Directory UMM :Data Elmu:jurnal:E:Economics Letters:Vol67.Issue2.May2000:

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Non-causality in VAR-ECM models with purely exogeneous

long-run paths

*

Christophe Rault

ˆ ´

Department of Economics, University of Paris I, Pantheon-Sorbonne, 106-112 bd. de L’Hopital, 75647 Paris Cedex 13, France

Received 23 March 1999; accepted 20 September 1999

Abstract

We propose a canonical representation of the long run matrix, which can constitute a basis for non-causality testing. This representation requires to determine a sub-matrix rank, which can be done using a test procedure, whose properties are analysed with Monte-Carlo experiments.  2000 Elsevier Science S.A. All rights reserved.

Keywords: Cointegration; Non-causality; Structural hypothesis; Monte Carlo JEL classification: C15; C22

1. Introduction

As it is now well established, maximum likelihood procedures for the analysis of multivariate autoregressive models with a cointegrated structure proposed by Johansen (1988, 1991), produce consistent estimates of the dimension of the cointegrating space and of the space itself, permitting

2

valid hypotheses testing about long run parameters to be conducted using asymptoticx criteria (see,

for instance, Johansen, 1995; Johansen and Juselius, 1992).

Here we are concerned with the parametric Granger non-causality conditions proposed by Mosconi and Giannini (1992), who suggest we can achieve an efficiency gain by imposing the cointegrating constraints under both the null and alternative hypothesis, while testing for non-causality in cointegrated systems using a likelihood ratio test (LR). However, a close examination of their paper reveals that Giannini and Mosconi did not in fact clearly distinguish in their theorems (Theorems 1

*Tel.:133-1-5543-4213; fax: 133-1-5543-4202. E-mail address: chrault@hotmail.com (C. Rault)

0165-1765 / 00 / $ – see front matter  2000 Elsevier Science S.A. All rights reserved. P I I : S 0 1 6 5 - 1 7 6 5 ( 9 9 ) 0 0 2 6 7 - 0

1

and 2 ) the arbitrary part (namely the nullity of some parameters blocks) we can always achieve without any loss of generality, of the nullity of the parameters blocks resulting from the non-causality property. Thus, the aim of this paper is to propose a framework, based on a canonical representation

of the long run matrix usually denoted P, which can constitute a basis for Granger non-causality

2

testing. This hypothesis expresses itself as minimum conditions on this canonical representation, in which the nullity of some parameters blocks do not imply any loss of generality.

This paper is organised as follows: Section 2 introduces the canonical representation of the long run

matrixP, which requires to determine a specific rank, denoted r , namely the number of cointegrating₁

relations related to both ‘exogenous’ and ‘endogenous’ variables. Section 3 proposes a sequential test

procedure to determine this rank r , and Section 4 reports some Monte Carlo results. Finally,₁

concluding remarks are presented in Section 5.

2. Canonical decomposition of the _P matrix

Consider an n-dimensional VAR-ECM (p) process hX_tj, generated by

P21

DXt5

O

GiDXt2i1ab9Xt211´t, t51, . . . ,T (1)

i51

whereG_i,a,b are, respectively (n, n), (n, r), (n, r), 0,r,n matrices such thatP5ab9;´_t is an i.i.d

normal distributed vector of errors, with a zero mean and a positive definite covariance matrixS; and

p is a constant integer. To conform to Mosconi and Giannini (1992), we omit deterministic

P21 i

components. It is assumed in addition that (i) u(In2oi51 Giz )(12z)1ab9zu50 implies either

p

9

uzu.1 or z51, and that (ii) the matrixa_'(I_n2o_i51 G_i) b_' is invertible, where a_' andb_' are both full rank (n, n2r) matrices satisfyinga9a_'50 andb9b_'50. These two conditions ensure thathXtj,

and hb9X_tj, are respectively I(1) and I(0) and that the Granger theorem (1987) is satisfied.

Let us now consider the partition of X andt Gi, respectively, into

G G

Yt YY,i YZ,i g k

and , Y[R , Z[R ,

S D F

_{G G}

G

t t

Zt ZY,i ZZ,i

with g1k5n, where Y is a vector of g variables that we will call ‘endogenous’, and Z a vector_{t t}

containing the k remaining other variables that we will call ‘exogenous’. Then we can state the following result:

Theorem 1. Let P5ab9 be a (n, n) reduced rank matrix of rank r (0,r,n) and partition b into

bY

.

F G

b_Z

1

Note that Theorem 1 has also been used by Hunter (1992) for his concept of cointegrating exogeneity.

2

This canonical decomposition is not only related to non-causality testing, since various other hypotheses such as weak exogeneity, strong exogeneity for instance, can also be tested in this framework (Rault, 1998). Nevertheless as our main concern is non-causality, these other hypotheses are out of scope for this paper.

3

If we define r15rank (bY) with max(0, r2k)#r1#min( g, r) , then the a and b matrices can

always be reparametrised as follows:

.

Taking this new writing into account, the P matrix can now be written as:

9

9

9

a bY 1 Y 1 a bY 1 Z 11a bY 2 Z 2

]]] ]]]]]]

P5ab9 5

F

_{a b}

₉

UU

_{a b}

₉

_{1a b}

₉

G

.

Z 1 Y 1 Z 1 Z 1 Z 2 Z 2

The proof is given in Appendix A.

