Time Series Analysis: Nonstationary and Noninvertible Distribution Theory

By Katsuto Tanaka

Reflects the developments and new directions in the field since the publication of the first successful edition and contains a complete set of problems and solutions

This revised and expanded edition reflects the developments and new directions in the field since the publication of the first edition. In particular, sections on nonstationary panel data analysis and a discussion on the distinction between deterministic and stochastic trends have been added. Three new chapters on long-memory discrete-time and continuous-time processes have also been created, whereas some chapters have been merged and some sections deleted. The first eleven chapters of the first edition have been compressed into ten chapters, with a chapter on nonstationary panel added and located under Part I: Analysis of Non-fractional Time Series. Chapters 12 to 14 have been newly written under Part II: Analysis of Fractional Time Series. Chapter 12 discusses the basic theory of long-memory processes by introducing ARFIMA models and the fractional Brownian motion (fBm). Chapter 13 is concerned with the computation of distributions of quadratic functionals of the fBm and its ratio. Next, Chapter 14 introduces the fractional Ornstein–Uhlenbeck process, on which the statistical inference is discussed. Finally, Chapter 15 gives a complete set of solutions to problems posed at the end of most sections. This new edition features:

• Sections to discuss nonstationary panel data analysis, the problem of differentiating between deterministic and stochastic trends, and nonstationary processes of local deviations from a unit root

• Consideration of the maximum likelihood estimator of the drift parameter, as well as asymptotics as the sampling span increases

• Discussions on not only nonstationary but also noninvertible time series from a theoretical viewpoint

• New topics such as the computation of limiting local powers of panel unit root tests, the derivation of the fractional unit root distribution, and unit root tests under the fBm error

Time Series Analysis: Nonstationary and Noninvertible Distribution Theory, Second Edition, is a reference for graduate students in econometrics or 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.

• Language: English
• Publisher: Wiley-Interscience
• Release date: Mar 27, 2017
• ISBN: 9781119132134

Book preview

Preface to the Second Edition

The first edition of this book was published in 1996. The book was written from a theoretical viewpoint of time series econometrics, where the main theme was to describe nonstandard theory for linear time series models that are nonstationary and/or noninvertible. I also proposed methods for computing numerically the distributions of nonstandard statistics arising from such processes.

The main theme of the present edition remains the same and reflects the developments and new directions in the field since the publication of the first edition. In particular, the discussion on nonstationary panel data analysis has been added and new chapters on long-memory discrete-time and continuous-time processes have been created, whereas some chapters have been merged and some sections deleted.

This edition is divided into two parts: Part I: Analysis of Non Fractional Time Series and Part II: Analysis of Fractional Time Series, where Part I consists of Chapters 1 through 11 while Part II consists of Chapters 12 through 14. The distinction between non fractional and fractional time series is concerned with the integration order of nonstationary time series. Part I assumes the integration order to be a positive integer, whereas Part II relaxes that assumption to allow the integration order to be any positive real number.

Chapter 1 is essentially the same as the first edition, except for the addition of an introductory description on nonstationary panels, and is a prelude to subsequent chapters. The three approaches, which I call the eigenvalue, stochastic process, and Fredholm approaches, to the analysis of non fractional time series are introduced through simple examples. Chapter 2 merged Chapters 2 and 3 of the first edition and discusses the Brownian motion, the Ito integral, the functional central limit theorem, and so on.

Chapters 3 and 4 discuss fully the stochastic process approach and the Fredholm approach, respectively. These approaches are used to derive limiting characteristic functions of nonstandard statistics that are quadratic functionals of the Brownian motion or its ratio. Chapter 5 is concerned with numerical integration for computing distribution functions via inversion of characteristic functions derived from the stochastic process approach or the Fredholm approach. Chapters 6 through 11 deal with unit root and cointegration problems. Chapters 1 through 11 except Chapter 10 were main chapters of the first edition. New topics such as unit root tests under structural breaks, differences between stochastic and deterministic trends, have been added. Chapter 10 has been added to discuss nonstationary panel data models, where our main concern is to compute limiting local powers of various panel unit root tests. For that purpose the moving average models are also considered in addition to autoregressive models.

Chapters 12 through 14 have been written newly under Part II: Analysis of Fractional Time Series. Chapter 12 discusses the basic theory of long-memory processes by introducing ARFIMA models and the fractional Brownian motion (fBm). The wavelet method is also introduced to deal with ARFIMA models and the fBm. Chapter 13 is concerned with the computation of distributions of quadratic functionals of the fBm and its ratio, where the computation of the fractional unit root distribution remains to be done, whereas an approximation to the true distribution is proposed and computed. Chapter 14 introduces the fractional Ornstein–Uhlenbeck process, on which the statistical inference is discussed. In particular, the maximum likelihood estimator of the drift parameter is considered, and asymptotics as the sampling span increases are discussed.

Chapter 15, the last chapter, gives a complete set of solutions to problems posed at the end of most sections.

There are about 140 figures and 60 tables. Most of these are of limiting distributions of nonstandard statistics. They are all produced by the methods described in this edition and include many distributions, which have never appeared in the literature.

The present edition is dedicated to my wife, Yoshiko, who died in 1999.

