A small sample correction for tests of hypotheses on cointegrating vectors

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E U R O P E A N U N IV E R S IT Y IN S T IT U T E , F L O R E N C E

ECONOM ICS D EPARTM EN T

WP

EUR

3 3 0

EUI Working Paper E C O No. 99/9

A Small Sample Correction

for Tests of Hypotheses on the Cointegrating Vectors

S0REN JOHANSEN

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All rights reserved.

N o part o f this paper may be reproduced in any form

without permission o f the author.

© 1999 Spren Johansen

Printed in Italy in March 1999

European University Institute

Badia Fiesolana

I - 50016 San D om enico (FI)

Italy

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A small sample correction for tests of

hypotheses on the cointegrating vectors

Soren Johansen,

Economics Department,

European University Institute, Florence

February 25, 1999

A b stract

A correction factor, depending on sample size and parameters, is found for the like­ lihood ratio test for some linear hypotheses on the cointegrating space in a vector au­ toregressive model, where the adjustment coefficients are known. The main idea is to condition on the common trends when making inference on the cointegrating coefficients in order to calculate the Bartlett correction factor. Some simulation experiments illus­ trate the findings.

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1

Introduction

W e consider the n —dimensional vector autoregressive model for cointegration fc-i

A X , = a(3'Xt-! + r , A X e_ , + <bd, +

•=i

where et arc i.i.d. N„(0, Q), with the purpose of finding a small sample correction for the likelihood ratio test for hypotheses on the cointegrating space spanned by p. It is known that the likelihood ratio test is x 2 asymptotically distributed despite the fact that the asymptotic- distribution of the estimator is mixed Gaussian; see Johansen (1988, 1996) and Aim and Reinsel (1990). M any simulation studies indicate that there can be considerable size distor­ tions when the x 2 tables are used for inference, see for instance, Fachin (1997), Jacobsen and Gredenhoff (1998), and Jacobson, Vredin, and Warne (1998). W e derive here a correction term to the likelihood ratio test statistic with the purpose o f improving the approximation to the asymptotic x 2 distribution. The correction is the so-called Bartlett correction, Bartlett (1937), and cm overview of this type o f correction can be found in Cribaro-Neto and Cordeiro (1996).

The actual distribution of the test statistic for p in the cointegrated vector autore­ gressive model depends on the sample size and on the many parameters of the model. Even though asymptotically this dependence vanishes, it is still important for finite T, and the simulations show that if the adjustment is slow the approximation can be very bad.

W e derive here a correction factor that depends on sample size and the parameters in the model, such that it can be decided analytically when the approximation to the asymptotic X2 distribution is good and when a correction will improve the approximation. W e face the usual problem with correction factors. If the factor is sufficiently close to one, we need not correct, and the asymptotic distribution is a good approximation, and if the factor is large, then the approximation by the asymptotic distribution is not very good, and the next order term may be needed. Hence the correction may not work. In between, there is an area where the correction factor is of moderate size and may be useful. Thus many simulations arc needed to assess the usefulness of such a correction factor.

T he problem to be solved is rather complicated and what follows is only a first attem pt of a solution of a special case.

This paper is based upon the following ideas and observations

1. Since inference on p is asymptotically independent of inference on a the calculations will be done in the model where this parameter is fixed and known.

2. Since 3 is asymptotically mixed Gaussian and the discussion of asymptotic inference involves a conditioning argument on the asymptotic common trends, we condition throughout on the common trends when making inference.

In order to illustrate the conditioning idea consider the simple bivariate regression

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model in triangular form

Yt = 0 X t + £k

A X t = e2t,

where et are i.i.d. Ni(0 , /î). In this situation it is well known that

0 - 0

e l , x ? '

has a mixed Gaussian distribution. T he likelihood ratio test is equivalent to a test on

T Q = { p - 0 ) * Y , X 1’

t

which is exactly x 2 (l)> however. T he reason for this is, that although the distribution o f 0 has heavy tails and the value of the estimator calculated will often be extreme, this phenomenon is always followed by a small value of the information Ylt=\ such that the normalized contribution to the test statistic is not extreme. T hat is, the extreme observations in 0 are not extreme if measured by their ’’ asymptotic conditional variance” X 2j . A wav o f avoiding the issue of mixed Gaussian distribution is to consider conditional inference, that is, we consider the process X t fixed and known.

Often, in the calculation of Bartlett corrections, one meets a factor of the form s~2 =

(T~1 Ylt= i A,2) “ 1, for some process X t. If X t is a stationary process such that s2 —> E (s2) =

it2, say, then we expand as follows

1 1 I s 2 I s 2

(T2 + (s 2 - <T2) ~ a2- ~ (T2 V 2 1 + <72 <T2 l ) 2 - . . .

In this way negative powers can be replaced by positive powers, which greatly facilitates the calculations. In case the process X t is a random walk, the limit T ~ 2 y ^ J X 2 is not a constant and the above expansion does not help, since a random term appears in all numerators. Obviously, conditioning on X t, will in this case fix all numerators and avoid the calculation of moments involving the random limit.