The theorem is proved using a basis change in the cointegrating space, which permits to separate the cointegrating vectors related to the exogenous variables alone, from those related to both

endogenous and exogenous variables: r₁ is thus defined as the rank of b_{Y 1}. To this partition

4

corresponds a newa partitioned intoa1Aa2. Unlike Giannini and Mosconi’s theorem 1 (1992), our

theorem implies no loss of generality, and only requires the determination of the rank r of the upper₁

block of the beta matrix, denoted bY. Furthermore, once this rank has been determined, Granger

non-causality hypothesis from Y to Z hG_i, i51, . . . p21 (short run non-causality) and a_{Z 1}50 (long

run non causality)j can easily be tested using conventional tests proposed by Johansen and Juselius

(1992), since long run non-causality expresses itself as minimum conditions on this canonical

5

representation. We propose in the next section a simple sequential test procedure to determine this specific rank r .₁

3. A sequential test procedure

The sequential procedure proposed in this section is based on an extensive use of a special class of test of structural hypotheses proposed by Johansen and Juselius (1992), expressed as linear restrictions

3

This condition ensures that theb matrix is of range r.

4

This topic has also been analysed in Rault (1998), who proposed a more general decomposition of thePmatrix which rests upon a second basis change on the adjustment space. This decomposition can give rise to I(2) behaviour of the endogenous variables Y in case of non-causality from Y to Z. Here we focus exclusively on the I(1) case.

5 2

Non-causality test statistics in VAR-ECM models usually require special conditions to be asymptoticallyx distributed: 9

this is because long run non-causality from Y to Zha b 5_{Z Y} 0jimplies non-linear constraints on long run parameters. What differs here from the usual case, is that given the canonical decomposition proposed in Theorem 1, long run non-causality

2

on the long run coefficientsb. The model is defined by the restrictions b5(H2 w,C), where H is a2

(N, s) known matrix and w and C are, respectively, (s, r ), (N, r ) matrices of parameters to be_{a b}

estimated, with ra#s#N, ra1rb5r. In other words we impose p2s restrictions on one set of ra

cointegrating vectors, the remaining r vectors varying freely. This test turns out to be asymptotically_b

2

x distributed and for detailed discussions the reader is referred to Johansen and Juselius (1992). As

our aim is to determine the rank r of₁ b_Y, the only restrictions considered here are zero restrictions.

More precisely, let us define m15min( g, r), m25max(0, r2k) and consider the following

sequences of null hypotheses:

0 ( g,k)

H :0,1 hthere exists a basis of cointegrating vectors such thatb 5(H2w,c), with H25

S D

and ra51, namely rank (bY)#m121j.

Ik



0

( g,k)

H :0,2 hthere exists a basis of cointegrating vectors such thatb 5(H2w,c), with H25

S D

and ra52, namely rank (bY)#m122j.

Ik



0

( g,k)

H :0, j hthere exists a basis of cointegrating vectors such thatb 5(H2w,c), with H25

S D

and ra5j, namely rank (bY)#m12jj,

Ik



with m2#rank(bY)#m .1

6

To test these different hypotheses, we adopt the following sequential test procedure:

2

Step 1: test H_0,1with the statisticc₁at thea₁level and refuse H_0,1ifc₁sx (v₁)⇒rank(b_Y)5m₁21 12a1

2

Step 2: test H with the statisticc at thea level and refuse H ifc sx (v ₎⇒_rank(b₎5_m 2₂

0,2 2 2 0,2 2 _12a2 2 Y 1

2

Step j: test H with the statisticc at thealevel and refuse H ifcsx (v)⇒rank(b )51,

0, j j j 0, j j 12aj j Y

5

2

else ifc_jax (v_j)⇒rank(b_Y)50. 12aj

Note that we only pass to step j if the null hypothesis H_{0, j21} has been accepted in step j21. Each

statistic is a likelihood ratio test:

ra rb r

˜ ˆ

ˆ

C_j5 22 ln Q(H /H )_{2 1} 5T

F

O

ln(12r_j)1

O

ln(12l_j)2

O

ln(12l_j)

G

(2)

i51 i51 i51 2

which is asymptotically distributed as x (v) with v 5(N2s2r )r degrees of freedom: H

j j b a 1

12a_j

˜ _ˆ ˆ

corresponds to the cointegrating hypothesis P5ab9, and lj, rj, lj denotes the eigenvalues of,

respectively, the unrestricted VAR-ECM, the r restricted long run relations and the r unrestricted_{a b}

relations.

It should be emphasised that as this statistic is derived under asymptotic arguments it seems to provide a bad approximation to the limit distribution in finite samples, since it tends to over-reject true nulls in small samples (see Psaradakis, 1994). Therefore, as one often encounters small size samples in empirical applications, we also consider the adjusted LR tests statistics, which is given by replacing

T by T2[(m /n)10.5(n2r(n2s) /(n11))] in Eq. (2), where m denotes the number of parameters to

6

Let’s point out that this procedure performs quite well in terms ofbYrank selection, even if somebYcolumns are a linear combination of others. This will be shown in Monte Carlo experiments.

be estimated in Eq. (1). This small sample correction performs better in terms of size distortion when testing for linear restrictions on a multivariate Gaussian model (see Anderson, 1984, Chapter 8).

