A distance measure between cointegration spaces

Transcription

1 Economics Letters 70 (001) locate/ econbase A distance measure between cointegration spaces Rolf Larsson, Mattias Villani* Department of Statistics, Stockholm University S Stockholm, Sweden Received 1 October 1; accepted 15 July 000 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. 001 Elsevier Science B.V. All rights reserved. Keywords: Cointegration; Distance measure; Simulation studies JEL classification: C15; C 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 (15) 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 (10); Gonzalo (14) and Jacobson (15). To measure the performance of an estimator of the unknown cointegration space, a measure of the distance between two cointegration *Corresponding author. Tel.: ; fax: address: mattias.villani@stat.su.se (M. Villani) / 01/ $ see front matter 001 Elsevier Science B.V. All rights reserved. PII: S (00)0034-

2 R. Larsson, M. Villani / Economics Letters 70 (001) 1 7 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.. Motivation Consider the error correction model (ECM) k1 DX 5 abx 1O G DX 1 FD 1, (1) t t1 i ti t t i51 where t 5 1,...,T, hxtj is a p-dimensional process, h tj are independent p-dimensional normal errors with expectation zero and covariance matrix V, the parameter matrices a and b are p 3 r, G 1,...,Gk1 are p 3 p, F is p 3 m and the dummy matrices hd t j are m 3 1. 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 number of times. For each generated process, an estimate of b, denoted by ˆ b in general, is computed by each estimation method. A distance measure, m(b,b ˆ ), 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 b 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 b can be written ib1 bi 5 (b1 b )(b1 b ) 5 (1 cos u ), since b1b15 bb5 1 and b1b5 cos u. Thus, the Euclidean distance is strictly increasing as u varies between 0 and p and therefore has the awkward consequence that as u approaches p, and thus sp(b ) approaches sp(b 1), ib1 bi 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 ). The importance of a distance measure between cointegration spaces goes far beyond its use in simulation studies, see e.g. Villani (000), where the metric presented in the next section is used to derive posterior location and variation measures for a Bayesian analysis of cointegration.

3 3. An alternative distance measure R. Larsson, M. Villani / Economics Letters 70 (001) Let b1 and b be two arbitrary orthonormal p 3 r 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 and b is based on the following decomposition of b b 5 bg 1 b g, () 1 1 where g is r 3 r and g is and s 1 1d b1b5 b1b g 5 b b 1 s d g 5 b b b b 5 b b, sp rd3 r. Explicitly, where b is the p 3 ( p r) orthogonal complement of b 1, which is also normalized by b b 5 I pr. In some sense, b is as far as we can get from b1 and it seems reasonable therefore to base the distance measure between b1 and b on some measure of the size of g, see (Golub and van Loan, 16, p. 76) for a similar idea. Note that, because of the normalizations, Ir 5 g 1g11 g g, 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, 17, chapter 6) 1/ iai;tr AA, s d and a natural distance measure between two cointegration spaces is therefore d b,b ;ig i;tr g g 1/ s 1 d s d. Thus, we suggest the following definition. Definition 1. The distance between sp(b 1) and sp(b ) is d b,b ; tr bb b b 1/ s d s d, 1 where all matrices involved have been made orthonormal. Note that the definition of dsb 1,bd is based on the decomposition of b in () when it could equally well have been based on the decomposition of b 1. It is therefore essential to prove that this choice is without consequence for dsb 1,b d, 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. d(b,b ) has the following properties: 1 (i) dsb 1,bd is invariant under the choice of different orthonormal versions of b. (ii) d(b 1,b ) 5 0 if.f. b 1[ sp(b ). (iii) d(b 1,b ) 5 d(b,b 1).

4 4 R. Larsson, M. Villani / Economics Letters 70 (001) 1 7 (iv) d(b 1,b 3) # d(b 1,b ) 1 d(b,b 3). (v) d(b 1,b ) 5 d(b,b ' ). (vi) 0 # d(b 1,b ) # minsr, p r d. (vii) For r # p r, maximum of d(b 1,b ) is obtained if.f. b [ sp(b ). For r. p r, the maximum is reached if.f. b [ sp(b ). Thus, the upper bound in (vi) is always attainable. (viii) r r 1 ij i51 j51 d (b,b ) 5 r OOcos u, where u is the angle between the ith column of b and the jth column of b. ij 1 Proof. (i) Any other orthonormal version of b may be written b 5 b A, where A is p r -square and of full rank. Further, because b and b are orthonormal, we have s d I 5 b b 5 Ab b A 5 AA, pr 1 and so, we must have A5A. In other words, A is orthogonal and g ; b b fulfills g g 5 b b b b 5 bb AAb b 5 bb b b 5 gg, which proves the result. (ii) Assume that b [ sp(b 1). Then, b5 bg 1 1 and thus g5 0, which implies that d(b 1,b ) 5 0. Conversely, assume that d(b 1,b ) 5 0, then g5 0 and therefore b5 bg. 1 1 Thus, b [ sp(b 1). (iii) From the definition we have d(b,b ) 5 tr bb b b 1/ s d. Using the relation bb 1 11 b b 5 Ip we can write 1 1/ 1/ 1/ d(b 1,b ) 5 trfb (I p bb 1 1)b g 5 tr(ir bb1b1b ) 5 tr(ir b1bbb 1) 5 d(b,b 1). (iv) We have to show the inequality ib b3i # ib bi 1 ib' b3i. (3) The idea is to square both sides of (3) to get ib b i # ib b i 1 ib b iib b i 1 ib b i, (4) 3 ' 3 ' 3 and then to show this inequality. Now, using Ip 5 bb 1 b' b ', we get implying b b35 b bbb31 b b' b' b 3,