November 2016

Katsuto Tanaka

Tokyo, Japan

Preface to the First Edition

This book attempts to describe nonstandard theory for linear time series models, which are nonstationary and/or noninvertible. Nonstandard aspects of the departure from stationarity or invertibility have attracted much attention in the field of time series econometrics during the last 10 years. Since there seem few books concerned with the theory for such nonstandard aspects, I have been at liberty to choose my way. Throughout this book, attention is oriented toward the most interesting theoretical issue, that is, the asymptotic distributional aspect of nonstandard statistics. The subtitle of the book reflects this.

Chapter 1 is a prelude to the main theme. By using simple examples, various asymptotic distributions of nonstandard statistics are derived by a classical approach, which I call the eigenvalue approach. It turns out that, if more complicated problems are to be dealt with, the eigenvalue approach breaks down and the introduction of notions such as the Brownian motion, the Ito integral, the functional central limit theorem, and so on is inevitable. These notions are now developed very deeply in probability theory. In this book, however, a knowledge of such probability notions is required only at a moderate level, which I explain in Chapters 2 and 3 in an easily accessible way.

Probability theory, in particular the functional central limit theorem, enables us to establish weak convergence of nonstandard statistics and to realize that limiting forms can be expressed by functionals of the Brownian motion. However, more important from a statistical point of view is how to compute limiting distributions of those statistics. For this purpose I do not simply resort to simulations but employ numerical integration. To make the computation possible, we first need to derive limiting characteristic functions of nonstandard statistics. To this end, two approaches are presented. Chapter 4 discusses one approach, which I call the stochastic process approach, while Chapter 5 discusses the other, which I call the Fredholm approach. The two approaches originate from quite different mathematical theories, which I explain fully, indicating the advantage and disadvantage of each approach for judicious use.

Chapter 6 discusses and illustrates numerical integration for computing distribution functions via inversion of characteristic functions. This chapter is necessary because a direct application of any computer package for integration cannot do a proper job. We overcome the difficulty by employing Simpson's rule, which can be executed on a desktop computer. The necessity for accurate computation based on numerical integration is recognized, for instance, when close comparison has to be made between limiting local powers of competing nonstandard tests.

Chapters 7 through 11 deal with statistical and econometric problems to which the nonstandard theory discussed in previous chapters applies. Chapter 7 considers the estimation problems associated with nonstationary autoregressive models, while Chapter 8 considers those with noninvertible moving average models. The corresponding testing problems, called the unit root tests, are discussed in Chapters 9 and 10, respectively. Chapter 11 is concerned with cointegration, which is a stochastic collinearity relationship among multiple nonstationary time series. The problems discussed in these chapters originate in time series econometrics. I describe in detail how to derive and compute limiting nonstandard distributions of various estimators and test statistics.

Chapter 12, the last chapter, gives a complete set of solutions to problems posed at the end of most sections of each chapter. Most of the problems are concerned with corroborating the results described in the text, so that one can gain a better understanding of details of the discussions.

There are about 90 figures and 50 tables. Most of these are of limiting distributions of nonstandard statistics. They are all produced by the methods described in this book and include many distributions, which have never appeared in the literature. Among these are limiting powers and power envelopes of various nonstandard tests under a sequence of local alternatives.

This book may be used as a textbook for graduate students majoring in econometrics or time series analysis. A general knowledge of mathematical statistics, including the theory of stationary processes, is presupposed, although the necessary material is offered in the text and problems of this book. Some knowledge of a programming language like FORTRAN and computerized algebra like REDUCE is also useful.

The late Professor E. J. Hannan gave me valuable comments on the early version of my manuscript. I would like to thank him for his kindness and for pleasant memories extending over the years since my student days. This book grew out of joint work with Professor S. Nabeya, another respected teacher of mine. He read substantial parts of the manuscript and corrected a number of errors in its preliminary stages, for which I am most grateful. I am also grateful to Professors C. W. Helstrom, S. Kusuoka, and P. Saikkonen for helpful discussions and to Professor G. S. Watson for help of various kinds. Most of the manuscript was keyboarded, many times over, by Ms. M. Yuasa, and some parts were done by Ms. Y. Fukushima, to both of whom I am greatly indebted. Finally, I thank my wife, Yoshiko, who has always been a source of encouragement.

January 1996

Katsuto Tanaka

Tokyo, Japan

Part I

Analysis of Non Fractional Time Series

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:

1.1

equation

where c01-math-002 are independent and identically distributed with common mean 0 and variance 1, which is abbreviated as c01-math-003 i.i.d. c01-math-004 . 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):

1.2

equation

where c01-math-006 . 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 → ∞. In fact, noting that c01-math-009 , we have

equationequation

It holds [Fuller (1996, p. 220)] that

equation

where c01-math-010 means that, for every ε > 0, there exists a positive number Tε such that c01-math-013 for all T. It is anticipated that c01-math-015 and c01-math-016 have different nondegenerate limiting distributions.

We now attempt to derive the limiting distributions of c01-math-017 and c01-math-018 . 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 c01-math-019 . We assume that c01-math-020 are independent and identically normally distributed with common mean 0 and variance 1, which is abbreviated as c01-math-021 NID c01-math-022 .

We also need to compute the eigenvalues of the matrices appearing in quadratic forms. To see this the observation vector c01-math-023 may be expressed as

1.3 equation

where the matrix C and its inverse c01-math-026 are given by

1.4

equation

The matrix C may be called the random walk generating matrix and play an important role in subsequent discussions.

