In this note an extension of the traditional definition of weak exogeneity when the variables are I(1) and cointegrated is presented. In particular, the concept of long-run weak exogeneity is introduced when the parameters of interest are the elements of the cointegrating vectors. This definition extends some previously available definitions in this framework.

Dolado, Juan José

English

Research Papers in Economics

Econornics Letters 38 (1992) 139-143 
North-Holland 
A note on weak exogeneity 
in  V AR cointegrated models 
Juan J.  Dolado * 
Bank of Spain,  Madrid,  Spain 
Received 11  October 1991 
Accepted 6 Decernber 1991 
139 
In  this  note an  extension  of the traditional definition  of weak exogeneity when  the variables are 1(1)  and cointegrated is 
presented.  In  particular, the concept of long-run weak exogeneity is  introduced when  the pararneters of interest are the 
elernents of the cointegrating vectors. This definition extends sorne previously available definitions in  this frarnework. 
1.  Introduction 
This note addresses the issue of which definition of exogeneity is  needed to carry out efficient 
estimation in the context of single equation conditional dynamic models estimated by least squares. 
In particular, we  examine if the concept of weak exogeneity, introduced by Engle et al.  (1983) as  a 
sufficient  condition  for  valid  inference  on  the  parameters  of  a  conditional  distribution  in  a 
framework of 1(0) variables, still holds when variables are 1(1) and there is cointegration. It is shown 
that the traditional definition is too strong and sorne examples are provided where there is no weak 
exogeneity but conditional  inference  is  stilI  efficient.  This  is  an  interesting issue  since  economic 
systems often have so many poteniially useful variables that the system gets very large and then it 
may be worth to consider a single equation framework, where a variable is modelled conditional on 
the remaining variables. 
The paper is  organized as  follows.  Section 1 discusses the data generation process (DGP) and 
introduces the relevant assumptions and notation. Section 2 analyses the concept of long-run weak 
exogeneity associated to the parameters of the cointegrating vector.  Finally, Section 3 extends the 
traditional definition of weak exogeneity to a nonstationary framework. 
2.  The vector autoregressive model 
Let  us  consider  an  n  dimensional  vector  time series {XI'  t =  1,  2 ...  },  which  foIlows  a  vector 
autoregressive (VAR) process of order k, 
k 
XI =  L IIjx l _j + El' 
j~l 
Correspondence to: Juan J.  Dolado, Research Departrnent, Bank of Spain, Alcalá 50,  28014-Madrid, Spain. 
* I arn  grateful to Manuel Arellano, Peter Boswijk and Soren Johansen for helpful cornrnents. 
0165-1765/92/$05.00 ©  1992  - Elsevier Science Publishers B.Y.  AH  rights reserved 
(1 ) 
1140  J.J.  Dolado  /  Weak exogeneity in VAR cointegrated models 
where {E}  is  a Gaussian innovation process relative to the past history of {Xt}  with variance matrix 
D.  We define the matrix polynomial 
(2) 
Following Johansen (1990), we  state: 
Assumption (Cointegration).  System (1) verifies 
(i)  II(z) I =  O has all its roots outside the unit circle or equal to 1. 
(ji)  II(1) =  -exf3', where  ex  and  f3  are n  X r  matrices of rank r, O  ~  r ~  n, and rank II(1) = r. 
(¡ii)  rank  ex~ (II¡(1))f3 1- =  n - r, where  ex  1- and  f3  1- are  the  n  X (n - r) orthogonal complement 
matrices of ex  and  f3,  of rank n - r, such that ex  ~  ex  =  O,  f3'1- f3  =  O,  and  II¡ (z ) =  aII(z)  /az. 
