Multicointegration, polynomial cointegration

In this paper we model the multicointegration relation, allowing for one structural break. Since multicointegration is a particular case of polynomial or I(2) cointegration, our proposal can also be applied in these cases. The paper proposes the use of a residualbased Dickey-Fuller class of statistic that accounts for one known or unknown structural break. Finite sample performance of the proposed statistic is investigated by using Monte Carlo simulations, which reveals that the statistic shows good properties in terms of empirical size and power. We complete the study with an empirical application of the sustainability of the US external deficit. Contrary to existing evidence, the consideration of one structural break leads to conclude in favour of the sustainability of the US external deficit. Keywords: I(2) processes, multicointegration, polynomial cointegration, structural break, sustainability of external deficit JEL Codes: C12, C22 1 The authors gratefully acknowledge the financial support of the Ministerio de Ciencia y Tecnología from the grant SEJ2005-08646/ECON. AQR-IREA Research Group. Department of Econometrics, Statistics and Spanish Economy. University of Barcelona. Av. Diagonal, 690. 08034 Barcelona. Tel: +34 934021826, fax: +34 934021821, and e-mails: [email protected] and [email protected]. 1 Introduction It is generally true that any linear combination of I(1) series is also I(1) by the dominant property of the stochastic trends. However, it is possible that linear combinations of I(1) stochastic processes generate an I(0) time series, in which case the time series are said to be cointegrated in the sense of Engle and Granger (1987) –hereafter, the so-called …rst level of cointegration. Moreover, it could also be possible that the cumulated cointegrated residuals, I(1) by de…nition, cointegrate with the I(1) original variables. In this case, a deeper level of cointegration –henceforth, second level of cointegration –occurs between the original time series. Granger and Lee (1989) denote this sort of cointegration as “multicointegration”and show that it is a long-run relationship that might be expected to occur in economics. One appealing example is given by the consumption life-cycle hypothesis. If income and consumption are I(1) variables the economic theory establishes that saving must be stationary, so that income and consumption have to be cointegrated. Furthermore, if we cumulate saving, which can be seen as the stock of consumers’wealth, the life-cycle hypothesis predicts that wealth must cointegrate with consumption –see Siliverstovs (2003) for an application of multicointegration to the analysis of the lifecycle hypothesis of consumption for the US. Other relevant applications can be found in the literature including the analysis of production and sales –Granger and Lee (1989) – housing units started and new housing units completed –see Lee (1992) and Engsted and Haldrup (1999) –the Ricardian Equivalence –see Leachman (1996) –the twin de…cits hypothesis –see Leachman and Francis (2002) –and the sustainability of …scal practices –see Leachman, Bester, Rosas and Lange (2005). The common feature that all these empirical applications share is that they involve stock-‡ow relationships among economic variables. The current de…nition of multicointegration assumes invariant parameters. However, the longer the time period the higher the probability that structural breaks appear affecting the behaviour of time series. Nowadays, it is well known that this feature may cause bias in the statistical inference that is obtained when dealing with time series – Perron (1989). To this end, unit root and stationarity tests have been designed to accom2 modate the presence of structural breaks. Besides, cointegration testing procedures have also been devised to account for the presence of instability either in the deterministic and/or in the cointegrating vector of the long-run relationship. Perron (2006) o¤ers a comprehensive overview of the …eld. As described above, the concept of multicointegration is naturally derived from the de…nition of cointegration, which in turn might be subject to the e¤ects of structural breaks –see Gregory and Hansen (1996). Therefore, the standard framework of analysis de…ned in Granger and Lee (1989), and Engsted, Gonzalo and Haldrup (1997) should be extended to accommodate the presence of structural breaks. In this paper we have addressed this issue dealing with the di¤erent speci…cations that may arise depending on whether the structural break a¤ects only the …rst and/or the second level of cointegration. Once the models are de…ned, we propose to test the null hypothesis of non-multicointegration without structural break against the alternative hypothesis of multicointegration with a structural break using a residual-based Dickey-Fuller class statistic. It is worth mentioning that though the set-up that is used in this paper is bases on the multicointegration framework, one appealing feature of our proposal is that it can be used to analyze longrun relationships among I(2) stochastic processes, provided that multicointegration can be understood as I(2) cointegration or polynomial cointegration. To the best of our knowledge, this has not been previously addressed in the literature. The paper provides di¤erent sets of critical values depending on the number and nature of the stochastic regressors involved in the model and on the deterministic speci…cation that is used. Our proposal is illustrated through the investigation of the sustainability of the external de…cit in the US. This topic has attracted the attention of economists both from a theoretical and empirical point of view, especially in recent times where the external de…cit of the US economy has been increasing. Previous empirical approaches in Leachman and Francis (2000, 2002) have used the multicointegration framework to analyze the relationship between ‡ow variables (exports and imports) and the stock variable (foreign debt). However, none of them has considered the presence of structural breaks in the analysis, which is shown to lead to misleading conclusions. The statistics that have been