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FORECASTING FOR ECONOMICS. AND BUSINESS. Gloria Gonzalez-Rivera. University of California-Riverside. PEARSON. Boston Col

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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

- -"

«

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

'

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

-„

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

'

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

CHAPTER 9

160

202 202 205 205 206 209 211 211 212 213 213 215 215 221 221 222

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

"

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

-,

Forecasting the Long Term: Deterministic and Stochastic Trends

255

Deterministic Trends 10.1.1 Trend Shapes 10.1.2 Trend Stationarity 10.1.3 Optimal Forecast Stochastic Trends 10.2.1 Trend Shapes 10.2.2 Stationarity Properties 10.2.3 Optimal Forecast x

Key Words Exercises CHAPTER 11

257 258 261 262 270 270 272 279 291 291

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

'

Key Words Appendix Exercises CHAPTER 15 15.1

360 362 368 370 378 380 390 390 393

Financial Applications of Time-Varying Volatility

Risk Management 75.7.7 Value-at-Risk (VaR) 15.1.2 Expected Shortfall (ES)

,

395 395 396 400

Contents 15.2 Portfolio Allocation 15.3 Asset Pricing 15.4 Option Pricing

xv 401 404 406

.

Key Words Appendix Exercises

411 411 412

V

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 .

414 414 417 419 420 421 422 427 436 440 440 441

Keywords Appendix Exercises

444 444 445

Appendix A: Review of Probability and Statistics

447

Appendix B: Statistical Tables

463

Glossary

472

"

References Index

,

•'

.

481 483

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