EKONOMI OCH SAMHÄLLE

Skrifter utgivna vid Svenska handelshögskolanPublications of the Swedish School of Economics

and Business Administration

Nr 110

NIKLAS AHLGREN

INFERENCE ON COINTEGRATIONIN VECTOR AUTOREGRESSIVE MODELS

Helsingfors 2002

Inference on Cointegration in Vector Autoregressive Models

Key words: Cointegration, Rank determination, Small sample properties, VectorAutoregressive models, Bootstrap

© Swedish School of Economics and Business Administration & Niklas Ahlgren

Niklas AhlgrenSwedish School of Economics and Business AdministrationDepartment of Finance and StatisticsP.O.Box 47900101 Helsinki, Finland

Distributor:

LibrarySwedish School of Economics and Business AdministrationP.O.Box 47900101 Helsinki, Finland

Telephone: +358-9-431 33 376, +358-9-431 33 265Fax: +358-9-431 33 425E-mail: publ@shh.fihttp://www.shh.fi/services/biblio/papers/index.htm

ISBN 951-555-748-8 (printed)ISBN 951-555-749-6 (PDF)ISSN 0424-7256

Universitetstryckeriet, Helsingfors 2002

ACKNOWLEDGMENTS

I would like to thank my supervisor, Professor Jukka Nyblom (University ofJoensuu, Senior Distinguished Fellow of the Swedish School of Economics), forhis help and advice. He suggested the problem of the first paper, which weworked on jointly. Jukka Nyblom read several drafts of the other papers andgave many useful comments. His unassuming and mild manner belies his deepknowledge of statistical theory and econometrics. I feel fortunate for havinghim as my supervisor. I am grateful to my second supervisor, Professor GunnarRosenqvist, for his help and advice, and his way of managing the department. Hehas supported me in many ways during the course of my studies, and I greatlyappreciate his fine sense of duty. Gunnar Rosenqvist handled the official process,and patiently listened to my complaints about the red tape. I would like tothank Professor Johan Fellman for his advice and useful discussions, and JanAntell, my coauthor of the fourth paper in the thesis, for useful discussions. I amgrateful to Professor Katarina Juselius, who allowed me to take part in the project’Macroeconomic transmission mechanisms in Europe. Empirical applications andeconometric methods’. Some of the topics discussed in the thesis are derived fromthese workshops and conferences. Katarina Juselius’ enthusiasm for research intime series econometrics, and upcoming workshops and conferences sustained methrough bad days.

I would like to thank my official examiners, Professors Søren Johansen andPentti Saikkonen. They both read the manuscript very carefully, pointed outmany errors and misunderstandings, and suggested many improvements. Someof their suggestions and ideas lead to substantial revisions of the manuscript andgreatly improved the final manuscript.

During the course of my studies, I spent the autumn term 1998 as a visitingPh.D. student at the School of Economics and Management, University of Aarhus.I would like to thank Professor Svend Hylleberg for arranging the stay, and inAarhus Professors Niels Haldrup and Allan Wurtz. During Lent Term 2002, I wasan Unofficial Visiting Scholar in the Faculty of Economics and Politics, Universityof Cambridge. I would like to thank Professor Andrew Harvey for inviting me toCambridge, and in Cambridge Dr. Melwyn Weeks and Mrs. Joan Salvesen.

Many have helped me with the book. I would like to thank Jan-Erik Ekbergfor advice on LATEX and for solving some difficult typesetting problems. BarbaraCavonius helped me to turn the manuscript into a book and handled the printingprocess.

Finally, I would like to thank my sister Lotte and her husband Kim, my fatherJohan and my grandmother Hjordis for their support and encouragement.

Financial support from the Foundation of the Swedish School of Economics,The Jenny and Antti Wihuri Foundation, Nordisk Forskerutdanningsakademi(NorFA), Broderna Lars och Ernst Krogius forskningsfond, Waldemar vonFrenckells stiftelse and Letterstedtska foreningen (Finlandska avdelningen) aregratefully acknowledged.

Niklas AhlgrenHelsingforsNovember 2002

iii

PREFACE

In the thesis we consider inference for cointegration in vector autoregressive(VAR) models. The thesis consists of an introduction and four papers. Thefirst paper proposes a new test for cointegration in VAR models that is directlybased on the eigenvalues of the least squares (LS) estimate of the autoregressivematrix. In the second paper we compare a small sample correction for the like-lihood ratio (LR) test of cointegrating rank and the bootstrap. The simulationexperiments show that the bootstrap works very well in practice and dominatesthe correction factor. The tests are applied to international stock prices data,and the finite sample performance of the tests are investigated by simulating thedata. The third paper studies the demand for money in Sweden 1970—2000 usingthe I(2) model. In the fourth paper we re-examine the evidence of cointegrationbetween international stock prices. The paper shows that some of the previousempirical results can be explained by the small-sample bias and size distortion ofJohansen’s LR tests for cointegration.

In all papers we work with two data sets. The first data set is a Swedishmoney demand data set with observations on the money stock, the consumerprice index, gross domestic product (GDP), the short-term interest rate andthe long-term interest rate. The data are quarterly and the sample period is1970(1)—2000(1). The second data set consists of month-end stock market indexobservations for Finland, France, Germany, Sweden, the United Kingdom and theUnited States from 1980(1) to 1997(2). Both data sets are typical of the samplesizes encountered in economic data, and the applications illustrate the usefulnessof the models and tests discussed in the thesis.

v

CONTENTS

1 Introduction 12 Integrated Processes 23 Cointegration 44 Cointegration in the Framework of the Linear Process 55 Cointegration in Vector Autoregressive Models 6

5.1 Representation 65.2 Testing 7

6 Asymptotic Theory for Integrated Processes 87 The Likelihood Ratio Test for Cointegrating Rank 10

and Small Sample Correctiuons8 Bootstrap Tests for Cointegration 119 The I(2) Model 1210 Conclusions 13References 14

The Papers

vii

THE PAPERS

[1] Ahlgren, N. and Nyblom, J. (2002), ’A Direct Test for Cointegration inVector Autoregressive Models’, Manuscript, Swedish School of Economics.