4. Monte Carlo experiments

In order to analyse the properties of the sequential test procedure discussed in the previous section, a Monte Carlo experiment was conducted. Five data generation processes (DGPs) that share the

following characteristics were included in the experiments: we consider 11 dimensionalhXtj processes

( g55, k56), integrated of order 1, cointegrated of order 4 (four long run relationships exist),

expressed in VAR-ECM forms with no short run dynamics. Each DGP only differs from the other by

the rank r₁ of b_Y, which varies from 0 to 4. Table 1 presents the five models (data generation

processes) used in this study.

In all cases, 10 000 samples of size T11001p were generated with the hundred first observations

discarded, where p denotes the lag length in the estimated VAR-ECM. The vector of innovations ´t

was a Gaussian 11-dimensional white noise, with zero mean and covariance matrix I . The initial₁₁

values (t50) have been set to zero for all variables in the model, that is X050 , and X11 15´1|

N(0 , I ). All simulations were carried out on a 233 Pentium II, using the matrix programming_{11 11}

languageGAUSS, the´_t were generated by the function ‘RNDN’ and the nominal level of all tests was

5%.

For each DGP, five sample sizes were included; T550, 100, 200, 500, 1000, and the adjusted LR

tests statistics was used for T#100. In each replication, the lag length ‘p’ and the dimension of the

cointegrating rank ‘r’ were supposed to have been correctly determined previously, so that we can exclusively focus on the performance of our sequential test procedure. The tabulated results of the experiments are reported in Tables 2 and 3: Table 2 contains the estimated empirical size and power of the H_{0, j}null hypothesis tests ( j51, . . . ,4), while Table 3 presents the global sample empirical size of the sequential test procedure. The numbers in the body of Tables 2 and 3 are respectively the

percentage of rejections of the null hypotheses and the percentage of acceptance of the true rank r of₁

bY at the 5% level.

All H_{0, j} null hypothesis tests ( j51, . . . ,4) suffer from size distortion in small samples (T550,

100). As the sample size increases they approximate quite well the correct size. It must be emphasised that the asymptotic is reached all the later as the number of tested restrictions is important: for the

least restricted null hypothesis (H0,1), the empirical size is close to the nominal size of the tests for

samples of size larger or equal to 500 (5.03% for T5500), whereas for the most restricted null

hypothesis (H0,4) the empirical size is still 5.15% for samples of size 1000.

Furthermore, the percentage of null hypothesis rejection when they are not true goes to 100% for all sample sizes considered in the experiment, indicating both finite distance and asymptotic power equal to one.

As far as the sequential test procedure is concerned, the multiplicity of tests can lead to a global size problem. Our simulations effectively show this phenomenon which is however more patent for

smaller samples (T550, 100), since the sequential procedure estimated global size turns out to be

highly dependent on the number of tests necessary to conclude (respectively, 6.16%, 7.05%, 9.43%,

10.6% for r153, . . . ,0 and T5100). Thus as the sequential procedure suffers from size distortion,

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Table 1

a

Data generation processes (DGP) (n511, g55, k56)

DGP (1): r₁54 DGP (2): r₁53 DGP (3): r₁52 DGP (4): r₁51 DGP (5): r₁50

Beta Beta Beta beta beta

1.00 21.00 0.20 21.00 1.00 21.00 0.20 0.00 1.00 21.00 21.00 0.00 21.50 20.50 21.00 0.00 0.00 0.00 0.00 0.00

22.00 4.00 24.00 20.60 22.00 4.00 24.00 0.00 22.00 4.00 4.00 0.00 6.00 2.00 4.00 0.00 0.00 0.00 0.00 0.00

23.00 20.90 2.00 0.90 23.00 20.90 2.00 0.00 23.00 20.90 20.90 0.00 21.35 20.45 20.90 0.00 0.00 0.00 0.00 0.00

21.00 1.00 0.80 0.50 21.00 1.00 0.80 0.00 21.00 1.00 1.00 0.00 1.50 0.50 1.00 0.00 0.00 0.00 0.00 0.00

20.90 21.00 20.20 0.30 20.90 21.00 20.20 0.00 20.90 21.00 21.00 0.00 21.50 20.50 21.00 0.00 0.00 0.00 0.00 0.00 0.20 20.10 20.20 0.10 0.20 20.10 0.20 0.40 0.20 20.10 0.20 0.40 0.20 0.10 20.20 20.40 20.20 20.10 0.20 20.40

0.80 0.20 0.10 0.30 0.80 0.20 0.10 0.30 0.80 0.20 0.10 0.30 0.80 0.20 0.10 0.30 3.00 20.90 0.10 0.50

0.10 0.30 20.30 0.60 0.10 0.30 20.30 0.60 0.10 0.30 20.30 0.60 0.10 0.30 20.30 0.60 0.10 0.70 20.30 0.60

0.20 20.50 0.70 20.10 0.20 20.50 0.70 20.10 0.20 20.50 0.70 20.10 0.20 20.50 0.70 20.10 0.20 20.50 0.70 1.20

0.40 20.20 0.50 0.00 0.40 20.20 0.50 0.00 0.40 20.20 0.50 0.00 0.40 20.20 0.50 0.00 0.40 20.20 0.50 0.00

0.10 20.30 0.40 0.20 0.10 20.30 0.40 0.20 0.10 20.30 0.40 0.20 0.10 20.30 0.40 0.20 0.10 20.30 22.00 0.20

alpha alpha alpha alpha alpha

0.00 0.50 0.00 0.60 0.00 0.50 0.00 0.60 0.00 0.50 20.10 0.60 0.00 0.50 0.00 0.60 0.00 0.50 0.00 0.60

