## Forecasting with Exponential Smoothing: The State Space ApproachExponential smoothing methods have been around since the 1950s, and are still the most popular forecasting methods used in business and industry. However, a modeling framework incorporating stochastic models, likelihood calculation, prediction intervals and procedures for model selection, was not developed until recently. This book brings together all of the important new results on the state space framework for exponential smoothing. It will be of interest to 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. Part 1 provides an introduction to exponential smoothing and the underlying models. The essential details are given in Part 2, which also provide links to the most important papers in the literature. More advanced topics are covered in Part 3, including the mathematical properties of the models and extensions of the models for specific problems. Applications to particular domains are discussed in Part 4. |

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### Contents

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22 Classiﬁcation of Exponential Smoothing Methods | 11 |

23 Point Forecasts for the BestKnown Methods | 12 |

24 Point Forecasts for All Methods | 17 |

121 Innovations State Space Models with a Random Seed Vector | 180 |

122 Estimation | 182 |

123 Information Filter | 185 |

124 Prediction | 193 |

125 Model Selection | 194 |

126 Smoothing Time Series | 195 |

127 Kalman Filter | 197 |

128 Exercises | 200 |

26 Initialization and Estimation | 23 |

27 Assessing Forecast Accuracy | 25 |

28 Model Selection | 27 |

29 Exercises | 28 |

Essentials | 30 |

Linear Innovations State Space Models | 33 |

32 Innovations and OneStepAhead Forecasts | 35 |

33 Model Properties | 36 |

34 Basic Special Cases | 38 |

35 Variations on the Common Models | 47 |

36 Exercises | 51 |

Nonlinear and Heteroscedastic Innovations State Space Models | 52 |

42 Basic Special Cases | 56 |

43 Nonlinear Seasonal Models | 61 |

44 Variations on the Common Models | 64 |

45 Exercises | 66 |

Estimation of Innovations State Space Models | 67 |

52 A Heuristic Approach to Estimation | 71 |

53 Exercises | 73 |

Prediction Distributions and Intervals | 75 |

61 Simulated Prediction Distributions and Intervals | 77 |

Linear Homoscedastic State Space Models | 80 |

Linear Heteroscedastic State Space Models | 83 |

65 Prediction Intervals | 88 |

66 LeadTime Demand Forecasts for Linear Homoscedastic Models | 90 |

67 Exercises | 94 |

Derivations Derivation of Results for Class 1 | 95 |

Selection of Models | 105 |

72 Choosing a Model Selection Procedure | 108 |

73 Implications for Model Selection Procedures | 116 |

74 Exercises | 117 |

Model Selection Algorithms The Linear Empirical Information Criterion | 118 |

Further Topics | 120 |

Normalizing Seasonal Components | 121 |

81 Normalizing Additive Seasonal Components | 124 |

82 Normalizing Multiplicative Seasonal Components | 128 |

Canadian Gas Production | 131 |

84 Exercises | 134 |

Derivations for Additive Seasonality | 135 |

Models with Regressor Variables | 137 |

91 The Linear Innovations Model with Regressors | 138 |

92 Some Examples | 139 |

93 Diagnostics for Regression Models | 143 |

94 Exercises | 147 |

Some Properties of Linear Models | 149 |

102 Stability and the Parameter Space | 152 |

103 Conclusions | 161 |

Reduced Forms and Relationships with ARIMA Models | 163 |

111 ARIMA Models | 164 |

112 Reduced Forms for Two Simple Cases | 168 |

113 Reduced Form for the General Linear Innovations Model | 170 |

114 Stationarity and Invertibility | 171 |

115 ARIMA Models in Innovations State Space Form | 173 |

116 Cyclical Models | 176 |

Linear Innovations State Space Models with Random Seed States | 178 |

Triangularization of Stochastic Equations | 203 |

Conventional State Space Models | 209 |

131 State Space Models | 210 |

132 Estimation | 212 |

133 Reduced Forms | 215 |

134 Comparison of State Space Models | 219 |

135 Smoothing and Filtering | 223 |

136 Exercises | 226 |

Maximizing the Size of the Parameter Space | 227 |

Time Series with Multiple Seasonal Patterns | 229 |

141 Exponential Smoothing for Seasonal Data | 231 |

142 Multiple Seasonal Processes | 234 |

143 An Application to Utility Data | 240 |

144 Analysis of Traffic Data | 246 |

145 Exercises | 250 |

Alternative Forms FirstOrder Form of the Model | 251 |

Nonlinear Models for Positive Data | 255 |

151 Problems with the Gaussian Model | 256 |

152 Multiplicative Error Models | 260 |

153 Distributional Results | 263 |

154 Implications for Statistical Inference | 266 |

155 Empirical Comparisons | 270 |

156 An Appraisal | 274 |

157 Exercises | 275 |

Models for Count Data | 277 |

161 Models for Nonstationary Count Time Series | 278 |

162 Crostons Method | 281 |

Car Parts | 283 |

164 Exercises | 286 |

Vector Exponential Smoothing | 287 |

171 The Vector Exponential Smoothing Framework | 288 |

172 Local Trend Models | 290 |

174 Other Multivariate Models | 293 |

Exchange Rates | 296 |

176 Forecasting Experiment | 299 |

Applications | 301 |

Inventory Control Applications | 302 |

181 Forecasting Demand Using Sales Data | 304 |

182 Inventory Systems | 308 |

183 Exercises | 315 |

Conditional Heteroscedasticity and Applications in Finance | 317 |

191 The BlackScholes Model | 318 |

192 Autoregressive Conditional Heteroscedastic Models | 319 |

193 Forecasting | 322 |

194 Exercises | 324 |

Economic Applications The BeveridgeNelson Decomposition | 325 |

201 The BeveridgeNelson Decomposition | 328 |

202 State Space Form and Applications | 330 |

203 Extensions of the BeveridgeNelson Decomposition to Nonlinear Processes | 334 |

204 Conclusion | 336 |

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