Forecasting with Exponential Smoothing: the State Space Approach
Rob J Hyndman, Anne B Koehler, J Keith Ord, Ralph D Snyder (Springer, 2008).
Exponential smoothing methods have been around since the 1950s, and are still the most popular forecasting methods used in business and industry. However, a modelling framework incorporating stochastic models, likelihood calculation, prediction intervals and procedures for model selection, was not developed until relatively recently. Two key papers were Ord, Koehler and Snyder (JASA, 1997) and Hyndman, Koehler, Snyder and Grose (IJF, 2002) although there have been many others filling in some of the details.
As a result, the area of exponential smoothing has undergone a substantial revolution in the past ten years. The new “state space framework” for exponential smoothing is discussed in numerous journal articles but there has been no systematic explanation and development of the ideas. Furthermore, the notation used in the journal articles tends to change from paper to paper. Consequently, researchers and practitioners struggle to use the new models in applications.
In this book we try to bring together all of the important results in a coherent manner with consistent notation. We have written it for people wanting to apply the methods in their own area of interest as well as for researchers wanting to take the ideas in new directions.
The readership is assumed to have a statistical background at about honours level in the UK/Australian/NZ system and Masters level in the US system.
R packages and data
The R packages forecast and fable both implement the methods described in the book. The expsmooth package contains the data for the exercises and most of the examples in the book.
You can also download individual csv files for each data set.
Table of contents
Part I: Introduction
 Basic concepts [R code]
 Getting started [R code] [Exercise solutions]
Part II: Essentials
 Linear innovations state space models [R code] [Exercise solutions]
 Nonlinear and heteroscedastic innovations state space models [Exercise solutions]
 Estimation of innovations state space models [R code] [Exercise solutions]
 Prediction distributions and intervals [R code] [Exercise solutions]
 Selection of models
Part III: Further topics
 Normalizing seasonal components [R code]
 Models with regressor variables [R code]
 Some properties of linear models
 Reduced forms and relationships with ARIMA models
 Linear innovations state space models with random seed states
 Conventional state space models
 Time series with multiple seasonal patterns (with Phillip Gould) [R code]
 Nonlinear models for positive data (with Muhammad Akram) [R code]
 Models for count data [R code]
 Vector exponential smoothing (with Ashton de Silva)
Part IV: Applications
Reviews
Errata
This is the list of all errors that we know about. If you think you’ve spotted a new one, please let us know!
 p15
 In the heading for section 2.3.3, “A_{d},A” should read “A_{d},N”
 p47
 The discount matrix for the damped local level model is D=\phi(1\alpha) and the model is stable provided \phi(1\alpha)<1, or equivalently \phi<1 and 0 < \alpha < 2.
 p51

Exercise 3.2. Change “stable” to “forecastable”.
Exercise 3.3. The variance should be \sigma^2(1+(t1)\alpha^2).  p55

The third equation should have a plus sign:
\boldsymbol{x}_{t} = \boldsymbol{D}(\boldsymbol{x}_{t1}) + g(\boldsymbol{x}_{t1})\frac{y_t}{r(\boldsymbol{x}_{t1})}.
 p76, line4
 Replace “Classes 14” with “Classes 15”
 p85
 For the ETS(M,A_{d},M) model, \boldsymbol{w}_1 = [1 ~ \phi]^\prime and the matrix \boldsymbol{G}_1 = \left[\begin{array}{ll} \alpha & \alpha\phi \\ \beta & \beta\phi\end{array}\right].
 p90
 last line: the last term should be just \varepsilon_{n+j} not \varepsilon q_{n+j}
 p92
 the sum on the first line should be multiplied by \sigma^2.
 p102
 For the ETS(M,A_{d},M) model, \boldsymbol{w}_1 = [1 ~ \phi]^\prime and the matrix \boldsymbol{G}_1 = \left[\begin{array}{ll} \alpha & \alpha\phi \\ \beta & \beta\phi\end{array}\right].
 p108: heading 7.2.1
 Change “Section” to “Selection”.
 p109: line 1

\hat{y}_{i,n_j}^{(i,j)} should be \hat{y}_{n_j}^{(i,j)}
 p118: line 4
 delete first “is”
 p150
 In the ETS(A,A_{d},N) model, \boldsymbol{w} = [1 ~ \phi]^\prime and \boldsymbol{F} = \left[\begin{array}{ll} 1 & \phi \\ 0 & \phi \end{array}\right]. Similarly, in the ETS(A,A_{d},A) model, the second element of \boldsymbol{w} should be \phi and the second element of the top row of \boldsymbol{F} should be \phi.
 p154
 In the ETS(A,A_{d},N) model, the second column of \boldsymbol{D} should contain \phi(1\alpha) and \phi(1\beta). Similarly, in the ETS(A,A_{d},A) model, the first two elements of the second column of \boldsymbol{D} should be \phi(1\alpha) and \phi(1\beta).
 p155
 In Table 10.1, the parameter range for \phi should be 1<\phi\le1.
 p158
 Last paragraph: the seasonal states should sum to zero, not one. So replace equation with s_{m,t} \equiv s_{0,t} = s_{1,t}  \dots  s_{m1,t}.
 p158
 Last sentence: the second element of \boldsymbol{w} should be \phi and the second element of the top row of \boldsymbol{F} should be \phi.
 p159
 The first two elements of the second column of \boldsymbol{D} should be \phi(1\alpha) and \phi(1\beta).
 p280: line 10
 Change “finite” to “nonzero”.