0.10 20.30 20.10 20.20 0.10 20.30 20.10 20.20 0.10 20.30 0.00 20.20 0.10 20.30 20.00 20.20 0.10 20.30 20.10 20.20

20.10 0.00 0.00 20.20 20.10 0.00 0.00 20.20 20.10 0.00 0.00 20.20 20.10 0.00 0.00 20.20 20.10 0.00 0.00 20.20 0.20 20.20 20.50 0.60 0.20 20.20 20.50 0.60 0.20 20.20 20.50 0.60 0.20 20.20 20.50 0.60 0.20 20.20 20.50 0.60

0.50 20.40 0.20 0.00 0.50 20.40 0.20 0.00 0.50 20.40 0.20 0.00 0.50 20.40 0.20 0.00 0.50 20.40 0.20 0.00

21.00 0.00 20.80 0.10 21.00 0.00 20.80 0.10 21.00 0.00 20.80 0.10 21.00 0.00 20.80 0.10 21.00 0.00 20.80 0.10

20.20 0.40 0.00 0.10 20.20 0.40 0.00 0.10 20.20 0.40 0.00 0.10 20.20 0.40 0.00 0.10 20.20 0.40 0.00 0.10

20.50 0.50 0.60 0.00 20.50 0.50 0.60 0.00 20.50 0.50 0.60 0.00 20.50 0.50 0.60 0.00 20.50 0.50 0.60 0.00

0.10 0.50 0.10 20.20 0.10 0.50 0.10 20.20 0.10 0.50 0.10 20.20 0.10 0.50 0.10 20.20 0.10 0.50 0.10 20.20

0.10 20.80 24.00 0.20 0.10 20.80 24.00 0.20 0.10 20.80 24.00 0.20 0.10 20.80 24.00 0.20 0.10 20.80 24.00 0.20

20.20 0.50 0.30 0.60 20.20 0.50 0.30 0.30 20.20 0.50 0.30 0.30 20.20 0.50 0.30 0.30 20.20 0.50 0.30 0.30

a

DGP (1), (2) and (5) can easily be seen to be respectively of rank r154, 1, 0. However the fact that DGP (3) and (4) are of rank r152 and 3 is less straightforward: it requires noticing that thebYcolumns of these two DGPs are not linearly independent since they are respectively linked by C25C , for DGP (3) and by C3 352C , C2 15C21C3 for DGP (4).

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Empirical size and power of the H0, jnull hypothesis tests ( j51, . . . ,4) (rejection per 100), with 10 000 replications at the 5% nominal level of significance

DGPS DGP (1): r₁54 DGP (2): r₁53 DGP (3): r₁52 DGP (4): r₁51 DGP (5): r₁50 Hypothesis tested

Sample size T 50 100 200 500 1000 50 100 200 500 1000 50 100 200 500 1000 50 100 200 500 1000 50 100 200 500 1000

b

a1W15C .1 A1 100 100 100 100 100 7.37 6.16 5.14 5.03 4.99 0.00 0.00 0.00 0.00 0.00 0.10 0.00 0.00 0.00 0.00 0.20 0.00 0.00 0.00 0.00 H0,1:hrang (bY)#3jagainsthrang (bY)54j a2W25C .2 A2 100 100 100 100 100 100 100 100 100 100 12.3 7.04 5.95 5.19 5.03 0.21 0.00 0.00 0.00 0.00 0.42 0.00 0.00 0.00 0.00 H0,2:hrang (bY)#2jagainsthrang (bY)$3j

a3W35C .3 A3 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 17.3 9.42 6.82 5.47 5.08 1.30 1.05 0.40 0.00 0.00 H0,3:hrang (bY)#1jagainsthrang (bY)$2j a4W45C .4 A4 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 19.5 9.68 6.96 5.62 5.15 H0,4:hrang (bY)50jagainsthrang (bY)$1j

a

The adjusted version of the test statistic was used for T550, 100.

b 2

A , ii 51, . . . ,4 denotes the critical value from thex distribution at the 5% level of significance.

Table 3

Empirical size of the sequential test procedure (rejection per 100), with 10 000 replications at the 5% nominal level of significance

DGPS DGP (2): r₁53 DGP (3): r₁52 DGP (4): r₁51 DGP (5): r₁50

] a ] ] ] ] ]

P (W )1 P (W W )1 2 P (W W W )1 2 3 P (W W W W )1 2 3 4

Sample size T 50 100 200 500 1000 50 100 200 500 1000 50 100 200 500 1000 50 100 200 500 1000

r estimated1 5r1 7.37 6.16 5.14 5.03 4.99 12.5 7.05 5.95 5.19 5.03 17.9 9.43 6.82 5.47 5.08 20.4 10.6 6.99 5.62 5.15

] ]

a

size of the procedure. On the contrary for larger samples (200, 500 or 1000), the estimated global size does not seem to vary a lot, indicating that the test procedure does not suffer from size distortion in

larger samples: for any possible r true rank, the estimated global size is always very close to 5%1

(respectively, 5.03%, 5.19%, 5.47%, 5.62% for r₁53, . . . ,0 and T5500). This result is due to the

fact that the H0, j null hypothesis tests ( j51, . . . ,4) are extremely powerful and never reject any null

hypothesis H_{0, j} when it’s true. Consequently if we need to perform up to j tests to determine the r₁

rank of bY, the global size of the sequential test procedure is simply given in larger samples by

a512(12a_j).

