Granger's representation theorem and multicointegration

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Economics Department

Granger's Representation Theorem

and Muiticointegration

To m En g sted

and

Sp r e n Jo h a n sen

ECO No. 97/15

EUI WORKING PAPERS

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EUROPEAN UNIVERSITY INSTITUTE 3 0 0 0 1 0 0 2 5 9 0 5 6 2 © The Author(s). European University Institute. version produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute Research Repository.

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E U R O P E A N U N IV E R SITY IN STITU TE, F L O R E N C E

ECONOMICS DEPARTMENT

EUI Working Paper E C O No. 97/15

G ranger’ s Representation Theorem and Multicointegration T O M ENGSTED and S0REN JOHANSEN B A D IA F IE S O L A N A , SAN D O M E N IC O (FI) © The Author(s). European University Institute. version produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute Research Repository.

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No part o f this paper may be reproduced in any form without permission o f the authors.

Badia Fiesolana I - 50016 San D om enico (FI)

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Granger’s representation theorem and

multicointegration

Tom Engsted, Aarhus Business School and

S0ren Johansen, European University Institute, Florence

January 1997

A bstract

We consider multicointegration in the sense of Granger and Lee (1990), that is, the cumulated equilibrium error cointegrates with the process it self. It is shown, that if the process is given by the cointegrated VAR model for 7(1) variables, then multicointegration cannot occur. If, how ever, the cumulated process satisfies an 7(2) model then multicointe gration may occur. Finally conditions are given on the moving average representation for the process to exhibit multicointegration. This result generalizes the analysis of Granger and Lee.

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1 Introduction

Since Engle and Granger’s (1987) seminal paper the concept of cointegration has developed progressively in several ways, and many extensions of the basic concept have been made. One such extension is the definition o f multicoin tegration which refers to the case where the cumulation o f equilibrium errors cointegrates with the original 7(1) variables of the system. Such situations arise naturally in economic models involving stock-flow relationships. One example, analysed in detail by Granger and Lee (1989), is the case where the two / (1) flow series production, X t, and sales, Yt, cointegrate such that inventory investment Zt = X t — Yt is 1(0). It follows that Zt is the level o f inventories (stock)

which might cointegrate with X t and Yt such that Ut = Xa=i Z, — aYt — bXt is 1(0). Thus, there are essentially two levels of cointegration between just two

7(1) time series. Other examples involve consumption, income, savings, and wealth, or new housing units started, new housing units completed, uncom pleted starts, and housing units under construction, see Lee (1992).

Granger (1986) anticipates the notion of multicointegration, and the con cept is formally developed in Granger and Lee (1989, 1990). In particular, they prove a representation theorem stating that multicointegrated time series are generated by an error-correction model which contains both Z ,_i and 77(_i as error-correction terms. In addition, they show that multicointegration can be derived from a standard linear quadratic adjustment cost framework often used in economics.

The papers by Granger and Lee constitute an important starting point for the analysis of multicointegrated time series, but they are somewhat limited in scope since they analyse only bivariate systems. Furthermore, in the estimation procedure they assume that the cointegration vector at the first level is known and, hence, does not have to be estimated. In estimating the cointegration vector at the second level they propose to use the simple OLS estimator which in general is not optimal and does not allow hypothesis testing using standard asymptotic theory.

The purpose o f the present paper is to provide a more detailed analysis of multicointegration in multivariate systems of 7(1) time series. It is shown that although the interest lies in the analysis of the 7(1) series one cannot achieve multicointegration in the cointegrated error correction model for 7(1) variables. If, however, we assume that the cumulated variables satisfy an error correction model for 7(2) variables then we have the possibility of modelling multicointegration for 7(1) variables.

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Engle and Yoo (1991) also suggest to relate multicointegration to 7(2) cointegration, and apply the Smith-McMillan decomposition to derive the VAR representation from the M A representation. Their results are discussed by Hal- drup and Salmon (1996) for multivariate processes. Equivalently one can say that the process satisfies an integral control mechanism, see Hendry and Von Ungern Sternberg (1981).