W e try in the following to see how far this idea can take us in the cointegrated vector autoregressive model with known a. W hen the parameter a is fixed, we can transform the model into a cointegrated regression model, see (2) and (3), of the type considered by Phillips (1991).

In Section 2 we show that the autoregressive model with known adjustment coefficients reduces to an ordinary regression model but with non-stationary regressors. W e derive likeli­ hood ratio tests for hypotheses on the cointegrating space and normalize the regressors using the parameters from the true data generating process in order to be able to calculate the Bartlett correction. Section 3 contains the main result which gives the correction term for a general regression model and in Section 4 the results are applied to tests in the autoregressive model. Section 5 contains some simulation results which are used to illustrate the findings. The proof o f the main result is given in an Appendix.

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2

The autoregressive model and the test statistic

W e show in this section that if a is known the model reduces to a regression model and we can formulate the test for a hypothesis on the cointegrating coefficients as a test on the coefficients to the non-stationary common stochastic trends. Since we condition on these the non-stationary regressors become fixed and deterministic, but they still satisfy some very special conditions, which are analogous to their stochastic properties when they are considered random.

W e give the results for tests of the form 3 = H<p, where H(n x s) is known and o(s x r) is a parameter to be estimated. W e treat first the case where s = r, corresponding to a known cointegrating space, and then show how the general situation can be treated the same way.

T he main result is the derivation of a general regression equation which contains all the above examples as special cases, and from which the likelihood ratio tests can easily be found. This implies that we can find the expansion of the likelihood ratio test which is needed for the calculation of the Bartlett correction.

2.1 The model equations

For notational reasons we focus here on the situation with two lags and mention when the results have to be modified for more lags. Consider the model

A X , = a d 'X i - 1 + r , A X , . , + W , + £ ,, t = 1 , . . . ,T. (1)

W e assume that e, are i.i.d. N„(0, f2), and for the derivation of the estimator and likelihood ratio test we assume that the initial values ,Y,o and X _ , are fixed. W e assume that the parameter a(n x r) is known and that 0(n x r ) , T i(n x n ), f !( n x n ), and 4>(n x d) are unknown and vary unrestrictedly. The deterministic terms dt may contain for instance a constant, a linear term or seasonal dummies. T he main property that we need here is that d,+1 is a linear function o f d ,, (dt+i = Mdt) a property that is satisfied in the above examples, and which implies that the processes 0 'X t and A X t have expectations that are linear functions o f dt.

W e exploit the fact that a is known to derive two equations from model (1). W e let O' = (a'a)~la‘ and find

a 'A X t = / J ' X , . . ] + 0 T i A X , _ , 4 0'4>d, 4 -O 's ,, (2)

a'± A X t = c*rj_T\AXt-i + oc'±$dt + ot'±£ti (3) and finally, with ui = O 'f i a j ^ a ^ f i a x ) - 1 , the conditional model of O 'A X , given a'x A X , and the past is dt A X t — utot^AXt 4- 01X t—\ 4* f i A X t —i 4- $ d , 4- £ ,, (4) where f j = (O' - u / a 'j J T i , 4> = (a ' è, = ( a ' - u / a 'x )e ,.

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Note that the errors are independent in (3) and (4). The errors a'± £t are the permanent shocks which will be kept fixed in the calculation of the Bartlett correction, and the errors ( o ' - aia'x )e't = ( a 'f i a ) - 1a 'f i ~ l e( are the transitory shocks which determine the random variation of the estimated 0 for given values o f the common trends. T he common trends in the process are £, and the analysis o f the distribution o f the test statistic will be conducted conditional on these in the following.

Since the parameter 0 enters the conditional model (4) only, we derive the tests based on that equation.

2.2

Test of a simple hypothesis on the cointegrating space in the

general model

W e consider in this subsection the test that the cointegrating space is given, that is, 0 =

Hip where 0 is (r x r ). It is of course very easy to derive the likelihood ratio test for a hypothesis on 0 in (4). W e derive here an expression for the test statistic that depends on the parameters o f the data generating process. This is convenient when we want to perform the very special calculations under a fixed probability measure specified by the parameters under the null hypothesis. These values we call the true values and denote them by (a °,0 ° = Hip0,r ° , $ ° , f l ° ) , and indicate by a superscript 0 any parameter derived from these.

W e assume that the process is 1(1), or equivalently that the roots o f the characteristic polynomial are either greater than one in absolute value or equal to 1, and for T ° = I„ — T®, we assume that has full rank or equivalently again that ( / 3 ° , r ° 'a ° ) has full rank; see Johansen (1996). W e define

<?° = Note that

( / „ - C°r°)0°± = 0,

such that ( / „ - C °r °)X t = ( / „ - Coro)0°0o'X t is a linear function of 0 °'X t, and C ° r ° X < is a linear function o f a ° 'r ° A 't . W e therefore apply the decomposition

x,_, = (/„ - c° r°)/

3

V % - i + CT0*,-,,

to decompose the process into a stationary and a non-stationary part. W e define new param­ eters k and 6 by

0

'x

t-1

=

0

'(in - c°r°)/

3

V % - i +

0

'c°r°x,-i

N ote that the (r x r) unknown parameters in a = 0'{In ~ C°T°)0° are the adjustment coefficients to the known cointegration vectors in /3°, and that the r x (n — r) parameters

S = / 3 X ( a O 'r y i . ) - 1 are the coefficient to the non-stationary variables. Thus the regression equation (4) is

a °'A X t = uia°[AXt + n (f'X t^ + 6aa[ T aX t. i + f x A X t- i + id , + et. (5)

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The hypothesis o f a fixed cointegrating space spanned by /3° is Ho : 6 = 0, and the likelihood ratio test is derived by a regression o f a 0' A X , on a ® T ° X , _ i corrected for the variables

a ° ' A X , , A a n d d,.