[2] Ahlgren, N. (2002), ’Comparison of a Small Sample Correction for the Testof Cointegrating Rank and the Bootstrap’, Manuscript, Swedish School of Eco-nomics.

[3] Ahlgren, N. (2002), ’An I(2) Model for Swedish Money Demand’, Manuscript,Swedish School of Economics.

[4] Ahlgren, N. and Antell, J. (2002), ’Testing for Cointegration between Inter-national Stock Prices’, forthcoming in Applied Financial Economics.

ix

1 Introduction

The notion of cointegration was introduced by Granger (1981, 1983), and Engleand Granger (1987) (see also Granger 1986 and Hendry 1986). The simple ideabehind cointegration is that sometimes the lack of stationarity of a multidimen-sional process is caused by common stochastic trends, which can be eliminatedby taking suitable linear combinations of the process, thereby making the linearcombination stationary (Johansen 1996a, 72). Seminal contributions in the liter-ature on cointegration were made by Stock and Watson (1988), Johansen (1988,1991, 1996b), and Johansen and Juselius (1990). Engle and Granger (1987), andStock and Watson (1988) considered cointegration in the framework of the linearprocess. Johansen (1988, 1991, 1996b) showed that cointegration can be definedas submodels of the vector autoregressive (VAR) model.

Johansen (1996a) notes that the reason why cointegration has been so popularin econometrics is that classical macroeconomic models are often formulated assimultaneous linear relations between variables following the Cowles Commissiontradition. This type of modelling has a long history in economics, going back tothe work of Tinbergen (1939) and Haavelmo (1944). For a modern rereading ofHaavelmo (1944), see Juselius (1993). These models were developed for stationaryprocesses, but in fact most economic variables are nonstationary. If we think ofthe classical economic relations as long-run relations, we can easily imagine thatsuch relations can be stationary even if the variables themselves are nonstationary.Cointegration is the mathematical formulation of this phenomenon (Johansen1996a, 73).

In the last two decades cointegration has been the topic of intensive researchin econometrics and statistics. There is a number of books and collections ofpapers dealing with cointegration (see Johansen 1996 and the references in thepreface therein). The book by Maddala and Kim (1998) is a review of unitroots and cointegration for economists. Mills (1999) and Chan (2002) considercointegration with a view towards financial time series. Hubrich, Lutkepohl andSaikkonen (2001) is a recent survey of system cointegration tests.

In the thesis we consider inference for cointegration in VAR models. Thethesis consists of an introduction and four papers. The papers are referred toby their numbers, [1], [2], [3] and [4]. Paper [1] (jointly with Jukka Nyblom)proposes a new test for cointegration in VAR models that is directly based onthe eigenvalues of the least squares (LS) estimate of the autoregressive matrix.In paper [2] we compare a small sample correction for the likelihood ratio (LR)test of cointegrating rank and the bootstrap. Paper [3] studies the demand formoney in Sweden 1970—2000 using the I(2) model. Paper [4] (jointly with JanAntell) tests for cointegration between international stock prices.

In all papers we work with two data sets. The first data set is a Swedishmoney demand data set with observations on the money stock, the consumerprice index, gross domestic product (GDP), the short-term interest rate and

1

the long-term interest rate. The data are quarterly and the sample period is1970(1)—2000(1), giving 117 quarterly observations. The second data set consistsof month-end stock market index observations for Finland, France, Germany,Sweden, the United Kingdom and the United States from 1980(1) to 1997(2),yielding 206 monthly observations. Both data sets are typical of the sample sizesencountered in economic data, and the applications illustrate the usefulness ofthe models and tests discussed in the thesis.

Section 2 of the introduction gives the basic definitions of integrated pro-cesses. Section 3 discusses the notion of cointegration and gives the basic defini-tions. Section 4 considers cointegration in the framework of the linear process.Representation and testing for cointegration in VAR models are considered inSection 5. Section 6 is a short introduction to the asymptotic theory for inte-grated processes. The likelihood ratio (LR) test for cointegrating rank, smallsample corrections, and bootstrap tests for cointegration are reviewed in Sections7 and 8, respectively. Section 9 gives an overview of the I(2) model. Section 10concludes and summarises the contributions of the thesis.

The following notation is used. The vector Xt = (X1t, . . . , Xpt)′ denotes an

observable p-dimensional time series. The sample size is denoted by T . Thelag operator is denoted by L and the differencing operator is denoted by ∆, i.e.∆Xt = (1 − L)Xt and ∆dXt = (1 − L)dXt. The symbol I(d) is used to denotea process which is integrated of order d, i.e. is stationary (or asymptoticallystationary) after differencing d times but is nonstationary after differencing only

d − 1 times. The symbolP→ signifies convergence in probability, and OP (·) and

oP (·) are the usual symbols for the order of convergence in probability of a se-

quence. The symbolD→ signifies convergence in distribution or more generally,

weak convergence. We abbreviate ’independent, identically distributed’ by iid.The normal distribution with mean µ and covariance matrix Ω is denoted byN(µ,Ω). We denote the (p × p) identity matrix by Ip. If A is a (p × r) ma-trix of full rank, we denote by A⊥ a p × (p − r) matrix of full rank such thatA′A⊥ = 0. If r = 0, we define A⊥ = Ip, and if p = r, we choose A⊥ = 0. Wedenote A = A(A′A)−1, such that PA = A(A′A)−1A′ = AA′ is the projectiononto the space spanned by A. We denote the norm of a matrix A by ‖A‖, i.e.‖A‖ =

√tr(A′A), where tr(A) denotes the trace of A. As a general convention,

a sum is defined to be zero if the lower bound of the summation index exceedsthe upper bound.

2 Integrated Processes

Let εt be a doubly infinite sequence of p-dimensional iid random variables withmean 0 and finite covariance matrix Ω. From these we construct linear ergodic

2

processes defined by

Xt =∞∑i=0

Ciεt−i, t = 1, . . . , T, (1)

where we assume that the coefficient matrices Ci decrease exponentially fast suchthat C(z) =

∑∞

i=0Cizi with C(0) = Ip is convergent for |z| ≤ 1 + δ for some

δ > 0. This guarantees the existence of the process Xt in the sense that theseries is convergent almost surely (see Johansen 1996b). Informally we can saythat a stationary sequence is ergodic if it has the property that a random eventinvolving every member of the sequence always has probability either 0 or 1, butsee Hannan (1970, 201—220) for a formal definition. For integrated processes werestrict the time index to t = 0, 1, . . . , where t = 0 determines the initial valueX0.

