Time Series Analysis

Katsuto Tanaka, PhD, is a professor in the Faculty of Economics at Gakushuin University and was previously a professor at Hitotsubashi University. He is a recipient of the Tjalling C. Koopmans Econometric Theory Prize (1996), the Japan Statistical Society Prize (1998), and the Econometric Theory Award (1999). Aside from the first edition of Time Series Analysis (Wiley, 1996), Dr. Tanaka had published five econometrics and statistics books in Japanese.

• Preface to the Second Edition xiPreface to the First Edition xiiiPart I Analysis of Non Fractional Time Series 11 Models for Nonstationarity and Noninvertibility 31.1 Statistics from the One-Dimensional Random Walk 31.1.1 Eigenvalue Approach 41.1.2 Stochastic Process Approach 111.1.3 The Fredholm Approach 121.1.4 An Overview of the Three Approaches 141.2 A Test Statistic from a Noninvertible Moving Average Model 161.3 The AR Unit Root Distribution 231.4 Various Statistics from the Two-Dimensional Random Walk 291.5 Statistics from the Cointegrated Process 411.6 Panel Unit Root Tests 472 Brownian Motion and Functional Central Limit Theorems 512.1 The Space L2 of Stochastic Processes 512.2 The Brownian Motion 552.3 Mean Square Integration 582.3.1 The Mean Square Riemann Integral 592.3.2 The Mean Square Riemann–Stieltjes Integral 622.3.3 The Mean Square Ito Integral 662.4 The Ito Calculus 722.5 Weak Convergence of Stochastic Processes 772.6 The Functional Central Limit Theorem 812.7 FCLT for Linear Processes 872.8 FCLT for Martingale Differences 912.9 Weak Convergence to the Integrated Brownian Motion 992.10 Weak Convergence to the Ornstein–Uhlenbeck Process 1032.11 Weak Convergence of Vector-Valued Stochastic Processes 1092.11.1 Space Cq 1092.11.2 Basic FCLT for Vector Processes 1102.11.3 FCLT for Martingale Differences 1122.11.4 FCLT for the Vector-Valued Integrated Brownian Motion 1152.12 Weak Convergence to the Ito Integral 1183 The Stochastic Process Approach 1273.1 Girsanov’s Theorem: O-U Processes 1273.2 Girsanov’s Theorem: Integrated Brownian Motion 1373.3 Girsanov’s Theorem: Vector-Valued Brownian Motion 1423.4 The Cameron–Martin Formula 1453.5 Advantages and Disadvantages of the Present Approach 1474 The Fredholm Approach 1494.1 Motivating Examples 1494.2 The Fredholm Theory: The Homogeneous Case 1554.3 The c.f. of the Quadratic Brownian Functional 1614.4 Various Fredholm Determinants 1714.5 The Fredholm Theory: The Nonhomogeneous Case 1904.5.1 Computation of the Resolvent – Case 1 1924.5.2 Computation of the Resolvent – Case 2 1994.6 Weak Convergence of Quadratic Forms 2035 Numerical Integration 2135.1 Introduction 2135.2 Numerical Integration: The Nonnegative Case 2145.3 Numerical Integration: The Oscillating Case 2205.4 Numerical Integration: The General Case 2285.5 Computation of Percent Points 2365.6 The Saddlepoint Approximation 2406 Estimation Problems in Nonstationary Autoregressive Models 2456.1 Nonstationary Autoregressive Models 2456.2 Convergence in Distribution of LSEs 2506.2.1 Model A 2516.2.2 Model B 2536.2.3 Model C 2556.2.4 Model D 2576.3 The c.f.s for the Limiting Distributions of LSEs 2606.3.1 The Fixed Initial Value Case 2616.3.2 The Stationary Case 2656.4 Tables and Figures of Limiting Distributions 2676.5 Approximations to the Distributions of the LSEs 2766.6 Nearly Nonstationary Seasonal AR Models 2816.7 Continuous Record Asymptotics 2896.8 Complex Roots on the Unit Circle 2926.9 Autoregressive Models with Multiple Unit Roots 3007 Estimation Problems in Noninvertible Moving Average Models 3117.1 Noninvertible Moving Average Models 3117.2 The Local MLE in the Stationary Case 3147.3 The Local MLE in the Conditional Case 3257.4 Noninvertible Seasonal Models 3307.4.1 The Stationary Case 3317.4.2 The Conditional Case 3337.4.3 Continuous Record Asymptotics 3357.5 The Pseudolocal MLE 3377.5.1 The Stationary Case 3377.5.2 The Conditional Case 3397.6 Probability of the Local MLE at Unity 3417.7 The Relationship with the State Space Model 3438 Unit Root Tests in Autoregressive Models 3498.1 Introduction 3498.2 Optimal Tests 3508.2.1 The LBI Test 3528.2.2 The LBIU Test 3538.3 Equivalence of the LM Test with the LBI or LBIU Test 3568.3.1 Equivalence with the LBI Test 3568.3.2 Equivalence with the LBIU Test 3588.4 Various Unit Root Tests 3608.5 Integral Expressions for the Limiting Powers 3628.5.1 Model A 3638.5.2 Model B 3648.5.3 Model C 3658.5.4 Model D 3678.6 Limiting Power