OXFORD BULLETIN OF ECONOMICS AND STATISTICS, 50,1(1988)0305-9049 $3.00

COINTEGRATION, ERROR CORRECTION, ANDAGGREGATION IN DYNAMIC MODELS:

A COMMENT*

Mananao Aoki

One of the ways the notion of cointegration is motivated in Granger (1983,1986) and Engle and Granger (1987) is to point to a better integrated treat-ment of short-run dynamics and longer-run equilibrium dynamics by an errorcorrecting model of Sargan (1964) in which the time difference of somevariable Ax1 is related, among other things, to the level variable z_1. Theinterpretation is that z1 refers to a combination of two (or more) levelvariables, x1 ay1, say, where x = ay is an equilibrium relation, and the errorcorrecting models allow deviations from the equilibrium relation to affect theshorter-run behaviour of x1. Basic in this scheme is the separation of dynamicmodes into fast-acting (shorter-run) and slower (longer-run) responses.Cointegration recognizes the fact that when (two or more) variables arecointegrated, then slower dynamic modes (i.e., longer-run responses) can beaggregated out from their dynamic behaviour by a suitable linear combina-tion of the original variables.

This note points out the relationship between the notion of cointegrationthat has recently become central in modelling time series with unit roots, asfor example in Stock and Watson (1986) and King, Plosser and Watson(1987), and the notion of dynamic aggregation introduced by Aoki (1968,1971). Aoki (1968) introduced the notion of dynamic aggregation to thecontrol literature as a dynamic generalization of the concept of static aggrega-tion in the economics literature. This notion was originally introduced as away of approximating complex (high-dimensional) dynamics by simpler(lower-dimensional) dynamics. Later, this notion was shown to produce exactlower-dimensional dynamics when dynamic matrices are of special structure,Aoki(1971).

The key idea of the dynamic aggregation in Aoki (1968) is to separate theeigenvalues of the dynamics into two mutually exclusive classes c1 and c2, andretain only eigenvalues of one class to build lower-dimensional dynamicmodels. Although the notion was introduced originally to continuous deter-ministic dynamics, there is a natural counterpart in discrete-time andstochastic dynamics systems. (See Aoki and Huddle (1967) for an indicationof this.) In typical applications, eigenvalues are classified into those near theunit circle and those near the origin. Since the eigenvalues near the origin

* Research supported in part by a grant from the National Science Foundation.

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correspond to fast-decaying dynamic modes, they may be ignored in anythingbut very short-run analysis. When this type of classification is made for adynamic system with unit root components, the natural classification foraggregation is to separate eigenvalues with unit magnitude from those strictlyinside the unit disk.

In Aoki (1971) it is shown that when an n X n matrix A is of rank k, andhas the structure A=DEC, where D, E and Care nXk, kxk and kXnrespectively, then the dynamic system

x,1=Ax+Be, (1)

can be aggregated exactly using the aggregation matrix C into

Cx,1 =(CDE)Cx1+ CBe

or

Z,+1 =Fz,+ CBe1,

where F= CDE, and hence z = Cx is an aggregated state vector. The condi-tion for exact aggregation of dynamic matrices is, therefore, CA = FC. Thiscondition was mentioned by Rosenblatt (1956) in a static context and wasused in Aoki (1968) as one of the conditions for dynamic aggregation.

Let P be an n X k matrix of linearly independent columns forming a basisfor a right invariant subspace of A associated with the eigenvalues in class c1,and normalize by P'P= 'k Let Sbe an n X(n - k) matrix of linearly indepen-dent columns, normalized by S'S = - k' forming a left invariant subspace ofA associated with class c2. They satisfy

AP=PA and S'A=NS'. (2)

When there are k linearly independent column eigenvectors and (n - k)linearly independent row eigenvectors of A, corresponding to eigenvalue inclasses c1, and c2 respectively, then A and N are both diagonal matrices witheigenvalues on the main diagonal lines.

Multiply the first relation from the left by S' and the second from the rightby P and subtract the latter from the former to obtain O = S'PA - NS'P. SinceA and Nhave no common eigenvalue by construction, we see that

s'P=O. (3)

Change the coordinate system of the system described by (1) to its com-ponents based on P and S:

X, = Sz, + P ii,.

Notice that S'x= z, and P'x,= ij, from (1) and (3). In the new coordinatesystem, the dynamic system (1) is represented as

Sz,+1+Pij,1 =A(Sz,+ Pi1 ,)+Be,

=ASz,+PAi1,+Be,, (4)

COINTEGRATION, ERROR CORRECTION, AND AGGREGATION 91

where (2) is used. Then multiply (4) from the left by S' and use (3) and (2) toextract from (1) a subsystem having eigenvalues of c2 only as

=Nz,+S'Be, (5)

where z,( = S'x,) is the aggregated state vector. Note that it is relation (3) thatpermits this derivation.