We can now rewrite c01-math-029 and c01-math-030 as

1.5

equation

1.6

equationequation

where

equation

Let us compute the eigenvalues and eigenvectors of c01-math-033 and c01-math-034 . The eigenvalues of c01-math-035 were obtained by Rutherford (1946) (see also Problem 1.1 in this chapter) by computing those of

equation

The jth largest eigenvalue λj of c01-math-038 is found to be

1.7

equation

There exists an orthogonal matrix P such that c01-math-041 , where the kth column of P is an eigenvector corresponding to λk. It can be shown [Dickey and Fuller (1979)] that the c01-math-045 th component of P is given by

equation

On the other hand, c01-math-047 is evidently singular because the vector e is the first column of C and c01-math-050 so that the first column of c01-math-051 is a zero vector. In fact, it holds that

1.8 equation

where the c01-math-053 matrix G∗ is given by

1.9

equation

Here C∗ and c01-math-057 are the last c01-math-058 and c01-math-059 submatrices of C and e, respectively, whereas c01-math-062 . The eigenvalues of

equation

can be easily obtained (Problem 1.2). We also have c01-math-063 . Then the jth largest eigenvalue γj of G∗ is found to be

1.10

equation

There exists an orthogonal matrix Q of size c01-math-069 such that c01-math-070 , where the kth column of Q is an eigenvector corresponding to γk. It can be shown [Anderson (1971, p. 293)] that the c01-math-074 th component of Q is given by

equation

We now have the following relations:

equation

where c01-math-076 , c01-math-077 and c01-math-078 . Noting that c01-math-079 and c01-math-080 , we can compute the exact distributions of c01-math-081 and c01-math-082 by deriving the characteristic functions (c.f.s) as

equation

Then the densities of c01-math-083 and c01-math-084 can be computed numerically following the inversion formula

1.11

equation

where c01-math-086 is the real part of z. These densities will be drawn later together with the limiting densities.

Because of the properties of the eigenvalues λj and γj, it roughly holds that, as T → ∞,

equation

In fact, it can be shown (Problem 1.3) that

equationequation

which leads us to derive

1.12

equation

where c01-math-092 signifies convergence in distribution.

The limiting distributions can be computed by deriving the c.f.s of S1 and S2. We have

1.13

equation

1.14

equation

where we have used the following expansion formulas for c01-math-097 and c01-math-098 functions:

1.15

equation

Figure 1.1 draws the densities of c01-math-100 for c01-math-101 , and ∞. 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.

A plot of probability densities of S1T with four curves plotted for different T values and a legend inset.

Figure 1.1 Probability densities of ST1.

Figure 1.2 draws the densities of c01-math-104 for c01-math-105 , and ∞. These were computed in the same way as those of c01-math-107 . 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 = ∞, while it is not as close in Figure 1.1.

A plot of probability densities of S2T with three curves plotted for different T values and a legend inset.

Figure 1.2 Probability densities of ST2.

The normalizer c01-math-111 for c01-math-112 , 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).

1.16

equationequation

It is noticed that the expansion contains no term of c01-math-115 . This is not the case if we use T2 as a normalizer. On the other hand, we have (Problem 1.5)

1.17

equationequation

Note that this expansion does not contain the term of c01-math-118 , which explains rapid convergence of c01-math-119 to the limiting distribution.

Table 1.1 reports percent points and means for distributions of c01-math-120 for c01-math-121 , and ∞, 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 c01-math-123 , 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 = ∞ 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 c01-math-126

Table 1.2 Percent points for distributions of c01-math-127

It is an easy matter to compute moments of these distributions. Let c01-math-128 be the jth order cumulant for the distribution of c01-math-130 based on the asymptotic expansion in (1.16). Define c01-math-131 similarly for the distribution of c01-math-132 based on the asymptotic expansion in (1.17). Then we have (Problem 1.6), up to c01-math-133 ,

1.18

equationequation

Cumulants for the limiting distributions are given (Problem 1.7) by

equation

where Bj's are the Bernoulli numbers: c01-math-136 , c01-math-137 , c01-math-138 , c01-math-139 , and so on. The skewness c01-math-140 and kurtosis c01-math-141 are 2.771 and 8.657, respectively, while c01-math-142 and c01-math-143 .

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 c01-math-144 and c01-math-145 given in (1.2), where we do not assume normality on c01-math-146 , but just assume c01-math-147 .

The stochastic process approach, which will be fully discussed in Chapter 3, starts with constructing a continuous time process c01-math-148 defined on c01-math-149 . The process c01-math-150 is defined, for c01-math-151 , by

1.19

equationequation

where c01-math-153 and c01-math-154 . The process c01-math-155 is called the partial sum process, which is continuous and belongs to the space of continuous functions defined on c01-math-156 . Then the process c01-math-157 converges weakly to the standard Brownianmotion (Bm) c01-math-158 , which we write as c01-math-159 . The convergence of this mode entails the notion of the weak convergence of stochastic processes called the functional central limit theorem (FCLT). Related materials will be discussed in Chapter 2.

Using the partial sum process c01-math-160 , the two statistics c01-math-161 and c01-math-162 may be rewritten as

equation

where c01-math-163 and c01-math-164 are remainder terms that converge to 0 in probability. This will be proved in Chapter 2 together with the following weak convergence:

1.20 equation

1.21

equation

The above results can be obtained via the FCLT and the continuous mapping theorem (CMT). This last theorem is an extension of the case of continuous functions to that of continuous functionals and is discussed in Chapter 2. The stochastic process

1.22 equation

is called the demeaned Bm, which has mean 0 on the interval c01-math-168 .

