Testing Linear Restrictions on Cointegration Vectors: Sizes and Powers of Wald Tests in Finite Samples

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Haug, Alfred A.

Working Paper Testing linear restrictions on cointegration vectors: Sizes and powers of Wald tests in finite samples

Technical Report, No. 1999,04

Provided in Cooperation with: Collaborative Research Center 'Reduction of Complexity in Multivariate Data Structures' (SFB 475), University of Dortmund

Suggested Citation: Haug, Alfred A. (1999) : Testing linear restrictions on cointegration vectors: Sizes and powers of Wald tests in finite samples, Technical Report, No. 1999,04, Universität Dortmund, Sonderforschungsbereich 475 - Komplexitätsreduktion in Multivariaten Datenstrukturen, Dortmund

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Testing linear restrictions on cointegrating vectors

Sizes and powers of Wald tests in nite samples

Alfred A Haug

University of Canterbury and York University

The Wald test for linear restrictions on cointegrating vectors is compared in

nite samples using the Monte Carlo metho d The Wald test within the vector error

correction based metho ds of Bewley et al and of Johansen the canon

ical cointegration metho d of Park the dynamic ordinary least squares metho d

of Phillips and Loretan Saikkonen and Sto ck and Watson the

fully mo died ordinary least squares metho d of Phillips and Hansen and the

band sp ectral techniques of Phillips are considered In terms of test size Jo

hansens metho d seems to be preferred and in terms of test power it is Parks and

Phillips However the relatively poor results in the context of cointegrating regres

sions suggest that improvements on the p erformance of the Wald tests considered

here are needed

Running title Performance of Wald tests in nite samples

Address Department of Economics UniversityofCanterbury Private Bag Christchurch

New Zealand Email ahaugeconcanterburyacnz Phone Financial

supp ort from the So cial Sciences and Humanities Research Council of Canada under grant

is gratefully acknowledged The author thanks without implicating participants at the

Canadian Economics Asso ciation Meetings the Econometric So ciety Australasian Meetings

the New Zealand Econometrics Study Group Meetings and the Workshop on Fractional

Intergration and Cointegration at the UniversityofDortmund for helpful comments on an earlier

draft

Intro duction

Cointegration techniques have b een applied widely in empirical economics in recent

years Numerous tests for cointegration and estimation metho ds for cointegrating

vectors have b een suggested in the literature Almost all results are based on asymp

totic theory and the p erformance in nite samples can dier substantially across tests

and estimation metho ds even though metho ds might be asymptotically equivalent

and ecient Cheung and Lai Gregory To da and Haug

among others provided Monte Carlo comparisons of size distortions and of p owers for

various tests for cointegration Sto ck and Watson Gonzalo Kitamura

and Phillips and Ho and Sorensen among others compared with the

Monte Carlo metho d the p erformance of estimators in terms of eg bias in median

by the interquartile range and disp ersion as measured

The purp ose of this pap er is to study the p erformance in nite samples of

tests for parameter restrictions on cointegrating vectors The Monte Carlo metho d

is employed for these purp oses Testing hyp otheses suggested by economic theory

is a central concern of econometrics and testing hyp otheses ab out restrictions on

parameters in cointegrating vectors is no exception The goal is to apply tests that

have close to correct size and high p ower

Wald tests have b een prop osed for testing linear restrictions on coin tegrating

vectors for dierent though asymptotically equivalent estimation metho ds This

Monte Carlo analysis studies the eects of varying the estimation technique on cal

culating the Wald test The Wald test statistics are distributed as under the null

hyp othesis and reduce to a t statistic when only one cointegrating vector is present

and only a single parameter is involved The t statistic is then distributed asymptot

ically as normal

The asymptotically ecient estimation metho ds considered for the Wald or t

tests in this pap er are in alphab etical order of the chosen abbreviations Bewley et

als BoxTiao canonical variates based metho d BWLY Parks canon

ical cointegration regression metho d CCR Phillips and Loretan Saikko

nen and Sto ck and Watsons dynamic ordinary least squares metho d

DOLS Phillips and Hansens fully mo died ordinary least squares metho d

FM Johansens maximum likeliho o d metho d JOH and Phillips

band sp ectral regression metho ds PH The most p opular metho d in empir

ical applications seems to be JOH Less often used are CCR DOLS FM and PH

Other metho ds have been suggested in the literature BWLY has been prop osed

it may outp erform JOH point more recently and is included in this study b ecause

estimates in certain cases as demonstrated by Bewley et al The ab ove metho ds

are applied to several data generating pro cesses DGPs of practical relevance The