From (3) we find that in the regression (5) we can replace a ® 'A X , by a®'£, since we are correcting for A X , _ i and d, in the regression anyway, and by summation o f (3) we find

Q“' r ° X , _ 1 = - a ° ' r ° A X , _ , + a®' + <J>°d,) + a * { X 0 - T ? X - i ) . (6 ) t= l

This shows that in the regression (5) we can replace a ° ' r ° X , _ i by a ? _ , = a®' } (e, +

$°di) + a ® '(X o — T j X _ i ) since we are correcting for A X , _ i in the regression. Note that if d,

contains a constant we can drop the initial values in the definition o f a°_ j . In the calculations in Section 3 we keep a ° '£ fixed (and hence a®_j) and we define the normalized deterministic regressors

b, = (a®'fi°a® ) - * a f e „ (7)

and let at~\ be but orthogonalized to dt and 6t, and moreover normalized, such that

T T T

^ ^ — 0 , ^ ^ — 0 , ~ ^n~ r ■

t=i t=i <=i

Similarly we define the normalized errors as

U, = ( a 0'n 0 “ 1a 0) - * a 0' n ° - 1£ , = ( a ° 'n ® - l a ® r * ( a ' - u V j r , . (8 ) W e define the unconditional variance

From the Granger representation of X , , see Johansen (1996),

x, = c° £ > , + $>od,) + j r c®(£t_( +

(

9

)

1=1 1=0

we find with dt~i = M ~ ldt, that

E t f x , )

=

=

C ?* ° A /-)d «

Thus the mean E (X [0°, AX',) is a linear function of the deterministic terms. W e define the normalized process

y = r o - i ( f X t - E { f X t) \

' V A X , - E (A X t) )■ (10)

If there are more than two lags in the model we have to define the process Yt by stacking 0 ° ' X „ A X , , . . . , A X , _ * + 2 , which becomes of dimension m = r + (k — l) n .

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T he regression equation (5) can then be written in the form

Zt = £1 a t - i + £2^ 1-1 + ^3 b, + £ 4dt + £ 5Ut, ( 11)

( r ) ( n - r ) (m ) ( n - r ) (d) (r) with Zt = a 'A X t, and

Y, = £ ° - * ( X t'/3° - E(X[(P), A X ; - E{ A X i ) , A X [ _ k+2 - E ( A X ',.k+2))' (12) and suitable coefficients Ç j , . . . ,£ 5 . Note that centering Yt~\ to have mean zero introduces a correction to the coefficient of dt since the mean o f (3'Xt and A X t are linear functions o f dt.

T he test that the cointegrating space is sp(/?°) is the test that = 0. Below the variables are indicated their dimensions. The maximum likelihood estimator or regression estimator for satisfies

^ 5 ^ u a . y , b , d = ( £ 1 — £ 1 ) S a a .y ,b ,d i

where we have used the notation for the product moments of (C/t, Yt-\,at-\,bt*dt) :

u, \

_{( Ut \}

Yt-1 Yt- i SyU Syy SyQ Syf) Syd

O't- 1 at- i = Sau Say In-r 0 0

b, b, Sbu Sfry 0 Sf)b Sfjd

d, t _{\ d‘ ^} \ Sdu Sdy 0 Sdb Sdd )

Let Vt , Wt , and Ft be any o f these processes then we also need the conditional product moments

S\jxU.f — S%)W S v f S f f S f y j .

W ith this notation we find the likelihood ratio test o f the hypothesis that the cointegrating space is given by sp(/3°) = s p(H) can be tested as f , = 0 by the likelihood ratio test

-2\og LR(0 = = - T l o g l‘^uu.y.q,6.d| |5u„.j,,6,d|

(13)

2.3

Test for linear restrictions on the cointegrating space

W e consider in this section the likelihood ratio test for the hypothesis (3 = H<t>, for H(n x s) and 0 ( s x r ) ( r < s < n). W e let L (0 ,^ ,T i,Q ) denote the Gaussian likelihood and define the concentrated likelihood

L(0) = m ax I ( / 3 , $ , r i , n )

' r ,,n ,* ' ’ '

and use the multiplicative property of likelihood ratio test to see that for /3° = H<t>°,

m axg = w L (/?) L(0°) TQ3°) maxL((3) max L(j3) ' m ax/j= H* L(0)

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and hence

- 2 log LR(i3 = H<t>)= 2logLR(3 = A 3 = H o)-2\ og L R (i3 = i f ) . (14)

T he second term was found in (13) and we investigate here the test o f a simple hypoth­ esis under the assumption that 3 = H<p and find that ( 11) still holds with a small modification o f the dimensions.

