ABSTRACT
CONTENTS 9.1 Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523 9.2 Linear Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523 9.3 Types of Stochastic Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524
9.3.1 Tools for the Analysis of Time-Series Data . . . . . . . . . . . . . . . . . . . . . . . . . . 525 9.3.2 Properties and Features of Autocovariances and ACs . . . . . . . . . . . . . . . . . . . 527
9.3.2.1 Estimation of Autocovariance and ACFs . . . . . . . . . . . . . . . . . . . . . 528 9.3.2.2 Standard Errors of AC Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . 528 9.3.2.3 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 528 9.3.2.4 Linear-Stationary Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 529 9.3.2.5 AR Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 530 9.3.2.6 Stationarity and Invertibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 531 9.3.2.7 ACF of an AR(p) Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 531 9.3.2.8 MA Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535 9.3.2.9 Mixed AR and MA Process: ARMA Process . . . . . . . . . . . . . . . . . . 537 9.3.2.10 Partial ACs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 538 9.3.2.11 Partial ACs of an AR Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 539 9.3.2.12 Alternative Derivation of the Partial ACs . . . . . . . . . . . . . . . . . . . . . 539 9.3.2.13 Estimation of the Partial ACF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 540 9.3.2.14 Partial AC of an MA Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 541 9.3.2.15 Autocovariance-Generating Function . . . . . . . . . . . . . . . . . . . . . . . . 541
9.3.3 Forecasting: Principles and Practical Aspects . . . . . . . . . . . . . . . . . . . . . . . . . 542 9.3.3.1 Principles of Forecasting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543 9.3.3.2 Forecasting ARMA Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543 9.3.3.3 Forecasting Particular Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 544 9.3.3.4 Confidence Intervals for the Forecasts . . . . . . . . . . . . . . . . . . . . . . . 545 9.3.3.5 Evaluating Forecasts, Forecast Accuracy, and Combination . . . . . . . . . 546 9.3.3.6 Evaluating a Single Forecast . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 546 9.3.3.7 Testing the Rationality of Multistep Forecasts . . . . . . . . . . . . . . . . . . 548 9.3.3.8 Accuracy Measures and Tests of a Comparative Forecast Accuracy . . . 549 9.3.3.9 Morgan-Granger-Newbold Test . . . . . . . . . . . . . . . . . . . . . . . . . . . 549 9.3.3.10 Meese-Rogoff Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 550 9.3.3.11 Diebold-Mariano Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 551 9.3.3.12 Modified DM Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 552
9.3.4 MLE and Prediction-Error Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . 555 9.3.4.1 State-Space Form and ARMA Models . . . . . . . . . . . . . . . . . . . . . . . 557
9.3.4.2 KF Recursions and the Estimation of ARMA Models . . . . . . . . . . . . . 559 9.3.4.3 KF Recursions: Main Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 559 9.3.4.4 Kalman Smoothing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 562
9.3.5 Time-Series Modeling: Practical Aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . 564 9.3.5.1 Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 567
9.3.6 Modeling Nonstationary Processes: Unit Roots, Testing, and Implications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 569 9.3.6.1 Classical Trend Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 570 9.3.6.2 Trend-Stationary and Difference-Stationary Models . . . . . . . . . . . . . . 572 9.3.6.3 Processes with Unit Roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 575 9.3.6.4 Unit-Root Tests for a Wider Class of Errors . . . . . . . . . . . . . . . . . . . 580 9.3.6.5 Efficient Unit-Root Tests and GLS Detrending . . . . . . . . . . . . . . . . . 582 9.3.6.6 Unit Roots versus Trend-Break Alternative . . . . . . . . . . . . . . . . . . . . 587 9.3.6.7 Extensions to Perron’s Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 589 9.3.6.8 Application of Unit-Root Test Procedures . . . . . . . . . . . . . . . . . . . . . 591 9.3.6.9 Right-Tail Unit-Root Test and Test for Bubbles . . . . . . . . . . . . . . . . . 598 9.3.6.10 Other Approaches to Unit-Root Tests: Variance-Ratio Test . . . . . . . . . 599
9.4 Spectral or Frequency-Domain Analysis: An Overview . . . . . . . . . . . . . . . . . . . . . . . 607 9.4.1 Uses of Spectral Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 607 9.4.2 Search for Periodicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 608 9.4.3 Power Spectrum and Its Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 608 9.4.4 Estimation of the Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 610 9.4.5 Application of Spectral Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 611 9.4.6 A Note on Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613
9.4.6.1 Complex Spectral Representation . . . . . . . . . . . . . . . . . . . . . . . . . . 613 9.4.6.2 ARMA in the Frequency Domain . . . . . . . . . . . . . . . . . . . . . . . . . . 613
9.4.7 Properties of Linear Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 615 9.4.7.1 Gain and Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 615 9.4.7.2 Spurious Cyclical Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 617
9.4.8 A Note on Analyzing Seasonality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 618 9.4.8.1 Deterministic Seasonality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 618 9.4.8.2 Stochastic Seasonality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 620 9.4.8.3 Canova-Hansen Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 621 9.4.8.4 HEGY Test Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 622
9.5 Recent Developments and Research Trends . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 625 9.6 Information on Software Packages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 626 9.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 627 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 627
ABSTRACT In this chapter, the aim is to analyze the basics of a univariate time-series model building, estimation, and forecasting in detail. We start with the nonseasonal time series and introduce the stationary models based on Wold’s decomposition, explain the salient features of Box-Jenkins methodology to a time-series model identification, and the estimation of such models using the exact maximum likelihood procedure. The practical aspects in estimating such models are also discussed with the help of examples. The section on forecasting discusses the principles and the practical aspects of forecasting a time series, using symmetric and asymmetric loss functions, besides introducing the idea of forecast combination. Relaxing the assumption of stationarity, we explain the differences between the stationary and nonstationary processes and also outline the
various econometric tests of nonstationarity. Next, we show how the frequency-domain techniques or spectral-analytic tools can be used to determine the periodicity of the cyclical component of a time series. An associated section explains the properties of filtering in general and highlights the distortion that such filtering may induce. The chapter concludes with a short note on the seasonality and the issues involved in testing for the deterministic and stochastic seasonality. We also include a brief discussion on the testing for unit roots at seasonal frequencies. A number of examples will highlight the important practical issues involved.