The following distributional equivalence should now hold.

1.23

equation

where c01-math-170 .

We have already obtained, by the eigenvalue approach, the c.f.s of the limiting distributions of c01-math-171 and c01-math-172 . This was possible because the eigenvalues were explicitly known. Suppose that we have no knowledge about the eigenvalues. In that case Girsanov's theorem, which transforms the measure on a function space to that on another and will be discussed in Chapter 3, enables us to compute the c.f.s of the distributions of the resulting statistics.

1.1.3 The Fredholm Approach

The second moment statistics c01-math-173 and c01-math-174 have three kinds of expressions described in (1.5) and (1.6), respectively. Here we use the last expressions of these, that is,

1.24

equation

1.25

equation

where we only assume c01-math-177 . These statistics can be seen to be characterized by

1.26 equation

where there exists a continuous and symmetric function c01-math-179 defined on c01-math-180 that satisfies

1.27

equation

Then, as is discussed in Chapter 4, it holds that

equation

where c01-math-182 is the Bm, whereas the integral is the Riemann–Stieltjes double integral with respect to the Bm to be discussed in Chapter 2. Thus it can be shown that

1.28

equation

1.29

equationequation

The c.f.s of the limiting random variables of the previous form can be derived by finding the Fredholm determinant (FD) of the function c01-math-185 , from which the present approach originates. Details will be discussed in Chapter 4.

So far we have obtained the following distributional relations:

1.30

equation

1.31

equation

We note in passing that the following relation holds:

equation

which will be shown in Chapter 3 by deriving the c.f.s by the stochastic process approach. The stochastic process

1.32 equation

is called the Brownian bridge (Bb), which has the property that c01-math-189 .

It also holds that

equation

which will be shown in Chapter 4 by deriving the c.f.s by the Fredholm approach.

1.1.4 An Overview of the Three Approaches

In the previous subsections we discussed the three approaches to deriving asymptotic distributions of second moment statistics arising from the random walk. Although details are postponed until later chapters, it may be of some help to give an overview of the three approaches here, making comparisons with each other. For this purpose we take up the statistic

equation

Then the three approaches may be summarized in terms of the following viewpoints, where (A), (B), and (C) refer to the eigenvalue, stochastic process, and Fredholm approaches, respectively.

Expressions for the statistic:

A.

c01-math-190

.

B.

c01-math-191

.

C.

c01-math-192

.

Assumptions and theorems necessary for convergence in distribution:

A. Distributional assumptions need to be imposed on c01-math-193 and knowledge of the eigenvalues is required.

B. The partial sum process needs to be constructed, and the FCLT and CMT are used to establish weak convergence.

C. The kernel function needs to be found and the usual CLT is required to establish weak convergence.

Derivation of the c.f.:

A. The limiting expression is an infinite, weighted sum of independent c01-math-194 random variables. Its c.f. can be easily derived.

B. The limiting expression is a simple Riemann integral of the squared Bm. Its c.f. can be derived via Girsanov's theorem.

C. The limiting expression is a Riemann–Stieltjes double integral of a symmetric and continuous kernel with respect to the Bm. Its c.f. can be derived by finding the Fredholm determinant of the kernel.

Problems

In the problems below it is assumed that C is the random walk generating matrix defined in (1.4), whereas c01-math-196 and c01-math-197 . We also assume that c01-math-198 and

equation

1.1 Show that the eigenvalues of c01-math-199 are c01-math-200 c01-math-201 .

1.2 Show that the nonzero eigenvalues of c01-math-202 are γk c01-math-204 .

1.3 Prove that

equationequation

1.4 Derive the following expansion:

equation

1.5 Derive the following expansion:

equation

1.6 Using the asymptotic expansions obtained in Problems 1.4 and 1.5, compute cumulants given in (1.18).

1.7 Show that the distribution with the c.f. c01-math-205 has the jth order cumulant given by

equation

where Bj is the Bernoulli number. Show also that the distribution with the c.f. c01-math-208 has the jth order cumulant given by

equation

1.8 Using the fact that the inverse Laplace transform of c01-math-210 is c01-math-211 for c > 0, show that

equation

where c01-math-213 and Φ is the distribution function of N c01-math-215 .

1.2 A Test Statistic from a Noninvertible Moving Average Model

Let us next consider the first-order moving average [MA(1)] model:

1.33 equation

where c01-math-217 , are c01-math-218 random variables. The parameter α is restricted to be c01-math-220 because of the identifiability condition. The MA(1) model (1.33) is said to be noninvertible when c01-math-221 . Various inference problems associated with the noninvertible case will be discussed in Chapter 7.

Let us consider here testing if the MA(1) model is noninvertible, that is, to test

equation

For this purpose we conduct a score or Lagrange multiplier (LM) type test. The log-likelihood c01-math-222 for c01-math-223 is given by

equation

where

equation

Let us put c01-math-224 . It is noticed that the matrix Ω is exactly the same, except for size, as c01-math-226 given in (1.8). For later purposes we also note that

1.34

equationequation

where C is the random walk generating matrix defined in (1.4), whereas

equation

Then the maximum likelihood estimators (MLEs) of α and σ2 under H0 are c01-math-232 and c01-math-233 . It can be checked that

equation

where we have used the formula

equation

These yield

equationequation

where we have used the facts that tr c01-math-234 , which follows from (1.34), and

equation

The LM test considered here rejects H0 if the second derivative of the log-likelihood under H0 is large, that is, if

1.35 equation

takes large values.

