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
xiv  Contents
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
xv
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
xvi  Contents
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
xviii
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