This page intentionally left blank CONTENTS OF VOLUME 1 Introduction to the Series v Contents of the Handbook vii PART 1: FORECASTING METHODOLOGY Chapter 1 Bayesian Forecasting JOHN GEWEKE AND CHARLES WHITEMAN 3 Abstract 4 Keywords 4 1. Introduction 6 2. Bayesian inference and forecasting: A primer 7 2.1. Models for observables 7 2.2. Model completion with prior distributions 10 2.3. Model combination and evaluation 14 2.4. Forecasting 19 3. Posterior simulation methods 25 3.1. Simulation methods before 1990 25 3.2. Markov chain Monte Carlo 30 3.3. The full Monte 36 4. ’Twas not always so easy: A historical perspective 41 4.1. In the beginning, there was diffuseness, conjugacy, and analytic work 41 4.2. The dynamic linear model 43 4.3. The Minnesota revolution 44 4.4. After Minnesota: Subsequent developments 49 5. Some Bayesian forecasting models 53 5.1. Autoregressive leading indicator models 54 5.2. Stationary linear models 56 5.3. Fractional integration 59 5.4. Cointegration and error correction 61 5.5. Stochastic volatility 64 6. Practical experience with Bayesian forecasts 68 6.1. National BVAR forecasts: The Federal Reserve Bank of Minneapolis 69 6.2. Regional BVAR forecasts: Economic conditions in Iowa 70 References 73 xi xii Contents of Volume 1 Chapter 2 Forecasting and Decision Theory CLIVE W.J. GRANGER AND MARK J. MACHINA 81 Abstract 82 Keywords 82 Preface 83 1. History of the field 83 1.1. Introduction 83 1.2. The Cambridge papers 84 1.3. Forecasting versus statistical hypothesis testing and estimation 87 2. Forecasting with decision-based loss functions 87 2.1. Background 87 2.2. Framework and basic analysis 88 2.3. Recovery of decision problems from loss functions 93 2.4. Location-dependent loss functions 96 2.5. Distribution-forecast and distribution-realization loss functions 97 References 98 Chapter 3 Forecast Evaluation KENNETH D. WEST 99 Abstract 100 Keywords 100 1. Introduction 101 2. A brief history 102 3. A small number of nonnested models, Part I 104 4. A small number of nonnested models, Part II 106 5. A small number of nonnested models, Part III 111 6. A small number of models, nested: MPSE 117 7. A small number of models, nested, Part II 122 8. Summary on small number of models 125 9. Large number of models 125 10. Conclusions 131 Acknowledgements 132 References 132 Chapter 4 Forecast Combinations ALLAN TIMMERMANN 135 Abstract 136 Keywords 136 1. Introduction 137 2. The forecast combination problem 140 Contents of Volume 1 xiii 2.1. Specification of loss function 141 2.2. Construction of a super model – pooling information 143 2.3. Linear forecast combinations under MSE loss 144 2.4. Optimality of equal weights – general case 148 2.5. Optimal combinations under asymmetric loss 150 2.6. Combining as a hedge against non-stationarities 154 3. Estimation 156 3.1. To combine or not to combine 156 3.2. Least squares estimators of the weights 158 3.3. Relative performance weights 159 3.4. Moment estimators 160 3.5. Nonparametric combination schemes 160 3.6. Pooling, clustering and trimming 162 4. Time-varying and nonlinear combination methods 165 4.1. Time-varying weights 165 4.2. Nonlinear combination schemes 169 5. Shrinkage methods 170 5.1. Shrinkage and factor structure 172 5.2. Constraints on combination weights 174 6. Combination of interval and probability distribution forecasts 176 6.1. The combination decision 176 6.2. Combinations of probability density forecasts 177 6.3. Bayesian methods 178 6.4. Combinations of quantile forecasts 179 7. Empirical evidence 181 7.1. Simple combination schemes are hard to beat 181 7.2. Choosing the single forecast with the best track record is often a bad idea 182 7.3. Trimming of the worst models often improves performance 183 7.4. Shrinkage often improves performance 184 7.5. Limited time-variation in the combination weights may be helpful 185 7.6. Empirical application 186 8. Conclusion 193 Acknowledgements 193 References 194 Chapter 5 Predictive Density Evaluation VALENTINA CORRADI AND NORMAN R. SWANSON 197 Abstract 198 Keywords 199 Part I: Introduction 200 1. Estimation, specification testing, and model evaluation 200 Part II: Testing for Correct Specification of Conditional Distributions 207 xiv Contents of Volume 1 2. Specification testing and model evaluation in-sample 207 2.1. Diebold, Gunther and Tay approach – probability integral transform 208 2.2. Bai approach – martingalization 208 2.3. Hong and Li approach – a nonparametric test 210 2.4. Corradi and Swanson approach 212 2.5. Bootstrap critical values for the V 1T and V 2T tests 216 2.6. Other related work 220 3. Specification testing and model selection