The limiting distribution of ST under H0 can be derived by the eigenvalue approach as follows. Put c01-math-240 so that c01-math-241 and

equation

where δj is the jth largest eigenvalue of c01-math-244 , which can be given, from (1.10), by

1.36

equation

It can now be seen that the limiting distribution of ST is the same as that of c01-math-247 discussed in the last section. Thus it holds that, as T → ∞ under H0,

1.37

equation

The stochastic process approach deals with c01-math-251 in the following way. Defining a random walk c01-math-252 with c01-math-253 , we have

equation

where RT is the remainder term of c01-math-255 (Problem 2.1). Thus, it follows from (1.21) that

1.38

equation

The Fredholm approach uses the expression

equation

which leads us, from (1.27), to

1.39

equation

We can extend the problem by considering the limiting distribution of ST under the local alternative

1.40 equation

where c is a nonnegative constant. Noting that

equation

and putting c01-math-261 so that c01-math-262 , we have (Problem 2.2)

equation

Thus it holds that, as T → ∞ under c01-math-264 ,

1.41

equation

The stochastic process approach gives the following expression, which will be discussed in Chapter 3:

equation

where c01-math-266 is the demeaned Bm defined in (1.22), whereas the stochastic process

1.42

equationequation

is called the integrated demeaned Bm, which has the property that c01-math-268 .

The Fredholm approach gives

equation

where c01-math-269 and c01-math-270 is called the iterated kernel defined by

1.43 equation

It can be shown (Problem 2.3) that the c.f. of S(c) in (1.41) is given by

1.44

equationequation

Note here that, when c = 0, the above c.f. reduces to c01-math-275 .

Figure 1.3 presents the probability densities of S(c) for various values of c. These were computed by using the inversion formula in (1.11). The density of S(0) gives the null distribution, whereas the densities of S(c) for c > 0 are those under the local alternative. The local power of the test at the c01-math-281 level can be computed using

1.45

equation

where xγ is the upper c01-math-284 point of the distribution of S(0) whereas Re c01-math-286 is the real part of z. The MA unit root test will be discussed in detail in Chapter 9.

A plot of probability densities of S(c) with three curves plotted for different c values and a legend inset.

Figure 1.3 Probability densities of c01-math-288 .

It is important to note that the assumption on the initial value ε0 is very crucial. If we assume c01-math-290 , which may be referred to as the conditional case, so that c01-math-291 is not stationary, the LM test becomes different, so is the limiting distribution of the LM statistic (Problem 2.4).

An MA unit root is often caused by overdifferencing of the original time series. From this point of view, Saikkonen and Luukkonen (1993a) suggested the following model:

1.46

equation

where μ is a constant and c01-math-294 . Then the null hypothesis c01-math-295 implies overdifferencing. If μ is known and is assumed to be zero, the LM test rejects H0 when c01-math-298 takes large values (Problem 2.5).

Suppose that the constant μ in (1.46) is unknown. Then the LM test rejects H0 for large values of

1.47 equation

where c01-math-302 with c01-math-303 and C is the random walk generating matrix [Saikkonen and Luukkonen (1993a) and Problem 2.6]. It can be shown (Problem 2.7) that c01-math-305 in (1.47) is rewritten as

1.48 equation

where

c01-math-307

and Ω∗ is the first c01-math-309 submatrix of Ω. Comparing c01-math-311 with ST in (1.37), we can conclude that the LM statistic for the model (1.46) is derived completely in the same way as in (1.37) just by disregarding the first equation in (1.46) and replacing yj by c01-math-314 c01-math-315 .

Nonetheless the formulation (1.46) is meaningful in connection with the determination of the order of integration of c01-math-316 , that is, the order of the AR unit root. If c01-math-317 is found to have an MA unit root, while c01-math-318 is not, then the order of integration of c01-math-319 is supposed to be d. The MA unit root test may be useful for that purpose.

Problems

In the problems below it is assumed that c01-math-321 , where C is the random walk generating matrix defined in (1.4), c01-math-323 and c01-math-324 . We also assume that c01-math-325 is the Bm, c01-math-326 and c01-math-327 .

2.1 Show that

equation

where c01-math-328 and c01-math-329 .

2.2 Suppose that y is a c01-math-331 observation vector from an MA(1) model (1.33) with coefficient c01-math-332 . Show that

equation

2.3 Derive the c.f. of the following statistic:

equation

2.4 Show that the LM test for testing c01-math-333 versus c01-math-334 in the MA(1) model (1.33) with c01-math-335 rejects H0 when c01-math-337 takes large values. Derive the asymptotic distribution of ST under H0.

2.5 Show that the LM test for testing c01-math-340 versus c01-math-341 in the model (1.46) with μ = 0 rejects H0 for large values of c01-math-344 . Derive the asymptotic distribution of ST under H0.

