 # Time-varying cointegration, identification, and cointegration spaces

• and Vasco J. Gabriel

## Abstract

We derive the conditions under which time-varying cointegration leads to cointegration spaces that may be time-invariant or, in contrast, time-varying. The model of interest is a vector error correction model with arbitrary time-varying cointegration parameters. We clarify the role of identification and normalization restrictions and show that structural breaks in error-correction models may actually correspond to stable long-run economic relationships, as opposed to a single-equation setup, in which an identification restriction is imposed. Moreover, we show that, in a time-varying cointegrating relationship with a given number of variables and cointegration rank, there is a minimum number of orthogonal Fourier functions that most likely guarantees time-varying cointegrating spaces.

Corresponding author: Luis Filipe Martins, ISCTE – Business School, Av-das Forças Armadas, 1649-026 Lisboa, Portugal

## 4 Appendix: Proofs of Lemma 1 and Theorems

Proof of Lemma 1. For a pair t, s, the null space is of dimension dim(Nts(ξ))=(m+1)rrank(ξ)<(m+1)r because ξ≠0. See Strang (1988), for example. Hence, rank(ξ)<(m+1)r and the result follows noting that 0<rank(ξ) ≤ min {k, (m+1)r}.

So now the question is whether any of these solutions of Lemma 1 correspond to λt, λs≠0 that results in To prove Theorems 1 and 2 we need some new definitions and two auxiliary lemmas. Take as a generic solution to the system and let the number of free variables in xts≠0 be l=(m+1)rrank(ξ). In the case of k<(m+1)r, we have l=(m+1)rk, …,(m+1)r–1; whereas in the case of k≥(m+1)r and rank(ξ)<(m+1)r, we have l=1, …,(m+1)r–1. The system is now Pts λts=xts where xts≠ 0 is is 2r×1 and is a matrix (m+1)r × 2r, with r× 2r elements By defining the solutions λt, λs are found from the equations with in ℜr for i=1, …,m, where is given.

LEMMA A.1. For all ts, t, s=1, …,T, P1,T(t)≠ P1,T(s). When I=2, …,m<T, Pi,T(t)=Pi,T(s) for at least one pair t, s with ts. Moreover, for all t≠ s,t,s=1,…,T and(i=1, …,m – 1, Pi,T(t)≠ Pi+1,T(s).

PROOF: The time periods such that Pi,T(t)=Pi,T(s) are or with p=1,2, …, that is, or with p=1,2, …. These are integer numbers as long as If i=1 then which rules out all cases since |st| and s+t cannot be greater or equal than 2T. The number of pairs t, s that satisfy Pi,T(t)=Pi,T(s) increases with i. If i=2, then Pi,T(t)=Pi,T(s) for all s+t=1+T. If i≥3, then or with p=1,2, …, such that The last result follows from 0.5/i ∉ℵ’ for any i integer. Note that, as a corollary, if λt, λs≠ 0 then but, if then λt, λs are not necessarily both different from zero.

Lemma A.2. For any ts and fixed r,m > 0,rank(Pts)=2r

PROOF: Without loss of generality, take r=m=1. Then, has rank equal to 2 because P1,T(t)≠ P1,T(s) for any ts (see Lemma A.1.).

Proof of Theorem 1. In the system Pts λts=xts, where xts≠ 0, the number of unknowns does not exceed the number of equations, (m + 1)r≥2r and rank(Pts)=2r for all ts, m≥1, by Lemma A.2. By the same reasoning, rank(Pts, xts)=rank(Pts)=2r for all ts, if m=1. Thus, when m=1, in the cases of k<(m+1)r=2r and k≥(m+1)r=2r with rank(ξ)<(m+1)r=2r there is one solution to the system. Consequently, if both λt, λs≠ 0 with λt,≠ λs or if either λt or λs equals zero. By (21), λt=0 for some t,s,t≠s, if where ηts ∈{P1,T(s), P1,T(t)} and Clearly, when and (where λt=λs≠ 0). On the contrary, λt, λs≠0 with λtλs, for any t,s,ts, if and and Note that, in the previous case, can have up to r free variables. When the number of free variables in xts is at most r, whereas if for all r × r matrices ϒts the number of free variables in xts is at most 2r. Given that for ηts ∉ {P1,t(s), P1,t(t)} implies βt(1)≠ 0 with rank(ξ) ≤ r the result then follows from the necessary conditions in Lemma 1 for the system Proof of Theorem 2. With m≥2,rank(Pts, xts) is either 2r or 2r+1 Whenever rank(Pts, xts)=2r=rank(Pts) there is one solution to the system and consequently if both λt, λs≠ 0 with λtλs or if either λt or λs equals zero. Whenever rank(Pts, xts)=2r+1>rank(Pts) there is no solution to the system and therefore So that rank(Pts, xts)=2r, we need for some with at least one different from zero for each i. With (21), we conclude that depend on the free variables and with for m≥2. Hence, rank(Pts, xts)=2r, if for some non zero r × r matrices and and by setting and Here, and are given by (14). Similar to the previous Theorem, λt=0 for some t, s where ts, if where ηts ∈ {P1,T(s), P1,T(t)} and Hence, given that the solutions (24) imply βt(m)≠ 0 and that depend on the free variable for TI spaces, we have rank(ξ) ≤ r and the result follows from the necessary conditions in Lemma 1 for the system 1. 1