5 3 3 3 R. Larsson, M. Villani / Economics Letters 70 (001) ib b i 5 trhsb b dsb b dj 5 trhb bbb3b3bb b j1 trhb bbb3b3b' b' b j 1 trhb b b b bb b b j. (5) ' ' 3 3 ' ' Here, because bb 3 35 Ip b3' b 3', the first term on the r.h.s. may be expressed as trhb bbsi p b3' b3' dbbb j5 trhb bbb j trhb bbb3' b3' bbb j # trhb b bb j5 ib b i, and it can similarly be shown that the third term on the r.h.s. of (5) is bounded by ib remains to show that b i. Thus, it trhb b bb bb b b j# ib b iib b i. (6) 3 3 ' ' ' 3 To this end, the Cauchy-Schwarz inequality (Harville (17), chapter 6) implies trsab d# iaiibi trhsb b bb dsbb b b dj # ibb bb iibb b b i. (7) 3 3 ' ' 3 3 ' ' But by definition, ib3bbb i equals the first term on the r.h.s. of (5),which we have shown to be bounded by ib bi. Similarly, we may relate ib3b' b' b i to the third term on the r.h.s. of (5) to see that it is bounded by ib b i. Thus, by (7), ' 3 trhsb b bb dsbb b b dj # ib b iib b i, 3 3 ' ' ' 3 and the proof of (6), and hence of (4), is completed. (v) Since (b )' 5 b1 d(b,b ) 5 tr b b bb 1/5tr bb b b 1/ s d s d 5 d(b,b ) 5 d(b,b ). ' ' 1 1 ' 1 ' ' (vi) As above, we have the representation b5 bg 1 11 b g where g5 b b. As usual, b 1, b and their orthogonal complements are assumed to be of full rank and orthonormal. It follows that r 5 trsbb d5 trsgg d1 trsgg d, 1 1 and so, d(b 1,b ) ; trsgg d# r with equality if.f. g15 0, which is possible if.f. r # p r (otherwise, b 5 b g is not of full rank). On the other hand, we may write b 5 b' h1 1 bh where h 5 bb 5 g. Hence, p r 5 trsb b d5 trshh d1 trshh d, 1 1 implying d(b 1,b ) ; trsgg d5 trshh d# p r with equality if.f. h15 0, which is possible if.f. p r # r. ' 3

6 6 R. Larsson, M. Villani / Economics Letters 70 (001) 1 7 (vii) From (vi), if r # p r, the maximum is obtained if.f g15 0, which implies that b [ sp(b ). For r. p r, the maximum is reached if.f. h15 0, which implies that b [ sp(b ). (viii) From the definition Define b d (b,b ) 5 tr(bb b b ) 5 tr[b (I bb )b ] 5 r tr(bb bb ). (8) ij 1 p as the jth column of b, i 5 1,. Then i b11b1 b11 b??? b11 br cosu11 cosu 1??? cosu 1 1r b b b b??? b br cosu1 cosu??? cosur b1b 5? 5?, () : :?? : : :?? : b b b b??? b b cosu cosu??? cosu 1r 1 1r 1r r r1 r rr since ab 5 iaiibicosu, for any vectors a and b with angle u and ib i 5 1 for i 5 1, and j 5 1,...,r. ij Thus, from (8) r r 1 ij i51 j51 d (b,b ) 5 r OOcos u. h Note that properties (ii), (iii) and (iv) in Theorem together imply that d(b 1,b ) is a metric. It is interesting to compare property (viii) to the Euclidean distance between b and b ib b i 5 tr[(b b )(b b )] 5 r tr(bb ), since tr(b1b 1) 5 tr(bb ) 5 r. Using () we obtain S r 1 ii i51 D ib b i 5 r O cos u. 1 Thus, the Euclidean measure only takes the angles between b1i and b i, i 5 1,,...,r, into account, whereas our measure considers the angles between all pairs of columns of b1 and b. In addition, the way the angles enter the two measures differ considerably; ib1 bi is strictly increasing in each of the u ii, whereas d(b 1,b ) 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 the uij 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, where u was the angle between b1 and b. From Theorem (viii), we know that d(b,b ) 5Π]]] 1 cos u 5 sin u. Thus, d(b,b ) approaches zero as u approaches either 0 or 1 1 p, which makes sense since in both cases sp(b ) approaches sp(b ). Note also that d(b,b ) attains 1 1 its maximum for u 5p/, which is the angle that makes b and b orthogonal Discussion We have proposed a way of measuring distances between cointegration spaces, and shown that this measure fulfills many desired properties. If the proposed distance measure is applied to a system of time series with widely differing scales

7 R. Larsson, M. Villani / Economics Letters 70 (001) then a normalization of time series, for example by the transformation Yt5 V X t, may be needed. 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, 184). 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. 1/ 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., 10. Estimation for partially nonstationary multivariate autoregressive models. J. Am. Statist. Assoc. 85, Anderson, T.W., 184. An Introduction To Multivariate Statistical Analysis. Wiley, New York. Golub, G.H., van Loan, C.F., 16. Matrix Computations, 3rd Edition. John Hopkins University Press, Baltimore. Gonzalo, J., 14. Five alternative methods of estimating long-run equilibrium relationships. Journal of Econometrics 60, Harville, D.A., 17. Matrix Algebra From A Statistician s Perspective. Springer-Verlag, New York. Jacobson, T., 15. Simulating small sample properties of the maximum likelihood cointegration model: estimation and testing. Finnish Economic Papers 8, Johansen, S., 15. Likelihood-Based Inference in Cointegrated Vector Autoregressive Models. Oxford University Press, Oxford. Villani, M., 000. Aspects of Bayesian Cointegration. Ph.D. Thesis, Department of Statistics, Stockholm University.