Elements of Forecasting

PART I: GETTING STARTED
Chapter 1: Introduction to Forecasting: Applications,
Methods, Books, Journals, and Software I
1. Forecasting in Action 1
2. Forecasting Methods: An Overview of the Book 3
3. Useful Books, Journals, Software, and Online Information 6
4. Looking Ahead 9
Exercises, Problems, and Complements 9
Forecasting in daily life: Wc are all forecasting, all the time 9
Forecasting in business, finance, economics, and government 9
The basic forecasting framework 10
Degrees of forecastability 10
Data on the web 10
Univariate and multivariate forecasting models 10
Concepts for Review 11
References and Additional Readings 11
Chapter 2: A Brief Review of Probability, Statistics,
and Regression for Forecasting 13
1. Why This Chapter? 13
2. Random Variables, Distributions, and Moments 14
3. Multivariate Random Variables 15
4. Statistics 16
5. Regression Analysis 18
Exercises, Problems, and Complements 30
Interpreting distributions and densities 30
Covariance and correlation 30
Conditional expectations versus linear projections 30
Conditional mean and variance 30
Scatterplots and regression lines 30
Desired values of regression diagnostic statistics 31
Mechanics of fitting a linear regression 31
Regression with and without a constant term 31
Interpreting coefficients and variables 31
Nonlinear least squares 31
Regression semantics 32
Bibliographical and Computational Notes 32
Concepts for Review 32
References and Additional Readings 33
Chapter 3: Six Considerations Basic to Successful
Forecasting 3 4
1. The Decision Environment and Loss Function 35
2. The Forecast Object 39
3. The Forecast Statement 40
4. The Forecast Horizon 43
5. The Information Set 45
6. Methods and Complexity, the Parsimony Principle,
and the Shrinkage Principle 46
7. Concluding Remarks 47
Exercises, Problems, and Complements 47
Data and forecast timing conventions 47
Properties of loss functions 47
Relationships among point, interval, and density forecasts 47
Forecasting at short through long horizons 47
Forecasting as an ongoing process in organizations 48
Assessing forecasting situations 48
Bibliographical and Computational Notes 49
Concepts for Review 49
References and Additional Readings 50
PART II: BUILDING USING AND EVALUATING
FORECASTING MDDELS
Chapter 4. Statistical Graphics for Forecasting 51
1. The Power of Statistical Graphics 51
2. Simple Graphical Techniques 55
3. Elements of Graphical Style 59
Contents xiii
4. Application: Graphing Four Components of Real GDP 63
5. Concluding Remarks 06
Exercises, Problems, and Complements 67
Outliers 67
Simple versus partial correlation 67
Graphical regression diagnostic 1: time series plot of y t , y,, and e t 67
Graphical regression diagnostic 2: lime series plot of e[ or \e t \ 68
Graphical regression diagnostic 3: scatterplot of e t versus x, 68
Graphical analysis of foreign exchange rate data 68
Common scales 69
Graphing real GDP, continued from Section 4 69
Color 69
Regression, regression diagnostics, and regression graphics in action 69
Bibliographical and Computational Notes 70
Concepts for Review 71
References and Additional Readings 71
Chapter 5: Modeling and Forecasting Trend 7 2
1. Modeling Trend 72
2. Estimating Trend Models 80
3. Forecasting Trend 81
4. Selecting Forecasting Models Using the Akaike and Schwarz Criteria 82
5. Application: Forecasting Retail Sales 87
Exercises, Problems, and Complements 94
Calculating forecasts from trend models 94
Identifying anrl testing trend models 94
Understanding model selection criteria 94
Mechanics of trend estimation and forecasting 95
Properties of polynomial trends 95
Specialized nonlinear trends 95
Moving average smoothing for trend estimation 95
Bias corrections when forecasting from logarithmic models 96
Model selection for long-horizon forecasting 97
The variety of "information criteria" reported across software packages 97
Bibliographical and Computational Notes 97
Concepts for Review 98
References and Additional Readings 98
Chapter G: Modeling and Forecasting Seasonality 3 9
1. The Nature and Sources of Seasonality 99
2. Modeling Seasonality 101
3. Forecasting Seasonal Series 103
4. Application: Forecasting Housing Starts 104
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Exercises, Problems, and Complements 108
Log transformations in seasonal models 108
Seasonal adjustment 108
Selecting forecasting models involving calendar effects 108
Testing for seasonality' 109
Seasonal regressions with an intercept and v— 1 seasonal dummies 109
Applied trend and seasonal modeling 109