The basic definitions of integrated processes are the following (Johansen 1996b).

Definition 1 A linear p-dimensional process Xt =∑

∞

i=0Ciεt−i is called inte-grated of order zero, I(0), if

∑∞

i=0Ci = 0.

The concept of I(0) is defined without allowing deterministic terms like a meanor a trend. The condition that

∑∞

i=0Ci = 0 guarantees that if Xt is stationary,then the cumulated process

∑ti=1Xt is nonstationary, and the so-called long-run

variance is positive.

Definition 2 A p-dimensional stochastic process Xt is called integrated of orderone, I(1), if ∆(Xt − E(Xt)) is I(0).

Note that the property of being integrated is related to the stochastic partof the process since we have subtracted the expectation in the definition. Thesimplest example of an I(1) process is a random walk. Many economic variablescan be thought of as being I(1) or at least well approximated by I(1) processes.In practice unit root tests can be applied to determine the degree of integrationof an economic time series. In the unit root literature the classic references areDickey and Fuller (1979, 1981), and Fuller (1976, 2nd edition 1996).

Definition 2 can be generalised to processes integrated of order d.

Definition 3 A p-dimensional stochastic process Xt is called integrated of orderd, I(d), d = 0, 1, 2, . . . , if ∆d(Xt − E(Xt)) is I(0).

Note that the definitions take the order of integration of the vector Xt to bethe maximum order of integration of the single components Xit, i = 1, . . . , p. Inpractice d = 0, d = 1, and possibly d = 2.

It is possible to classify a linear trend as an I(1) process and a quadratic trendas an I(2) process (see Paruolo 1996 and Yoo 1986), but we do not want to doso here since we are mainly interested in stochastic trends.

3

Juselius (1999a) argues that treating an economic variable as being integratedof order d in empirical applications is only an approximation and depends on thelength of the time period considered. See Juselius (1999a) for empirical examples.

In paper [4] (jointly with Jan Antell) we treat logarithmic stock prices log ptas being I(1), and consequently logarithmic returns log pt− log pt−1 as being I(0),where pt is the stock price. This has a long tradition in finance, going back morethan 100 years to Bachelier (1900) (see Berglund 1986). Some economic timeseries behave as I(2) processes. Examples include the nominal money stock andthe price level, implying that the inflation rate is I(1). In paper [3] we estimatean I(2) model for Swedish money demand.

3 Cointegration

A formal definition of cointegration is

Definition 4 If Xt is integrated of order one but some linear combination β′Xt,β = 0, can be made stationary by a suitable choice of its initial distribution β′X0,then Xt is called cointegrated with cointegrating vector β. The cointegrating rankis the number of linearly independent cointegrating vectors, and the space spannedby the cointegrating vectors is the cointegrating space.

Note that we are disregarding deterministic terms in the definition of cointe-gration. If Xt contains deterministic terms like a constant and a linear trend, wesubtract the expectation in the definition. Thus, if β′X−E(β′X) is stationary,Xt is said to be cointegrated. The I(1) process Xt is called cointegrated even ifβ′Xt can only be made trend-stationary.

The geometric interpretation is the following (see Paruolo 1996). The p-dimensional process Xt is cointegrated if there exists a subspace R0 of Rp suchthat for any β ∈ R0, the process β′Xt is I(0). R0 is the cointegrating space,and the dimension of the subspace R0, dim(R0), is the cointegrating rank. Thenumber of common trends is the dimension of the orthogonal complement R1 inRp of the cointegrating space, dim(R1). By the definition, R0 and R1 provide an

orthogonal direct sum decomposition of Rp, and dim(Rp) = dim(R0) + dim(R1).The following example from Johansen (1996a, 71) illustrates the concept of

cointegration.

Example 1 We define a three-dimensional process by

X1t =t∑

i=1

ε1i + ε2t, (2)

X2t =1

2

t∑i=1

ε1i + ε3t, (3)

X3t = ε4t, t = 0, 1, . . . , T, (4)

4

where the εjt, j = 1, . . . , 4, t = 0, 1, . . . , T , are iid with mean zero and finitevariance. It is seen that Xt is nonstationary, and that ∆Xt is stationary andI(0). Thus Xt is an I(1) process, which can be made stationary by differencing.It is also seen that X1t − 2X2t is stationary, and we say that Xt is cointegratedwith (1,−2, 0)′ as a cointegrating vector. In fact, X3t is also stationary and I(0),and with a slight abuse of language the unit vector (0, 0, 1)′ is a cointegratingvector. The cointegrating rank is thus 2. The process

∑ti=1 ε1t is called a common

stochastic trend. In the example the dimension of Xt is p = 3, the cointegratingrank is r = 2 and the number of common stochastic trends is p− r = 1.

We have defined the class CI(1, 1) of integrated processes that are I(1) andwhich cointegrate to I(0). The generalisation to CI(d, b) leads to the followingdefinition.

Definition 5 The I(d) process Xt is called cointegrated CI(d, b) with cointegrat-ing vector β = 0 if β′Xt is I(d− b), b = 1, . . . , d, d = 1, . . .

4 Cointegration in the Framework of the Linear

Process

Engle and Granger (1987), and Stock and Watson (1988) consider cointegrationin the framework of the p-dimensional linear process

∆Xt = C(L)εt,∞∑j=1

j ‖Cj‖ < ∞, t = 1, . . . , T, (5)

whereC(z) =∑

∞

i=0Cizi with C(0) = Ip, and εt is iid with mean 0 and covariance

matrix Ω. The 1-summability condition in (5) ensures that the linear process isstationary and ergodic. The 1-summability condition is stronger than the squaresummability condition

∑∞

j=1 ‖Cj‖2 < ∞ in Theorem 3 of Hannan (1970, 204).

The stronger assumption is needed to prove large sample results (see e.g. Hayashi2000).