Wald or t statistic for a linear restriction on the cointegrating vector is computed

from the parameter and variance estimates of each metho d Then empirical sizes

and powers of these tests are calculated and compared The Monte Carlo metho d is

used in connection with a DGP that allows for endogenous weakly exogenous and

sense of Engle et al strongly exogenous regressors in the

In previous research Sto ckandWatson compared nite sample critical values

of the t statistic for parameter restrictions on cointegrating vectors of ve of the six

metho ds considered ab ove Their DGP revealed relatively mo dest size distortions

Further Li and Maddala suggested to use the moving blo ck b o otstrap to

correct size distortions for the t statistic for three of the ab ove six metho ds However

these studies did not rep ort results on test powers of the t tests On the other hand

Inder rep orted results for p owers of t tests for one of the ab ove metho ds FM

and other metho ds not considered in my pap er His preferred choice was atwostage

metho d combining an errorcorrection regression with the FM metho d

arious estimation metho ds used in the Monte Section briey outlines the v

Carlo study In Section the Monte Carlo design is explained and results are dis

cussed Section concludes

The Wald test in cointegrated systems

The BoxTiao Metho d of Bewley et al BWLY

Bossaerts and Bewley in several pap ers suggested a metho d for coin

tegrated systems of equations based on the levels canonical correlation analysis sug

gested byBox and Tiao This is in contrast to Johansens well known metho d

which relates levels to rst dierences and do es therefore incorp orate information

on the presence of unit ro ots into the estimation Bewley et al used the Monte

Carlo metho d to compare Johansens estimators to theirs and found for a bivari

ant cases less disp ersed ate rstorder mo del that their estimator is in several relev

and lepto curtic in small samples than Johansens Gonzalo derived for the bivariate

rst order mo del the asymptotic distribution of Bewleys BoxTiao estimator The

distribution is nonsymmetric and nonstandard Also it includes terms that lead

to nitesample bias in the median Despite these asymptotic problems hyp othesis

tests on cointegrating vectors in small samples with this metho d may outp erform

those with Johansens metho d parallel to the ndings of Bewley et al for prop erties

of the two estimators

Following Yang and Bewley consider a pdimensional vector autoregressive

representation of order k for the cointegrating relationship

Y Y v t T Y Y

t k tk t t t

with v distributed I IN Y is a p vector of variables integrated of order one

t t

denoted by I and is a vector of constants is the rst dierence op erator and

is a p p matrix It is assumed that r p Then is a full rank p r matrix

of errorcorrection vectors and is a full rank p r matrix of r cointegrating vectors

such that Y is integrated of order zero denoted I

See Bewley and Orden Bewley et al Bewley and Yang and Yang and

Bewley The last two pap ers describ e cointegration tests within this system

See Phillips and Sto ck and Watson for a theoretical and an empirical study

resp ectively for Johansens metho d Phillips results also apply to Bewleys BoxTiao estimator

