Time Series Analysis : Nonstationary and Noninvertible Distribution Theory.

Cover -- Title Page -- Copyright -- Contents -- Preface to the Second Edition -- Preface to the First Edition -- Part I Analysis of Non Fractional Time Series -- Chapter 1 Models for Nonstationarity and Noninvertibility -- 1.1 Statistics from the One-Dimensional Random Walk -- 1.1.1 Eigenvalue Approach -- 1.1.2 Stochastic Process Approach -- 1.1.3 The Fredholm Approach -- 1.1.4 An Overview of the Three Approaches -- 1.2 A Test Statistic from a Noninvertible Moving Average Model -- 1.3 The AR Unit Root Distribution -- 1.4 Various Statistics from the Two-Dimensional Random Walk -- 1.5 Statistics from the Cointegrated Process -- 1.6 Panel Unit Root Tests -- Chapter 2 Brownian Motion and Functional Central Limit Theorems -- 2.1 The Space L2 of Stochastic Processes -- 2.2 The Brownian Motion -- 2.3 Mean Square Integration -- 2.3.1 The Mean Square Riemann Integral -- 2.3.2 The Mean Square Riemann-Stieltjes Integral -- 2.3.3 The Mean Square Ito Integral -- 2.4 The Ito Calculus -- 2.5 Weak Convergence of Stochastic Processes -- 2.6 The Functional Central Limit Theorem -- 2.7 FCLT for Linear Processes -- 2.8 FCLT for Martingale Differences -- 2.9 Weak Convergence to the Integrated Brownian Motion -- 2.10 Weak Convergence to the Ornstein-Uhlenbeck Process -- 2.11 Weak Convergence of Vector-Valued Stochastic Processes -- 2.11.1 Space Cq -- 2.11.2 Basic FCLT for Vector Processes -- 2.11.3 FCLT for Martingale Differences -- 2.11.4 FCLT for the Vector-Valued Integrated Brownian Motion -- 2.12 Weak Convergence to the Ito Integral -- Chapter 3 The Stochastic Process Approach -- 3.1 Girsanov's Theorem: O-U Processes -- 3.2 Girsanov's Theorem: Integrated Brownian Motion -- 3.3 Girsanov's Theorem: Vector-Valued Brownian Motion -- 3.4 The Cameron-Martin Formula -- 3.5 Advantages and Disadvantages of the Present Approach -- Chapter 4 The Fredholm Approach.

4.1 Motivating Examples -- 4.2 The Fredholm Theory: The Homogeneous Case -- 4.3 The c.f. of the Quadratic Brownian Functional -- 4.4 Various Fredholm Determinants -- 4.5 The Fredholm Theory: The Nonhomogeneous Case -- 4.5.1 Computation of the Resolvent-Case 1 -- 4.5.2 Computation of the Resolvent-Case 2 -- 4.6 Weak Convergence of Quadratic Forms -- Chapter 5 Numerical Integration -- 5.1 Introduction -- 5.2 Numerical Integration: The Nonnegative Case -- 5.3 Numerical Integration: The Oscillating Case -- 5.4 Numerical Integration: The General Case -- 5.5 Computation of Percent Points -- 5.6 The Saddlepoint Approximation -- Chapter 6 Estimation Problems in Nonstationary Autoregressive Models -- 6.1 Nonstationary Autoregressive Models -- 6.2 Convergence in Distribution of LSEs -- 6.2.1 Model A -- 6.2.2 Model B -- 6.2.3 Model C -- 6.2.4 Model D -- 6.3 The c.f.s for the Limiting Distributions of LSEs -- 6.3.1 The Fixed Initial Value Case -- 6.3.2 The Stationary Case -- 6.4 Tables and Figures of Limiting Distributions -- 6.5 Approximations to the Distributions of the LSEs -- 6.6 Nearly Nonstationary Seasonal AR Models -- 6.7 Continuous Record Asymptotics -- 6.8 Complex Roots on the Unit Circle -- 6.9 Autoregressive Models with Multiple Unit Roots -- Chapter 7 Estimation Problems in Noninvertible Moving Average Models -- 7.1 Noninvertible Moving Average Models -- 7.2 The Local MLE in the Stationary Case -- 7.3 The Local MLE in the Conditional Case -- 7.4 Noninvertible Seasonal Models -- 7.4.1 The Stationary Case -- 7.4.2 The Conditional Case -- 7.4.3 Continuous Record Asymptotics -- 7.5 The Pseudolocal MLE -- 7.5.1 The Stationary Case -- 7.5.2 The Conditional Case -- 7.6 Probability of the Local MLE at Unity -- 7.7 The Relationship with the State Space Model -- Chapter 8 Unit Root Tests in Autoregressive Models -- 8.1 Introduction -- 8.2 Optimal Tests.