The contents of the present paper is the following. First we give a the orem about the inversion o f matrix valued functions which is the essence of the Granger representation theorem. We then discuss multicointegration in the usual error correction model for 7(1) variables, and show that this phe nomenon cannot occur. If, however, we assume that the cumulated process satisfies an 7(2) model, then the results about this model can be phrased in terms of multicointegration. Finally we show how the general formulation of Granger’s theorem solves the problem of deriving the 1(2) model from the mov

ing average representation. This generalizes the results o f Granger and Lee (1990). The statistical analysis o f the multicointegrated 7(2) model is given in Johansen (19966). Throughout we assume that the equations generating the process have no deterministic terms. The representation results given are easily generalized, but the statistical analysis becomes more complicated, see Johansen (1995. 19966) and Paruolo (1996).

2 A general formulation of Granger’s represen

tation theorem

In this section we consider n x n matrix valued functions A(z) with entries that

are power series in a complex argument z. Let |A(z)| denote the determinant

and adjA(z) the adjoint matrix.

A s s u m p t i o n 1 The power series

OO

A(z) = ' £ A iz'

i=0

is convergent for \z\ < 1 + 6, and satisfies the condition that if |A(z)| = 0, then

either |z| > 1 + 7 or z = 1. Here 0 < 7 < 6. We assume further that A (z) = I

for z = 0.

We are concerned with the power series for the function A~1(z). This

function will have a power series expansion in a neighborhood o f the origin.

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since A(0) ■= 7 implies that \A(z)\ ^ 0 for z sufficiently small, and hence A~l(z) exists.

We give a theorem that summarizes the Granger representation theorems for 1(0), 7(1) and 7(2) variables given in Johansen (1992). We give the results a

purely analytic formulation without involving any probability theory, since the basic structure is then more transparent. The result allows a direct identification of the relevant coefficients of the inverse function in terms of the coefficients of the matrix function, and gives conditions for the presence o f poles o f the order 0,1, and 2 respectively. The result will be applied below to derive the autoregressive representation from the moving average representation and vice versa, and the explicit formulae allow one to discuss the coefficients in the moving average representation in terms of the estimated coefficients from the autoregressive model.

We expand the function A(z) around £ = 1 and define the coefficients A (l) and A (l) by the expansion

A(z) = A( 1) + ( z - l) À ( l) + ~(z - 1)2A (1) + • • •,

which is convergent for \z — 1| < <5. Thus

i ( l ) = dA(z)

dz A{ 1) = d2A(z

d2z

For any n x m matrix a of full rank m < n, we denote by an n x (n — m)

matrix of rank n — m such that a'aj_ = 0. We define a = a(a'a)-1 , such that a'a = 7, and aa' is the projection of Rn onto the space spanned by the columns

of a. For notational convenience we let a± = I if a = 0, and m = 0.

T h e o re m 1 Let A(z) be a matrix power series which satisfies Assumption 1. Then the following results hold for the function A~x(z).

1. If z = 1 is not a root, then A -1(z) is a power series with exponentially

decreasing coefficients.

2. If z = 1 is a root then A (l) is of reduced rank m < n, and A (l) = £?/, where £ and 77 are of dimension n x m and rank m. If further

\^À(l)r,L\ÏO,

(

1

)

then A~1( z ) = C - ^ z + C '(z), 1 — z © The Author(s). European University Institute. version produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute Research Repository.

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where C *(2) is a power series with exponentially decreasing coefficients, and

where

c = v ± ( ? x M i ) v ± ) V ±

-3. If z = 1 is a root such that .4(1) = (rf and if

Z'LA(l)r,1 = <K\

is of reduced rank, where cp and £ are (n — m) x k matrices o f rank k < n — m, and if # o , ₍

₂

₎ then A ~\z) = C2 (1 - z ï + C\ (1 - 2)+ C ” (z)

where C " (z ) is a power series vjith exponentially decreasing coefficients. Ex pressions for the coefficients Ci and C2 can be found in Johansen (1992, The orem 3). Here we give the expression

C2 = VxC± (V ± £l( ^ ( 1 ) - À ( m ' À ( l ) ) V±C i) 1

4>'xC-The proof can be found in the above mentioned reference for the case when A(z) is a polynomial. The proof for the general case where infinitely

many terms are allowed is the same. Note that it is not enough to assume that the roots are outside the unit circle or equal to 1, since we could have infinitely many roots converging to the unit circle, which would ruin the proof. Hence we assume that the roots are bounded away from the unit disk or equal to 1. If 2 = 1 is a root then A~i (z) has a pole at the point 2 = 1 , since