5. Conclusion

In this paper we have provided a framework based on a canonical representation of the long run matrix, which can constitute a basis for Granger non-causality testing, and that unlike Mosconi and Giannini (1992), permits to clearly distinguish the nullity of some parameter blocks we can always achieve without any loss of generality using basis changes, of the nullity of the parameter blocks resulting from the non-causality property. Furthermore, Granger non-causality can be tested using

2

asymptotic x tests.

This canonical representation requires to determine a specific rank r of a particular sub-matrix,₁

which can be done using a simple sequential test procedure, whose properties have been analysed in small and large samples with Monte Carlo experiments. The results can be summarised as follows: for

smaller samples (T550, 100), the sequential procedure estimated global size turns out to be highly

dependent on the number of tests necessary to conclude, whereas for larger samples (T5200, 500,

1000) it does not suffer from size distortion. This permits easy control of the global empirical size of this procedure in large samples.

Acknowledgements

I am grateful to Jacqueline Pradel for her suggestions. I also wish to thank Søren Johansen for helpful comments and discussions on an earlier version of this paper. All remaining errors and insufficiencies are my own.

Appendix A. Proof of Theorem 1

The theorem is proved as follows: first note that if we make a basis change in the cointegrating

space such as b*5bP, where P is a non singular (r, r) matrix, then the a* matrix is completely

determined by:

ab9 5a*b9*⇔(a*P9 2a)b9 50

⇔a*P9 2a50, becauseb9is of full rank column r

21

n

Next consider a b basis of the cointegrating space of rank r, EC5hv[R : a b 1 . . . 1a b 5

1 1 r r

0

n

v.a [Rj, and let U be a vectorial subspace of R spanned by the set of vectors of the form

F G

. U

i k I k

k

intersection with the cointegrating space is also a vectorial subspace. This implies that a EC basis can

be determined in completing a b₂ basis of EC>U with_k b₁ vectors. We define r as₁ b₁ rank and

r2r as1 b2 rank. Moreover sinceb1 is also a supplementary space basis, it follows thatbY 1 is of full

rank column r : in other words, if a linear combination of the₁ b_{Y 1} columns that produces a column of

zeros existed, this would mean that a vector of U spanned by thek b1 vectors would exist, which has

been excluded by construction.

It is easily shown that the transformation matrix from the b basis to the b*5bP basis can be

21

written as and then one deduces a*5a(P9) .

Finally the expressions of the reparametrised matrices b and a are given by

This completes the proof of Theorem 1.

References

Anderson, T.W., 1984. In: 2nd Edition, An Introduction to Multivariate Statistical Analysis, John Wiley, New York. Hunter, J., 1992. Tests of cointegrating exogeneity for ppp and uncovered interest rate parity in the United Kingdom. Journal

of Policy Modeling 4 (14), 453–463.

Johansen, S., 1988. Statistical analysis of cointegrated vectors. Journal of Economic Dynamics and Control 12, 231–534. Johansen, S., 1991. Estimation and hypothesis testing of co-integration vectors in Gaussian vectors autoregressive models.

Econometrica 6, 1551–1580.

Johansen, S., 1995. In: Likelihood-based Inference in Co-integrated Vector Autoregressive Models, Oxford University Press, p. 267.

Johansen, S., Juselius, K., 1992. Testing structural hypotheses in a multivariate cointegration analysis of the PPP and UIP for UK. Journal of Econometrics 53, 211–244.

Mosconi, R., Giannini, X., 1992. Non-causality in cointegrated systems: representation estimation and testing. Oxford Bulletin of Economics and Statistics 54 (3), 399–417.

Psaradakis, Z., 1994. A comparison of tests of linear hypotheses in cointegrated vector autoregressive models. Economic Letters 45, 137–144.

´ ´ ´ ` `

Rault, C., 1998. L’exogeneite dans les modeles VAR-ECM avec des sentiers de long terme purement exogenes. In: Cahiers

´ ´

124 C. Rault / Economics Letters 67 (2000) 121 –129

on the long run coefficientsb. The model is defined by the restrictions b5(H2 w,C), where H is a2 (N, s) known matrix and w and C are, respectively, (s, r ), (N, r ) matrices of parameters to be_{a b} estimated, with ra#s#N, ra1rb5r. In other words we impose p2s restrictions on one set of ra

cointegrating vectors, the remaining r vectors varying freely. This test turns out to be asymptotically_b 2

x distributed and for detailed discussions the reader is referred to Johansen and Juselius (1992). As our aim is to determine the rank r of₁ b_Y, the only restrictions considered here are zero restrictions. More precisely, let us define m15min( g, r), m25max(0, r2k) and consider the following

sequences of null hypotheses:

0 ( g,k)

H :0,1 hthere exists a basis of cointegrating vectors such thatb 5(H2w,c), with H25

S D

and ra51, namely rank (bY)#m121j.

Ik



0

( g,k)

H :0,2 hthere exists a basis of cointegrating vectors such thatb 5(H2w,c), with H25

S D

and ra52, namely rank (bY)#m122j.

Ik



0

( g,k)

H :0, j hthere exists a basis of cointegrating vectors such thatb 5(H2w,c), with H25

S D

and ra5j, namely rank (bY)#m12jj, Ik



with m2#rank(bY)#m .1

6 To test these different hypotheses, we adopt the following sequential test procedure:

2

Step 1: test H_0,1with the statisticc₁at thea₁level and refuse H_0,1ifc₁sx (v₁)⇒rank(b_Y)5m₁21

12a1 2

Step 2: test H with the statisticc at thea level and refuse H ifc sx (v ₎⇒_rank(b₎5_m 2₂

0,2 2 2 0,2 2 12a2 2 Y 1

2

Step j: test H with the statisticc at thealevel and refuse H ifcsx (v)⇒rank(b )51,

0, j j j 0, j j 12aj j Y

5

2

else ifc_jax (v_j)⇒rank(b_Y)50.