Together  these  assumptions  imply  that  the  components  of  x t  are  1(1),  whereas  the  linear 
combinations  f3' x tare stationary. Hence the components of x tare cointegrated with cointegrating 
matrix  f3  and cointegrating rank  r.  Moreover, system (1) can be expressed as  an error correction 
model (E  CM), 
with 
k-l 
L1Xt  =  L IjL1xt_j  + exf3'xt- k + El' 
¡ 
(3) 
In order to discuss the problem of exogeneity in the context of a single equation, let us use the 
(1  X n) vector e; = (1,  O, ... , O)  to select, without loss  of generality, the first  equation and  E~ be a 
(n + 1 X n)  marix  of  vectors  orthogonal  to  el'  which  selects  the  remaining  (n - 1)  variables. 
Partitioning Et  and D conformably, given that {Et}  is  a Gaussian process, we have 
where  Ft_¡  is  the  a-field generated by  {E2P  Xt- f ,  j = 1, 2 ...  } and  a12  is  an (n - 1 X  1)  column 
vector. 
Therefore, the conditional distribution for the first equation can be written as 
(4) 
with  7]t - nid(O,  a¡u)· 
2J.J. Dolado  /  Weak exogeneity in VAR cointegrated models 
The marginal distribution for the remaining (n - 1)  elements in  Llxt  is  given by 
k~l 
LlX2t=E~Llxt= L E~riLlxt~1 +E~af3'xt~k+E~ef' 
1 
i.e.,  the distribution of x 2t  given  Xlt~l' X2t~l' etc. 
3.  Long-ron weak exogeneity 
141 
(5) 
Engle, Hendry and Richard (1983)  introduced the concept of weak exogeneity with respect to 
the parameters of the conditional distribution (4) as  a condition which allows to carry out optimal 
inference  with  respect  to  a  set  of  parameters  disregarding  the  information  on  the  process 
generating the  marginal  distribution. If D(x ti  Xt  ~ l' A)  denotes  the joint distribution of the  x;  s 
conditional on the past history  Xt~1 =  {Xt~j; j = 1,  2 ...  },  then D  can be factorized as follows: 
(6) 
Then, Engle et al. (1983) define x 2t  as weakly exogenous for a set of parameters of interest P  if 
and only if there exists a partition (Al' A2)  of A such that: (j) '"  is a function of Al  alone, and (ji) Al 
and A2  are variation free, so that there are no restrictions linking  Al  with  A2 . 
Following  the  previous  definition,  Boswijk  (1991)  and  Johansen  (1990)  find  x2t  as  weakly 
exogenous for  P  =  (a, (3)  if  E~a  =  O,  that is, 
(a) I  I  a= 
O  n~l' 
Because  E~a  =  O,  the  marginal  model  of  x2t  conditional  on  past  history  does  not  contain 
information on the cointegrating vectors in  f3  and the error correction coefficient a l' 
However,  as  we  will  prove  in  what  follows,  the previous condition  is  too  strong.  To prove  a 
weaker  condition,  we  rely  upon  a  recent  result  by  Phillips  (1991)  on  optimal  inference  in 
cointegrated systems. Phillips' Theorem 2 can be used to show that the ML estimator of f3  in (4) is 
asymptotically  equivalent  to  the  FIML estimator  in  (3)  iff  n\2 =  O,  where  nI  is  the  long-run 
variance  between  Tlt  and  Llx2t.  Hence,  there  is  no  loss  of efficiency  in  estimating  f3  from  the 
conditional model. To obtain the off-diagonal terms in the long-run variance-covariance matrix, we 
use the MA representation of the cointegrated system, as derived by Johansen (1989, Theorem 3.1), 
(7) 
where c(1) =  f3  1- (a'1- IJ¡(1)f3  1-)~la'1- which exists by assumption (ji). 
Since  Tlt = (el - E2nZ21(T12)'Et  then n\2 = O implies that 
(8) 
and this is a necessary and sufficient condition for x 2t  being weakly exogenous with respect to  f3  in 
this framework. We will  denote this property as  long-run weak exogeneity. 
3142  J.J.  Dolado  /  Weak exogeneity in VAR cointegrated models 
