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A distance measure between cointegration spaces
*
Rolf Larsson, Mattias Villani
Department of Statistics, Stockholm University S-106 91 Stockholm, Sweden Received 19 October 1999; accepted 15 July 2000
Abstract
A distinguishing feature of cointegration models, and many other multivariate models, is that only spaces spanned by parameter vectors are identified. We point out that traditional distance measures, such as the Euclidean measure, are not reasonable to use when measuring distances between spaces. This point has been either missed or ignored in many simulation studies where inappropriate distance measures have been used. We propose a simple measure based on the idea that the space spanned by the orthogonal complement of a matrix lies as far away as possible from the space spanned by the matrix itself. Several properties of this measure are derived. 2001 Elsevier Science B.V. All rights reserved.
Keywords: Cointegration; Distance measure; Simulation studies
JEL classification: C15; C22
1. Introduction
The analysis of cointegration in multivariate time series has been the object of an impressively large body of both theoretical and empirical research in econometrics during the last decade. The possibility to empirically estimate long run equilibrium relationships, the so called cointegration vectors, is one of the most important developments in modern macroeconometrics. Many estimators of the space spanned by the cointegration vectors (the cointegration space), which is what we can estimate uniquely, have been suggested and the maximum likelihood estimator of Johansen (1995) is the most widely used. Only large sample results are available for these estimators, however, and many simulation studies have therefore been conducted to compare their properties in smaller samples, see e.g. Ahn and Reinsel (1990); Gonzalo (1994) and Jacobson (1995). To measure the performance of an estimator of the unknown cointegration space, a measure of the distance between two cointegration
*Corresponding author. Tel.: 146-8-162578; fax:146-8-167511.
E-mail address: [email protected] (M. Villani).
spaces is clearly needed. Our purpose here is twofold; first, we point out that traditionally used distance measures, like the Euclidean metric, are not appropriate for this problem. Second, an alternative measure with desirable properties is proposed.
where t51, . . . ,T, X is a p-dimensional process,h jt h j´t are independent p-dimensional normal errors with expectation zero and covariance matrixV, the parameter matricesa andb are p3r,G1, . . . ,Gk21
are p3p, F is p3m and the dummy matrices D are mh jt 31. It is well known that a and b are unidentified without restrictions and that only sp(a) and sp(b) are estimable.
Estimation of sp(b), the cointegration space, is the central part in the statistical analysis of cointegration models. Analytical expressions for the (small sample) bias and standard error of the estimators of sp(b) are very rare and we are therefore referred to simulation studies if such estimators are to be compared. A typical simulation study proceeds as follows. All parameters of the ECM are given fixed values by the investigator and a sequence of processes are then generated a specified
ˆ
number of times. For each generated process, an estimate ofb, denoted by b in general, is computed
ˆ
by each estimation method. A distance measure, m(b,b), which measures the closeness of an estimate to the true, and known, b is computed for each method and then averaged over all generated processes. The estimation method which produces the smallest average distance is preferred over its competitors, everything else equal. There is always some controversy about the correct distance measure to use in simulation studies, but, as we will argue, the fact that only spaces spanned by vectors is estimable poses a new problem that has not received its deserved attention.
To see why the Euclidean metric is inappropriate for measuring the distance between spaces, let b1 and b2 be two p-dimensional vectors of unit length with an angle u between them, where 0#u, p. The squared Euclidean distance between b1 and b2 can be written
2
ib12b2i 5(b12b2)9(b12b2)52(12cosu),
9
9
9
since b b1 15b b2 251 and b b1 25cos u. Thus, the Euclidean distance is strictly increasing as u varies between 0 andpand therefore has the awkward consequence that asu approachesp, and thus
sp(b2) approaches sp(b1), ib12b2i does not approach zero but instead approaches its maximal value.
Since most simulation studies have been based on the Euclidean metric (or other distance measures with the same deficiency), it is likely that their results have been distorted by focusing directly on estimates of the elements in b instead of what we actually can estimate, or even interpret, which is
sp(b).
3. An alternative distance measure
Letb1 andb2 be two arbitrary orthonormal p3r matrices of full rank. Since any arbitrary full rank
matrix, b, can be made orthonormal and still belong to sp(b), the restriction to orthonormal matrices causes no reduction in generality. Our distance measure between b1 andb2 is based on the following decomposition of b2
b25b g1 11b g1' 2, (2)
where g1 is r3r and g2 is ps 2rd3r. Explicitly,
21
9
9
9
g15sb b1 1d b b1 25b b1 2
and
21
9
9
9
g25sb1'b1'd b1'b25b1'b2,
9
where b1' is the p3( p2r) orthogonal complement of b1, which is also normalized by b1'b1'5
Ip2r. In some sense, b1'is as far as we can get from b1 and it seems reasonable therefore to base the distance measure between b1 andb2 on some measure of the ‘size’ ofg2, see (Golub and van Loan,
9
9
1996, p. 76) for a similar idea. Note that, because of the normalizations, Ir5g g1 11g g2 2, and so, there is no need to take g1 into account. The most common size measure for a matrix is the (Frobenius) matrix norm (Harville, 1997, chapter 6)
1 / 2
iAi ;tr As 9Ad ,
and a natural distance measure between two cointegration spaces is therefore
1 / 2
9
dsb1,b2d; ig2i ;trsg g2 2d .