Engle and Granger (1987) showed that if the (p × 1) vector Xt given in (5)is cointegrated with d = 1, b = 1 and with cointegrating rank r, then C(1) isof rank p − r, and there exists a (p × r) matrix β such that β′C(1) = 0. Thecolumns of β are the cointegrating vectors. The theorem is known as Granger’srepresentation theorem (Engle and Granger 1987). Note that Stock and Watson(1988) allow for a possible drift in Xt.

5

5 Cointegration in Vector Autoregressive Mod-

els

5.1 Representation

Consider a general vector autoregressive (VAR) model for the p-dimensional pro-cess Xt defined by the equations

Xt = Π1Xt−1 + · · ·+ΠkXt−k +ΦDt + εt, t = 1, . . . , T, (6)

where εt, t = 1, . . . , T , are p-dimensional independent Gaussian with mean 0and covariance matrix Ω. The values X−k+1, . . . ,X0 are assumed fixed. Thedeterministic terms Dt can contain a constant, a linear trend, seasonal dummiesand intervention dummies, and Dt is an (m× 1) vector of deterministic terms.

The parameters are(Π1, . . . ,Πk,Φ,Ω),

where Π1 . . . ,Πk, are (p × p) parameter matrices, Φ is a (p × m) parametermatrix and Ω is a (p× p) positive definite covariance matrix.

The properties of the matrix polynomial

Π(z) = Ip −Π1z − · · · −Πkzk (7)

determine the properties of the process. This is a well known result in multivariatetime series analysis (see e.g. Anderson 1971). We let |Π(z)| denote the determi-nant of Π(z), and define the matrices Π = −Π(1) and Γ = −dΠ(z)/dz |z=1 +Π.We make the following assumption:

Assumption 1 The roots of|Π(z)| = 0

are either outside the unit circle or at z = 1.

Johansen (1991) proved Granger’s representation theorem in the vector au-toregressive representation. The theorem is known as the representation theoremfor I(1) variables. In order to state the theorem, we need the notation α⊥ andβ⊥ for the p × (p − r) orthogonal complements of α and β of rank p − r suchthat α′α⊥ = 0 and β′β

⊥= 0.

Theorem 1 (Johansen 1991, Theorem 4.1, 1559) If Xt is given by (6) andif Assumption 1 holds, then Xt is I(1) if and only if

Π = αβ′, (8)

where α, β are (p× r) matrices of full rank r < p, and

α′

⊥Γβ⊥ (9)

6

has full rank. In this case ∆Xt−E(∆Xt) and β′Xt−E(β′Xt) can be given initialdistributions such that they become I(0). The process has the representation

Xt = Ct∑

i=1

(εi +ΦDi) +C(L)(εt +ΦDt) + Pβ⊥X0, (10)

where C = β⊥(α′

⊥Γβ⊥)

−1α′

⊥. Thus Xt is a cointegrated I(1) process with coin-

tegrating vectors β.

Without reference to the initial distributions, we can say that ∆Xt−E(∆Xt)and β′Xt − E(β′Xt) are asymptotically stationary and I(0).

Under the reduced rank condition (8), the process has the reduced form errorcorrection representation

∆Xt = αβ′Xt−1 +k−1∑i=1

Γi∆Xt−i +ΦDt + εt, (11)

where Γi = −∑k

j=i+1Πj and the parameters (α,β,Γ1, . . . ,Γk−1,Φ,Ω) varyfreely (see Johansen 1996b, Definition 5.1, 70).

Model (11) defined by the reduced rank condition (8) and the full rank of(9) is known as the I(1) model. The full rank condition (9) excludes processesintegrated of higher order than one. In particular, the condition excludes I(2)processes, which is the practical meaning of condition (9) (see Section 9).

5.2 Testing

Assume that Xt is a (p × 1) vector of time series that has a finite order VARrepresentation

Xt = Π1Xt−1 + · · ·+ΠkXt−k +ΦDt + εt, t = 1, . . . , T. (12)

The error correction representation is

∆Xt = ΠXt−1 +k−1∑i=1

Γi∆Xt−i +ΦDt + εt, (13)

where Π =∑k

i=1Πi − Ip, Γi = −∑k

j=i+1Πj, and for later use we denote Γ =

Ip−∑k

i=1Γi, as defined below equation (7). The hypothesis of cointegration canbe formulated in model (13) as

H0 : rank(Π) = r against H1 : rank(Π) > r, r ≤ p− 1. (14)

The rank of Π is the cointegrating rank and is the number of linearly independentcointegrating vectors.

7

Several tests for cointegration in VAR models have been suggested. Stockand Watson (1988) suggested a test for the number of unit roots or commontrends based on principal components. Johansen (1988, 1991, 1992a, 1994, 1996b)derived the likelihood ratio (LR) test for the hypotheses (14). The LR test isbased on the canonical correlations between ∆Xt and Xt−1. Bewley and Yang(1995), and Yang and Bewley (1996) proposed a test for cointegration, whichis based on the levels canonical correlations between Xt and Xt−1. Hubrich,Lutkepohl and Saikkonen (2001) is a recent review of system cointegration tests.

6 Asymptotic Theory for Integrated Processes

In this section Xt denotes a (p× 1) vector of random variables and β is a (p× 1)parameter vector. In the linear regression model

yt = X′

tβ + εt, t = 1, . . . , T, (15)

where εt are iid with mean 0 and finite variance σ2, the standard proof of con-sistency of the least squares (LS) estimator β of β uses the assumption that

T−1∑T

t=1XtX′

tP→ M, where M is a positive definite matrix of full rank. Under

some further assumptions (see e.g. Proposition 2.1 of Hayashi 2000, 113), the LSestimator of β has a limiting distribution which is normal

T 1/2(β − β)D→ N(0, σ2M−1). (16)

If the regressors Xt are nonstationary and integrated, the usual asymptoticnormality does not hold for estimators. Stock (1987) proved that

T 1−δ(β − β)P→ 0, δ > 0, (17)

under the assumption that the regressors Xt are I(1) processes and that (yt,X′

t)′

is cointegrated with cointegrating vector (1,−β′)′, i.e. yt − X′

tβ is stationary.