The mo died BoxTiao pro cedure describ ed in Bewley and Orden uses the

least squares residuals g and h from regressing Y on Y and Y and

t t t t tk

from regressing Y on the same set of regressors resp ectively The sp ecication

considered in this pap er allows for a constant in the cointegrating vector only In other

words the constant is restricted Next

G g g

and

H h h

are formed and the eigenproblem

i h

H H H H G e G G G

i i

c c

and eigenvectors e ordered so that is solved for p pairs of eigenvalues

i i

In a mo del with r cointegrating vectors the estimator of is asso ciated

with the r smallest eigenvectors

b b

e e

Parallel to Johansen and Juselius and Johansen a Wald test

for linear restrictions is applicable The null hyp othesis for linear restrictions on the

cointegrating vectors is

where K is a p p s matrix The Wald teststatistic involves normalizing

by its standard deviation K

h i

b b

BWLY T trace fK D I K gK ee K

See Johansen on the role of the constant in equation

Yang has recently suggested mo dications that can b e applied to any systems estimator

of a cointegrated pro cess with variables integrated of order one Wald like tests not considered here

are suggested based on estimators mo died to achieve asymptotic eciency and asymptotic mixed

normality so that the test is asymptotically distributed

c c c c

with e the eigenvectors corresp onding to and D diag

r p r

The BWLY statistic is asymptotically distributed as with r p s degrees of

freedom In the case of r theWald statistic reduces to a statistic that is asymp

totically distributed as normal

c b

K e BWLYT K

The Bewley estimator involves cho osing the unknown autoregressive order k

Reimers compared various data based lag selection criteria in cointegrated

vector autoregressive systems using the Monte Carlo metho d and recommended the

Schwarz or HannanQuinn criterion These are consistent estimators of the lag order

whereas Akaikes criterion is not Therefore I will employ the Schwarz criterion

Parks canonical cointegration regression metho d CCR

Park derived a canonical cointegration regression estimator for

the cointegrating vector normalized in the following single equation cointegration

mo del

x u y

t t t

where u is I with mean zero Parks canonical regression pro cedure is based on the

idea that cointegrating vectors are not unique and transformations using stationary

comp onents of the mo del do not alter the cointegrating relation Nonparametric

data transformations are used to remove asymptotically the cross serial correlations

between the regression errors and the innovations of the regressors

It is assumed that x is one and x v i m so that p m

t it it

and with the v representing meanzero random errors Dene z u v

it t

t t

T T

X X

lim T E z z

T t

X X

lim T E z z

The matrices and are partitioned in conformitywithz

uu uv

vu vv

and

uu uv

vu vv

Next y and x are mo died in order to eliminate nuisance parameters

t t

b b

y y z

t t

b b

and

h i

b b

x x z

t t

where

b b b

vu vv

b b

z z T

and is the least squares estimate from equation The next step is to apply least

squares estimation to equation with y and x instead of y and x in order to

t t

get the asymptotically ecient estimators and the asso ciated variancecovariance

matrix

h i

b b b b

x x

uu vu

vu vv t t

The Wald statistic for H h with H h of full rank q the number

of restrictions is

h i

b b b b

h CCR h H C C H

where

uv uu uv vu

and is the longrun variance and its estimate C is the design matrix of the

CCR The statistic has a limiting distribution with q degrees of freedom When

only one parameter is involved and r this test reduces again to a t statistic with

an asymptotic normal distribution

The estimations of the longrun variancecovariance matrix and are car

ried out using nonparametric metho ds The metho d of Andrews is used

to calculate the teststatistic denoted by CCRA A quadratic sp ectral kernel with

the asso ciated automatic datadep endent plugin bandwidth estimator is employed

Also this kernel estimator is prewhitened with a rst order vector autoregression as

suggested by Andrews and Monahan Furthermore to provide a comparison

for the p erformance of Andrews estimators the Bartlett window with four lags is

used instead to calculate the variances and covariances denoted by CCRB

Phillips and Loretan Saikkonen and Sto ck and Watsons

dynamic ordinary least squares metho d DOLS

k and Watson Phillips and Loretan Saikkonen and Sto c

suggested the dynamic ordinary least squares DOLS metho d for estimating coin

tegrating vectors Sto ck and Watson compared dierent asymptotically ecient es

timators and recommended based on a limited Monte Carlo study for US money

demand the DOLS estimator If the variables are I and there are r cointegrating

vectors among the p variables then there are r least squares regressions Each re

gression has p r regressors in levels a constant contemp oraneous values leads

The DOLS estimator has a mixture and lags of the rst dierence of each regressor

normal distribution and the Wald statistic for restrictions on the parameters in the

