Nothing in nature is unbeautiful. Alfred, Lord Tennyson
Idea Transcript
FORECASTING FOR ECONOMICS AND BUSINESS
Gloria Gonzalez-Rivera University of California-Riverside
PEARSON Boston Columbus Indianapolis New York San Francisco Upper Saddle River Amsterdam Cape Town Dubai London Madrid Milan Munich Paris Montreal Toronto Delhi Mexico City Sao Paulo Sydney Hong Kong Seoul Singapore Taipei Tokyo
Contents
Preface
MODULE I
v
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«
xvi
STATISTICS AND TIME SERIES
CHAPTER 1 Introduction and Context
1
1.1
1 1 2 3 4 4 4 5 5
1.2
1.3
1.4
1.5 1.6
What Is Forecasting? 1.1.1 The First Forecaster in History: The Delphi Oracle 1.1.2 Examples of Modern Forecasts 1.1.3 Definition of Forecasting 1.1.4 Two Types of Forecasts Who Are the Users of Forecasts? . 1.2.1 Firms 1.2.2 Consumers and Investors 1.2.3 Government Becoming Familiar with Economic Time Series: Features of a Time Series 1.3.1 Trends 1.3.2 Cycles 1.3.3 Seasonality Basic Notation and the Objective of the Forecaster 1.4.1 Basic Notation 1.4.2 The Forecaster's Objective A Road Map for This Forecasting Book Resources
Key Words Exercises CHAPTER 2 2.1 2.2
. ';
6 7 8 9 11 11 12 13 14 16 17
Review of the Linear Regression Model
Conditional Density and Conditional Moments Linear Regression Model
24 24 27
IX
Contents 2.3
2.4
Estimation: Ordinary Least Squares 2.3.1 R-squared and Adjusted R-squared 2.3.2 Linearity and OLS 2.3.3 Assumptions of OLS: The Gauss-Markov Theorem 2.3.4 An Example: House Prices and Interest Rates Hypothesis Testing in a Regression Model *-' 2.4.1 The t-ratio 2.4.2 The F-test
Key Words Appendix Exercises CHAPTER 3
3.1
3.2
3.3
3.4
K
»
CHAPTER 4
4.3
4
46 47 9 52
Stochastic Process and Time Series 3.1.1 Stochastic Process 3.1.2 Time Series The Interpretation of a Time Average 3.2.1 Stationarity 3.2.2 Useful Transformations of Nonstationary Processes A New Tool of Analysis: The Autocorrelation Functions 3.3.1 Partial Autocorrelation 3.3.2 Statistical Tests for Autocorrelation Coefficients Conditional Moments and Time Series: What Lies Ahead
MODULE II
4.2
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Statistics a n d Time Series
Key Words Appendix Exercises
4.1
29 32 33 35 38 41 41 44
54 55 56 57 58 62 65 69 71 73
^
74 74 76
MODELING LINEAR DEPENDENCE FORECASTING WITH TIME SERIES MODELS Tools of the Forecaster
The Information Set 4.1.1 Some Information Sets Are More Valuable Than Others 4.1.2 Some Time Series Are More Forecastable Than Others The Forecast Horizon 4.2.1 Forecasting Environments The Loss Function 4.3.1 Some Examples of Loss Functions
79
80 82 84 84 86 89 91
Contents 4.3.2 4.3.3 Key Words Appendix Exercises
Examples Optimal Forecast: An Introduction
xi 91 93
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96 97 98
A PAUSE Where Are We and Where Are We Going?