Consider equation (2), in particular the term

/ ? % _ i = r ° X , _ j + k/ 3 ° % _ 1,

where k = <t>'H'(In - C °r°)3°(r x r ). If ( f = H<t>°, is the true value o f the parameter, then = (H±,H<t>°x ), such that

4>'H'c° r 0 * , - ! = tf, / r ( f f x , t f * i ) K i V ! . ) _ 1« i r 0* « - i = d>'d>Ox ( 0 (. - r) x ( „ - » ) , / - r ) ( a ' x r o/3ox ) - 1a'x r ° A :( - 1

= bAa'±T°Xt-\,

where 6 = <t> <t>°± of dimension r x (s - r) is a parameter and

A = (0(. _ r)x(n_ . ) , / . - r)(Q'x r o/3°x ) - 1

is a known (s — r) x (n — r) matrix, such that the number of common trends that enter the equation is only s — r not n — r as in the previous section. Thus the regression equation (5) still holds with Qx r ° X t_ ] replaced by ,4a'x r 0.X ( _ i. W e can, as before, replace a'x A X f with

ax e , and Qx r ° X , _ i by a x possibly with initial values, but the matrix A

reduces the number o f trends, so we define in this case a t_ i = ,4a'x J3! =i(£* + corrected for bt and dt and normalized such that a t - i a l - i = h -r - Thus equation (11) holds but with the dimension o f a reduced to s — r.

T he test for a simple hypothesis on /3, when we assume that 3 = H<t>, can be tested as = 0 and gives the statistic (13) only with a of dimension s — r, and (14) gives an expression for - 2 1 o g T r t ( /? = H<j>).

2.4

An expansion of the likelihood ratio test

In order to cover the two examples in a general equation, and in order to apply the results in the further analyses o f the Bartlett correction, we formulate the main result for the equation:

Zt = + £2^ 1-1 + £3 f>t + 4 4 d( + £ 5 ^»- (15) (n » ) (n „ ) (nv ) (nt ) ( n j) ( n „ )

This formulation covers the cases discussed in subsections 2.2 and 2.3 by suitable choices of

Zt, at, Yt, bt, dt, and t /t . W e assume that Zt is a linear function o f A X t, Yt is the stacked process ( /J 'X t - i , A X (_ i , . . . , A X (_jt+2) normalized to have mean zero and variance J„B, O i_i is a linear function o f a x £, possibly modified by deterministic terms and bt is a linear function of q'x £(. Finally dt+1 = Mdt.

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W e derive an expansion for the likelihood ratio test that = 0. W e find from (13)

- 2 1 0 6 1 ^ = 0 ) = - r i o g f e ^ Ml

= - 7 T o g | /n„ - T - l ( T -'S uu.y.b.d) - ' S ua.v,b.dS ^ y l>dSau y.b.d\

= —X lo g |/„u — T - 1<5| = t r { Q } + jjp tr {Q 2} ,

where the notation = indicates that we have kept terms o f the order T ~‘ , s = 0 , 1 , . . . , and

Q ~ { T S u u .y ,b ,d ) S u a .y ,b ,d S a a . y tl,,d S a u .y ,b ,d € O p ( 1 ) . ( 1 6 )

In order to simplify the expression for Q we note that since a (_ i is orthogonal to b, and dt, we get

where

Baa.y,b,d — Saa.b,d Say.b,dSyy'b,dSya-b,d

~ Baa ~ SaySyy\dSya = Inu ~ T 1 B (17)

B<xu.y,b,d — Bau.b,d ~ Say.b.dByy ^ dSyU.b,d

= Sau ~ SaySyy b dSyu.b,d = N — T 2 A, (18)

N = S au (19)

B = Sa y( T 1SyV'b,d) * By a > (2 0 )

A = Say( T - l S yy.b,d) - l ( T - t Sv u .b,d ) 1 (2 1 ) are Op( 1) and N is distributed as AfnaXn „ ( 0 , / „ o when we condition on a'± e.

W e find

r

& u u .y ,b ,d — 1 & u u .b ,d 1 ‘^ u y.6 ,d * -? y y .5 )£f*^ yu .6 ,d

_ rp — 1 C r p —\ q n - l o 7^ -1 c C1— 1 c rr — 1 C — 1 ‘-'■uu •* ‘Jud&dd ^du •Jub.d&bbd&bu.d 1 ^u;

= +

(T~1SUU

T~'(SudSM Sdu + Sub.dSbb.dSbu d

Suy.b.dS~^bdSy

+ T - ^ D 1 - T ~ 1D2.

y.b,dSyy.b.d^yut>-d

W e then find from (1 6 ), (17), and (18) that

= t r { ( /„ „ + T ~ i D i - T - 1D 2)~ 1( N - T - l A ) ' { I n' - T ~ 'B )~ l {N - T ~ iA )}

= tr{N 'N - T ~^(N 'A + A 'N + D\N'N)

+ T - l{D2N 'N + D lN 'N + D i(N 'A + A'N) + N 'B N + A'A )),

where we have kept terms o f order T ~ l . T he notation has been chosen such that the power o f T in front indicates the order o f the various terms.