out-of-sample 220 3.1. Estimation and parameter estimation error in recursive and rolling estimation schemes – West as well as West and McCracken results 221 3.2. Out-of-sample implementation of Bai as well as Hong and Li tests 223 3.3. Out-of-sample implementation of Corradi and Swanson tests 225 3.4. Bootstrap critical for the V 1P,J and V 2P,J tests under recursive estimation 228 3.5. Bootstrap critical for the V 1P,J and V 2P,J tests under rolling estimation 233 Part III: Evaluation of (Multiple) Misspecified Predictive Models 234 4. Pointwise comparison of (multiple) misspecified predictive models 234 4.1. Comparison of two nonnested models: Diebold and Mariano test 235 4.2. Comparison of two nested models 238 4.3. Comparison of multiple models: The reality check 242 4.4. A predictive accuracy test that is consistent against generic alternatives 249 5. Comparison of (multiple) misspecified predictive density models 253 5.1. The Kullback–Leibler information criterion approach 253 5.2. A predictive density accuracy test for comparing multiple misspecified models 254 Acknowledgements 271 Part IV: Appendices and References 271 Appendix A: Assumptions 271 Appendix B: Proofs 275 References 280 PART 2: FORECASTING MODELS Chapter 6 Forecasting with VARMA Models HELMUT LÜTKEPOHL 287 Abstract 288 Keywords 288 1. Introduction and overview 289 1.1. Historical notes 290 1.2. Notation, terminology, abbreviations 291 2. VARMA processes 292 2.1. Stationary processes 292 2.2. Cointegrated I(1) processes 294 2.3. Linear transformations of VARMA processes 294 Contents of Volume 1 xv 2.4. Forecasting 296 2.5. Extensions 305 3. Specifying and estimating VARMA models 306 3.1. The echelon form 306 3.2. Estimation of VARMA models for given lag orders and cointegrating rank 311 3.3. Testing for the cointegrating rank 313 3.4. Specifying the lag orders and Kronecker indices 314 3.5. Diagnostic checking 316 4. Forecasting with estimated processes 316 4.1. General results 316 4.2. Aggregated processes 318 5. Conclusions 319 Acknowledgements 321 References 321 Chapter 7 Forecasting with Unobserved Components Time Series Models ANDREW HARVEY 327 Abstract 330 Keywords 330 1. Introduction 331 1.1. Historical background 331 1.2. Forecasting performance 333 1.3. State space and beyond 334 2. Structural time series models 335 2.1. Exponential smoothing 336 2.2. Local level model 337 2.3. Trends 339 2.4. Nowcasting 340 2.5. Surveys and measurement error 343 2.6. Cycles 343 2.7. Forecasting components 344 2.8. Convergence models 347 3. ARIMA and autoregressive models 348 3.1. ARIMA models and the reduced form 348 3.2. Autoregressive models 350 3.3. Model selection in ARIMA, autoregressive and structural time series models 350 3.4. Correlated components 351 4. Explanatory variables and interventions 352 4.1. Interventions 354 4.2. Time-varying parameters 355 5. Seasonality 355 5.1. Trigonometric seasonal 356 xvi Contents of Volume 1 5.2. Reduced form 357 5.3. Nowcasting 358 5.4. Holt–Winters 358 5.5. Seasonal ARIMA models 358 5.6. Extensions 360 6. State space form 361 6.1. Kalman filter 361 6.2. Prediction 363 6.3. Innovations 364 6.4. Time-invariant models 364 6.5. Maximum likelihood estimation and the prediction error decomposition 368 6.6. Missing observations, temporal aggregation and mixed frequency 369 6.7. Bayesian methods 369 7. Multivariate models 370 7.1. Seemingly unrelated times series equation models 370 7.2. Reduced form and multivariate ARIMA models 371 7.3. Dynamic common factors 372 7.4. Convergence 376 7.5. Forecasting and nowcasting with auxiliary series 379 8. Continuous time 383 8.1. Transition equations 383 8.2. Stock variables 385 8.3. Flow variables 387 9. Nonlinear and non-Gaussian models 391 9.1. General state space model 392 9.2. Conditionally Gaussian models 394 9.3. Count data and qualitative observations 394 9.4. Heavy-tailed distributions and robustness 399 9.5. Switching regimes 401 10. Stochastic volatility 403 10.1. Basic specification and properties 404 10.2. Estimation 405 10.3. Comparison with GARCH 405 10.4. Multivariate models 406 11. Conclusions 406 Acknowledgements 407 References 408 Chapter 8 Forecasting Economic Variables with Nonlinear Models TIMO TERÄSVIRTA 413 Abstract 414 Keywords 415 Contents of Volume 1 xvii 1. Introduction 416 2. Nonlinear models 416 2.1. General 416 2.2. Nonlinear dynamic regression model 417 2.3. Smooth transition regression model 418 2.4. Switching regression and threshold autoregressive model 420 2.5. Markov-switching model 421 2.6. Artificial neural network model 422 2.7. Time-varying regression model 423 2.8. Nonlinear moving average models 424 3. Building nonlinear models 425 3.1. Testing linearity 426 3.2. Building STR models 428 3.3. Building switching regression models 429 3.4. Building Markov-switching regression models 431 4. Forecasting with nonlinear models 431 4.1. Analytical point forecasts 431 4.2. Numerical techniques in forecasting 433 4.3. Forecasting using recursion formulas 436 4.4. Accounting for estimation uncertainty 437 4.5. Interval and density forecasts 438 4.6. Combining forecasts 438 4.7. Different models for different forecast horizons? 