Periodic models 109
Interpreting dummy variables 110
Constructing seasonal models 110
Calendar effects 110
Bibliographical and Computational Notes 111
Concepts for Review 111
References and Additional Readings 111
C h a p t e r 7 C H s r a c t p . n i r t n g C y c l e s 112
1. Covariance Stationary Time Series 113
2. White Noise 117
3. The Lag Operator 123
4. Wold's Theorem, the General Linear Process,
and Rational Distributed l-ags 124
5. Estimation and Inference for the Mean, Autocorrelation, and Partial
Autocorrelation Functions 127
6. Application: Characterizing Canadian Employment Dynamics 130
Exercises, Problems, and Complements 132
Lag operator expressions 1 132
Lag operator expressions 2 133
Autocorrelation functions of covariance stationary series 133
Autocorrelation vs. partial autocorrelation 133
Conditional and unconditional means 133
White noise residuals 133
Selecting an employment forecasting model with the AIC and SIC 134
Simulation of a time series process 134
Sample autocorrelation functions for trending series 134
Sample autocorrelation functions for seasonal series 134
Volatility dynamics: correlograms of squares 135
Bibliographical and Computational Notes 135
Concepts for Review 135
References and Additional Readings 136
C h a p t e r B: M a d p l i n g C y c l e s MA AR
a n d APMA M o d e l ? 137
1. Moving Average (MA) Models 138
2. Autoregressive (AR) Models 145
3. Autoregressive Moving Average (ARMA) Models 152
Contents
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4. Application: Specifying and Estimating Models
for Employment Forecasting 154
Exercises, Problems, and Complements 163
ARMA lag inclusion 163
Shapes of correlograms 163
The autocovariance function of the MA(1) process, revisited 163
ARMA algebra 163
Diagnostic checking of model residuals 163
Mechanics of fitting ARMA models 165
Modeling cyclical dynamics 165
Aggregation and disaggregation: top-down forcasting model
vs. bottom-up forecasting model 165
Nonlinear forecasting models: regime switching 165
Difficulties with nonlinear optimization 166
Bibliographical and Computational Notes 167
Concepts for Review 168
References and Additional Readings 169
Chapter 9: Forecasting Cycles 171
1. Optimal Forecasts 171
2. Forecasting Moving Average Processes 172
3. Making the Forecasts Operational 176
4. The Chain Rule of Forecasting 177
5. Application: Forecasting Employment 180
Exercises, Problems, and Complements 184
Forecast accuracy across horizons 184
Mechanics of forecasting with ARMA models: Bankwire continued 184
Forecasting an AR(1) process with known and unknown parameters 185
Forecasting an ARMA(2, 2) process 185
Optimal forecasting under asymmetric loss 186
Truncation of infinite distributed lags, state space representations,
and the Kalman filter 187
Point and interval forecasts allowing for serial correlation—
Nile.com continued 187
Bootstrapping simulation to acknowledge innovation distribution
uncertainty and parameter estimation uncertainty 188
Bibliographical and Computational Notes 189
Concepts for Review 190
References and Additional Readings 190
Chapter ID: Putting It All Together: A Forecasting
Model with Trend. Seasonal, and Cyclical Components 191
1. Assembling What We've Learned 191
2. Application: Forecasting Liquor Sales 193
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3. Recursive Estimation Procedures for Diagnosing
and Selecting Forecasting Models 207
4. Liquor Sales, Continued 212
Exercises, Problems, and Complements 214
Serially correlated disturbances vs. lagged dependent variables 214
Assessing the adequacy of the liquor sales forecasting model
trend specification 214
Improving nontrend aspects of the liquor sales forecasting model 214
CUSUM analvsis of the housing starts model 215
Model selection based on simulated forecasting performance 215
Seasonal models with time-varying parameters: forecasting
AirSpeed passenger-miles 215
Formal models of unobserved components 216
The restrictions associated with unobserved-components structures 216
Additive unobserved-components decomposition and multiplicative
unobserved-components decomposition 217
Signal, noise, and over fit ting 217
Bibliographical and Computational Notes 217
Concepts for Review 218
References and Additional Readings 218
Chapter II: Forecasting with Regression Models 219
1. Conditional Forecasting Models and Scenario Analysis 220
2. Accounting for Parameter Uncertainty in Confidence
Intervals for Conditional Forecasts 220
3. Unconditional Forecasting Models 223