12aj

Note that we only pass to step j if the null hypothesis H_{0, j21} has been accepted in step j21. Each statistic is a likelihood ratio test:

ra rb r

˜ ˆ

ˆ

C_j5 22 ln Q(H /H )_{2 1} 5T

F

O

ln(12r_j)1

O

ln(12l_j)2

O

ln(12l_j)

G

(2)

i51 i51 i51

2

which is asymptotically distributed as x (v) with v 5(N2s2r )r degrees of freedom: H

j j b a 1

12a_j

˜ _ˆ ˆ

corresponds to the cointegrating hypothesis P5ab9, and lj, rj, lj denotes the eigenvalues of, respectively, the unrestricted VAR-ECM, the r restricted long run relations and the r unrestricted_{a b} relations.

It should be emphasised that as this statistic is derived under asymptotic arguments it seems to provide a bad approximation to the limit distribution in finite samples, since it tends to over-reject true nulls in small samples (see Psaradakis, 1994). Therefore, as one often encounters small size samples in empirical applications, we also consider the adjusted LR tests statistics, which is given by replacing

T by T2[(m /n)10.5(n2r(n2s) /(n11))] in Eq. (2), where m denotes the number of parameters to 6

Let’s point out that this procedure performs quite well in terms ofbYrank selection, even if somebYcolumns are a linear combination of others. This will be shown in Monte Carlo experiments.

4. Monte Carlo experiments

In order to analyse the properties of the sequential test procedure discussed in the previous section, a Monte Carlo experiment was conducted. Five data generation processes (DGPs) that share the following characteristics were included in the experiments: we consider 11 dimensionalhXtj processes ( g55, k56), integrated of order 1, cointegrated of order 4 (four long run relationships exist), expressed in VAR-ECM forms with no short run dynamics. Each DGP only differs from the other by the rank r₁ of b_Y, which varies from 0 to 4. Table 1 presents the five models (data generation processes) used in this study.

In all cases, 10 000 samples of size T11001p were generated with the hundred first observations

discarded, where p denotes the lag length in the estimated VAR-ECM. The vector of innovations ´t

was a Gaussian 11-dimensional white noise, with zero mean and covariance matrix I . The initial₁₁ values (t50) have been set to zero for all variables in the model, that is X050 , and X11 15´1|

N(0 , I ). All simulations were carried out on a 233 Pentium II, using the matrix programming_{11 11}

languageGAUSS, the´_t were generated by the function ‘RNDN’ and the nominal level of all tests was 5%.

For each DGP, five sample sizes were included; T550, 100, 200, 500, 1000, and the adjusted LR tests statistics was used for T#100. In each replication, the lag length ‘p’ and the dimension of the cointegrating rank ‘r’ were supposed to have been correctly determined previously, so that we can exclusively focus on the performance of our sequential test procedure. The tabulated results of the experiments are reported in Tables 2 and 3: Table 2 contains the estimated empirical size and power of the H_{0, j}null hypothesis tests ( j51, . . . ,4), while Table 3 presents the global sample empirical size of the sequential test procedure. The numbers in the body of Tables 2 and 3 are respectively the percentage of rejections of the null hypotheses and the percentage of acceptance of the true rank r of₁ bY at the 5% level.

All H_{0, j} null hypothesis tests ( j51, . . . ,4) suffer from size distortion in small samples (T550, 100). As the sample size increases they approximate quite well the correct size. It must be emphasised that the asymptotic is reached all the later as the number of tested restrictions is important: for the least restricted null hypothesis (H0,1), the empirical size is close to the nominal size of the tests for samples of size larger or equal to 500 (5.03% for T5500), whereas for the most restricted null hypothesis (H0,4) the empirical size is still 5.15% for samples of size 1000.

Furthermore, the percentage of null hypothesis rejection when they are not true goes to 100% for all sample sizes considered in the experiment, indicating both finite distance and asymptotic power equal to one.

As far as the sequential test procedure is concerned, the multiplicity of tests can lead to a global size problem. Our simulations effectively show this phenomenon which is however more patent for smaller samples (T550, 100), since the sequential procedure estimated global size turns out to be highly dependent on the number of tests necessary to conclude (respectively, 6.16%, 7.05%, 9.43%, 10.6% for r153, . . . ,0 and T5100). Thus as the sequential procedure suffers from size distortion, we have to control the partial nominal levels of the different tests in a heuristic way to get a 5% global

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Table 1

a

Data generation processes (DGP) (n511, g55, k56)

DGP (1): r₁54 DGP (2): r₁53 DGP (3): r₁52 DGP (4): r₁51 DGP (5): r₁50

Beta Beta Beta beta beta

1.00 21.00 0.20 21.00 1.00 21.00 0.20 0.00 1.00 21.00 21.00 0.00 21.50 20.50 21.00 0.00 0.00 0.00 0.00 0.00

22.00 4.00 24.00 20.60 22.00 4.00 24.00 0.00 22.00 4.00 4.00 0.00 6.00 2.00 4.00 0.00 0.00 0.00 0.00 0.00