Similarly we find t r { Q 2} £ tr{(N 'N )2}.

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The asymptotic properties in the usual analysis, where s is considered random, are given in terms of a Brownian motion W(u) :

[Tu\

r - i

^

w(u)-i = l

W e define the independent standard Brownian motions Bi(u) = (a '^ n a j. ) _ i a '± U ’ (u) and

B 2(u) =

The asymptotic distribution of - 2 log LR, for the hypothesis that the cointegrating space is spanned by /3, is given by

- 2 lo gLR = t r { Q } ^ t r { J ^ ( d B 2)F ‘ ^ FF'du^j F(dB2)‘ ), (23)

where F is constructed from B\ depending on the deterministic terms. If d, = 0. then F = B\.

and if for instance = p0 , then a'x £> + a'±ho(t ~ 1) *s corrected for the mean. The process is linear in one direction, a'± n0, and a random walk in n — r — 1 directions such that

Fi(u) = Bi(u) - f ‘ Bi(s)ds,i = 1 , . . . , n - r - 1, . . Fn_ r ( « ) = « * - $ .

For fixed value of F the distribution of /„ ’ F(dB2)' is Gaussian with mean zero and variance matrix fg FF'du 0 I„u such that the limit distribution o f Q for fixed F is \2 with degrees of freedom n un „. Since this distribution does not depend on the conditioning random variable, the marginal distribution is also x 2(nun 0 ).

The idea in the following is to take the consequence of this conditioning argument, and condition already from the beginning on the common trends a'± e,, which give rise to the Brownian motion B\ and F. T he paper by Rothenberg (1988) contains another conditioning idea, due to Cavanagh (1983), which in the present context would mean conditioning on the zero order term N 'N and work out the moments of the remaining terms in this conditional distribution. This will not be attempted here.

Bartlett (1937) proposed to improve the approximation to the limit distribution of the likelihood ratio test by adjusting the finite sample distribution to have the same mean as the limit distribution. This simple correction turns out in a number o f cases to have a good effect on the approximation, and we shall therefore try to calculate the expectation

E[-2\ogLR\a'x e\. The exact calculation is not possible but we can derive an expression for the first order approximation. It turns out that this approximation depends on the parameters and thus will have to be estimated in practice, which gives some extra uncertainty.

Thus we want to calculate

E[—2 log LR\a'x e] ± F [tr {Q }| a '± £ ] + ± E [ t r { Q 2}\a'± s}, (25) where

F [tr{Q }|Q 'l £ ]

= f r { E [ A r'A r | a ' i £ ]} -

T-^tr{E[(2A'N +

N'N)\a'±e}}

+ T ~ 1tr{E[{D2N'N + D2N'N

2DiN’A + N'BN

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and

£ [tr{Q 2}|a'±£] £

2.5

The fixed regressors

W e insert a few comments on the fixed regressors a t_ i and bt. defined in terms of a x and a x £ ,.

B y the normalization chosen above we have

T T T

Sad =

=

Sab ~

at —

Saa —

^at —

lat — l = ^ na

t

W h en we do not condition on a'± X^i=i £i we have the following relations

SMS ^ S db 6 O p (l),

(28)

T

y}[2 at - ia't-\-k — In., for all k, t=\

(29)

r l E w ; . ^ {

t

Finally we have that Ylt=i &tat converges weakly to a limit The limit depends on the deterministic terms, such that if dt = 0 , then

£ bta't ^ ( / „ , + j \ d B j ) B [ ) Q f ' BiB\duj = l ba,

whereas if <&dt = p0 then

2Ü 0 (nt_ 1 )x l) ' + j \ d B i ) F ) ^ F F ' d u ^ j

- i

(32)

(33)

with F given by (24).

W hen conditioning on the sequence a'± e we assume that such relations hold for the sequence we are fixing. T hat is, we shall replace ° ( - i n< -i-fc by Ylt=1 a t - i a ! _ i = In.,

X -1 btb't_ k by / „ _ r or 0 , and, for k > 0 , tzj by the m atrix Xbtt in the limit results. It turns out that although l ba enters some o f the intermediate results, the final result does not involve l ab, such that there is no need to calculate the expectation when it appears.

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T he papers by Kiviet and Phillips (1997a,1997b) deal with a regression model which resembles (15):

Vt = Apt-1 + l ' xt + f t .

and derive approximations for the bias of the least squares estimator of (A, 7 ). The calculations are very similar to those needed here, but the scope differs in a number of aspects. W e are analysing a multivariate situation rather than a univariate, and want an approximation to the likelihood ratio test statistic rather than the bias. W e are interested only in the first order term, which implies that many of the formulae look simpler, and involve smaller matrices, since the above relations for the regressors a t_ i and b, allow some simplifications o f the general terms.

3

The main results on the Bartlett correction

W e introduce here the coefficients that are used to express the main result in Proposition 1, which gives an approximation to the various terms entering the expression for the conditional expectation of the likelihood ratio test of the hypothesis = 0 in (15).