4. Distributed Lags, Polynomial Distributed Lags,
and Rational Distributed Lags 224
5. Regressions with Lagged Dependent Variables, Regressions with
ARM\ Disturbances, and Transfer Function Models 225
6. Vector Autoregressions 228
7. Predictive Causality 230
8. Impulse-Response Functions and Variance Decompositions 231
9. Application: Housing Starts and Completions 235
Exercises, Problems, and Complements 249
Econometrics, time series analysis, and forecasting 249
Forecasting crop yields 249
Regression forecasting models with expectations, or anticipatory, data 249
Business cycle analysis and forecasting: expansions, contractions,
turning points, and leading indicators 250
Subjective information, Bayesian VARs, and the Minnesota prior 251
Housing starts and completions, continued 251
Nonlinear regression models 1: functional form and Ramsey's test 251
Nonlinear regression models 2: logarithmic regression models 252
Nonlinear regression models 3: neural networks 252
Spurious regression 253
Comparative forecasting performance of VAR and univariate models 254
Contents
Bibliographical and Computational Notes 254
Concepts for Review 255
References and Additional Readings 255
Chapter 12 Evaluating and Combining Forecasts 2 5 7
1. Evaluating a Single Forecast 257
2. Evaluating Two or More Forecasts: Comparing Forecast Accuracy 260
3. Forecast Encompassing and Forecast Combination 263
4. Application: OverSea Shipping Volume
on the Atlantic East Trade Lane 268
Exercises, Problems, and Complements 280
Forecast evaluation in action 280
Forecast error analysis 280
Combining forecasts 280
Quantitative forecasting, judgmental forecasting, forecast
combination, and shrinkage 281
The algebra of forecast combination 281
The mechanics of practical forecast evaluation and combination 282
What arc we forecasting? Preliminary series, revised series,
and the limits to forecast accuracy 282
Ex post versus real-time forecast evaluation 283
What do we know about the accuracy of macroeconomic forecasts? 283
Forecast evaluation when realizations are unobserved 283
Forecast error variances in models with estimated parameters 283
The empirical success of forecast combination 284
Forecast combination and the Box-Jenkins paradigm 284
Consensus forecasts 285
The Delphi method for combining experts' forecasts 285
Bibliographical and Computational Notes 285
Concepts for Review 286
References and Additional Readings 286
PART Ml M O P E A D V A N C E D TOPICS
Chnpter 13 Unit Pout*. Star.hjytic Trends, APIMA
ForRCRF-Tinu MULIRI^ ^nri Smoothing 2 S S
1. Stochastic Trends and Forecasting 288
2. Unit Roots: Estimation and Testing 295
3. Application: Modeling and Forecasting the Yen/Dollar Exchange Rate 302
4. Smoothing 312
5. Exchange Rates, Continued 318
Exercises, Problems, and Complements 320
Modeling and forecasting the deutschemark/dollar
(DEM/USD) exchange rate 320
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Contents
Housing starts and completions, continued 320
ARIMA models, smoothers, and shrinkage 320
Using stochastic trend unobserved-components models to
implement smoothing techniques in a probabilistic framework 320
Automatic ARIMA modeling 321
The multiplicative seasonal ARIMA(/>, rf, q) x {P, D. Q) model 321
The Dickey-Fuller regression in the AR(2) case 321
Holt-Winters smoothing with multiplicative seasonality 322
Cointegration 323
Error correction 323
Forecast encompassing tests for 7(1) series 324
Evaluating forecasts of integrated series 324
Theil's cAstaustic 324
Bibliographical and Computational Notes 325
Concepts for Review 326
References and Additional Readings 326
Chapter 14: Volatility Measurement,
Modeling, and Forecasting 3 2 9
1. The Basic ARCH Process 330
2. The GARCH Process 333
3. Extensions of ARCH and GARCH Models 337
4. Estimating, Forecasting, and Diagnosing GARCH Models 340
5. Application: Stock Market Volatility 341
Exercises, Problems, and Complements 349
Removing conditional mean dynamics before modeling
volatility dynamics 349
Variations on the basic ARCH and GARCH models 349
Empirical performance of pure ARCH models as approximations
to volatility dynamics 349
Direct modeling of volatility proxies 350
GARCH volatility forecasting 350
Assessing volatility dynamics in observed returns and in
standardized returns 350
Allowing for leptokurtic conditional densities 351
Optimal prediction under asymmetric loss 351
Multivariate GARCH models 351
Bibliographical and Computational Notes 352
Concepts for Review 352
References and Additional Readings 353
Bibliography 355
Name Index 361
Subject Index 363