439 5. Forecast accuracy 440 5.1. Comparing point forecasts 440 6. Lessons from a simulation study 444 7. Empirical forecast comparisons 445 7.1. Relevant issues 445 7.2. Comparing linear and nonlinear models 447 7.3. Large forecast comparisons 448 8. Final remarks 451 Acknowledgements 452 References 453 Chapter 9 Approximate Nonlinear Forecasting Methods HALBERT WHITE 459 Abstract 460 Keywords 460 1. Introduction 461 2. Linearity and nonlinearity 463 2.1. Linearity 463 2.2. Nonlinearity 466 xviii Contents of Volume 1 3. Linear, nonlinear, and highly nonlinear approximation 467 4. Artificial neural networks 474 4.1. General considerations 474 4.2. Generically comprehensively revealing activation functions 475 5. QuickNet 476 5.1. A prototype QuickNet algorithm 477 5.2. Constructing  m 479 5.3. Controlling overfit 480 6. Interpretational issues 484 6.1. Interpreting approximation-based forecasts 485 6.2. Explaining remarkable forecast outcomes 485 6.3. Explaining adverse forecast outcomes 490 7. Empirical examples 492 7.1. Estimating nonlinear forecasting models 492 7.2. Explaining forecast outcomes 505 8. Summary and concluding remarks 509 Acknowledgements 510 References 510 PART 3: FORECASTING WITH PARTICULAR DATA STRUCTURES Chapter 10 Forecasting with Many Predictors JAMES H. STOCK AND MARK W. WATSON 515 Abstract 516 Keywords 516 1. Introduction 517 1.1. Many predictors: Opportunities and challenges 517 1.2. Coverage of this chapter 518 2. The forecasting environment and pitfalls of standard forecasting methods 518 2.1. Notation and assumptions 518 2.2. Pitfalls of using standard forecasting methods when n is large 519 3. Forecast combination 520 3.1. Forecast combining setup and notation 521 3.2. Large-n forecast combining methods 522 3.3. Survey of the empirical literature 523 4. Dynamic factor models and principal components analysis 524 4.1. The dynamic factor model 525 4.2. DFM estimation by maximum likelihood 527 4.3. DFM estimation by principal components analysis 528 4.4. DFM estimation by dynamic principal components analysis 532 4.5. DFM estimation by Bayes methods 533 4.6. Survey of the empirical literature 533 Contents of Volume 1 xix 5. Bayesian model averaging 535 5.1. Fundamentals of Bayesian model averaging 536 5.2. Survey of the empirical literature 541 6. Empirical Bayes methods 542 6.1. Empirical Bayes methods for large-n linear forecasting 543 7. Empirical illustration 545 7.1. Forecasting methods 545 7.2. Data and comparison methodology 547 7.3. Empirical results 547 8. Discussion 549 References 550 Chapter 11 Forecasting with Trending Data GRAHAM ELLIOTT 555 Abstract 556 Keywords 556 1. Introduction 557 2. Model specification and estimation 559 3. Univariate models 563 3.1. Short horizons 565 3.2. Long run forecasts 575 4. Cointegration and short run forecasts 581 5. Near cointegrating models 586 6. Predicting noisy variables with trending regressors 591 7. Forecast evaluation with unit or near unit roots 596 7.1. Evaluating and comparing expected losses 596 7.2. Orthogonality and unbiasedness regressions 598 7.3. Cointegration of forecasts and outcomes 599 8. Conclusion 600 References 601 Chapter 12 Forecasting with Breaks MICHAEL P. CLEMENTS AND DAVID F. HENDRY 605 Abstract 606 Keywords 606 1. Introduction 607 2. Forecast-error taxonomies 609 2.1. General (model-free) forecast-error taxonomy 609 2.2. VAR model forecast-error taxonomy 613 3. Breaks in variance 614 3.1. Conditional variance processes 614 . 28 7 Abstract 28 8 Keywords 28 8 1. Introduction and overview 28 9 1.1. Historical notes 29 0 1 .2. Notation, terminology, abbreviations 29 1 2. VARMA processes 29 2 2. 1. Stationary processes 29 2 2. 2 misspecified models 25 4 Acknowledgements 27 1 Part IV: Appendices and References 27 1 Appendix A: Assumptions 27 1 Appendix B: Proofs 27 5 References 28 0 PART 2: FORECASTING MODELS Chapter 6 Forecasting. results 22 1 3 .2. Out -of- sample implementation of Bai as well as Hong and Li tests 22 3 3.3. Out -of- sample implementation of Corradi and Swanson tests 22 5 3.4. Bootstrap critical for the V 1P,J and V 2P,J tests