The identification restrictions can be made more general, given the usual notation, and taking any k and any r:

2. 2

One such case is the Purchasing Power Parity TV cointegration analysis in Bierens and Martins (2010). It most probably has TV cointegrating spaces since r=1, k=3 and m was always >5 according to the Hannan-Quinn criterion.

3. 3

This will form a unique solution by applying the transversality condition 4. 4

US data (from 1900 to 2006) is available from Robert Shiller’s webpage (www.econ.yale.edu/126shiller), where stock prices are January values for the Standard and Poor Composite Index, dividends are year-averages and both series are deflated by January values of the producer price index. Following several other authors, we do not include the latest available sampling period, as the deviations from the implied relationship are unusually large and persistent, albeit temporary.

## References

Bierens, H. J., and L. Martins. 2010. “Time Varying Cointegration.” Econometric Theory 26: 1453–1490.10.1017/S0266466609990648Search in Google Scholar

Donald, S. G., N. Fortuna, and V. Pipiras. 2007. “On Rank Estimation in Symmetric Matrices: The Case of Indefinite Matrix Estimators.” Econometric Theory 23: 1217–1232.10.1017/S0266466607070478Search in Google Scholar

Froot, K., and M. Obstfeld. 1991. “Intrinsic bubbles: the case of stock prices.” American Economic Review 81: 1189–1214.Search in Google Scholar

Granger, C. W. J. 2008. “Non-Linear Models: Where Do We Go Next - Time Varying Parameter Models?” Studies in Nonlinear Dynamics & Econometrics 12(3): 1–9.10.2202/1558-3708.1639Search in Google Scholar

Hansen, B. E. 1992. “Tests for Parameter Instability in Regressions with I(1) Processes.” Journal of Business and Economic Statistics 10: 321–335.Search in Google Scholar

Hansen, P. R. 2003. “Structural Changes in the Cointegrated Vector Autoregressive Model.” Journal of Econometrics 114: 261–295.10.1016/S0304-4076(03)00085-XSearch in Google Scholar

Johansen, S. 1995. “Likelihood-Based Inference in Cointegrated Vector Autoregressive Models.” Oxford: Oxford University Press.10.1093/0198774508.001.0001Search in Google Scholar

Kleibergen, F., and R. Paap. 2006. “Generalized Reduced Rank Tests using the Singular Value Decomposition.” Journal of Econometrics 133: 97–126.10.1016/j.jeconom.2005.02.011Search in Google Scholar

Maddala, G. S., and I-M. Kim. 1998. “Unit Roots, Cointegration and Structural Change.” Cambridge: Cambridge University Press.10.1017/CBO9780511751974Search in Google Scholar

Park, J. Y., and S. B. Hahn. 1999. “Cointegrating Regressions with Time Varying Coefficients.” Econometric Theory 15: 664–703.10.1017/S0266466699155026Search in Google Scholar

Strang, G. 1988. Linear Algebra and its Applications. 3rd ed. New York: Saunders HBJ.Search in Google Scholar

Villani, M. 2006. “Bayesian Point Estimation of the Cointegration Space.” Journal of Econometrics 134: 645–664.10.1016/j.jeconom.2005.07.008Search in Google Scholar

Published Online: 2013-04-11

©2013 by Walter de Gruyter Berlin Boston