cointegrating vectors is distributed as Again the test reduces to a t statistic with

a limiting normal distribution when r and only one parameter is involved I

use Schwarzs criterion in order to determine the appropriate lead and lag lengths for

the DOLS regressions I calculate for the Wald or t statistics the variances again

with the quadratic kernel estimator of Andrews denoted by DOLSA The Bartlett

metho d is used to o denoted by DOLSB

See Sto ck and Watson

Phillips and Hansens fully mo died regression metho d

The pro cedure of Phillips and Hansen is similar to Parks It is also a twostep

pro cedure and the asymptotic distributions of the two estimators are identical Parks

pro cedure is to correct both y and x b efore applying least squares In contrast

t t

Phillips and Hansen rst mo died y to get y and then corrected the least squares

estimates from the regression of y on x in order to eliminate nuisance parameters

leading to Phillips and Hansens metho d employs semiparametric corrections

that also lead to asymptotically medianunbiased estimates

Phillips and Hansens pro cedure applies least squares to equation to get

b b

Dene the residuals z u x

t t

vu vu vv

vv vu

Next the variancecovariance matrices are estimated again with Andrews pro cedure

The term represents the bias due to endogeneity of the regressors x The fully

mo died estimator of is given by

T T

X X

y x x x

t t vu t

t t

where

b b

y y x

t uv t

t vv

The Wald test is

h i

b b b b

h H x x H h FMA

t uv

The test statistic with the Bartlett window instead is denoted FMB The asymp

totic corrections of the least squares estimator and of are equivalent Both

estimators eliminate nuisance parameters asymptotically

Johansens maximum likeliho o d metho d JOH

Johansen derived the Wald test within the following vector autoregres

sive representation of order k

Y Y Y v

t t k tk t

The system is rewritten as an errorcorrection mo del as in equation

Y Y Y Y v

t t k tk tk t

where

I i k

i i

and

The rank of the matrix determines the number r of cointegrating vectors among

the variables in Y rank r p If r then and all variables

app ear only in rst dierences in the mo del and there are no cointegrating vectors

If r p then the matrix and Y is I

are calcu For this pro cedure the ordinary least squares residuals R and R

t kt

lated from regressions of Y on Y andY andofY on the same

t tk t

set of regressors resp ectively to purge the system of shortrun dynamics Reduced

rank regressions are then employed to estimate the cointegrating vectors The ma jor

dierence to Bewleys metho d is that it relates in the canonical correlation analysis

the levels of lagged Y instead of rst dierences to Y Bewleys metho d extracts

t t

rst the most nonstationary comp onents and then the stationary canonical variates

ariates Bewley whereas Johansens metho d extracts rst the stationary canonical v

et al argued that their metho d ensures small sample in addition to asymptotic

orthogonalitybetween the estimated stationary and most nonstationary variates

The cross correlation matrix of the residuals is given by

S T R R

ij it

c c

where i j k The eigenvalues are the solutions of

S S S S

k kk k

and represent the squared canonical correlations

For a given r the cointegrating vectors in are given by the eigenvectors asso

c c

ciated with the r largest eigenvalues and these are the reduced rank

estimators of It can be shown that this estimator is equivalent to the maximum

likeliho o d estimator when errors are Gaussian Johansen and Juselius and Jo

hansen suggested to use aWald test for the linear restrictions as describ ed in

Section For the JOH statistic e is replaced by the eigenvectors corresp onding to

c c c c

and D diag The JOH statistic is asymptotically

r p r

distributed as with r p s degrees of freedom In the case of r the Wald

statistic reduces again to a t statistic that is asymptotically distributed as normal