8.2.1 The LBI Test -- 8.2.2 The LBIU Test -- 8.3 Equivalence of the LM Test with the LBI or LBIU Test -- 8.3.1 Equivalence with the LBI Test -- 8.3.2 Equivalence with the LBIU Test -- 8.4 Various Unit Root Tests -- 8.5 Integral Expressions for the Limiting Powers -- 8.5.1 Model A -- 8.5.2 Model B -- 8.5.3 Model C -- 8.5.4 Model D -- 8.6 Limiting Power Envelopes and Point Optimal Tests -- 8.7 Computation of the Limiting Powers -- 8.8 Seasonal Unit Root Tests -- 8.9 Unit Root Tests in the Dependent Case -- 8.10 The Unit Root Testing Problem Revisited -- 8.11 Unit Root Tests with Structural Breaks -- 8.12 Stochastic Trends Versus Deterministic Trends -- 8.12.1 Case of Integrated Processes -- 8.12.2 Case of Near-Integrated Processes -- 8.12.3 Some Simulations -- Chapter 9 Unit Root Tests in Moving Average Models -- 9.1 Introduction -- 9.2 The LBI and LBIU Tests -- 9.2.1 The Conditional Case -- 9.2.2 The Stationary Case -- 9.3 The Relationship with the Test Statistics in Differenced Form -- 9.4 Performance of the LBI and LBIU Tests -- 9.4.1 The Conditional Case -- 9.4.2 The Stationary Case -- 9.5 Seasonal Unit Root Tests -- 9.5.1 The Conditional Case -- 9.5.2 The Stationary Case -- 9.5.3 Power Properties -- 9.6 Unit Root Tests in the Dependent Case -- 9.6.1 The Conditional Case -- 9.6.2 The Stationary Case -- 9.7 The Relationship with Testing in the State Space Model -- 9.7.1 Case (I) -- 9.7.2 Case (II) -- 9.7.3 Case (III) -- 9.7.4 The Case of the Initial Value Known -- Chapter 10 Asymptotic Properties of Nonstationary Panel Unit Root Tests -- 10.1 Introduction -- 10.2 Panel Autoregressive Models -- 10.2.1 Tests Based on the OLSE -- 10.2.2 Tests Based on the GLSE -- 10.2.3 Some Other Tests -- 10.2.4 Limiting Power Envelopes -- 10.2.5 Graphical Comparison -- 10.3 Panel Moving Average Models -- 10.3.1 Conditional Case -- 10.3.2 Stationary Case.

10.3.3 Power Envelope -- 10.3.4 Graphical Comparison -- 10.4 Panel Stationarity Tests -- 10.4.1 Limiting Local Powers -- 10.4.2 Power Envelope -- 10.4.3 Graphical Comparison -- 10.5 Concluding Remarks -- Chapter 11 Statistical Analysis of Cointegration -- 11.1 Introduction -- 11.2 Case of No Cointegration -- 11.3 Cointegration Distributions: The Independent Case -- 11.4 Cointegration Distributions: The Dependent Case -- 11.5 The Sampling Behavior of Cointegration Distributions -- 11.6 Testing for Cointegration -- 11.6.1 Tests for the Null of No Cointegration -- 11.6.2 Tests for the Null of Cointegration -- 11.7 Determination of the Cointegration Rank -- 11.8 Higher Order Cointegration -- 11.8.1 Cointegration in the I(d) Case -- 11.8.2 Seasonal Cointegration -- Part II Analysis of Fractional Time Series -- Chapter 12 ARFIMA Models and the Fractional Brownian Motion -- 12.1 Nonstationary Fractional Time Series -- 12.1.1 Case of Case of d = 1/2 -- 12.1.2 Case of Case of d &gt -- 1/2 -- 12.2 Testing for the Fractional Integration Order -- 12.2.1 i.i.d. Case -- 12.2.2 Dependent Case -- 12.3 Estimation for the Fractional Integration Order -- 12.3.1 i.i.d. Case -- 12.3.2 Dependent Case -- 12.4 Stationary Long-Memory Processes -- 12.5 The Fractional Brownian Motion -- 12.6 FCLT for Long-Memory Processes -- 12.7 Fractional Cointegration -- 12.7.1 Spurious Regression in the Fractional Case -- 12.7.2 Cointegrating Regression in the Fractional Case -- 12.7.3 Testing for Fractional Cointegration -- 12.8 The Wavelet Method for ARFIMA Models and the fBm -- 12.8.1 Basic Theory of the Wavelet Transform -- 12.8.2 Some Advantages of the Wavelet Transform -- 12.8.3 Some Applications of the Wavelet Analysis -- 12.8.3.1 Testing for d in ARFIMA Models -- 12.8.3.2 Testing for the Existence of Noise -- 12.8.3.3 Testing for Fractional Cointegration -- 12.8.3.4 Unit Root Tests.

Chapter 13 Statistical Inference Associated with the Fractional Brownian Motion -- 13.1 Introduction -- 13.2 A Simple Continuous-Time Model Driven by the fBm -- 13.3 Quadratic Functionals of the Brownian Motion -- 13.4 Derivation of the c.f. -- 13.4.1 Stochastic Process Approach via Girsanov's Theorem -- 13.4.1.1 Case of H = 1/2 -- 13.4.1.2 Case of H &gt -- 1/2 -- 13.4.2 Fredholm Approach via the Fredholm Determinant -- 13.4.2.1 Case of H = 1/2 -- 13.4.2.2 Case of H &gt -- 1/2 -- 13.5 Martingale Approximation to the fBm -- 13.6 The Fractional Unit Root Distribution -- 13.6.1 The FD Associated with the Approximate Distribution -- 13.6.2 An Interesting Moment Property -- 13.7 The Unit Root Test Under the fBm Error -- Chapter 14 Maximum Likelihood Estimation for the Fractional Ornstein-Uhlenbeck Process -- 14.1 Introduction -- 14.2 Estimation of the Drift: Ergodic Case -- 14.2.1 Asymptotic Properties of the OLSEs -- 14.2.2 The MLE and MCE -- 14.3 Estimation of the Drift: Non-ergodic Case -- 14.3.1 Asymptotic Properties of the OLSE -- 14.3.2 The MLE -- 14.4 Estimation of the Drift: Boundary Case -- 14.4.1 Asymptotic Properties of the OLSEs -- 14.4.2 The MLE and MCE -- 14.5 Computation of Distributions and Moments of the MLE and MCE -- 14.6 The MLE-based Unit Root Test Under the fBm Error -- 14.7 Concluding Remarks -- Chapter 15 Solutions to Problems -- References -- Author Index -- Subject Index -- EULA.

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Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2024. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.