Time Series Analysis
Description
More details
Other editions
Additional editions
Person
Content
Preface to the Second Edition xi
Preface to the First Edition xiii
Part I Analysis of Non Fractional Time Series 1
1 Models for Nonstationarity and Noninvertibility 3
1.1 Statistics from the One-Dimensional Random Walk 3
1.1.1 Eigenvalue Approach 4
1.1.2 Stochastic Process Approach 11
1.1.3 The Fredholm Approach 12
1.1.4 An Overview of the Three Approaches 14
1.2 A Test Statistic from a Noninvertible Moving Average Model 16
1.3 The AR Unit Root Distribution 23
1.4 Various Statistics from the Two-Dimensional Random Walk 29
1.5 Statistics from the Cointegrated Process 41
1.6 Panel Unit Root Tests 47
2 Brownian Motion and Functional Central Limit Theorems 51
2.1 The Space L2 of Stochastic Processes 51
2.2 The Brownian Motion 55
2.3 Mean Square Integration 58
2.3.1 The Mean Square Riemann Integral 59
2.3.2 The Mean Square Riemann-Stieltjes Integral 62
2.3.3 The Mean Square Ito Integral 66
2.4 The Ito Calculus 72
2.5 Weak Convergence of Stochastic Processes 77
2.6 The Functional Central Limit Theorem 81
2.7 FCLT for Linear Processes 87
2.8 FCLT for Martingale Differences 91
2.9 Weak Convergence to the Integrated Brownian Motion 99
2.10 Weak Convergence to the Ornstein-Uhlenbeck Process 103
2.11 Weak Convergence of Vector-Valued Stochastic Processes 109
2.11.1 Space Cq 109
2.11.2 Basic FCLT for Vector Processes 110
2.11.3 FCLT for Martingale Differences 112
2.11.4 FCLT for the Vector-Valued Integrated Brownian Motion 115
2.12 Weak Convergence to the Ito Integral 118
3 The Stochastic Process Approach 127
3.1 Girsanov's Theorem: O-U Processes 127
3.2 Girsanov's Theorem: Integrated Brownian Motion 137
3.3 Girsanov's Theorem: Vector-Valued Brownian Motion 142
3.4 The Cameron-Martin Formula 145
3.5 Advantages and Disadvantages of the Present Approach 147
4 The Fredholm Approach 149
4.1 Motivating Examples 149
4.2 The Fredholm Theory: The Homogeneous Case 155
4.3 The c.f. of the Quadratic Brownian Functional 161
4.4 Various Fredholm Determinants 171
4.5 The Fredholm Theory: The Nonhomogeneous Case 190
4.5.1 Computation of the Resolvent - Case 1 192
4.5.2 Computation of the Resolvent - Case 2 199
4.6 Weak Convergence of Quadratic Forms 203
5 Numerical Integration 213
5.1 Introduction 213
5.2 Numerical Integration: The Nonnegative Case 214
5.3 Numerical Integration: The Oscillating Case 220
5.4 Numerical Integration: The General Case 228
5.5 Computation of Percent Points 236
5.6 The Saddlepoint Approximation 240
6 Estimation Problems in Nonstationary Autoregressive Models 245
6.1 Nonstationary Autoregressive Models 245
6.2 Convergence in Distribution of LSEs 250
6.2.1 Model A 251
6.2.2 Model B 253
6.2.3 Model C 255
6.2.4 Model D 257
6.3 The c.f.s for the Limiting Distributions of LSEs 260
6.3.1 The Fixed Initial Value Case 261
6.3.2 The Stationary Case 265
6.4 Tables and Figures of Limiting Distributions 267
6.5 Approximations to the Distributions of the LSEs 276
6.6 Nearly Nonstationary Seasonal AR Models 281
6.7 Continuous Record Asymptotics 289
6.8 Complex Roots on the Unit Circle 292
6.9 Autoregressive Models with Multiple Unit Roots 300
7 Estimation Problems in Noninvertible Moving Average Models 311
7.1 Noninvertible Moving Average Models 311
7.2 The Local MLE in the Stationary Case 314
7.3 The Local MLE in the Conditional Case 325
7.4 Noninvertible Seasonal Models 330
7.4.1 The Stationary Case 331
7.4.2 The Conditional Case 333
7.4.3 Continuous Record Asymptotics 335
7.5 The Pseudolocal MLE 337
7.5.1 The Stationary Case 337
7.5.2 The Conditional Case 339
7.6 Probability of the Local MLE at Unity 341
7.7 The Relationship with the State Space Model 343