Next, multiply (4) from the left by P' and use (2) and (3) again to derive thedynamic equation for ij as

in which eigenvalues belonging to classes c, and c2 are separately displayedas block diagonal matrices and in which dynamics having eigenvalues in c2 acton the dynamics with modes in class c1, but are not acted on by the modes incl.

Any dichotomy of the eigenvalues of the dynamic matrix results in such arecursive representation. If N contains fast-acting eigenvalues, i.e., those withsmall magnitude, and A collects eigenvalues of slower dynamic modes, thenthe z, in (5) approaches zero much faster than and (6) may be approxi-mated by

=Ai,+P'Be,.This approximation would be a good representation of (1) for anything butthe very short-run. This is how the notion of dynamic aggregation was intro-duced in Aoki (1968) as a way of retaining 'significant' dynamic modes ofsystem responses.

When A contains only eigenvalues with unit magnitude, an importantspecial case obtains since (6) then represents random walk dynamics. Typi-cally k = 1 in such a case. State space representations of time series consist ofa dynamic equation for the state vector (1) and an observation or dataequation'

y,=Hx,+e,, (8)

which relates the data (observation) vector y, to the state vector x,. In thecoordinate system which converts (1) into (7), (8) becomes

The fact that the same vector e, appears in (1) and (8) is no restriction. The vector e, is theinnovation in y,. Then substituting e, = y, - Cx, into (1), we see that (1) is in the form of aKalman filter. Alternatively, the spectral decomposition can be used to justify the presence ofthe same e, in (1) and (8). See Aoki (1987a, p.67).

where the aggregatedEquations (5) and

vector(6) show

71t+i

zt+1

is now ij,( =that (1) can

A P'ASO N

P 'x,).

z,

be written

S'

in a recursive form as:

Be,,

(6)

(7)

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y,=HSz,+HPi,+e,. (9)

In this special case, , is a nonstationary process since its dynamic represen-tation (6) has one (or more) unit root(s). Any vector y which is in the nullspace of P'H' is a cointegrating factor introduced in Granger (1983) andelaborated in Engle and Granger (1987) because

v'y,= v'Hz,+ Ve, (10)

has no unit root dynamics associated with it but only the dynamics for z,, i.e.(5), which has only asymptotically stable eigenvalues in the matrix N. Whenthe dimension of the vector y, is larger than k (the dimension of the vector ij,)then we can call rj, the common (random walk) factor of the data series.

Equation (9) shows that in disaggregating the state vector z, of a short-runmodel into y,, there is a disaggregation error HPi11. Conversely (10) showsthat v'y, (where y is a cointegrating factor) can be disaggregated exactly fromthe short-run model except for random noise.

When (7) is viewed from the point of focusing on short-run dynamicsneglecting longer-run effects, then (5) is the appropriate repesentation since itretains only those eigenvalues with smaller magnitudes belonging to class c2and z, S'x, is the aggregated state vector. In practice it is desirable to put alleigenvalues of magnitude p or greater into c1 for some p close to but notequal to 1 to allow for a small sample situation in which it is not statisticallypossible to reject the hypothesis that some eigenvalues with magnitude lessthan 1 may indeed have a unit magnitude.

When A contains only unit roots, (7) shows that short-run disturbances'feed' into longer-run or random trends phenomena. We can reverse the rolesof eigenvalues in classes e1 and e2. Then the same procedure produces adynamic representation which focuses on the effects that equilibrium (orlonger-run) relations have on shorter-run dynamics. Now let R form a basisfor the right invariant subspace associated with N, i.e., AR = RN, R'R 'n-kand let Q be such that Q'A=AA', Q'Q=Ik. They satisfy Q'R=O. Nowintroduce the coordinate change x, = Rz, + Q,. The model (1) becomes

and (8) is given by

y,=HRz,+HQ,+ e,.In (11), the role of N and A is reversed from that in (7). The longer-rundynamic vector , now 'feeds' into the shorter run dynamics Since , = Q'x,aggregates out shorter-run dynamic phenomena, , =0 represents the long-run equilibrium and any non-zero , indicates deviation from the long-runequilibrium. Viewed this way, the equation for z,

z,41 = Nz,+ R'AQ,+ R'Be,

(z I N R'AQl[zIR'BlA [Q'B]'

COINTEGRATION, ERROR CORRECTION, AND AGGREGATION 93

is in the form of an error correction model. In the literature, the error correc-tion model is usually put as

Az1 = z1 1z1(N l)z1+R'AQ1+R'Be1.