2.6 Show that the LM statistic for testing c01-math-347 versus c01-math-348 in the model (1.46) with μ being unknown rejects H0 for large values of

equation

2.7 Show that the statistic c01-math-351 in Problem 2.6 can be rewritten as

equation

where c01-math-352 and Ω∗ is the first c01-math-354 submatrix of Ω. On the basis of this expression, derive the asymptotic distribution of c01-math-356 under H0.

1.3 The AR Unit Root Distribution

Let us consider the following AR(1) model:

1.49

equation

where we assume that the true value of ρ is unity so that the model reduces to the random walk or the unit root model discussed in Section 1.1.

Suppose here that ρ is unknown and we estimate it using

1.50 equation

where δ is a fixed constant. It is seen that c01-math-363 is the least squares estimator (LSE) and becomes the MLE under the normality assumption on c01-math-364 , whereas c01-math-365 is the Yule–Walker estimator. If c01-math-366 , the asymptotic distribution of c01-math-367 does not depend on δ and tends to N c01-math-369 , but that of c01-math-370 does depend on δ when ρ = 1, which we shall show by deriving the limiting distribution of a suitably normalized quantity of c01-math-373 .

White (1958) first obtained the limiting c.f. associated with c01-math-374 as T → ∞ under c01-math-376 . Here we continue to assume that ρ = 1 and follow his approach, assuming that c01-math-378 .

Let us consider

equation

where

equation

Here the following expression for UT is useful.

equation

We now compute the limit of

c01-math-380

for any real x. Then we have (Problem 3.1)

1.51 equation

equation

where c01-math-383 , c01-math-384 and C is the random walk generating matrix defined in (1.4). It can be shown [White (1958) and Problem 3.2] that the moment generating function (m.g.f.) of XT is given by

1.52

equation

where

equation

The m.g.f. c01-math-388 may be expanded [Knight and Satchell (1993) and Problem 3.3], up to c01-math-389 , as

1.53

equationequation

where c01-math-391 , and thus the limiting c.f. c01-math-392 of XT is given by

1.54

equation

We now have, by Imhof's formula [Imhof (1961)],

1.55 equation

equation

where c01-math-396 is the imaginary part of z. The limiting probability density c01-math-398 of c01-math-399 is computed as c01-math-400 , which is much involved, unlike the previous cases. Once the distribution function is obtained, we can use a numerical derivative to compute the density. This will be discussed in Chapter 5.

The following equivalent expressions will be obtained in later chapters for the weak convergence of c01-math-401 :

1.56 equation

equation

where c01-math-403 and c01-math-404 is a sequence of solutions to

equation

while the integral c01-math-405 is called the Ito integral, which will be discussed in Chapter 2 and has the following property:

equation

We also have the following expressions for the weak convergence of c01-math-406 :

equation

In particular,

equation

is called the unit root distribution.

Moments of the limiting distribution of c01-math-407 , that is, moments of c01-math-408 , can be derived following Evans and Savin (1981b). Put

c01-math-409

. Then the kth order raw moment c01-math-411 of c01-math-412 is given (Problem 3.4) by

1.57

equationequation

where

equation

Figure 1.4 draws the limiting probability densities c01-math-414 of c01-math-415 for c01-math-416 , and 1. It is seen that c01-math-417 is located to the left of c01-math-418 , as is expected from the definition of c01-math-419 . Table 1.3 reports percent points, means, and standard deviations (SDs) of the limiting distributions of c01-math-420 for various values of δ. The limiting distribution of c01-math-422 was earlier tabulated in Fuller (1976) by simulations, while tables based on numerical integration were provided by Evans and Savin (1981a), Bobkoski (1983), Perron (1989a), and Nabeya and Tanaka (1990a).

Image described by caption and surrounding text.

Figure 1.4 Limiting probability densities of c01-math-423 .

Table 1.3 Percent points for limiting distributions of c01-math-424

A closer look at the values of means and SDs in Table 1.3 leads us to conclude the following: Let the limit in distribution of c01-math-473 be c01-math-474 . Then it holds (Problem 3.5) that

1.58

equation

1.59 equation

Thus the LSE and the Yule–Walker estimator, if normalized, have asymptotically the same variance, although the mean of the former is too larger than the latter. We also have (Problem 3.5)

1.60 equation

The unit root distribution can be extended to cover more general models allowing for mean, trend, seasonality, multiple unit roots, and so on. For example, the fitted-mean estimator c01-math-478 of ρ in the model (1.49) follows c01-math-480 , where

equation

It is shown by using the joint m.g.f. that U2 and V2 are uncorrelated, whereas

equation

The unit root distribution can also be extended to deal with the near unit root case where c01-math-483 with c being a constant. These topics will be discussed in Chapter 6. The estimator c01-math-485 may be used to test the unit root hypothesis c01-math-486 versus c01-math-487 . The limiting local powers under c01-math-488 will be computed in Chapter 8 by numerical integration for various values of δ and it will be found that the test based on the LSE (δ = 0) is the best of all the tests based on c01-math-491 . We shall also explore better tests than the test based on c01-math-492 .

Problems

3.1 Express c01-math-493 in quadratic form as in (1.51).

3.2 Show that the m.g.f. c01-math-494 of XT in (1.51) is given by (1.52).

3.3 Derive the asymptotic expansion of c01-math-496 given in (1.53).

3.4 Prove that, for random variables U and V with V > 0, the following formula holds:

equation

where k is a nonnegative integer and α is a positive number whereas the c01-math-502 is the joint m.g.f. of U and V.