Phillips sp ectral regression metho d PH

Phillips prop osed to employ a blo ck triangular representation of the

the regression errors cointegrated system and to apply nonparametric metho ds to

from the system The advantage of this approach is that it is not necessary to be

explicit ab out the generating mechanism of the errors Phillips suggested to use so

called Hannanecient sp ectral regressions Because cointegration is concerned with

longrun relationships it is p ossible to fo cus on the most relevant frequency by using

band sp ectral regression at zero frequency In other words the regressors are I

pro cesses whose power is concentrated at the origin Full frequency band regression

is not needed for ecient estimation in large samples However it may be useful

leads to cointegration in small samples Furthermore the system sp ectral metho d

estimators that are asymptotically median unbiased and symmetrically distributed

and an optimal theory of inference applies Hyp othesis tests can b e carried out using

asymptotic tests Also full sp ectral estimation is asymptotically equivalent to

maximum likeliho o d

The blo ck triangular error correction representation is given by

Y Y

t t t

with Further

t t

and

with Y an I variable and Y avector of m variables each I so that p m

t t

and is I

t t t

The rst step is to apply least squares to equation to get the residuals

Y Y

t t t

Next nite Fourier transforms are calculated

T Y exp

Y exp T

t Y

T exp

for Y Y Y and partitioned con

t t Y

Next the smo othed formably with Y Also it holds that

Y t

p erio dogram estimates are computed The ecient weight function

s j

The Co oleyTukey Fast Fourier algorithm in GAUSSisused

Instead of the smo othed p erio dogram estimates other conventional sp ectral estimates could b e used

is used for all

B j

j s j

M M

for a frequency band with width so that M when

M T

b b

j s Y s s Y s

B j

j s s

B j

s s j

B j

The estimate is consistent so that

f f

as T The full sp ectrum estimator of is

M M

X X

b b b b

f f f f

j j j j

M M

j M j M

Nonlinear estimation is not necessary b ecause is known The sp ectral estimator at

the origin zero frequency is

i h h

b b b b

f f f f

The usual Wald statistic denoted PHzerois constructed for with the variance

dened by

h i

b b

V f f

The Monte Carlo Design and Results

The DGP used in my Monte Carlo study is similar to the one used by among

many others Gonzalo It is given for p by

y x

t t t

Results for the PH statistic calculated with the full sp ectrum estimators denoted PHfull are

not rep orted in all Tables b ecause in general PHzero p erforms b etter

a y a x

t t t

t t t t t

and

iidN

Gonzalo showed that this DGP can be expressed as a DGP with moving average

errors or alternatively as an errorcorrection mo del The DGP can be extended to

p The parameter space exp erimented with is

The pseudonormal variates u and e are generated by the RNDN function in GAUSS

t t

A sample of size T is generated and replications are used for every exp eri

ment Istart at u and e and discard the rst observations to mitigate

eects The parameter value choices for the DGP are motivated by cho os startup

ing realistic values so that the DGP should come close to actual pro cesses found in

nonarticial data

I used GAUSS COINT and own co de for all simulations When a

See Haug fo otnote for details on one p ossibility

See Haug for more details

x is weakly exogenous with resp ect to the parameter of interest and with a it

is endogenous It is strongly exogenous when a and

The inclusion of stationary autoregressive errors AR at a long lag twelve

lags in the ab ove DGP is motivated by a study by Rossana and Seater who

demonstrated this to b e a feature of many macro economic timeseries at a disaggre

gated level This case will be considered only in Table and will be set to zero

otherwise

the tests for a nominal level two Table rep orts the empirical sizes of

sided t test when For BWLY and JOH the distributions of the t statistic are

invarianttochanges in the value of the sign of and to whether a or a

Various values for T and are considered In general size distortions increase

as the sample size decreases except for BWLY where distortions change little For

T the t statistic calculated with Johansens metho d has overall the most stable

size with the least distortion across the various values of and when a It has

the least size distortion of all metho ds in seven out of the nine cases considered In

wo other cases it ranked second and third The size distortion of JOH ranges the t

from to for the nominal level test size This distortion is not trivial

however compared to the other tests which reach empirical sizes in the range

it has rather go o d size prop erties When a the preferred tests in terms of size

are CCRA and JOH followed by FMA Dierences are not very large

Table also gives results for T and T The relative rankings change

verall JOH is still the preferred test with the least somewhat for a However o

size distortions On the other hand the FMA test is preferred when a It leads

to much less size distortions in smaller samples than CCRA and JOH

In general BWLY do es not p erform b etter in Table than JOH Also CCR

DOLS and FM p erform in all cases considered b etter with Andrews metho d A than

with the Bartlett window B In summary the results for size distortions suggest to