In the following we assume that the parameters ( a , /3, T ] , . . . , Tk-i , fl) are so cho­ sen that the process is / ( 1). Thus the non-stationary part of the process is started in the initial values, but the stationary part of the process, f3'Xt - E (3 'X t) and AXt — E (A X t),

are given their invariant distribution. W e do not use the notation with a superscript 0 since we need not distinguish between the parameter value and the true value in this section.

This stationarity o f 01 X t — E(fl'Xt) and A X t — E (A X t) of course holds when we do not condition on a'x s , and then Yt, see (12), has the representation

oo oo

Yt = J 2 C .e t- „ = + V ^ A -v ) , (34)

i/=0 i/=0

see (7) and (8 ) for suitable matrices see (47). W hen we condition on a'x £( , however, then Yt is no longer stationary, since its mean depends on t.

Note that the unconditional autocorrelation function is

OO

C o v (y t , y t+h) = 7 (h) = + VvVC +a). 7 (0 ) = v (35)

i/= 0 W e define the coefficients

oo oc

6 = = ( 36)

v=0 i/=0

such that long-run variance of the stationary process Yt is 99' + whereas the long-run variance o f Yt conditional on the common trends is 99'.

W e can now formulate the main result on the test that = 0 in the regression (15). T he result is given as a first order approximation of E[-2\o%LR\a'Le\ expressed in terms of the coefficients 9u,i>„, and 7(h) and the dimensions nu, na, ns, n ,n d) and n v, and M. First, however, we give a technical result which is the basis for the various applications later.

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P r o p o s it io n 1 In the regression equation (15) we have the results

tr{E [(./V 'A r)2|a'x s]} = nan u(na + nu + 1) (37)

t r { £ [ A r'N|a'x £]} = nan „ (38)

T iir {.E [A /V '| a 'x £]}

— n „ (n u + 1 )tr{99'} + nutr {lbal abi('ip} - " a (3 9 )

E Ü l o M 7 (" + 1 ) ^ } +

+ l)}tr{00'„}]

tr{E\A' A\a'Le\} —> non ut r { 0 0 '} + n „ i r { î tQI 0|,t/)'V’} (40)

tr{£[AT'BA^|Qx £]} —* n „ (n a + n u + l ) t r { 0 0 '} + nut r { I 6aXo(,v>V } (41)

ritr{£[Z?i7V 'A 7|Q 'x £]} - nanu(nu + 1) (42)

t r { £ [ ( D 2lV'lV|a'x £]} —* nanu(nd + n b + ny) (43)

t r { £ [ D l W'i4|a'x £]} 0 (44)

tr(E[D\ N'./V|a'x e]} —> nan u(n u + 1) (45)

where the parameters <?„, ipy, 9, if, and 7(h) are given in (34), (35), and (36). The definition of (N, A, B, D\, D2 ) are given in (19), (20), (21), and (22).

Note that the correction terms depend on the conditioning variable through the quan­ tity l ab, see (32) and (33). T he proof o f this result is given in the Appendix.

P r o p o s it io n 2 The first order approximation to the conditional expectation of the likelihood ratio test statistic for = 0 in (15) is given by

£ [ —21ogZ,Æ|a'x £] = nuna + + na + 1) + (nd + nb + nv)] + - l) u + 2 (c + cd)]. with v = tr{90'} 00 cd = i/= 0 00

c = y ~ ] (t r { 7 (i/ + 1 )99'y ) + tr{f(i/ + 1 )}tr{flfl[,}) v= 0

The proof follows from Proposition 1 by using (25 ), (26), and (27). Note that the correction term to the likelihood ratio test statistic does not depend on the conditioning random variables in the sense that Tab is no longer present. In the following we give more explicit formulae using the properties o f the underlying process.

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T he process Yt, see (10) and (34), satisfies the equation D i r , = P L *Y ,-i + B e t, where / 7r + /3'a B 'T , •■■ /3 T * - 2 /3 T t _ , \ / ^ \ P (n v x nv) = a r , • 0 / „ • f\ - 2 r*._ j 0 0 , B ( n y x n) = ^ 0 0 In 0 / 0 l o / W e find the representation

OO

Y, = Y , Z ~ ipV B et - ‘'

t/=0 oo

i/ = 0

and use it to calculate the covariance matrix £

oo oo

tiec(E) = u e c ( ^ P l,B n B 'P " / ) = ^ ( P ® P ) ,'t> e c (B n B ')

i/=0 i/=0

= ( / „ v ® / „ t - P ® P ) _ 1t 'e c ( B f l B ') . Comparing (47) with (34) we find

0„ = X r i P ‘' B a ( a ' r r 1a r * , W = E - i p ‘'B n a 1 (Q'± n Q ± ) “ i .