8 Unit Root Tests in Autoregressive Models 349
8.1 Introduction 349
8.2 Optimal Tests 350
8.2.1 The LBI Test 352
8.2.2 The LBIU Test 353
8.3 Equivalence of the LM Test with the LBI or LBIU Test 356
8.3.1 Equivalence with the LBI Test 356
8.3.2 Equivalence with the LBIU Test 358
8.4 Various Unit Root Tests 360
8.5 Integral Expressions for the Limiting Powers 362
8.5.1 Model A 363
8.5.2 Model B 364
8.5.3 Model C 365
8.5.4 Model D 367
8.6 Limiting Power Envelopes and Point Optimal Tests 369
8.7 Computation of the Limiting Powers 372
8.8 Seasonal Unit Root Tests 382
8.9 Unit Root Tests in the Dependent Case 389
8.10 The Unit Root Testing Problem Revisited 395
8.11 Unit Root Tests with Structural Breaks 398
8.12 Stochastic Trends Versus Deterministic Trends 402
8.12.1 Case of Integrated Processes 403
8.12.2 Case of Near-Integrated Processes 406
8.12.3 Some Simulations 409
9 Unit Root Tests in Moving Average Models 415
9.1 Introduction 415
9.2 The LBI and LBIU Tests 416
9.2.1 The Conditional Case 417
9.2.2 The Stationary Case 419
9.3 The Relationship with the Test Statistics in Differenced Form 424
9.4 Performance of the LBI and LBIU Tests 427
9.4.1 The Conditional Case 427
9.4.2 The Stationary Case 430
9.5 Seasonal Unit Root Tests 434
9.5.1 The Conditional Case 434
9.5.2 The Stationary Case 436
9.5.3 Power Properties 438
9.6 Unit Root Tests in the Dependent Case 444
9.6.1 The Conditional Case 444
9.6.2 The Stationary Case 446
9.7 The Relationship with Testing in the State Space Model 447
9.7.1 Case (I) 449
9.7.2 Case (II) 450
9.7.3 Case (III) 452
9.7.4 The Case of the Initial Value Known 454
10 Asymptotic Properties of Nonstationary Panel Unit Root Tests 459
10.1 Introduction 459
10.2 Panel Autoregressive Models 461
10.2.1 Tests Based on the OLSE 463
10.2.2 Tests Based on the GLSE 471
10.2.3 Some Other Tests 475
10.2.4 Limiting Power Envelopes 480
10.2.5 Graphical Comparison 485
10.3 Panel Moving Average Models 488
10.3.1 Conditional Case 490
10.3.2 Stationary Case 494
10.3.3 Power Envelope 499
10.3.4 Graphical Comparison 502
10.4 Panel Stationarity Tests 507
10.4.1 Limiting Local Powers 508
10.4.2 Power Envelope 512
10.4.3 Graphical Comparison 514
10.5 Concluding Remarks 515
11 Statistical Analysis of Cointegration 517
11.1 Introduction 517
11.2 Case of No Cointegration 519
11.3 Cointegration Distributions: The Independent Case 524
11.4 Cointegration Distributions: The Dependent Case 532
11.5 The Sampling Behavior of Cointegration Distributions 537
11.6 Testing for Cointegration 544
11.6.1 Tests for the Null of No Cointegration 544
11.6.2 Tests for the Null of Cointegration 547
11.7 Determination of the Cointegration Rank 552
11.8 Higher Order Cointegration 556
11.8.1 Cointegration in the I(d) Case 556
11.8.2 Seasonal Cointegration 559
Part II Analysis of Fractional Time Series 567
12 ARFIMA Models and the Fractional Brownian Motion 569
12.1 Nonstationary Fractional Time Series 569
12.1.1 Case of d = ¿ 570
> ¿ 572
12.2 Testing for the Fractional Integration Order 575
12.2.1 i.i.d. Case 575
12.2.2 Dependent Case 581
12.3 Estimation for the Fractional Integration Order 584
12.3.1 i.i.d. Case 584
12.3.2 Dependent Case 586
12.4 Stationary Long-Memory Processes 591
12.5 The Fractional Brownian Motion 597
12.6 FCLT for Long-Memory Processes 603
12.7 Fractional Cointegration 608
12.7.1 Spurious Regression in the Fractional Case 609
12.7.2 Cointegrating Regression in the Fractional Case 610
12.7.3 Testing for Fractional Cointegration 614
12.8 The Wavelet Method for ARFIMA Models and the fBm 614
12.8.1 Basic Theory of the Wavelet Transform 615
12.8.2 Some Advantages of the Wavelet Transform 618
12.8.3 Some Applications of the Wavelet Analysis 625