A~\z) adjA(z)

^ ) T

and adjAfz) is a matrix valued power series with exponentially decreasing co

efficients. The / ( l ) condition (1) is necessary and sufficient for the pole to be of order 1. The function C-A-z has a pole of order 1 at 2 = 1 and the theorem says that the difference is a convergent power series. Thus the pole can be removed by subtracting the function Cy^j. The 1(2) condition (2) is necessary

and sufficient for the pole to be o f order 2, in which case it can be removed by subtracting the function + C ^y^, which also has a pole o f order 2.

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In order to apply this result in the autoregressive model

X t = £ H X (_, + e „ 1=1

define A(z) to be the matrix polynomial

A(z) = i - ' £ n izi.

i = l

Then A~l(z) gives the solution to the equations, that is, the coefficients in the

expansion for A 1 (z ) determine X, as a function of the errors e(. The translation o f the result is via the lag operator, such that for a function C(z) =

with exponentially decreasing coefficients and a sequence o f i.i.d. variables e(. we define the stationary process

C(L)et = y CjEt-j.

i = 0

For the expression yLy, we use the interpretation

(1 - L )_1£( = A _1£t = 1=1 and -1- , is translated into

( 1 Z )

( 1 _ l) 2£( = a 2et = y y * £,.

j =1 i = l

The result o f Theorem 1 can be used to check whether a given example o f an autoregressive process is 7(0), 7(1) or 7(2). It is the fundamental tool in building 7(1) and 7(2) models for autoregressive processes as we shall show below.

3 Multicointegration in the 1(1) model

In the following we apply these results to discuss the problem of multicointe gration as defined by Granger and Lee (1989, 1990).

D e f i n i t i o n 1 The n — dimensional 7 (1 ) process X t is said to be multicointe- grated with coefficient r if r ’ X t is stationary and if the process r'X t coin tegrates with Xt, such that there exist coefficients p and ip, that is, p' r 'Xi + ip'Xt is stationary. 5 © The Author(s). European University Institute. version produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute Research Repository.

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We want to prove that multicointegration cannot take place in the error correction model for / ( l ) variables

k

- 1

A X , = a ffX t-x + Y X t-i + £ , , « = 1 , T, (3) i=i

where a and (3 are n x r, where r < n.

T h e o re m 2 Multicointegration cannot appear in the 7(1) model (3) if the process X t is 7(1), that is, if

K ( 7 - E r <)0i I / O . (4) t=l

P r o o f. We apply Theorem 1 to the polynomial

A(z) = (1 - z)I - a ffz - Y r t(l ~

z)z'-t = l

Here ,4(1) = —a ft and ,4(1) = - a f t —I + Yli=i b., such that the 7(1) condition

(1) becomes the condition (4). The inverse polynomial has the expression as

A~\z) = C j ^ + C W

= C ^ —z + C ' + (l-z )C J (* ), such that the process has the representation

Xt — C Yj 4- C*e, + Ah', + A, (5) i=l

where A depends on initial conditions, 0 A = 0, and Yt = C[(L)£, is a stationary

process, see Johansen (1995). The matrix C is given by

C = /?x

fc-i

a f d - Y ^ L ) <*[

The cumulated equilibrium error has the form

Y P'x i = 0 'C + 0 Y t - ftY0.

i = l i = 1

The common trends in the expression for X t are of the form and the common trends in the cumulated equilibrium error are of the form ftC* £ '= i

Ei-© The Author(s). European University Institute. version produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute Research Repository.

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In order to see if these cointegrate we have to find the matrix 3 ' C . From the

relation A ( r ) A _1(,i) = / we find

! = 1

+ C " + ( 1 - * ) C 7 ( Z ) ) = / .

or

C + A(z)(C1 + ( l - z ) C ‘1(z)) = I.