3.5 Show that

equation

1.4 Various Statistics from the Two-Dimensional Random Walk

As a sequel to Section 1.1, which dealt with the one-dimensional random walk, we consider the two-dimensional random walk

1.61

equation

where c01-math-506 and c01-math-507 . Under this last assumption c01-math-508 and c01-math-509 are independent of each other so that Cov c01-math-510 for any j and k. It then holds that any linear combination of c01-math-513 and c01-math-514 yields a one-dimensional random walk or the unit root process. There is no linear combination that becomes stationary. This is an ordinary case referred to as no cointegration. There are, however, some cases called cointegration where linear combinations of random walks produce stationarity, which we shall discuss briefly in the next section.

For subsequent discussions we rewrite the model (1.61) in two ways. One is

equation

where c01-math-515 and C is the random walk generating matrix defined in (1.4) whereas c01-math-517 is the Kronecker product. Note that the Kronecker product of matrices A c01-math-519 and B c01-math-521 is a c01-math-522 matrix defined by

equation

Note also the following properties of the Kronecker product:

equation

It also holds that, if A and B are square matrices of order m and p, respectively, the eigenvalues of A ⊗ B are c01-math-528 c01-math-529 , where c01-math-530 is the ith eigenvalue of A and c01-math-533 is the jth eigenvalue of B.

The other expression for y is

equation

where c01-math-537 , c01-math-538 c01-math-539 , and c01-math-540 .

The nonstandard nature of statistics arising from the model in (1.61) can be best seen from the following example. Consider the sample covariance

1.62

equationequation

where

equation

We shall show that ST has a nondegenerate limiting distribution, although c01-math-543 and c01-math-544 are independent. The c.f. c01-math-545 of ST is given (Problem 4.1) by

1.63

equation

where λj is the jth largest eigenvalue of c01-math-550 given in (1.7). It is noted that the distribution of ST is symmetric about the origin since c01-math-552 is real. From the expression in (1.63), we have

equation

where c01-math-553 and thus

1.64 equation

1.65

equationequation

Therefore the sample covariance ST has a nondegenerate limiting distribution even if c01-math-557 and c01-math-558 are independent of each other. Note that the limiting distribution is also symmetric about the origin.

The limiting expression S in (1.64) is based on the eigenvalue approach. We also have the other two expressions based on the stochastic process approach and the Fredholm approach.

For the stochastic process approach, we construct the two-dimensional partial sum process

equation

Then the vector version of the FCLT yields c01-math-560 , where c01-math-561 is the two-dimensional Bm. We now have, by the vector version of the CMT,

equation

Details of the previous discussion will be given in Chapter 2.

For the Fredholm approach we consider

equation

where c01-math-562 is the c01-math-563 th element of c01-math-564 , that is, c01-math-565 . Then, as is shown in Chapter 4, it holds that

equation

We now have the following equivalent expressions for the limiting random variable S.

1.66 equation

equation

As the next example we consider

1.67

equationequation

The c.f. of UT is given by

equation

As T → ∞ it holds that (Problem 4.2)

1.68

equation

where the integral is the Ito integral to be introduced in Chapter 2. In the present case we cannot express the limit in distribution in (1.68) using a double integral with a continuous kernel as in the last expression in (1.66). It, however, holds that the conditional distribution of c01-math-572 given c01-math-573 has the following property:

equation

which leads us to

equation

where the second last equality comes from Section 1.1 and the arguments discussed in Chapter 4.

If we consider, instead of UT in (1.67),

1.69 equation

it holds (Problem 4.2) that

1.70 equation

equation

where c01-math-577 and the single integral is again the Ito integral. The double integral expression is possible in the present case, although an additive constant term emerges.

As the third example let us consider

1.71

equation

1.72

equation

which are mixed versions of UT in (1.67). The c.f.s of c01-math-581 are given (Problem 4.3) by

1.73

equation

1.74

equation

Therefore the distributions of c01-math-584 and c01-math-585 are symmetric about the origin, as was anticipated. It evidently holds that

equation

We shall have the following equivalent expressions:

1.75 equation

equation

1.76 equation

equation

where c01-math-588 and c01-math-589 .

The limiting distributions of c01-math-590 and c01-math-591 are also symmetric about the origin. The former can be interpreted from the first expression in (1.75) as the distribution of the difference of two independent c01-math-592 random variables, while the latter is known as the distribution of Lévy's stochastic area [Hida (1980)]. In the latter case the double integral expression is not possible, unlike in the former. We, however, have that, given c01-math-593 , the conditional distribution of

equation

is normal with mean 0 and variance

equation

Thus we can also derive the c.f. of the limiting distribution of c01-math-594 from

equation

whose derivation will be presented in Chapters 3 and 4.

We note in passing that, by comparing (1.76) with (1.68), the following relation is seen to hold:

equation

where c01-math-595 is the four-dimensional Bm.

It is easy to obtain cumulants c01-math-596 and c01-math-597 for the limiting distributions of c01-math-598 and c01-math-599 , respectively. We have (Problem 4.4)

1.77 equation

1.78 equation

where Bℓ is the Bernoulli number.