13 Statistical Inference Associated with the Fractional Brownian Motion 629
13.1 Introduction 629
13.2 A Simple Continuous-Time Model Driven by the fBm 632
13.3 Quadratic Functionals of the Brownian Motion 641
13.4 Derivation of the c.f. 645
13.4.1 Stochastic Process Approach via Girsanov's Theorem 645
13.4.2 Fredholm Approach via the Fredholm Determinant 647
13.5 Martingale Approximation to the fBm 651
13.6 The Fractional Unit Root Distribution 659
13.6.1 The FD Associated with the Approximate Distribution 659
13.6.2 An Interesting Moment Property 664
13.7 The Unit Root Test Under the fBm Error 669
14 Maximum Likelihood Estimation for the Fractional Ornstein-Uhlenbeck Process 673
14.1 Introduction 673
14.2 Estimation of the Drift: Ergodic Case 677
14.2.1 Asymptotic Properties of the OLSEs 677
14.2.2 The MLE and MCE 679
14.3 Estimation of the Drift: Non-ergodic Case 687
14.3.1 Asymptotic Properties of the OLSE 687
14.3.2 The MLE 687
14.4 Estimation of the Drift: Boundary Case 692
14.4.1 Asymptotic Properties of the OLSEs 692
14.4.2 The MLE and MCE 693
14.5 Computation of Distributions and Moments of the MLE and MCE 695
14.6 The MLE-based Unit Root Test Under the fBm Error 703
14.7 Concluding Remarks 707
15 Solutions to Problems 709
References 865
Author Index 879
Subject Index 883
Chapter 1
Models for Nonstationarity and Noninvertibility
We deal with linear time series models on which stationarity or invertibility is not imposed. Using simple examples arising from estimation and testing problems, we indicate nonstandard aspects of the departure from stationarity or invertibility. In particular, asymptotic distributions of various statistics are derived by the eigenvalue approach under the normality assumption on the underlying processes. As a prelude to discussions in later chapters, we also present equivalent expressions for limiting random variables based on the other two approaches, which I call the stochastic process approach and the Fredholm approach.
1.1 Statistics from the One-Dimensional Random Walk
Let us consider the following simple nonstationary model:
1.1where are independent and identically distributed with common mean 0 and variance 1, which is abbreviated as i.i.d.. The model (1.1) is usually referred to as the random walk. It is also called the unit root process in the econometrics literature.
Let us deal with the following two statistics arising from the model (1.1):
1.2where . Each second moment statistic has a normalizer T2, which is different from the stationary case, and is necessary to discuss the limiting distribution as T 8. In fact, noting that , we have
It holds [Fuller (1996, p. 220)] that
where means that, for every e > 0, there exists a positive number Te such that for all T. It is anticipated that and have different nondegenerate limiting distributions.
We now attempt to derive the limiting distributions of and . There are three approaches for this purpose, which I call the eigenvalue approach, the stochastic process approach, and the Fredholm approach. The first approach is described here in detail, whereas the second and third are only briefly described and the details are discussed in later chapters.
1.1.1 Eigenvalue Approach
The eigenvalue approach requires a distributional assumption on . We assume that are independent and identically normally distributed with common mean 0 and variance 1, which is abbreviated as NID.
We also need to compute the eigenvalues of the matrices appearing in quadratic forms. To see this the observation vector may be expressed as
1.3where the matrix C and its inverse are given by
1.4The matrix C may be called the random walk generating matrix and play an important role in subsequent discussions.
