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and b1a4 for appropriate matrices ai , i. 1,2,3,4+ This solves the identification problem for t+. NOTE. 1+ An excellent partial solution based on a state space ...
Econometric Theory, 20, 2004, 639–640+ Printed in the United States of America+ DOI: 10+10170S0266466604203103

PROBLEMS AND SOLUTIONS PROBLEM 04.3.1 An I (2) Model for VAR(1) Processes Paolo Paruolo University of Insubria, Varese, Italy

This problem discusses an I~2! model in the VAR~1! case+ The I ~2! representation theorem of Johansen ~1992! ~JRT! holds also for VAR~1! processes+ The I ~2! model for VAR~k! processes has been discussed for k $ 2 in Johansen ~1996, Ch+ 9; 1997!+ We here discuss a parametrization for the I ~2! case of VAR~1! that differs from the VAR~k! model+ Consider the VAR~1! process with no deterministics, DX t 5 PX t21 1 et , with et i+i+d+ N~0, V! of dimension p, V positive definite+ It is assumed that 6I 2 ~I 1 P!z6 5 0 has solutions either at z 5 1 or 6z6 . 1 ~Assumption A!+ Under Assumption A, the JRT states that X t is I ~2! iff the following three conditions hold+ ~a! P 5 ab ' , with a, b, full rank p 3 r matrices, r , p, ~b! Pa4 Pb4 5 a1 b1' for a1 , b1 p 3 s full column rank matrices, s , p 2 r, and ~c! a2' ub2 is square and invertible, where u :5 I 1 bN aT ', a2 :5 ~a : a1 !4 , b2 :5 ~ b : b1 !4 , aS :5 a~a ' a!21, Pa :5 aa S ' + In the following p2 :5 p 2 r 2 s+ 1+ Let a VAR~1! process be I ~2!, i+e+, let Assumption A and conditions ~a!–~c! hold+ Prove that p2 # r+ 2+ Define the I~1! model as the collection of VAR~1! processes indexed by the parameters t :5 ~a, b, V!+ Find the restrictions on t that ensure that ~b! holds+ Also find an explicit parametrization+ This model is called the I ~2! model+ 3+ Let Yt :5 b ' X t 1 d~ b2' b2 !21 b2' DX t describe the ~possibly multi-! cointegrating relations+ Show that d is of full column rank p2 and find d in terms of the parameters in part 2+ Indicate also within Yt the r 2 p2 polynomially cointegrating relations and the p2 CI ~2,2! relations+ 4+ X t has representation D2 X t 5 C2 ~L!et + Find the expression for C2 :5 C2 ~1! in terms of the parameters in part 2+ 5+ Discuss the identification of t+

© 2004 Cambridge University Press

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PROBLEMS AND SOLUTIONS

REFERENCES Johansen, S+ ~1992! A representation of vector autoregressive processes integrated of order 2+ Econometric Theory 8, 188–202+ Johansen, S+ ~1996! Likelihood-Based Inference in Cointegrated Vector Auto-regressive Models+ Oxford University Press+ Johansen, S+ ~1997! A likelihood analysis of the I ~2! model+ Scandinavian Journal of Statistics 24, 433– 462+

Econometric Theory, 21, 2005, 665–666+ Printed in the United States of America+ DOI: 10+10170S026646660505036X

SOLUTION TO PROBLEM Solution to Problem Posed in Volume 20(3) 04.3.1. An I(2) Model for VAR(1) Processes 1 — Solution Paolo Paruolo ~poser of the problem! University of Insubria, Varese, Italy Let a* be any basis of col~P! and b* be any basis of col~P ' !; we note col~a4 ! 5 col~a*4 !, col~ b4 ! 5 col~ b*4 ! so no distinction is made between a4 and a*4 , b4 and b*4 in the following discussion+ 1+ Condition ~b! is equivalent to rank~a4' b4 ! 5 s , p 2 r; this implies that A :5 col~a4 ! ù col~ b! has dimension p2 :5 p 2 r 2 s+ Because p2 5 dim~A! # min~r, p 2 r!, one has that p2 # r+ 2+ Condition ~b! can be written as a4' b4 5 jh ' , with j, h ~ p 2 r! 3 s full column rank matrices; a4 can thus be chosen as follows: a4 5 ~ bN * , bN 4 !

S D a

b1

0

b2

5 ~ bN * , bN 4 !

q p3~ p2r!