They further showed that temp oral aggregation can distort this underlying pro cess and lead to

an integrated moving average pro cess instead

employ the JOH test when a and the FMA test when a Alternatively

the results suggest that test sizes should be corrected The literature has suggested

b o otstrap techniques for this purp ose Li and Maddala suggested to use the

moving blo ck b o otstrap and showed that it works well for t statistics in cointegrated

systems

Tables and rep ort powers of t tests The null hyp othesis is H

and the alternative hyp othesis is H Powers dep end crucially on the

and larger values would pro duce muchhigherpowers and the reverse holds value of

for lower values For the power studies the untrue null hyp othesis that is

tested when the data are generated under the alternative and the Tables rep ort the

rejection frequency for twosided t tests at the signicance level Sizeadjusted

or empirical powers are rep orted throughout the pap er These powers are based on

critical values calculated as quantiles under the null hyp othesis for every sample size

and DGP used instead of using asymptotic critical values

For a sample of the BWLY Table depicts p owers for when a

statistic pro duces in several cases the highest p owers However it also pro duces out

liers with the lowest p owers among all test The same holds true for samples of T

and T however the p erformance of the BWLY test deteriorates somewhat rel

ative to the other tests as T falls Contrary to Bewley et als ndings the earlier

mentioned asymptotic problems of Bewleys estimator come to b ear when testing hy

p otheses on cointegrating vectors When T CCRB leads overall to the highest

and most stable powers across the various values of and When T FMB

p erforms relatively well followed closely by PHzero DOLSB and CCRB When

of DOLSB FMB CCRB and PHzero is very similar T the p erformance

Overall the CCRB test is preferable in terms of p ower when a In general the

See Banerjee et al Chapter on testing for a

Algebraic derivations of Edgeworth expansions for size corrections seem to b e to o cumb ersome

here

See also Davidson and MacKinnon on sizes of b o otstrap tests in general

Table considers H

Bartlett window B outp erforms Andrews metho d A in the power studies

Table studies the powers of the t tests for when a The BWLY

test p erforms well in many cases but again pro duces outliers with the lowest powers

among all tests Regardless of sample size the PHzero test leads overall to the

highest and most stable powers among all tests and it is the preferred test when

for instead of Finally Table studies the behavior of the tests

three instead of two cointegrated variables AR errors and instead of

under the alternativehyp othesis To save space Table rep orts results only for one

value of and Before discussing the eect of changing I will discuss the other

cases Increasing p leads to more size distortion and lower powers in general but

leaves the relative rankings of the tests unchanged Similarly the intro duction of AR

DGP increases size distortions and lowers powers in general without errors into the

changing relative rankings As exp ected a lower value of under the alternative

hyp othesis leads to substantially lower powers

For and a the results for powers in Table dier from those in

the other Tables where The parameter measures the sp eed of adjustment

sp eed to the equilibrium cointegrating relationship A high value indicates a slower

of adjustment and vice versa Bewley et al rep orted exp erimental results for

Johansens estimator that showed that it p erforms well when sp eeds of adjustment

are high and that it pro duces outliers when the sp eed of adjustment is slow Table

conrms these results for the t tests JOH pro duces the highest and most stable

powers when takes on low values of or not rep orted however it p erforms

worse than other tests when takes on larger values of or not rep orted On

the other hand when a changes in do not aect the previous results

Phillips provided a theoretical analysis showing that the nite sample distribution is lepto curtic in the general case

Conclusion

This pap er used the Monte Carlo metho d to study the p erformance of tests

of linear restrictions on cointegrating vectors The t statistics were calculated for

several cointegration estimators and size distortions and test p owers were compared