For z = 1 we find

which when multiplied by a1 gives

/

**3'C* = aT/?,. (a'±r**

where F = I — J2i=i FiZ1- This shows that the non-stationarity of the cumulated

equilibrium errors is given in part by the n — r common trends q'± £ {=1 e, of the process X t, and in part by r random walks a112-=1 £, which do not appear in

X t. Thus any linear combination of X t and Y)\=i P'X, will necessarily contain

the trends a1 £*=i £i and hence be non-stationary. ■

This shows that multicointegration can not appear in the / ( l ) model. Another way of formulating this result is that no process y!X t, where X t is

generated by the 7(1) model, will be / ( —!). In order to see this assume that there is a stationary process Zt, say, and a coefficient vector /i 6 Rn such that n'Xt = A Z f From (5) we find that we must have p 'C = p'C* = 0.

Hence /x = /3k for some vector k and /t'C*a = /t'/3'C*a = — n' = 0 shows

the impossibility. We therefore next discuss the 1(2) model for the cumulated

variables.

4 Multicointegration in the 1(2) model

Next we want to prove a more constructive result where we take as a starting point that the cumulated processes are generated by an 1(2) model, see Engle

and Yoo (1991). Thus we define

s t = E * -i=l 7 © The Author(s). European University Institute. version produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute Research Repository.

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and assume that this new process is given by an autoregressive model, restricted such that it generates 1(2) variables. This model can be parametrized in many

ways. A parametrization that allows freely varying parameter is given by

A2St = q(p't'S (_i + rjj'ASt-i) + Q a i(a 'xQ Q i) '/c 'r 'A S t-i + et. (6) The parameters in this model are (a,p,r,i/j,K, Q.) and it is assumed that all

parameters vary freely. This gives the possibility to derive the maximum likeli hood estimators and find their asymptotic distributions, see Johansen (19966). We can add a lag polynomial applied to A2St, to account for more short term

dynamics.

The characteristic polynomial is given by

A (z) = (1 — z )2/ - a (p V z + ip'(l - z)z) - fia x (a x f2ax )~1K;V( 1 - z)z, such that

A ( l ) = —a p V , A ( l ) = —a p V -I- aip1 + fta x (a'± f2a !i)- 1K;V. With 0 = rp we find that A ( l ) = —a0' is of reduced rank and that

a'j. M l )P± = K'T'0± = k'(PlP'x + PP'VPx = (k'p± )(p'±.t'0±)>

since p'r'0± = 0'0L — 0. This matrix is o f reduced rank, and we can define <\> = k'p± and £ = 0’x rpL. If further condition (2) is satisfied, the process St is 1(2), which implies that X t = A St is / ( l ) . It is a consequence of the

results in Johansen (1992) that r'A S f = t'X t is stationary, and furthermore

that p'r'St + ip'ASt = p' t'X, -|- ip'Xt is stationary. Thus we find that

expressed in terms o f the process X t we have multicointegration and the error

correction terms are exactly the integral correction term p' t' Xx+tp’X t and

the usual error correction term t' Xt.

Thus this model is a general version o f the error correction model derived by Granger and Lee (1990). The result shows that the general model for the 1(2)

variable S< can be formulated as an error correction model for X, = A St which

has both integral correction terms and equilibrium correction terms exactly as the model in Granger and Lee (1990). Model (6) can be written in this way as

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A X t = a(p' 52 r'Xi + ip'Xt-1) + fiaj.(a'xn a i ) -1K V X (_i + et. i

Note that when the cumulated X t satisfies an autoregressive error cor

rection model then X t itself does not, since the equations we find for X t by

differencing will have A et as an error term.

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5 Multicointegration and moving average mod

els

The formulation of the result of Granger and Lee (1990) starts with the MA representation o f the process and derives an (infinite) A R model for the process involving an integral correction term and an error correction term. We shall here show how Theorem 1 gives a necessary and sufficient condition for this construction to go through.

We consider the situation where we model the process in the usual form by its moving average form

A X t = C(L)et = C0Et + Cx A et + C*(L) A2et

= Co£t

+

C\&Et

+ A

2y(.