23.00 20.90 2.00 0.90 23.00 20.90 2.00 0.00 23.00 20.90 20.90 0.00 21.35 20.45 20.90 0.00 0.00 0.00 0.00 0.00

21.00 1.00 0.80 0.50 21.00 1.00 0.80 0.00 21.00 1.00 1.00 0.00 1.50 0.50 1.00 0.00 0.00 0.00 0.00 0.00

20.90 21.00 20.20 0.30 20.90 21.00 20.20 0.00 20.90 21.00 21.00 0.00 21.50 20.50 21.00 0.00 0.00 0.00 0.00 0.00 0.20 20.10 20.20 0.10 0.20 20.10 0.20 0.40 0.20 20.10 0.20 0.40 0.20 0.10 20.20 20.40 20.20 20.10 0.20 20.40 0.80 0.20 0.10 0.30 0.80 0.20 0.10 0.30 0.80 0.20 0.10 0.30 0.80 0.20 0.10 0.30 3.00 20.90 0.10 0.50 0.10 0.30 20.30 0.60 0.10 0.30 20.30 0.60 0.10 0.30 20.30 0.60 0.10 0.30 20.30 0.60 0.10 0.70 20.30 0.60 0.20 20.50 0.70 20.10 0.20 20.50 0.70 20.10 0.20 20.50 0.70 20.10 0.20 20.50 0.70 20.10 0.20 20.50 0.70 1.20 0.40 20.20 0.50 0.00 0.40 20.20 0.50 0.00 0.40 20.20 0.50 0.00 0.40 20.20 0.50 0.00 0.40 20.20 0.50 0.00 0.10 20.30 0.40 0.20 0.10 20.30 0.40 0.20 0.10 20.30 0.40 0.20 0.10 20.30 0.40 0.20 0.10 20.30 22.00 0.20 alpha alpha alpha alpha alpha

0.00 0.50 0.00 0.60 0.00 0.50 0.00 0.60 0.00 0.50 20.10 0.60 0.00 0.50 0.00 0.60 0.00 0.50 0.00 0.60 0.10 20.30 20.10 20.20 0.10 20.30 20.10 20.20 0.10 20.30 0.00 20.20 0.10 20.30 20.00 20.20 0.10 20.30 20.10 20.20

20.10 0.00 0.00 20.20 20.10 0.00 0.00 20.20 20.10 0.00 0.00 20.20 20.10 0.00 0.00 20.20 20.10 0.00 0.00 20.20 0.20 20.20 20.50 0.60 0.20 20.20 20.50 0.60 0.20 20.20 20.50 0.60 0.20 20.20 20.50 0.60 0.20 20.20 20.50 0.60 0.50 20.40 0.20 0.00 0.50 20.40 0.20 0.00 0.50 20.40 0.20 0.00 0.50 20.40 0.20 0.00 0.50 20.40 0.20 0.00

21.00 0.00 20.80 0.10 21.00 0.00 20.80 0.10 21.00 0.00 20.80 0.10 21.00 0.00 20.80 0.10 21.00 0.00 20.80 0.10

20.20 0.40 0.00 0.10 20.20 0.40 0.00 0.10 20.20 0.40 0.00 0.10 20.20 0.40 0.00 0.10 20.20 0.40 0.00 0.10

20.50 0.50 0.60 0.00 20.50 0.50 0.60 0.00 20.50 0.50 0.60 0.00 20.50 0.50 0.60 0.00 20.50 0.50 0.60 0.00

0.10 0.50 0.10 20.20 0.10 0.50 0.10 20.20 0.10 0.50 0.10 20.20 0.10 0.50 0.10 20.20 0.10 0.50 0.10 20.20 0.10 20.80 24.00 0.20 0.10 20.80 24.00 0.20 0.10 20.80 24.00 0.20 0.10 20.80 24.00 0.20 0.10 20.80 24.00 0.20

20.20 0.50 0.30 0.60 20.20 0.50 0.30 0.30 20.20 0.50 0.30 0.30 20.20 0.50 0.30 0.30 20.20 0.50 0.30 0.30

a

DGP (1), (2) and (5) can easily be seen to be respectively of rank r154, 1, 0. However the fact that DGP (3) and (4) are of rank r152 and 3 is less straightforward: it requires noticing that thebYcolumns of these two DGPs are not linearly independent since they are respectively linked by C25C , for DGP (3) and by C3 352C , C2 15C21C3

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Sample size T 50 100 200 500 1000 50 100 200 500 1000 50 100 200 500 1000 50 100 200 500 1000 50 100 200 500 1000

b

a1W15C .1 A1 100 100 100 100 100 7.37 6.16 5.14 5.03 4.99 0.00 0.00 0.00 0.00 0.00 0.10 0.00 0.00 0.00 0.00 0.20 0.00 0.00 0.00 0.00 H0,1:hrang (bY)#3jagainsthrang (bY)54j

a2W25C .2 A2 100 100 100 100 100 100 100 100 100 100 12.3 7.04 5.95 5.19 5.03 0.21 0.00 0.00 0.00 0.00 0.42 0.00 0.00 0.00 0.00 H0,2:hrang (bY)#2jagainsthrang (bY)$3j

a3W35C .3 A3 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 17.3 9.42 6.82 5.47 5.08 1.30 1.05 0.40 0.00 0.00 H0,3:hrang (bY)#1jagainsthrang (bY)$2j

a4W45C .4 A4 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 19.5 9.68 6.96 5.62 5.15 H0,4:hrang (bY)50jagainsthrang (bY)$1j

a

The adjusted version of the test statistic was used for T550, 100.

b 2

A , ii 51, . . . ,4 denotes the critical value from thex distribution at the 5% level of significance.