In terms of size distortions Johansens t test is preferred However the size is often

double of its nominal value and b o otstrap or other techniques should be considered

to correct sizes as suggested by Li and Maddala Johansen prop osed

recently a Bartlett typ e correction factor for the likeliho o d ratio instead of the Wald

test in cointegrating systems

In terms of sizeadjusted test p owers the JOH test p erformance dep ends crit

pro duces rel ically on the sp eed of adjustment to the cointegration equilibrium and

atively lowpowers when the adjustment sp eed is slow Instead the CCRB test is in

general preferred when regressors are not weakly exogenous and the PHzero when

they are weakly or strongly exogenous These results suggest to explore size correc

tions for the CCR and PH based tests Xiao and Phillips a develop ed recently

asymptotic expansions for Wald tests They prop osed a mo died Wald test that uses

a bandwidth selection criterion to minimize second order eects and is mo died by

using consistent estimates of second order terms

Overall the Monte Carlo results indicate serious size distortions of Wald tests

Also powers in samples of size and below are very low for the Wald tests in

cointegrated systems The pap er shows that the problems of Wald tests found in

stationary cases are comp ounded in the cointegrating cases

When a the FMA test has somewhat b etter size in samples of and observations

than JOH

See also Xiao and Phillips b on the issue of using second order expansions and mean

squared error approximations for ecient frequency domain regression estimators

See for example Bera et al

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systems using the Johansen test An illustration using data based Monte Carlo

simulations Review of Economics and Statistics

Inder B Estimating longrun relationships in economics Journal of Econo

metrics

Inder B Finite sample arguments for appropriate estimation of cointegrating

vectors Monash University Department of Econometrics

Johansen S A small sample correction for tests of hyp otheses on cointegrating

vectors Europ ean University Institute

Johansen S Estimation and hyp othesis testing of cointegration vectors in

Gaussian vector autoregressive mo dels Econometrica

Johansen S Statistical analysis of cointegration vectors Journal of Economic

Dynamics and Control

Johansen S and Juselius K Maximum likeliho o d estimation and inference

on cointegration With applications to the demand for money Oxford Bulletin

of Economics and Statistics

Kitamura Y and P C B Phillips Ecient IV estimation in nonstationary

regression Econometric Theory

Li H and G S Maddala Small sample inference in cointegrated systems

using b o otstrap metho ds pap er presented at the Econometric So ciety Summer

Meetings in Queb ec City

Park J Y Canonical cointegrating regressions Econometrica

Phillips P C B Some exact distribution theory for maximum likeliho o d

estimators of cointegrating co ecients in errorcorrection mo dels Econometrica

regression for cointegrated time series in Non Phillips P C B Sp ectral

parametric and semiparametric metho ds in economics and statistics ed by

W Barnett Cambridge University Press Cambridge

Phillips P C B and Hansen B E Statistical inference in instrumental

variable regression with I pro cesses Review of Economic Studies

Phillips P C B and M Loretan Estimating long run economic equilibria

Review of Economic Studies

Reimers H Comparison of tests for multivariate cointegration Statistical

Pap ers

Rossana R J and Seater J J Temp oral aggregation and economic time

series Journal of Business and Economic Statistics

Saikkonen P Asymptotically ecient estimation of cointegrating vectors

Econometric Theory

Sto ck J H and Watson M W A simple estimator of cointegrating vectors

in higher order integrated systems Econometrica

Toda H Y Finite sample p erformance of likeliho o d ratio tests for cointe

grating ranks in vector autoregressions Econometric Theory

Xiao Z and P C B Phillips a Higher order approximations for Wald statis

tics in cointegrating regressions Working Pap er available from

httpkororaeconyaleeduphillipshtm

Xiao Z and P C B Phillips b Higher order approximations for frequency

domain time series regressions Journal of Econometrics

Yang M System estimators of cointegrating matrix in absence of normalising

information Journal of Econometrics

Yang M and Bewley R On coin tegration tests for VAR mo dels with drift

Economics Letters