100
Where Are We Going from Here? How to Organize Your Reading of the Forthcoming Chapters
100 102
CHAPTER 5 5.1
5.2 5.3 5.4
A Understanding Linear Dependence: A Link to Economic Models
Price 5.1.1 5.1.2 5.1.3
Dynamics: The Cob-Web Model (Beginner Level) ' The Effect of Only One Supply Shock The Effect of Many Supply Shocks A Further Representation of the Dynamics in the Cob-Web Model 5.1.4 Simulation of the Model, pt=p*(l — ) + 4> pt-j + e,, and Autocorrelation Function Portfolio Returns and Nonsynchronous Trading (Intermediate Level) \ Asset Prices and the Bid-Ask Bounce (Advanced Level) Summary
Key Words Appendix Exercises CHAPTER 6 6.1 6.2
6.3
103 105 106 107 109 113 116 121 121 121 123
Forecasting with Moving Average (MA) Processes
A Model with No Dependence: White Noise 6.1.1 What Does This Process Look Like ? The Wold Decomposition Theorem: The Origin of AR and MA Models (Advanced Section) 6.2.1 Finite Representation of the Wold Decomposition Forecasting with Moving Average Models , 6.3.1 MA(1) Process 6.3.2 MA(q) Process
Key Words Appendix Exercises
103
125 125 126 129 131 133 135 147 157 157 158
xii
Contents CHAPTER 7 Forecasting with Autoregressive (AR) Processes 7.1 7.2
7.3
Cycles Autoregressive Models 7.2.1 The AR(1) Process 7.2.2 AR(2) Process v 7.2.3 AR(p) Process " 7.2.4 Chain Rule of Forecasting Seasonal Cycles 7.3.1 Deterministic and Stochastic Seasonal Cycles 7.3.2 Seasonal ARMA Models 7.3.3 Combining ARMA and Seasonal ARMA Models
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Key Words Exercises CHAPTER 8 8.1 8.2
8.3
Forecasting Practice I
Key Words Exercises
9.1
9.2
162 165 165 173 185 187 188 189 192 197 200 200
The Data: San Diego House Price Index Model Selection 8.2.1 Estimation: AR, MA, and ARMA Models 8.2.2 Is the Process Covariance-Stationary, and Is the Process Invertible ? 8.2.3 Are the Residuals White Noise? \ 8.2.4 Are the Parameters of the Model Statistically Significant? 8.2.5 Is the Model Explaining a Substantial Variation of the Variable of Interest? 8.2.6 Is It Possible to Select One Model Among Many? The Forecast 8.3.1 Who Are the Consumers of Forecasts? ; 8.3.2 Is It Possible To Have Different Forecasts from the Same Model? 8.3.3 What Is the Most Common Loss Function in Economics and Business? 8.3.4 Final Comments
Forecasting Practice II: Assessment of Forecasts and Combination of Forecasts
Optimal Forecast 9.1.1 Symmetric and Asymmetric Loss Functions 9.1.2 Testing the Optimally of the Forecast Assessment of Forecasts 9.2.1 Descriptive Evaluation of the Average Loss 9.2.2 Statistical Evaluation of the Average Loss
224 225 225 229 238 239 240
Contents 9.3
Combination of Forecasts 9.3.1 Simple Linear Combinations 9.3.2 Optimal Linear Combinations
Key Words Appendix Exercises
xiii 244 244 245
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247 248 250
A PAUSE Where Are We and Where Are We Going?
252
Where Are We Going from Here?
253
CHAPTER 10 10.1
10.2
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Forecasting the Long Term: Deterministic and Stochastic Trends
Forecasting with a System of Equations: Vector Autoregression
11.1 What Is Vector Autoregression (VAR)? 11.2 Estimation of VAR 11.3 Granger Causality 11.4 Impulse-Response Functions ' 11.5 Forecasting with VAR
293 :
Key Words* Exercises CHAPTER 12 12.1 12.2 12.3
309 309 Forecasting the Long Term and the Short Term Jointly
Finding a Long-Term Equilibrium Relationship Quantifying Short-Term Dynamics: Vector Error Correction Model Constructing the Forecast
Key Words Exercises
294 294 299 302 305
311 315 322 327 332 332
iv
Contents A PAUSE
Where Are We and Where Are We Going?
334
Where We Are Going from Here How to Organize Your Reading of the Forthcoming Chapters
335 336
MODULE III MODELING MORE COMPLEX DEPENDENCE CHAPTER 13 13.1
13.2 13.3 13.4 13.5
Forecasting Volatility I
337
Motivation 13.1.1 The World is Concerned About Uncertainty 13.1.2 Volatility Within the Context of Our Forecasting Problem 13.1.3 Setting the Objective Time-Varying Dispersion: Empirical Evidence Is There Time Dependence in Volatility? What Have We Learned So Far? Simple Specifications for the Conditional Variance 73.5.7 Rolling Window Volatility 13.5.2 Exponentially Weighted Moving Average (EWMA) Volatility
337 337 • 339 340 341 345 353 353 354 355
Keywords Exercises CHAPTER 14 14.1
14.2
357 357 Forecasting Volatility II
359
The ARCH Family 74.7.7 ARCH(l) 14.1.2 ARCH(p) 14.1.3 GARCH(l.l) 14.1.4 Estimation Issues for the ARCH Family Realized Volatility
CHAPTER 16 Forecasting with Nonlinear Models: An Introduction
413
16.1
Nonlinear Dependence — " » 16.1.1 Whdtlslt? 16.1.2 Is There Any Evidence of Nonlinear Dynamics in the Data? 16.1.3 Nonlinearity, Correlation, and Dependence 16.1.4 What Have We Learned So Far? 16.2 Nonlinear Models: An Introduction 16.2.1 Threshold Autoregressive Models (TAR) 16.2.2 Smooth Transition Models 16.2.3 Markov Regime-Switching Models: A Descriptive Introduction 16.3 Forecasting with Nonlinear Models 16.3.1 One-Step-Ahead Forecast 16.3.2 Multistep-Ahead Forecast .