The estimator β is said to be superconsistent.The distributions of estimators and test statistics for integrated processes are

nonstandard. The basic result used in the derivations of limiting distributions isDonsker’s theorem. Let εt be a sequence of p-dimensional iid random variableswith mean 0 and finite, positive definite covariance matrix Ω. Then

XT (u) = T−1/2[Tu]∑i=1

εiD→ W(u), u ∈ [0, 1], (18)

whereW(u) is a Brownian motion on C[0, 1], the space of continuous functions onthe unit interval [0, 1], with covariance function Cov(W(s),W(u)) = min(s, u)Ω,

s, u ∈ [0, 1], andD→ signifies weak convergence of probability measures in the

8

sense of Billingsley (1968, 2nd edition 1999). Note that XT (u) is not an elementof C[0, 1], but belongs to the space D[0, 1] of cadlag functions (for the definitionof cadlag functions, see Billingsley 1999, 121). The space C is a subset of D.Donsker’s theorem can be formulated for both C[0, 1] and D[0, 1] (see Billingsley1968, 1999).

The second important theorem used to derive limiting distributions is the

continuous mapping theorem (CMT). If XTD→ X on C[0, 1] and if g(·) is any

continuous functional on C[0, 1] with values in Rp or C[0, 1], then g(XT )

D→ g(X).

The basic reference for Donsker’s theorem and the CMT for univariate processesis Billingsley (1968, 1999). Phillips and Durlauf (1986) extended the results tomultivariate processes.

Define the function

XT (t/T ) = T−1/2t∑

i=1

εi, (19)

and interpolate continuously for all 0 ≤ t/T ≤ 1. Then by Donsker’s theorem

XTD→ W, where W is a Brownian motion with covariance matrix min(s, u)Ω.

We also have

T−2T∑t=1

XtX′

t = T−2T∑t=1

(t∑

i=1

εi)(t∑

i=1

εi)′

= T−1T∑t=1

XT (t/T )XT (t/T )′

=

∫ 1

0

XT (u)XT (u)′ du+ oP (1).

Applying the CMT to the continuous functional x →∫ 1

0x(u)x(u)′ du, we obtain

T−1T∑t=1

XT (t/T )XT (t/T )′ D→

∫ 1

0

W(u)W(u)′ du. (20)

The third important result is convergence to the stochastic integral

T−1T∑t=1

Xt−1ε′

t = T−1T∑t=1

(t−1∑i=1

εi)εt′ D→

∫ 1

0

W(u) dW′, (21)

proved in the paper by Chan and Wei (1988).The asymptotic theory for regression models with integrated processes was

developed by Phillips (1987, 1991), Phillips and Durlauf (1986), Park and Phillips(1988, 1989), and Phillips and Ouliaris (1990).

The limiting distributions of estimators and test statistics in the I(1) modelare functionals of Brownian motion (see Johansen 1996b). The limiting distri-bution of Johansen’s LR test statistic for cointegrating rank is given in paper[2].

9

7 The Likelihood Ratio Test for Cointegrating

Rank and Small Sample Corrections

The likelihood ratio (LR) trace statistic for the hypotheses (14) is

LR = −T

p∑i=r+1

ln(1− λi), (22)

where the eigenvalues 1 > λ1 > · · · > λp > 0 solve the eigenvalue problem

|λS11 − S10S−100 S01| = 0 (23)

and

Sij = T−1T∑t=1

RitR′

jt, i, j = 0, 1 (24)

are the product moment matrices of the residuals R0t and R1t from regressing∆Xt and Xt−1 on the lagged differences and the deterministic terms, respectively(see Johansen 1996b for details).

Monte Carlo studies have shown that in small samples the distribution of theLR test is not well approximated by the limiting distribution (see e.g. Cheungand Lai 1993, Gonzalo and Pitarakis 1999, Haug 1996, Ho and Sørensen 1996,and Toda 1995).

The total number of parameters in each equation in a p-dimensional VARmodel with k lags is kp, excluding the deterministic terms. Reinsel and Ahn(1992) suggested to use the small sample correction (T −kp)/T and the adjustedstatistic

LR = −(T − kp)

p∑i=r+1

ln(1− λi). (25)

Some other small sample corrections have also been suggested, but they are allbased on the same idea. The motivation for these degrees of freedom correc-tions is that Johansen’s LR test tends to overreject the null hypothesis of nocointegration, and the small sample correction corrects down the LR statistic.

In a simulation study Reimers (1992) finds that overspecification of the laglength induces a high rejection probability of the null hypothesis of no cointegra-tion. Ho and Sørensen (1996) examine the performance of the LR test in highdimensional systems, and find that Johansen’s ML estimator and LR test findtoo much cointegration when the lag length is overspecified. Their conclusionis that monotone degrees of freedom adjustments do not much improve the sizeof the test. Gonzalo and Pitarakis (1999) term this the dimensionality effect incointegration analysis, and propose various alternative solutions.

In a recent paper, Johansen (2002) proposes a correction factor to the LR testwhich improves the finite sample properties. The correction factor uses the idea of

10

the Bartlett correction, and involves finding an approximation to the expectationof the LR statistic. The LR statistic is then corrected to have the same meanas the limiting distribution. Let θ denote the parameters of model (13) underthe assumption that Π = αβ′. Johansen (2002) derives an approximation toEθ(−2 logLRΠ = αβ′), which is a function of θ and T under the assumptionof Gaussian errors.

In paper [4] we use the small sample correction in (25) when testing for coin-tegration between international stock prices. We find that the conclusions ofthe LR tests change if we use the small sample correction, and we do not findcointegration any longer if we use the small sample correction. The empiricalapplication in paper [2] considers the same problem and data set, but uses thecorrection factor in Johansen (2002) and the bootstrap.

8 Bootstrap Tests for Cointegration

The bootstrap was invented by Efron (1979). The principles of bootstrap method-ology are discussed in Chapter 1 in Hall (1992). A good general introductionto the bootstrap is Efron and Tibshirani (1993), and in econometrics Horowitz(2001). Li and Maddala (1996), and Berkowitz and Kilian (2000) discuss boot-strapping time series.