Now specialize and assume that there is only one eigenvalue ) = 1. Thenthe representation of the dynamic structure in (7) becomes, using the vector prather than the matrix P,

p'ASl Ím11p'B1e,

z1+,/ Lo N j LzJ [S'B]

where Ap = p. Use the lag operator L, e.g., Lz11 z1, after lagging the timeindex by 1, to rewrite(12) as

1 p'ASL

p'BLê

O N î S'B

where denotes a lag transform. Solve the above out for and £ to write

=L p'ASL p'B

Lê.î O NL S'B

Now use = Sî + pito separate out terms containing (1 - L)-1 as

= S(I NLY' S'BLê+ Ç'[I+A5L(I NL) 'S']Bê

S(INL)'S'BLêpp'A5 (INL)'(IN)'S'BLê

+11{J

A(I - N)' S']Bê,

where we use the relation

L(I NL)1 S'=(IN)1 S'+L(I NL)S'(IN)'S'=(I N)'S'+(L 1)(INL)'(I N)'S'

to extract a factor proportional to (1 - L) from the second term. The trans-form of the data is thus decomposed into two parts

9= H + ê={I+ HS(I NL)' S'BL Hpp'A(I NL)- 1(J_ N)' S'BL}ê

Hpp'+ [I+A(IN)S']BLê1L

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where the first term is the transform of a weakly stationary process which is acyclical component and the second represents a random walk (trend) com-ponent in y. The second component represents a random trend which is apredictable process since it contains only e1, e,2, ... at time t. As notedearlier, any vector y such that v'Hp = O is a co-integrating vector because v'LÇhas no random walk component.

The decomposition of (13) can then be used to calculate the random walkcomponent in y as shown in Aoki (1 987b). (The decomposition of (13) differsfrom that of Beveridge and Nelson (1981) since the latest innovation e, isassigned to the cyclical component. Beveridge and Nelson assign e to thecyclical and random trend component differently.)

Finally, state vectors are often formed by stacking lagged variables. Co-integration for state vectors, then, implies cointegration among laggedvariables, a generalization to the dynamic context of the original concept.

University of California, Los Angeles

Date of Receipt of Final Manuscript: July 1987

REFERENCES

Aoki, M, (1987a). State Space Modeling of Time Series, Springer Verlag, Heidelbergand New York.

Aoki, M. (1 987b). 'An Alternative Measure of Random Walk Components in TimeSeries', forthcoming in Economics Letters, 24, pp. 227-30.

Aoki, M. (1978). 'Control of Large-Scale Dynamic Systems by Aggregation', IEEET-A C, A C-13, pp. 246-53, June 1968, also reprinted in Distributed Control, R.Larson et al. (eds), IEEE Computer Society.

Aoki, M. (1971). 'Aggregation in Optimization Methods for Large-Scale Systems', D.Wismer (ed.), McGraw-Hill.

Aoki, M. and Huddle, J. R. (1967). 'Estimation of the State Vector of a LinearStochastic System with a Constrained Estimator', IEEE T-A C, AC-12, pp.432-33.

Beveridge, S. and Nelson, C. R. (1981). 'A New Approach to Decomposition ofEconomic Time Series into Permanent and Transitory Components with ParticularAttention to Measurement of the "Business Cycle" ',JME 7, pp. 151-74.

Engle, R.F. and Granger, C. W J. (1987). 'Co-integration and Error Correction:Representation, Estimation and Testing', Econometrica, Vol. 55, pp. 251-76.

Granger, C. W J. (1986). 'Developments in the Study of Co-integration EconomicVariables', BULLETIN, Vol.48, pp. 2 13-28.

Granger, C. W. J. (1983). 'Co-integrated Variables and Error-Correcting Models',University of California Department of Economics Working Paper, pp. 83-113.

King, R., Plosser, C. and Stock, J. 'Stochastic Trends and Economics fluctuations',unpublished MS, 1987.

Rosenblatt, M. (1956). 'On Aggregation and Consolidation in Linear Systems', Tech-nical Report C, Department of Statistics, American University, Washington, D.C.

COINTEGRATION, ERROR CORRECTION, AND AGGREGATION 95Sargan, J. D. (1964). 'Wages and Prices in the United Kingdom: A Study in Econo-

metric Methodology', in Hart et al.(eds), Econometric A nalysis for National Econo-mic Planning, Butterworths, London.

Stock, J. H. and Watson, M. W. (1984). 'Testing for Common Trends', DiscussionPaper 1222, Harvard Institute of Economic Research.