If we assume that Co is o f reduced rank, such that t'Cq = 0 for some r / 0.

then we find

Xt = W0 + Co £ £, + Ci(£, - e0) + AY( - AY0, »=l

r'AXt = r'C1A£( + r'A 2Yt,

r'Xt = t'Xo + t'C^s,. - E0) + r'(A Y - A Vo),

t r ' X i = r'C i ( è - too) + r'(Yt - Y 0 - tAY0).

1=1 1=1

In order to find examples which exhibit multicointegration we only have to construct the matrices Co and C x such that there are coefficients p and xp with

the property that

iP'Co - p'r'Cx = 0, since

xP'Xt -p 'T l ' t x i,

•=1

does not contain any random walk. Thus there by chossing Co and C\ appro

priately it is easy to find examples o f multicointegration. We shall now show how Theorem 1 generalizes the result o f Granger and Lee.

T h e o r e m 3 Let the n —dimensional process X t satisfy the equation

A Xt = C{L)eu 9 © The Author(s). European University Institute. version produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute Research Repository.

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where C(0) = 7, and we assume that, the roots of ^ ( z ) ! = 0 are either hounded away from the unit disk or equal to 1.

1. If z = 1 is not a root, then A X t satisfies an (infinite order) autoregres sive equation

C ~ l(L )A X t = et.

The process X, is 7(1) and does not cointegrate.

2. If z = 1 is a root, then C ( l ) = £r/ is of reduced rank m < n and if further (fLC(l)r}L has full rank then X t satisfies an (infinite order) 7(1) model

a p X t + A ' ( L ) A X t = et,

with a/31 = — Here A*(z) — C~l(z) - a/3'yrj has exponen tially decreasing coefficients.

3. If z — 1 is a root, such that C ( l ) = £?/ is o f reduced rank m < n and if further £'J_C(l)r]1 = <j>(f is of reduced rank k < n — m and condition (2) holds then X t satisfies an (infinite order) autoregressive model with integral and error correction terms.

P r o o f. 1. If the roots o f |C(z)| = 0 are all bounded away from the unit disk, then the power series o f C~1(z) = Atzl is convergent for \z\ < 1 + S. where

6 > 0. This means that the coefficients in C~l (z) are exponentially decreasing

such that the stationary process Yfflo A iA X t-i is well defined and equal to

Expanding the function C (z) around z = l w e find

C(z) — C ( l ) + (1 - z)CT(z),

such that when summing the original equation from s = 1 to s = t we find that

X ^ X o + C W E e . + V t -T o , i=i

where Yt = C ’ (L)et is stationary. Thus X t is an 7(1) process and since C ( l )

has full rank it does not cointegrate.

2. Now assume that z = 1 is a root such that C ( l ) = {r / is of reduced

rank, but £,'LC(l)riL has full rank. We then find from Theorem 1, that

( l - z ) C - l (z) = A + ( l - z ) A ' ( z ) , (7)

with A = Inserting this expression into (7) we find

+ A ' ( L ) A X t = C~\L) A X t = et. © The Author(s). European University Institute. version produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute Research Repository.

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This is the required result if we define ft = £x and « = —7/1 (f'x C (l)r/x ) *. 3. Finally assume that C{ 1) = £77' and that = 4>C is of reduced

rank. In this case we have from Theorem 1

This shows the occurrence of integral correction terms and error correction terms in the same model. This model can be expressed in terms o f freely varying parameters as (6) by using the explicit form for the matrices A2 and

Ai given in Johansen (1992). ■

6 An example

Consider the example given by Granger and Lee (1990)

(1 - z)2C ~\z) = A 2 + ( 1 - z)Ax + (1 - z)2A“ (z).

Insert this into the equation for St = ]T‘=1

A 2St = C(L)et,

and we find

A2St + Ai&St + A' ‘ (z) A2St = C~1( L) A2St = et.

Expressing this in terms o f X we find

A2 £ + A , X t + A“ ( z) AX t = et.

In this case the polynomial is

C(z) = a + (1 - z)(\ - a) —a + (1 — 2)(1 + a) with 11 © The Author(s). European University Institute. version produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute Research Repository.

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In this case we find

Condition (2) is satisfied, since (7(1) = 0 and we can take <j> = Ç = 0. In this

case the 7(1) condition reduces to

Thus the cumulated X t satisfies an 7(2) model which gives multicointegration.