For stationary data and pivotal test statistics, the bootstrap provides an ap-proximation to the Edgeworth expansion (Hall 1992, 1994). For integrated andcointegrated data there are no theoretical results on the bootstrap, but resultsfrom Monte Carlo simulations show that in practice the size of bootstrap testsare closer to the nominal significance level than the size of tests using asymptoticcritical values (Li and Maddala 1997, Maddala and Kim 1998, Chapter 10, andFachin 2000).

In paper [2] we compare the asymptotic and corrected LR tests for cointegrat-ing rank with the bootstrap test by Monte Carlo simulation. Bootstrap replica-tions of the LR statistic are generated by resampling the data and computingthe LR statistic for the bootstrap samples. The finite sample distribution of theLR statistic is then approximated by the empirical distribution of the bootstrapreplications of the LR statistic. A practical problem is how to choose the numberof bootstrap samples. Two recent proposals are Andrews and Buchinsky (2000,2001), and Davidson and MacKinnon (2000).

Without going into details about the bootstrap algorithm, we reject the nullhypothesis H0 : rank(Π) = r at the significance level α, if P

θ(LR∗ > LR) < α,

where LR denotes the computed value of the LR statistic and LR∗ denotes thebootstrap LR statistic. The bootstrap p-value is computed as

p∗ =1

B

B∑b=1

ILR∗ > LR, (26)

11

where I· is the indicator function and B is the number of bootstrap samples.As B → ∞, the estimated bootstrap p-value p∗ tends to the ideal bootstrap p-value p∗ (Davidson and MacKinnon 2000, 56). The ideal bootstrap p-value is theprobability that the test statistic is greater than the computed value of the teststatistic under the bootstrap data generation process (DGP), i.e. P

θ(LR > LR).

The simulation experiments reported in paper [2] suggest that the bootstrapworks very well in practice, and dominates the correction factor in Johansen(2002).

9 The I(2) Model

We consider again the p-dimensional VAR model in (6). For the statistical anal-ysis of the I(2) model it is convenient to write the model in error correctionform

∆2Xt = Γ∆Xt−1 +ΠXt−2 +k−2∑i=1

Γi∆2Xt−i +ΦDt + εt. (27)

The relation between the parameters (Γ,Π,Γ1, . . . ,Γk−2) and (Π1, . . . ,Πk) isfound by identifying coefficients of the lagged levels in the two different represen-tations (Johansen 1992b).

Johansen (1992b) proved the representation theorem for I(2) variables, andshowed that if Xt is I(2), then the space spanned by Xt can be decomposed intor stationary directions β and p − r nonstationary directions β

⊥. Further, β

⊥

can be decomposed into β1 and β2, where β1 is (p× s1) and β2 is (p× s2), ands1 + s2 = p− r.

The result of Theorem 1 shows that the condition (9) that α′

⊥Γβ

⊥is of full

rank is necessary for the process to be I(1). If this matrix has reduced rank, wehave processes that are integrated of higher order than one. In practice we areinterested in I(2) processes. If α′

⊥Γβ

⊥has reduced rank s, it holds that

α′

⊥Γβ

⊥= ξη′ (28)

for some matrices ξ and η of dimension (p−r)×s, and rank s, with s < p−r. Weintroduce the notation α = α′(α′α)−1 and β = β′(β′β)−1 such that α′α = Ipand β′β = Ip. We then decompose α⊥ and β

⊥as follows

α1 = α⊥ξ, α2 = α⊥ξ⊥, (29)

β1 = β⊥η, β2 = β

⊥η⊥. (30)

Note that α, α1 and α2 are mutually orthogonal, and (α,α1,α2) span Rp. The

same holds for β, β1 and β2. We need a special notation for the matrix

θ = Γβα′Γ+k−1∑i=1

iΓi, (31)

12

since it will appear in the conditions for I(2) below. With this notation we thenformulate the representation theorem for I(2) variables.

Theorem 2 (Johansen 1992b, Theorem 3, 194—195) If Xt is given by (6)and if Assumption 1 holds, then Xt is I(2) if and only if

Π = αβ′, (32)

where α, β are (p× r) matrices of full rank r < p,

α′

⊥Γβ⊥ = ξη′, (33)

where ξ, η are (p− r)× s matrices of full rank s < p− r, and

α′

2θβ2 (34)

has full rank p− r − s. In this case ∆2Xt − E(∆2Xt), β′

1∆Xt − E(β′1∆Xt) andβ′Xt−α′Γβ2β

′

2∆Xt−E(β′Xt−α′Γβ2β′

2∆Xt) can be given initial distributionssuch that they become I(0).

The solution of equation (27) or the common trends representation of Xt isgiven in Johansen (1992b).

The conditions for I(2) thus involve two reduced rank conditions (32) and(33), and the full rank of (34), to exclude I(3) and higher order processes. Themodel is the so called I(2) model.

Johansen’s (1995) two-step procedure can be used to determine the numberof common components integrated of a given order, I(0), I(1) and I(2). Paruolo(1996) called these the integration indices, and suggested a two-step procedurefor determining the integration indices in a VAR system in the presence of alinear trend. Rahbek et al. (1999) proposed an I(2) model which allows fortrend-stationary components and restricts the deterministic part of the processto have at most a linear trend. Johansen (1997) derived the ML estimator in theI(2) model. The full likelihood analysis of the I(2) model is fairly complicatedand has not been used in applications due to the lack of computer programsfor computing the ML estimator, but see the illustrative example in Section 9of Johansen (1997). For a nice application of the I(2) model and the two-stepprocedure, see Juselius (1999b). Paper [3] studies the demand for money inSweden 1970—2000 using the I(2) model.

10 Conclusions

In the thesis we consider inference for cointegration in VAR models. We believethat the VAR model is useful for summarising the variation in economic data,and as a framework for formulating and testing hypotheses of cointegration. In

13

small samples the distributions of test statistics are not well approximated by thelimiting distributions, and some modifications of the test statistics are requiredin small samples.

The main contributions of the thesis are to suggest a simple alternative tothe LR test for cointegrating rank in paper [1], to show that bootstrap inferencecan be used to improve the approximation provided by the limiting distributionof the LR statistic in small samples in paper [2], and to model some real lifeeconomic data using cointegration in papers [3] and [4].