Note that by the definition o f E OO

X ^ ( M L + V ' X ) = V ar(y ,) = /n v. i/=0

3.1

Calculation of p = <r{00'}

W e first calculate the conditional long-run variance of Y, as given by 0 0 '. W e find

O C OO 9 = = 5 2 z ~ i P ,'B a (a 'n -1a ) - t = £ " * ( / „ , - P )~ lB a (a 'C l-'a )-l, i/—0 u= 0 and hence 0 0 ' = £ - * ( / „ „ - P ) - 1B Q ( a 'n - 1Q ) - 1Q,B ' ( / n> - P ' J - ' E - i . Since ( / „ „ - P ) ( / B„ , 0 ... 0)' = - B a

■

(46)

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we find

W e define the coefficient matrix

V =

W

~* = ( (a'n 0‘ a) 1 ° ) E_1.

tr{6'0} = tr{ ( 7" “ ) E " 1 ( 7" “ ) ( a 'n ^ a ) - 1} = t r f E J ^ a 'l T 'a r 1}.

(51)

Ew = Var(/3%| AX«, A X (_ i , . .. , A X ,- t+2).

3.2

Calculation of c and c,i

W e note that ^ +l = e ; ( E i p ' - + i E - i ) , < + „ +1 = v - ; ( s i p ,‘' + 1E - i ) , such that OO 7 ( " + l ) = + ' M V * ' + l ) T7=0 = + ^ , ) ( = * ^ ,,+ l = ' 4 ) TJ=0 = s i p ' ^ ' E - * , since by (35) OO T7= 0 W e then find the expression,

OO ^ t r { 7 (i/ + i ) e e ^ } i/= 0 OO = ^ t r { E i p ' * ' + 1E - = E - i ( / „ > - P ) - 1B a ( a ' r r 1a ) - 1a 'B 'P ''/ E - * } i/= 0

= t r { E - l (/„„ - P ) - 1B a (a ,n - , a ) - , Q 'B '(/„1( - P ') “ ‘ (/nv + P ') - 'P '}

= tr{V'{Iny + P ’ r ' P ' } = t r {P (/n> + P ) - l V),

(fn„ — P ) ~ 1Ba = —( / n„ , 0 , . .. ,0)',

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where we have used (50). A different expression can be found if the m atrix P can be di­ agonalized. If we let A = diag(p1, . . . ,p n>) and K = (t'i, . . . , r „ v) denote eigenvalues and eigenvectors o f the matrix P, then P = A A A "- 1 . Note that these can be found from the eigenvalues for which p ^ 1 and the corresponding eigenvectors of the companion matrix.

In terms of these we have

oc n„

I > { 7(*' +

= tHPVn, + P ) 1V] = Y . T T ( K ~ ' V K ) "

In terms o f the eigenvectors and eigenvalues we can write

oo

i/=0

B y the same method we find

oo

Y , t r { M ^ } t r { e e 'u} = tr{(M ® ( / „ , - P)V)(I„d ® - ( A / ® P) ) ~ ' }.

i/= 0

3.3

The main result on a simple hypothesis

W e now formulate the main result for test for a simple hypothesis on f , in the equation (15)

T h e o r e m 3 If the underlying process X t is an AR(k) model with deterministic term $dt, the correction factor to the likelihood ratio test statistic for a simple hypothesis on the cointegrating space in (2.13) is given by £ [—2 log LR\a\ e] nun 0 = 1 + fi[(n<i + nj, + n l>)-(- i ( n u + n„ + 1)] + y ^ [ ( r ia - l) v + 2 (c + cd)\. £ t r { P " + 1} t r{ r t f * , - F )V ) u=0 Y Y ^ ( l - P i ) ( K - l VK)u

i/=0 j,i=l

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Here the constants v, c, and cd are given by

v = tr{V } (52)

Cd = tr{(M ® ( / n„ - P )V ){I„d ® - (M ®

P ) ) - 1 }

f r { [ P ® ( / n>- P ) V ] ( ( / nv® / ni, - ( P ® P ) r 1]}

T h e o r e m 4 where the matrix V is given by (51), P by (46), and M is defined by dt+\ = Mdt.

4

Results for the autoregressive model

This section contains the consequences for the cointegrated vector autoregressive model for the various tests for linear hypotheses. W e consider the autoregressive model with dimension n, lag length k, cointegrating rank r. In the following we restrict the deterministic term to be powers, like dt = 1, dt = ( l , t ) or even dt = ( l , t , t 2), since in this case one can see that

tr{M'/ } = n,i, the number of deterministic terms. W e find, see (53), cd = nd YITLo ^{0^8 } =

ndv. T he results can be modified if for instance a seasonal dum m y is included, in which case t r { M l/} is not constant but oscillates.

4.1 Test for simple hypothesis on the cointegrating space

T he simple hypothesis on the cointegrating space is formulated as 0 = H<t>, where H(n x r)

is known and <j>(r x r) is unknown.

C o r o lla r y 5 In the AR(k) model with power deterministic term 4>dt the correction term to the likelihood ratio test statistic for a simple hypothesis on the cointegrating space can be expressed in terms of the constants c and v, see (54) and (52):

E |-21ogX,n|<»'i c|

r ( n - r )

= 1 + if[(nd + kn) + j ( n + 1)] + j^ [ (n - r + 2nd - l)t> + 2c].

