugboat
• 10.6 Non-linearity
• 10.7 Exercises

11 Multivariate Models

• 11.1 Purpose
• 11.2 Spurious regression
• 11.3 Tests for unit roots
• 11.4 Cointegration
  • 11.4.1 Definition
  • 11.4.2 Exchange rate series
• 11.5 Bivariate and multivariate white noise
• 11.6 Vector autoregressive models
  • 11.6.1 VAR model fitted to US economic series
• 11.7 Summary of R commands
• 11.8 Exercises

12 State Space Models

• 12.1 Purpose
• 12.2 Linear state space models
  • 12.2.1 Dynamic linear model
  • 12.2.2 Filtering*
  • 12.2.3 Prediction*
  • 12.2.4 Smoothing*
• 12.3 Fitting to simulated univariate time series
  • 12.3.1 Random walk plus noise model
  • 12.3.2 Regression model with time-varying coefficients
• 12.4 Fitting to univariate time series
• 12.5 Bivariate time series – river salinity
• 12.6 Estimating the variance matrices
• 12.7 Discussion
• 12.8 Summary of additional commands used
• 12.9 Exercises

Sources

• Introductory Time Series with R — front matter (Springer, 2009) — the book's front matter, free from the publisher: title page, copyright page (ISBN, LCCN), the preface and the full table of contents
• Introductory Time Series with R (Springer Nature Link) — the publisher's record: authors, the Use R! series, softcover and eBook ISBNs, and the list of 12 chapters, which agrees with the contents
• CAS Exam MAS-II Content Outline (2025) — the citation (Springer, 2009) and the assigned scope