It is, of course, important to recognise the limitations of the VAR model.The VAR model is only an approximation, and in some cases the autoregressiveapproximation can be very poor, even if a large number of parameters is used(Harvey 1997).

References

[1] Anderson, T. W. (1971), The Statistical Analysis of Time Series (New York:Wiley).

[2] Andrews, D. W. K. and Buchinsky, M. (2000), ’A Three-Step Method forChoosing the Number of Bootstrap Repetitions’, Econometrica, 68, 23—51.

[3] Andrews, D. W. K. and Buchinsky, M. (2001), ’Evaluation of a Three-StepMethod for Choosing the Number of Bootstrap Repetitions’, Journal ofEconometrics, 103, 345—386.

[4] Bachelier, L. (1900), Theorie de la Speculation (Paris: Gauthier-Villars).

[5] Berglund, T. (1986), Anomalies in Stock Returns on a Thin Security Market,Publications of the Swedish School of Economics and Business Administra-tion, 37.

[6] Berkowitz, J. and Kilian, L. (2000), ’Recent Developments in BootstrappingTime Series’, Econometric Reviews, 19, 1—48.

[7] Bewley, R. and Yang, M. (1995), ’Tests for Cointegration Based on CanonicalCorrelation Analysis’, Journal of the American Statistical Association, 90,990—996.

[8] Billingsley, P. (1968), Convergence of Probability Measures, 1st edition (NewYork: Wiley).

[9] Billingsley, P. (1999), Convergence of Probability Measures, 2nd edition (NewYork: Wiley).

14

[10] Chan, N. H. (2002), Time Series. Applications to Finance (New York: Wi-ley).

[11] Chan, N. H. and Wei, C. Z. (1988), ’Limiting Distributions of Least SquaresEstimates of Unstable Autoregressive Processes’, Annals of Statistics, 16,367—401.

[12] Cheung, Y.-W. and Lai, K. S. (1993), ’Finite-Sample Sizes of Johansen’sLikelihood Ratio Tests for Cointegration’, Oxford Bulletin of Economics andStatistics, 55, 313—328.

[13] Davidson, R. and MacKinnon, J. G. (2000), ’Bootstrap Tests: How ManyBootstraps?’, Econometric Reviews, 19, 55—68.

[14] Dickey, D. A. and Fuller, W. A. (1979), ’Distribution of the Estimatorsfor Autoregressive Time Series with a Unit Root’, Journal of the AmericanStatistical Association, 74, 427—431.

[15] Dickey, D. A. and Fuller, W. A. (1981), ’Likelihood Ratio Statistics forAutoregressive Time Series with a Unit Root’, Econometrica, 49, 1057—1072.

[16] Engle, R. F. and Granger, C. W. J. (1987), ’Co-Integration and Error Correc-tion: Representation, Estimation, and Testing’, Econometrica, 55, 251—276.

[17] Efron, B. (1979), ’Bootstrap Methods: Another Look at the Jackknife’, An-nals of Statistics, 7, 1—26.

[18] Efron, B. and Tibshirani, R. J. (1993), An Introduction to the Bootstrap(New York: Chapman and Hall).

[19] Fachin, S. (2000), ’Bootstrap and Asymptotic Tests of Long-Run Relation-ships in Cointegrated Systems’, Oxford Bulletin of Economics and Statistics,62, 543—551.

[20] Fuller, W. A. (1976), Introduction to Statistical Time Series, 1st edition(New York: Wiley).

[21] Fuller, W. A. (1996), Introduction to Statistical Time Series, 2nd edition(New York: Wiley).

[22] Gonzalo, J. and Pitarakis, J.-Y. (1999), ’Dimensionality Effect in Cointegra-tion Analysis’, in Engle, R. F. andWhite, H. (eds.), Cointegration, Causality,and Forecasting. Festschrift in Honour of Clive W. J. Granger, Chapter 9,212—229 (Oxford: Oxford University Press).

[23] Granger, C. W. J. (1981), ’Some Properties of Time Series Data and TheirUse in Econometric Model Specification’, Journal of Econometrics, 16, 121—130.

15

[24] Granger, C. W. J. (1983), ’Co-Integrated Variables and Error-CorrectingModels’, Discussion Paper 83-13, University of California, San Diego.

[25] Granger, C. W. J. (1986), ’Developments in the Study of Cointegrated Eco-nomic Variables’, Oxford Bulletin of Economics and Statistics, 48, 213—228.

[26] Haavelmo, T. (1944), ’The Probability Approach in Econometrics’, Econo-metrica, 12, 1—118. Supplement.

[27] Hall, P. (1992), The Bootstrap and Edgeworth Expansion (New York:Springer-Verlag).

[28] Hall, P. (1994), ’Methodology and Theory for the Bootstrap’, in Engle, R.F. and McFadden, D. L. (eds.), Handbook of Econometrics, Vol. IV, Chapter39, 2341—2381 (Amsterdam: North-Holland), Chapter 39, 2341—2381.

[29] Hannan, E. J. (1970), Multiple Time Series (New York: Wiley).

[30] Harvey, A. C. (1997), ’Trends, Cycles and Autoregressions’, Economic Jour-nal, 107, 192—201.

[31] Haug, A. A. (1996), ’Tests for Cointegration: A Monte Carlo Comparison’,Journal of Econometrics, 71, 89—115.

[32] Hayashi, F. (2000), Econometrics (Princeton: Princeton University Press).

[33] Hendry, D. F. (1986), ’Econometric Modelling with Cointegrated Variables:An Overview’, Oxford Bulletin of Economics and Statistics, 48, 201—212.

[34] Ho, M. S. and Sørensen, B. E. (1996), ’Finding Cointegration Rank in HighDimensional Systems Using the Johansen Test: An Illustration Using DataBased Monte Carlo Simulations’, Review of Economics and Statistics, 78,726—732.

[35] Horowitz, J. L. (2001), ’The Bootstrap’, in Heckman, J. J. and Leamer, E.(eds.), Handbook of Econometrics, Vol. V, Chapter 52, 3159—3228 (Amster-dam: North-Holland).