P r o o f . T his follows from Theorem 3 with the values n „ = r, 7i(, = na = n - r, ny = r + (k — l ) n , and using the result that Cd = ndv. m

A s a particularly simple case which is convenient for the simulation we consider C o r o lla r y 6 For the autoregressive model in n dimensions with 1 lag and r — 1 and constant term we find the correction factor

£ [ —21ogX,P|a'± e] i 1 .3 ( n + l) a'(3[{2 + a'0)n + 4(1 + a'0)\

(n — 1) + r ‘ 2 p'SiPa'n-'a

If the constant is absent we have

E[-2\ogLR\a'± e] i 1 3n + 1 a'0[(2 + a'0)(n - 2) + 4(1 + a'/?)]

(n — 1) - 1 + T l 2 /3 'n (3 a , n - 1a

P r o o f . In the special case of an autoregressive model with 1 lag and r = 1 we find that P = 1 + (3'a and M — 1 or 0 , and that

a10(2 + a'13) na'0( l + a '0 )

v 0 ln 0ain - 1a ' c V f w n - w

(55)

T he result then follows from Corollary 5. ■

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4.2 Test for linear restrictions on

8

W e consider here hypotheses o f the form 0 = H4>, where H(n x s) is known and ®(s x r) is unknown.

Calculation o f the expectation o f the log-likelihood ratio statistic is simply performed by applying Proposition 2 twice. The expectation of the second term is given by Corollary 5 and the expectation o f the first can be found by replacing (n - r) by (s - r) as argued in subsection 2.3. W e then find

C o r o lla r y 7 The correction factor for the hypothesis 0 = H<t> in the vector autoregressive model with power determinstic terms is given by

= 1 + + kn) + j ( n + 1 + s - r)] + ^ [ ( n - 2r + s + 2nd - l ) t ’ + 2c]

5

A simulation experiment

W e simulate the distribution of the test statistic for an n -d im en sion a l autoregressive model with one lag and one cointegrating relation and constant term in order to investigate if the correction factor improves the finite sample approximation. W hen simulating the distribution of the test statistics in the model

we have to specify the value of the parameters

(a,0,p,Q),

For all n x n matrices L o f full rank, the transformation Y = L X leaves the statistic invariant and corresponds to a change o f the parameters into (La, L ~110, LQ.L' , Lp). W e can reduce the number of parameters in the simulation experiment, by first choosing L = such that c , = Qiet are i.i.d. iVn (0, / „ ) . W e can still transform by orthogonal transformations without changing the independence of the errors, and since there is no loss of generality in assuming that 0'0 = 1, we can rotate the coordinate system such that 0 = ( 1. 0 , . . . , 0 )' 6

Rn and finally we rotate the coordinates ( 2 , . . . ,n ) such that a = (rj.fe-] )' where e\ —

( 1 , 0 , . . . ,0 ) ' € Rn " 1. Finally we note that the values £ and — { give the same value of the test statistics and the correction. W e therefore choose both r) and £ non-positive. Finally by a further rotation we can reduce the number of parameters in /j to three. Thus only five parameters need to be chosen in the simulation experiment when generating the data under the null hypothesis.

A more formal way of making this transformation in model (56) is to choose vectors

t),

=0(0’Q0)-i,

E[ — 2 log LR\a\ c] r ( n - a )

AXt —

(

5

'

Xt

-1

+ + £f)

v2 = - ( f r 1 - 0(0'n 0)-'l0')a ( a ' f r ' a - a'0(0 'U0)~l0'a) * and t>3, . . . ,vn such that

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For the new variables Yt = v 'X ,, the constant term is fi = v'/j., and the errors are I , = v's,.

which are i.i.d.' . V „ ( 0 ,/n). W e get the equations

AT), =

r]Yit-i

+ Ai + 5u

£Yu-i + jj,

r, = (ffwrtpoctfw)*

In the simulations we construct the stationary process 0 'X , by starting it at 0, and discarding the first 200 observations. T he random walk a'x X t is started with initial value a'± X 0 = 0. Finally the process is constructed as

X , = a (p a )~ ll?X , + 0 ± (a'± 0 x )~ 1a'± Xt, t = l , . . . , T .

W e thus want to test — • • • = r n = 0 in the model

A X \ t —

+ ----

"

+

-

=

+

+ •

• •

+

+

+ £

tt

A X it

= M*+£it,i =

where

et

Nn(0,In).

Under the null hypothesis the system generates / ( l ) variables with one cointegrating vector if — 2 < r; < 0. If r; = 0 , f = 0 then we get an 7 (1) process with no cointegration and finally if = 0 and ( ^ 0 then the system will generate an 1(2) process.

The correction factor is given in (7) with r = l,fc = l , « ,, = ! , and

see (55), and is given by

77(2 + T7)

e + * ’

c

77(1

-I-

77

)

Note the special case of £ = 0, where the correction factor has a pole at 77 = 0 corre­ sponding to an extra unit root in the process. A t this boundary the limit distribution is no longer \2 and a different type of correction could be calculated under the local alternative that 77 —► 0, T —* 0 0 , and Tr) is fixed. Note also that if £ ^ 0 ,a n d 77 = 0 then there is no singularity in the expression even though the process is 1(2).

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