Note that  E~a = O is  a sufficient condition for (8) to hold,  since  E~a  = O implies  a2 = O.  Then 
a -L  is  any matrix of the form (O,  Q), where Q is full  rank, and the first three terms in (8) can be 
written as 
However, it is  not necessary since it could happen that  f3  ~  E2  has a deficient column rank and 
happens to lie in its null space. The latter restriction holds in the following example. Let n = 3 and 
f3  =  (0,1,1)';  a  =  (-0.1,0.1,0.1); 
1 
-1  -1)  (1  1  ;  a'-L  =  1 
2 
1  ~)  . 
It is  easy  to  check that this  example verifies  assumptions  (i)-(iii) and that (8)  holds  in spite  of 
a 2 =  O. 
4.  A new definition of weak exogeneity when variables are 1(1) 
In the light of the previous discussion,  the traditional definition of weak exogeneity in  an 1(0) 
framework can be amended in the following way.  Let us define two sets of parameters of interest: 
the long-run parameters PI =  {f3}  and the short-run parameters 
Retaining the  assumptions (i)-(iii) the conditional and marginal densities in  (6)  can be reformu-
lated as 
Then, x 21  is weakly exogenous with respect to P  if:  (i) PI is a function of Al  and satisfies long-run 
weak exogeneity and (iD  P s  is  a function of A1 and A1 and A2  are variation free. 
The following example serves as  an illustration of the previous concepts. Let the DGP be 
XII = f3x 21 + EII , 
with  Eit - nid(O,  a/), E(EltE2s) =  O. 
Then, the OLS estimator of  f3  in (9) is  given by 
(9) 
(10) 
( 11) 
4J.J.  Dolado  /  Weak exogeneity in  VAR cointegrated models  143 
where  B¡(r) (i =  1,  2)  are  unit variance  independent Brownian  motions  and '=' denotes weak 
convergence.  Phillips  (1991)  has  shown  that (11)  is  a  'mixture  of normals'  and  hence  the  OLS 
estimator of f3  in the conditional model is  asymptotically equivalent to the ML estimator of f3  in 
the ful!  system. Thus, no asymptotic efficiency is  lost.  However,  it  is  important to note that since 
0ILlElt_ 1  appears  in  the marginal  density,  x2t  is  not weakly  exogenous  in  the  traditional sense, 
though  it  is  long-run weakly  exogenous.  This  can  be  easily  seen  since  LlE It -1  vanishes  in  the 
long-run variance-covariance matrix and  hence  n\2 = 0,  as  required [see  Phillips  (1991,  p.  300, 
remark (m))]. Thus, ful!  'variation freeness' is  too strong. 
References 
Boswijk,  P.H., 1991, Testing for cointegration in  structural models, Report AE 7/91 (University of Amsterdam). 
Engle, R.F., D.F. Hendry and J.F.  Richard, 1983, Exogeneity, Econometrica 51,277-304. 
Johansen, S.,  1989,  Estimation and  hypothesis  testing of cointegration vectors  in  Gaussian vector  autoregressive  models, 
Preprint 1989, no.  3 Onstitute of Mathematical Statistics, University of Copenhagen); forthcoming in  Econometrica. 
Johansen, S.,  1990, Cointegration in  partial systems and the efficiency of single equation analysis, Preprint 1990 (Institute of 
Mathematical Statistics, University of Copenhagen). 
Phillips, P.c.B., 1991,  Optimal inference in  cointegrated systems, Econometrica 59,  283-306. 
5

Cointegration in partial systems and the efficiency of single equation analysis, Preprint

Estimation and hypothesis testing of cointegration vectors in Gaussian vector autoregressive models,

Optimal inference in cointegrated systems,

Testing for cointegration in structural models,

A Note on Weak Exogeneity in VAR Cointegrated Models.

http://e-archivo.uc3m.es/bitstream/10016/3318/1/weak_dolado_EL_1992_ps.pdf

A Note on Weak Exogeneity in VAR Cointegrated Models.

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