Envelopes and Point Optimal Tests 3698.7 Computation of the Limiting Powers 3728.8 Seasonal Unit Root Tests 3828.9 Unit Root Tests in the Dependent Case 3898.10 The Unit Root Testing Problem Revisited 3958.11 Unit Root Tests with Structural Breaks 3988.12 Stochastic Trends Versus Deterministic Trends 4028.12.1 Case of Integrated Processes 4038.12.2 Case of Near-Integrated Processes 4068.12.3 Some Simulations 4099 Unit Root Tests in Moving Average Models 4159.1 Introduction 4159.2 The LBI and LBIU Tests 4169.2.1 The Conditional Case 4179.2.2 The Stationary Case 4199.3 The Relationship with the Test Statistics in Differenced Form 4249.4 Performance of the LBI and LBIU Tests 4279.4.1 The Conditional Case 4279.4.2 The Stationary Case 4309.5 Seasonal Unit Root Tests 4349.5.1 The Conditional Case 4349.5.2 The Stationary Case 4369.5.3 Power Properties 4389.6 Unit Root Tests in the Dependent Case 4449.6.1 The Conditional Case 4449.6.2 The Stationary Case 4469.7 The Relationship with Testing in the State Space Model 4479.7.1 Case (I) 4499.7.2 Case (II) 4509.7.3 Case (III) 4529.7.4 The Case of the Initial Value Known 45410 Asymptotic Properties of Nonstationary Panel Unit Root Tests 45910.1 Introduction 45910.2 Panel Autoregressive Models 46110.2.1 Tests Based on the OLSE 46310.2.2 Tests Based on the GLSE 47110.2.3 Some Other Tests 47510.2.4 Limiting Power Envelopes 48010.2.5 Graphical Comparison 48510.3 Panel Moving Average Models 48810.3.1 Conditional Case 49010.3.2 Stationary Case 49410.3.3 Power Envelope 49910.3.4 Graphical Comparison 50210.4 Panel Stationarity Tests 50710.4.1 Limiting Local Powers 50810.4.2 Power Envelope 51210.4.3 Graphical Comparison 51410.5 Concluding Remarks 51511 Statistical Analysis of Cointegration 51711.1 Introduction 51711.2 Case of No Cointegration 51911.3 Cointegration Distributions: The Independent Case 52411.4 Cointegration Distributions: The Dependent Case 53211.5 The Sampling Behavior of Cointegration Distributions 53711.6 Testing for Cointegration 54411.6.1 Tests for the Null of No Cointegration 54411.6.2 Tests for the Null of Cointegration 54711.7 Determination of the Cointegration Rank 55211.8 Higher Order Cointegration 55611.8.1 Cointegration in the I(d) Case 55611.8.2 Seasonal Cointegration 559Part II Analysis of Fractional Time Series 56712 ARFIMA Models and the Fractional Brownian Motion 56912.1 Nonstationary Fractional Time Series 56912.1.1 Case of d = ½ 57012.1.2 Case of d > ½ 57212.2 Testing for the Fractional Integration Order 57512.2.1 i.i.d. Case 57512.2.2 Dependent Case 58112.3 Estimation for the Fractional Integration Order 58412.3.1 i.i.d. Case 58412.3.2 Dependent Case 58612.4 Stationary Long-Memory Processes 59112.5 The Fractional Brownian Motion 59712.6 FCLT for Long-Memory Processes 60312.7 Fractional Cointegration 60812.7.1 Spurious Regression in the Fractional Case 60912.7.2 Cointegrating Regression in the Fractional Case 61012.7.3 Testing for Fractional Cointegration 61412.8 The Wavelet Method for ARFIMA Models and the fBm 61412.8.1 Basic Theory of the Wavelet Transform 61512.8.2 Some Advantages of the Wavelet Transform 61812.8.3 Some Applications of the Wavelet Analysis 62513 Statistical Inference Associated with the Fractional Brownian Motion 62913.1 Introduction 62913.2 A Simple Continuous-Time Model Driven by the fBm 63213.3 Quadratic Functionals of the Brownian Motion 64113.4 Derivation of the c.f. 64513.4.1 Stochastic Process Approach via Girsanov’s Theorem 64513.4.2 Fredholm Approach via the Fredholm Determinant 64713.5 Martingale Approximation to the fBm 65113.6 The Fractional Unit Root Distribution 65913.6.1 The FD Associated with the Approximate Distribution 65913.6.2 An Interesting Moment Property 66413.7 The Unit Root Test Under the fBm Error 66914 Maximum Likelihood Estimation for the Fractional Ornstein–Uhlenbeck Process 67314.1 Introduction 67314.2 Estimation of the Drift: Ergodic Case 67714.2.1 Asymptotic Properties of the OLSEs 67714.2.2 The MLE and MCE 67914.3 Estimation of the Drift: Non-ergodic Case 68714.3.1 Asymptotic Properties of the OLSE 68714.3.2 The MLE 68714.4 Estimation of the Drift: Boundary Case 69214.4.1 Asymptotic Properties of the OLSEs 69214.4.2 The MLE and MCE 69314.5 Computation of Distributions and Moments of the MLE and MCE 69514.6 The MLE-based Unit Root Test Under the fBm Error 70314.7 Concluding Remarks 70715 Solutions to Problems 709References 865Author Index 879Subject Index 883