We can now rewrite and as
1.5 1.6where
Let us compute the eigenvalues and eigenvectors of and . The eigenvalues of were obtained by Rutherford (1946) (see also Problem 1.1 in this chapter) by computing those of
The jth largest eigenvalue ?j of is found to be
1.7There exists an orthogonal matrix P such that , where the kth column of P is an eigenvector corresponding to ?k. It can be shown [Dickey and Fuller (1979)] that the th component of P is given by
On the other hand, is evidently singular because the vector e is the first column of C and so that the first column of is a zero vector. In fact, it holds that
1.8where the matrix G* is given by
1.9Here C* and are the last and submatrices of C and e, respectively, whereas . The eigenvalues of
can be easily obtained (Problem 1.2). We also have . Then the jth largest eigenvalue ?j of G* is found to be
1.10There exists an orthogonal matrix Q of size such that , where the kth column of Q is an eigenvector corresponding to ?k. It can be shown [Anderson (1971, p. 293)] that the th component of Q is given by
We now have the following relations:
where , and . Noting that and , we can compute the exact distributions of and by deriving the characteristic functions (c.f.s) as
Then the densities of and can be computed numerically following the inversion formula
1.11where is the real part of z. These densities will be drawn later together with the limiting densities.
Because of the properties of the eigenvalues ?j and ?j, it roughly holds that, as T 8,
In fact, it can be shown (Problem 1.3) that
which leads us to derive
1.12where signifies convergence in distribution.
The limiting distributions can be computed by deriving the c.f.s of S1 and S2. We have
1.13 1.14where we have used the following expansion formulas for and functions:
1.15Figure 1.1 draws the densities of for , and 8. These were computed numerically following the inversion formula in (1.11). The numerical computation involves the square root of complex variables, and how to compute this together with numerical integration will be discussed in Chapter 5. It is seen from Figure 1.1 that the finite sample densities converge rapidly to the limiting density, although the former have a heavier right-hand tail.
Figure 1.1 Probability densities of ST1.
Figure 1.2 draws the densities of for , and 8. These were computed in the same way as those of . Note that Figure 1.2 does not contain the density for T = 50 because it was found to be very close to that for T = 8, while it is not as close in Figure 1.1.
Figure 1.2 Probability densities of ST2.
The normalizer for , instead of T2, could make finite sample densities closer to the limiting density. More specifically, we have the following expansion for the c.f. of the modified statistic (Problem 1.4).
1.16It is noticed that the expansion contains no term of . This is not the case if we use T2 as a normalizer. On the other hand, we have (Problem 1.5)
1.17Note that this expansion does not contain the term of , which explains rapid convergence of to the limiting distribution.
Table 1.1 reports percent points and means for distributions of for , and 8, where "E" stands for exact distributions while "A" for distributions based on the asymptotic expansion given in (1.16). Table 1.2 shows distributions of , where the asymptotic expansion "A" is based on (1.17). It is seen from these tables that the finite sample distributions are really close to the limiting distribution. Especially, percent points for T = 50 are identical with those for T = 8 within the deviation of 3/10,000. Asymptotic expansions also give a fairly good approximation to finite sample distributions. In most cases, they give a correct value up to the fourth decimal point.
Table 1.1 Percent points for distributions of
Table 1.2 Percent points for distributions of
It is an easy matter to compute moments of these distributions. Let be the jth order cumulant for the distribution of based on the asymptotic expansion in (1.16). Define similarly for the distribution of based on the asymptotic expansion in (1.17). Then we have (Problem 1.6), up to ,
1.18Cumulants for the limiting distributions are given (Problem 1.7) by
where Bj's are the Bernoulli numbers: , , , , and so on. The skewness and kurtosis are 2.771 and 8.657, respectively, while and .
The eigenvalue approach has been successful so far. This is because eigenvalues associated with quadratic forms can be explicitly computed, which is rarely possible in more complicated situations. The other two approaches, however, do not require such condition, which we will discuss next.
1.1.2 Stochastic Process Approach
We continue to deal with the random walk model (1.1) and consider the second moment statistics and given in (1.2), where we do not assume normality on , but just assume .
The stochastic process approach, which will be fully discussed in Chapter 3, starts with constructing a continuous time process defined on . The process is defined, for , by
1.19where and . The process is called the partial sum process, which is continuous and belongs to the space of continuous functions defined on . Then the process converges weakly to the standard...