Table 3

Empirical size of the sequential test procedure (rejection per 100), with 10 000 replications at the 5% nominal level of significance

DGPS DGP (2): r₁53 DGP (3): r₁52 DGP (4): r₁51 DGP (5): r₁50

] a ] ] ] ] ]

P (W )1 P (W W )1 2 P (W W W )1 2 3 P (W W W W )1 2 3 4

Sample size T 50 100 200 500 1000 50 100 200 500 1000 50 100 200 500 1000 50 100 200 500 1000

r estimated1 5r1 7.37 6.16 5.14 5.03 4.99 12.5 7.05 5.95 5.19 5.03 17.9 9.43 6.82 5.47 5.08 20.4 10.6 6.99 5.62 5.15

] ]

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size of the procedure. On the contrary for larger samples (200, 500 or 1000), the estimated global size does not seem to vary a lot, indicating that the test procedure does not suffer from size distortion in larger samples: for any possible r true rank, the estimated global size is always very close to 5%1 (respectively, 5.03%, 5.19%, 5.47%, 5.62% for r₁53, . . . ,0 and T5500). This result is due to the fact that the H0, j null hypothesis tests ( j51, . . . ,4) are extremely powerful and never reject any null hypothesis H_{0, j} when it’s true. Consequently if we need to perform up to j tests to determine the r₁ rank of bY, the global size of the sequential test procedure is simply given in larger samples by a512(12a_j).

5. Conclusion

In this paper we have provided a framework based on a canonical representation of the long run matrix, which can constitute a basis for Granger non-causality testing, and that unlike Mosconi and Giannini (1992), permits to clearly distinguish the nullity of some parameter blocks we can always achieve without any loss of generality using basis changes, of the nullity of the parameter blocks resulting from the non-causality property. Furthermore, Granger non-causality can be tested using

2 asymptotic x tests.

This canonical representation requires to determine a specific rank r of a particular sub-matrix,₁ which can be done using a simple sequential test procedure, whose properties have been analysed in small and large samples with Monte Carlo experiments. The results can be summarised as follows: for smaller samples (T550, 100), the sequential procedure estimated global size turns out to be highly dependent on the number of tests necessary to conclude, whereas for larger samples (T5200, 500, 1000) it does not suffer from size distortion. This permits easy control of the global empirical size of this procedure in large samples.

Acknowledgements

I am grateful to Jacqueline Pradel for her suggestions. I also wish to thank Søren Johansen for helpful comments and discussions on an earlier version of this paper. All remaining errors and insufficiencies are my own.

Appendix A. Proof of Theorem 1

The theorem is proved as follows: first note that if we make a basis change in the cointegrating space such as b*5bP, where P is a non singular (r, r) matrix, then the a* matrix is completely determined by:

ab9 5a*b9*⇔(a*P9 2a)b9 50

⇔a*P9 2a50, becauseb9is of full rank column r

21

v.a [Rj, and let U be a vectorial subspace of R spanned by the set of vectors of the form

F G

. U

i k I k

k

intersection with the cointegrating space is also a vectorial subspace. This implies that a EC basis can be determined in completing a b₂ basis of EC>U with_k b₁ vectors. We define r as₁ b₁ rank and

r2r as1 b2 rank. Moreover sinceb1 is also a supplementary space basis, it follows thatbY 1 is of full rank column r : in other words, if a linear combination of the₁ b_{Y 1} columns that produces a column of zeros existed, this would mean that a vector of U spanned by thek b1 vectors would exist, which has been excluded by construction.

It is easily shown that the transformation matrix from the b basis to the b*5bP basis can be

21

written as and then one deduces a*5a(P9) .

Finally the expressions of the reparametrised matrices b and a are given by

This completes the proof of Theorem 1.

References

Anderson, T.W., 1984. In: 2nd Edition, An Introduction to Multivariate Statistical Analysis, John Wiley, New York. Hunter, J., 1992. Tests of cointegrating exogeneity for ppp and uncovered interest rate parity in the United Kingdom. Journal

of Policy Modeling 4 (14), 453–463.

Johansen, S., 1988. Statistical analysis of cointegrated vectors. Journal of Economic Dynamics and Control 12, 231–534. Johansen, S., 1991. Estimation and hypothesis testing of co-integration vectors in Gaussian vectors autoregressive models.

Econometrica 6, 1551–1580.

Johansen, S., 1995. In: Likelihood-based Inference in Co-integrated Vector Autoregressive Models, Oxford University Press, p. 267.

Johansen, S., Juselius, K., 1992. Testing structural hypotheses in a multivariate cointegration analysis of the PPP and UIP for UK. Journal of Econometrics 53, 211–244.

Mosconi, R., Giannini, X., 1992. Non-causality in cointegrated systems: representation estimation and testing. Oxford Bulletin of Economics and Statistics 54 (3), 399–417.

Psaradakis, Z., 1994. A comparison of tests of linear hypotheses in cointegrated vector autoregressive models. Economic Letters 45, 137–144.

´ ´ ´ ` `

Rault, C., 1998. L’exogeneite dans les modeles VAR-ECM avec des sentiers de long terme purement exogenes. In: Cahiers

´ ´