Figure 1.5 draws the limiting probability densities c01-math-603 and c01-math-604 of c01-math-605 and c01-math-606 , respectively, together with the density of c01-math-607 . The three distributions have means 0 and variances 1. We computed c01-math-608 and c01-math-609 following

1.79

equation

1.80

equation

From the computational point of view we have difficulty in computing (1.79) since the integrand is oscillating and c01-math-612 approaches 0 rather slowly. Chapter 5 will suggest a method for overcoming this difficulty, from which Figure 1.5 has been produced. Percent points for the three distributions are tabulated in Table 1.4.

A plot of probability densities of (1.79), (1.80), and N(0, 1) with three curves plotted and a legend inset.

Figure 1.5 Probability densities of (1.79) (1.80), and N(0, 1).

Table 1.4 Percent points for distributions in (1.79) (1.80), and N(0,1)

Finally let us consider the ratio statistics:

1.81 equation

1.82 equation

where c01-math-618 and c01-math-619 are sample means of c01-math-620 and c01-math-621 , respectively. These statistics may be interpreted as the LSEs derived from the regression relations c01-math-622 and c01-math-623 , respectively. Because the two series c01-math-624 and c01-math-625 are independent, the regression relations should indicate such signals, but the reality is that the regressions produce seemingly meaningful results. These are called spurious regressions following Granger and Newbold (1974). In fact, the LSEs c01-math-626 and c01-math-627 have nondegenerate limiting distributions, which we show below.

We first deal with c01-math-628 . Let us put c01-math-629 , where

1.83 equation

equation

with

equation

The c.f. c01-math-631 of c01-math-632 is given (Problem 4.5) by

equation

where c01-math-633 and c01-math-634 , whereas λj is the jth largest eigenvalue of c01-math-637 given in (1.7). Arguing as before it is an easy matter to derive

equation

We can deal with c01-math-638 in (1.82) similarly. Let us put c01-math-639 , where

1.84

equationequation

with c01-math-641 and c01-math-642 . Then the c.f. c01-math-643 of c01-math-644 is given (Problem 4.6) by

equation

where c01-math-645 is the jth largest eigenvalue of c01-math-647 defined in (1.10). Then we have

equation

Figure 1.6 draws the limiting probability densities c01-math-648 of c01-math-649 , which were numerically computed from c01-math-650 , where c01-math-651 can be computed by Imhof's formula given in (1.55). Moments of c01-math-652 can also be computed following the formula (1.57). Let c01-math-653 be the jth order raw moment of c01-math-655 . Then we have (Problem 4.7)

1.85 equation

equation

It can also be shown (Problem 4.7) that c01-math-657 and c01-math-658 are both symmetric about the origin. Table 1.5 reports percent points for c01-math-659 .

A plot of limiting probability densities with two curves plotted for k values and a legend inset.

Figure 1.6 Limiting probability densities of c01-math-660 .

Table 1.5 Percent points for limiting distributions of c01-math-661

The following equivalent expressions will emerge for the weak convergence of c01-math-664 in (1.83):

equation

where c01-math-665 and c01-math-666 is the two-dimensional Bm. We shall also have

equation

where c01-math-667 is the two-dimensional demeaned Bm.

In terms of the weak convergence of c01-math-668 c01-math-669 , we shall have the following expressions:

1.86 equation

1.87

equationequation

The limiting distribution of c01-math-672 is dispersed more than that of c01-math-673 , as is shown in Figure 1.6 and Table 1.5.

Problems

In the problems below it is assumed that c01-math-674 c01-math-675 with c01-math-676 . We also assume that c01-math-677 and

equation

4.1 Show that the c.f. of c01-math-678 is given by

equation

4.2 Show that

equation

where c01-math-679 and c01-math-680 .

4.3 Prove that the c.f.s of the following statistics

equation

are given, respectively, by

equationequation

4.4 Prove that the jth order cumulants c01-math-682 and c01-math-683 of

equation

are given, respectively, by

equation

where Bℓ is the Bernoulli number.

4.5 Show that the c.f. of

equation

is given by

equation

4.6 Show that the c.f. of

equation

is given by

equation

4.7 Compute variances of the limiting distributions of the following statistics:

equationequation

Also show that the distributions of c01-math-685 and c01-math-686 and the limiting distributions are all symmetric about the origin.

1.5 Statistics from the Cointegrated Process

We continue discussions on the two-dimensional random walk, but, unlike the last section, we deal with the case where the random walks depend on each other. Let us consider the model

1.88 equation

equation

where β ≠ 0, c01-math-689 and c01-math-690 are observable and c01-math-691 follows c01-math-692 . This model may be regarded as a regression model, where c01-math-693 is regressed on the random walk c01-math-694 . The difference from the spurious regression discussed in the last section is that the regression is a true relationship and c01-math-695 is not a random walk because c01-math-696 is not independent, though stationary. The process whose difference follows a stationary process is called an integrated process of order 1, which is denoted as an c01-math-697 process.

The present model is a simplified version of the cointegrated system to be discussed in later chapters. The implication of (1.88) is that c01-math-698 , a linear combination of two c01-math-699 processes c01-math-700 and c01-math-701 , follows c01-math-702 . In general, a vector-valued process c01-math-703 is said to be integrated of order d and is denoted as an I c01-math-705 process if c01-math-706 is I c01-math-707 and c01-math-708 is stationary. Following Engle and Granger (1987), such a process is called a cointegrated process of order c01-math-709 if there exists a linear combination c01-math-710 , which is I c01-math-711 . The above model follows a