[36] Hubrich, K., Lutkepohl, H. and Saikkonen, P. (2001), ’A Review of SystemsCointegration Tests’, Econometric Reviews, 20, 247—318.

[37] Johansen, S. (1988), ’Statistical Analysis of Cointegration Vectors’, Journalof Economic Dynamics and Control, 12, 231—254.

[38] Johansen, S. (1991), ’Estimation and Hypothesis Testing of CointegrationVectors in Gaussian Vector Autoregressive Models’, Econometrica, 59, 1551—1580.

16

[39] Johansen, S. (1992a), ’Determination of Cointegration Rank in the Presenceof a Linear Trend’, Oxford Bulletin of Economics and Statistics, 54, 383—397.

[40] Johansen, S. (1992b), ’A Representation of Vector Autoregressive ProcessesIntegrated of Order 2’, Econometric Theory, 8, 188—202.

[41] Johansen, S. (1994), ’The Role of the Constant and Linear Terms in Coin-tegration Analysis of Non-Stationary Variables’, Econometric Reviews, 13,205—229

[42] Johansen, S. (1995), ’A Statistical Analysis of Cointegration for I(2) Vari-ables’, Econometric Theory, 11, 25—59.

[43] Johansen, S. (1996a), ’Likelihood-Based Inference for Cointegration of someNonstationary Time Series’, in Cox, D. R., Hinkley, D. V. and Barndorff-Nielsen, O. E. (eds.), Time Series Models: In Econometrics, Finance andother Fields, Chapter 2, 69—100 (London: Chapman and Hall).

[44] Johansen, S. (1996b), Likelihood-Based Inference in Cointegrated Vector Au-toregressive Models (Oxford: Oxford University Press).

[45] Johansen, S. (1997), ’Likelihood Analysis of the I(2) Model’, ScandinavianJournal of Statistics, 24, 433—462.

[46] Johansen, S. (2002), ’A Small Sample Correction for the Test of Cointegrat-ing Rank in the Vector Autoregressive Model’, Econometrica, 70, 1929—1961.

[47] Johansen, S. and Juselius, K. (1990), ’Maximum Likelihood Estimation andInference on Cointegration: With Applications to the Demand for Money’,Oxford Bulletin of Economics and Statistics, 52, 169—210.

[48] Juselius, K. (1993), ’VAR Modelling and Haavelmo’s Probability Approachto Macroeconomic Modelling’, Empirical Economics, 18, 595—622.

[49] Juselius, K. (1999a), ’Models and Relations in Economics and Econometrics’,Journal of Economic Methodology, 6, 259—290.

[50] Juselius, K. (1999b), ’Price Convergence in the Medium and Long Run: AnI(2) Analysis of Six Price Indices’, in Engle, R. F. and White, H. (eds.),Cointegration, Causality, and Forecasting. A Festschrift in Honour of CliveW. J. Granger, Chapter 13, 301—325 (Oxford: Oxford University Press).

[51] Li, H. and Maddala, G. S. (1996), ’Bootstrapping Time Series Models’,Econometric Reviews, 15, 115—158.

[52] Li, H. andMaddala, G. S. (1997), ’Bootstrapping Cointegrating Regressions’,Journal of Econometrics, 80, 297—318.

17

[53] Maddala, G. S. and Kim I.-M. (1998), Unit Roots, Cointegration, and Struc-tural Change (Cambridge: Cambridge University Press).

[54] Mills, T. C. (1999), The Econometric Modelling of Financial Time Series,2nd edition (Cambridge: Cambridge University Press).

[55] Park, J. Y. and Phillips, P. C. B. (1988),’Statistical Inference in Regressionswith Integrated Processes: Part 1, Econometric Theory, 4, 468—497.

[56] Park, J. Y. and Phillips, P. C. B. (1989),’Statistical Inference in Regressionswith Integrated Processes: Part 2, Econometric Theory, 5, 95—131.

[57] Paruolo, P. (1996), ’On the Determination of Integration Indices in I(2)Systems’, Journal of Econometrics, 72, 313—356.

[58] Phillips, P. C. B. (1987), ’Time Series Regression with a Unit Root’, Econo-metrica, 55, 277—301.

[59] Phillips, P. C. B. (1991), ’Optimal Inference in Cointegrated Systems’,Econometrica, 59, 283—306.

[60] Phillips, P. C. B. and Durlauf, S. N. (1986), ’Multiple Time Series Regressionwith Integrated Processes’, Review of Economic Studies, 53, 473—495.

[61] Phillips, P. C. B. and Ouliaris, S. (1990), ’Asymptotic Properties of ResidualBased Tests for Cointegration’, Econometrica, 58, 165—193.

[62] Rahbek, A., Kongsted, H. C. and Jørgensen, C. (1999): ’Trend Stationarityin the I(2) Cointegration Model’, Journal of Econometrics, 90, 265—289.

[63] Reimers, H.-E. (1992), ’Comparisons of Tests for Multivariate Cointegra-tion’, Statistical Papers, 33, 335—359.

[64] Reinsel, G. C. and Ahn, S. K. (1992), ’Vector Autoregressive Models withUnit Roots and Reduced Rank Structure: Estimation, Likelihood Ratio Test,and Forecasting’, Journal of Time Series Analysis, 13, 353—375.

[65] Stock, J. H. (1987), ’Asymptotic Properties of Least Squares Estimators ofCointegrating Vectors’, Econometrica, 55, 1035—1056.

[66] Stock, J. H. and Watson, M. W. (1988), ’Testing for Common Trends’,Journal of the American Statistical Association, 83, 1097—1107.

[67] Tinbergen, J. (1939), Business Cycles in the United States of America 1919—1932 (Geneva: League of Nations).

18

[68] Toda, H. Y. (1995), ’Finite Sample Performance of Likelihood Ratio Testsfor Cointegrating Ranks in Vector Autoregressions’, Econometric Theory,11, 1015—1032.

[69] Yang, M. and Bewley, R. (1996), ’On Cointegration Tests for VAR Modelswith Drift’, Economics Letters, 51, 45—50.

[70] Yoo, S. (1986), ’Multicointegrated Time Series and a Generalized Error Cor-rection Model’, Discussion Paper, University of California, San Diego.

19