,

where a is r 3 p2 , p2 # r, b :5 ~b1' : b2' ! ' is p 3 s, and q is of full column rank+ The matrix a4' b4 becomes a4' b4 5

SD SD 0

b2'

5

0

I p2

b2' ,

so that j 5 ~0 : Ip2 ! ' and h 5 b2 ; thus b1 5 bN 4 b2 , b2 5 b4 b24 , a1 5 aT 4 ~0 : Ip2 ! ' , a2 5 a4 ~Is : 0! ' 5 bN * a+ Using orthogonal projections, one can represent a* as a* 5 ~ b* , b4 !c; from a4' a 5 0 one deduces that c 5 q4 , where c 5 q4 can be chosen as follows: a* 5 ~ b* , b4 !

S

a4

0

2~ b4' b4 !21 b2 ~b2' ~ b4' b4 !21 b2 !21 b1' a4

b24

D

v

5 ~ b* a4 1 b1 v : b2 !v, where v is a nonsingular square matrix and v :5 2~b2' ~ b4' b4 !21 b2 !21 b1' a4 + Note that the second block of columns is equal to b2 5 b4 k, where we have named k :5 b24 + Now let a :5 ~ b* a4 1 b1 v : b2 ! and define b ' :5 aT ' P; because b and b* are two bases of col~P ' !, they are related by the relation bu 5 b* for a nonsingular matrix u+ Substituting within a one finds a :5 ~ bw 1 b1 v : b2 ! with w :5 ua4 © 2005 Cambridge University Press

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SOLUTION TO PROBLEM

of full column rank+ Summarizing, the coefficients ~a, b, V! are constrained as follows: a :5 ~ bw 1 b1 v : b2 !,

(1)

where b1 5 bN 4 k4 , b2 5 b4 k+ The I~2! model is thus indexed by the parameters t :5 ~ b, k, w, v, V!, which are freely varying+ 3+ From the I~2! representation theorem of Johansen ~1992!, one finds that d :5 aT ' b2 + By ~1! one sees that d 5 ~0, Ip2 ! ' , which is full rank p2 + Let b 1 :5 b~Ir2p2 : 0! ' and b 2 :5 b~0, Ip2 ! ' ; one sees that b 1' X t are the CI~2,2! relations and b 2' X t 1 ~ b2' b2 !21 b2' DX t are the multicointegrating relations+ 4+ By the I~2! representation theorem of Johansen ~1992!, C2 5 b2 ~a2' ub2 !21 a2' , where u :5 ~I 1 bN aT ' !+ The choice b is by construction conformable with a :5 ~ bw 1 b1 v : b2 !+ Going back to the representation of a2 one finds bN * 5 b~u N ' !21 and thus a2 5 bN * a 5 bw N 4 , where we note that ~u ' !21 a 5 w4 + Thus a2' ub2 5 w4' bN ' ~I 1 bN aT ' !b2 5 w4' bN ' bN aT ' b2 5 w4' bN ' bd, N given the structure ~1! of a, and hence

S

C2 5 b4 k w4' ~ b ' b!21

S DD 0

I p2

21

w4' bN '+

(2)

Note that C2 depends on b, k, w and it does not depend on v, V+ 5+ The likelihood function depends on t through the product ab ' and V+ Specifically ab ' 5 ~ bw 1 b1 v!b 1' 1 b4 kb 2' , where b 5 ~ b 1 : b 2 ! is partitioned conformably with a+ It is simple to see that t is not identified because one can insert two square invertible matrices fi , i 5 1,2, and their inverses within the two factors in ab ' ; i+e+, t :5 ~ b, k, w, v, V! and t * :5 ~ bf ' , kf221 , ~f ' ! 21 wf121 , vf121 , V! give the same likelihood, where f :5 diag~f1 ,f2!+ Moreover because b4 , b1 are any bases of col~ b4 ! and col~ b1 !, one also needs to identify them+ For any matrix g, let a be of the same dimensions such that a ' g is nonsingular; define ga :5 g~a ' g!21 ~see, e+g+, Johansen, 1996, ch+ 5!+ To achieve identification one can identify b 1, b 2, b4 , and b1 by defining ba11 , ba22 , b4a 3 , and b1a 4 for appropriate matrices a i , i 5 1,2,3,4+ This solves the identification problem for t+ NOTE 1+ An excellent partial solution based on a state space representation has been independently proposed by Dietmar Bauer, Vienna University of Technology ~Austria!+

REFERENCES Johansen, S+ ~1992! A representation of vector autoregressive processes integrated of order 2+ Econometric Theory 8, 188–202+ Johansen, S+ ~1996! Likelihood-Based Inference in Cointegrated Vector Auto-Regressive Models+ Oxford University Press+