7 References

Engle, R. F. and Yoo, S. B. (1991). Cointegrated economic time series: an overview with new results. In Engle, R. F. and Granger, C. W . J. (eds.) Long-run Economic Relations: Readings in Cointegration, pp. 237-266, Oxford University Press, Oxford.

Engle, R. F. and Granger, C. W. J. (1987). Co-integration end error correction: representation, estimation, and testing, Econometrica, 55 251-276.

Granger, C. W . J. (1986). Developments in the study o f cointegrated economic variables. Oxford Bulletin of Economics and Statistics, 48, 213-228.

Granger, C. W . J. and Lee, T.-H. (1990). Multicointegration. Advance in Econometrics: Cointegration, Spurious Regressions and unit Roots, 8, 71-84.

Granger, C. W . J. and Lee, T.-H. (1989). Investigation o f production sales and inventory relationships, using multicointegration and non symmetric error correction models. Journal o f Applied Econometrics, 4, Suppl. 145-159.

Haldrup, N. and Salmon, M. (1996). Representations of 1(2) cointegrated systems using the Smith-McMillan form. Journal o f Econometrics (forthcom

ing)-Hendry, D. F. and von Ungern Sternberg, T. (1981). Liquidity and in flation effects on consumer’s expenditure. In A. S. Deaton (ed.) Essays in the Theory and Measurement of Consumer’s Behaviour, pp. 237-261, Cambridge University Press, Cambridge.

Johansen, S. (1992). A representation o f vector autoregressive processes integrated of order 2. Econometric Theory, 8, 188-202.

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Johansen, S. (1995). A statistical analysis of cointegration for 1(2) vari ables. Econometric Theory, 11, 25-59.

Johansen, S. (1996a). Likelihood-based Inference in Cointegrated Vector Autoregressive Models, (2.nd ed.) Oxford University Press, Oxford.

Johansen, S. (19966). Likelihood inference in the 7(2) model, Scandina vian Journal o f Statistics, (forthcoming).

Lee, T.-H. (1992). Stock-flow relationships in housing construction, Ox ford Bulletin of Economics and Statistics, 54, 419-430.

Paruolo, P. (1996). On the determination of integration indices in 1(2)

systems, Journal of Econometrics, 72, 313-356.

13 © The Author(s). European University Institute. version produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute Research Repository.

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■ © The Author(s). European University Institute. version produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute Research Repository.

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Explaining the Dynamics o f Spanish Unemployment

ECO No. 96/15

Spyros VASSILAKIS

Accelerating New Product Development by Overcoming Complexity Constraints

ECO No. 96/16 Andrew LEWIS On Technological Differences in Oligopolistic Industries ECO No. 96/17 Christian BELZIL

Employment Reallocation, Wages and the Allocation o f Workers Between Expanding and Declining Firms

ECO No. 96/18

Christian BELZIL/Xuelin ZHANG Unemployment, Search and the Gender Wage Gap: A Structural Model

ECO No. 96/19

Christian BELZIL

The Dynamics o f Female Time Allocation upon a First Birth

ECO No. 96/20

Hans-Theo NORMANN

Endogenous Timing in a Duopoly Model with Incomplete Information

♦out of print © The Author(s). European University Institute. version produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute Research Repository.

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ECO No. 96/21

Ramon MARIMON/Fabrizio ZILIBOTT1 ‘ Actual’ Versus ‘ Virtual’ Employment in Europe: Is Spain Different?

ECO No. 96/22

Chiara MONFARDINI

Estimating Stochastic Volatility Models Through Indirect Inference

ECO No. 96/23

Luisa ZANFORLIN

Technological Diffusion, Learning and Growth: An Empirical Investigation o f a Set o f Developing Countries

ECO No. 96/24

Luisa ZANFORLIN

Technological Assimilation, Trade Patterns and Growth: An Empirical Investigation o f a Set o f Developing Countries

ECO No. 96/25

Giampiero M.GALLO/Massimiliano MARCELLINO

In Plato’ s Cave: Sharpening the Shadows o f Monetary Announcements

ECO No. 96/26

Dimitrios SIDERIS

The Wage-Price Spiral in Greece: An Application o f the LSE Methodology in Systems o f Nonstationary Variables