Thus, we suggest the following definition.
Definition 1. The distance between sp(b1) and sp(b2) is
1 / 2
9
9
dsb1,b2d;trsb b2 1'b1'b2d ,
where all matrices involved have been made orthonormal.
Note that the definition of dsb1,b2dis based on the decomposition ofb2 in (2) when it could equally well have been based on the decomposition ofb1. It is therefore essential to prove that this choice is without consequence for dsb1,b2d, or, in other words, that the proposed distance is symmetric in its two arguments. This and other properties are proved in the next theorem.
Theorem 2. d(b1,b2) has the following properties:
(i) dsb1,b2d is invariant under the choice of different orthonormal versions of b1'. (ii) d(b1,b2)50 if.f. b1[sp(b2).
(iv) d(b1,b3)#d(b1,b2)1d(b2,b3). (v) d(b1,b2)5d(b1',b2').
2
(vi) 0#d(b1,b2) #min r, ps 2r .d
(vii) For r#p2r, maximum of d(b1,b2) is obtained if.f. b2[sp(b1'). For r.p2r, the
maximum is reached if.f. b1'[sp(b2). Thus, the upper bound in (vi) is always attainable. (viii)
Proof. (i) Any other orthonormal version of b1' may be written
˜
The idea is to square both sides of (3) to get
2
To this end, the Cauchy-Schwarz inequality (Harville (1997), chapter 6)
tr As 9Bd#iAiiBi
and the proof of (6), and hence of (4), is completed. (v) Since (b1')'5b1
and their orthogonal complements are assumed to be of full rank and orthonormal. It follows that
(vii) From (vi), if r#p2r, the maximum is obtained if.fg150, which implies thatb2[sp(b1').
Note that properties (ii), (iii) and (iv) in Theorem 2 together imply that d(b1,b2) is a metric. It is interesting to compare property (viii) to the Euclidean distance between b1 and b2
2
9
Thus, the Euclidean measure only takes the angles between b1i and b2i, i51,2, . . . ,r, into account, whereas our measure considers the angles between all pairs of columns ofb1 andb2. In addition, the way the angles enter the two measures differ considerably;ib12b2i is strictly increasing in each of the uii, whereas d(b1,b2) decreases as any of the uij approaches either 0 or p, given that all other angles remain the same (which is not always possible since theuij are sometimes linked to each other). To see why the latter behavior is more appropriate, let us return to the case of a single cointegration vector discussed in Section 2, where]]] u was the angle betweenb1 andb2. From Theorem 2 (viii), we
2
We have proposed a way of measuring distances between cointegration spaces, and shown that this measure fulfills many desired properties.
21 / 2
then a normalization of time series, for example by the transformation Yt5V X , may be needed.t
If V is unavailable then an estimate can be used in its place.
No properties of the cointegration model have been used in the derivation of our measure, and so it is, of course, applicable to many other multivariate models where only the space spanned by a set of vectors is estimable, e.g. the common factor model (Anderson, 1984). However, in certain situations, for example if the cointegrating space is restricted, our measure will need to be modified. To work out such modifications, as well as to apply our measure empirically and in simulation studies, are interesting topics for future research.
Acknowledgements
The authors would like to thank Daniel Thorburn for valuable comments. Mattias Villani was financially supported by the Swedish Council of Research in Humanities and Social Sciences (HSFR).
References
Ahn, S.K., Reinsel, G.C., 1990. Estimation for partially nonstationary multivariate autoregressive models. J. Am. Statist. Assoc. 85, 813–823.
Anderson, T.W., 1984. An Introduction To Multivariate Statistical Analysis. Wiley, New York.
Golub, G.H., van Loan, C.F., 1996. Matrix Computations, 3rd Edition. John Hopkins University Press, Baltimore. Gonzalo, J., 1994. Five alternative methods of estimating long-run equilibrium relationships. Journal of Econometrics 60,
203–233.
Harville, D.A., 1997. Matrix Algebra From A Statistician’s Perspective. Springer-Verlag, New York.
Jacobson, T., 1995. Simulating small sample properties of the maximum likelihood cointegration model: estimation and testing. Finnish Economic Papers 8, 96–107.
Johansen, S., 1995. Likelihood-Based Inference in Cointegrated Vector Autoregressive Models. Oxford University Press, Oxford.