ECO No. 96/27

Andrei SAVKOV

The Optimal Sequence o f Privatization in Transitional Economies

ECO No. 96/28

Jacob LUNDQUIST/Dorte VERNER Optimal Allocation o f Foreign Debt Solved by a Multivariate GARCH Model Applied to Danish Data

ECO No. 96/29

Done VERNER

The Brazilian Growth Experience in the Light o f Old and New Growth Theories

ECO No. 96/30

Steffen HORNIG/Andrea LOFARO/ Louis PHLIPS

How Much to Collude Without Being Detected

ECO No. 96/31

Angel J. UBIDE

The International Transmission o f Shocks in an Imperfectly Competitive

International Business Cycle Model

ECO No. 96/32

Humberto LOPEZ/Angel J. UBIDE Demand, Supply, and Animal Spirits

ECO No. 96/33

Andrea LOFARO

On the Efficiency o f Bertrand and Cournot Competition with Incomplete Information

ECO No. 96/34

Anindya BANERJEE/David F. HENDRY/Grayham E. MIZON The Econometric Analysis o f Economic Policy

ECO No. 96/35

Christian SCHLUTER

On the Non-Stationarity o f German Income Mobility (and Some Observations on Poverty Dynamics)

ECO No. 96/36

Jian-Ming ZHOU

Proposals for Land Consolidation and Expansion in Japan

ECO No. 96/37

Susana GARCIA CERVERO

Skill Differentials in the Long and in the Short Run. A 4-Digit SIC Level U.S. Manufacturing Study

-& if-ECO No. 97/1

Jonathan SIMON

The Expected Value o f Lotto when not all Numbers are Equal

ECO No. 97/2

Bernhard WINKLER

O f Sticks and Carrots: Incentives and the Maastricht Road to EMU

ECO No. 97/3

James DOW/Rohit RAHI Informed Trading, Investment, and Welfare © The Author(s). European University Institute. version produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute Research Repository.

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ECO No. 97/4

Sandrine LABORY

Signalling Aspects o f Managers’ Incentives

ECO No. 97/5

Humberto L^IPEZ/Eva ORTEGA/Angel UBIDE

Dating and Forecasting the Spanish Business Cycle

ECO No. 97/6

Yadira GONZALEZ de LARA

Changes in Information and Optimal Debt Contracts: The Sea Loan

ECO No. 97/7

Sandrine LABORY

Organisational Dimensions o f Innovation

ECO No. 97/8

Sandrine LABORY

Firm Structure and Market Structure: A Case Study o f the Car Industry

ECO No. 97/9

Elena BARDASI/Chiara MONFARDINI The Choice o f the Working Sector in Italy: A Tri variate Probit Analysis

ECO No. 97/10

Bernhard WINKLER

Coordinating European Monetary Union

ECO No. 97/11

Alessandra PELLONI/Robert WALDMANN

Stability Properties in a Growth Model

ECO No. 97/12

Alessandra PELLONI/Robert WALDMANN

Can Waste Improve Welfare?

ECO No. 97/13

Christian DUSTMANN/Arthur van SOEST

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ECO No. 97/14

Spren JOHANSEN

Mathematical and Statistical Modelling o f Cointegration

ECO No. 97/15

Tom ENGSTED/Spren JOHANSEN Granger’ s Representation Theorem and Multicointegration out of print © The Author(s). European University Institute. version produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute Research Repository.

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Granger's representation theorem and multicointegration

Granger's Representation Theorem

and Muiticointegration

EUI WORKING PAPERS

Granger’s representation theorem and

multicointegration

1

Introduction

2

A general formulation of Granger’s represen­

tation theorem

\^À(l)r,L\ÏO,

(

)

2

3

Multicointegration in the 1(1) model

- 1

/

3'C* = aT/?,. (a'±r

4

Multicointegration in the 1(2) model

5

Multicointegration and moving average mod­

els

+

+ A

2y(.

6

An example

7

References

EUI

W O R K IN G

PAPERS

Publications of the European University Institute

A general formulation of Granger’s represen

₂

**3'C* = aT/?,. (a'±r**

Multicointegration and moving average mod