A REVIEW OF SYSTEMS COINTEGRATION TESTS

Abstract

The literature on systems cointegration tests is reviewed and the various sets of assumptions for the asymptotic validity of the tests are compared within a general unifying framework. The comparison includes likelihood ratio tests, Lagrange multiplier and Wald type tests, lag augmentation tests, tests based on canonical correlations, the Stock-Watson tests and Bierens' nonparametric tests. Asymptotic results regarding the power of these tests and previous small sample simulation studies are discussed. Further issues and proposals in the context of systems cointegration tests are also considered briefly. New simulations are presented to compare the tests under uniform conditions. Special emphasis is given to the sensitivity of the test performance with respect to the trending properties of the DGP.

Keywords:

• Systems cointegration tests
• LR tests
• Nonparametric tests
• Asymptotic power
• Small sample simulations

ACKNOWLEDGMENTS

We are grateful to Christian Müller, Carsten Trenkler and Ralf Brüggemann for helping with the computations and we thank Jesus Gonzalo, Jürgen Wolters, Jörg Breitung and two anonymous referees for commenting on earlier versions of the paper. Part of this research was carried out while the first author had a position at De Nederlandsche Bank, Amsterdam, and another part while the third author was visiting the Humboldt University in Berlin. Financial support by the Deutsche Forschungsgemeinschaft, Sonderforschungsbereich 373, and the European Commission under the Training and Mobility of Researchers Programme (contract No. ERBFMRXCT980213) is gratefully acknowledged.

Notes

¹The table has to be used with caution because of a flaw in the legend.

² LM _ta(1)(r ₀) and LM _ta(2)(r ₀) correspond to LM(r ₀) and LM _*(r ₀), respectively, in Lütkepohl and Saikkonen (2000).

³The table has to be used with caution because of a flaw in the legend.

⁴In the proof of her Theorem 1, Quintos makes use of the inverse of the matrix Ω _e·ε2 , the long-run covariance matrix of the process e _t = [e′_1t ,e′_2t ]′ given ε_2t . These processes are stationary by assumption and satisfy Δy _1t = β′αy _1t−1 + e _1t and ε_2t = β′_⊥αy _1t−1 + e _2t , where y _1t = β′y _t in the notation of our paper. If r > 0, the definitions and the assumed nonsingularity of the matrix β′α show that − β′_⊥ α(β′α)^{− 1} e _1t + e _2t − ε_2t = − β′_⊥ α (β′α)^{− 1} Δy _1t . By the stationarity of the process y _1t , the long-run covariance matrix of Δy _1t vanishes and it follows from the foregoing equation that the rank of the matrix Ω _{e · ε2} is at most n − r.

⁵The table has to be used cautiously because the legend is not fully clear and the user is expected to add minus signs to some of the entries.

⁶Further results on different sample sizes, variations in the parameters of the DGPs and results on further processes are available from the first author upon request.

⁷Version 1, 1991.

⁸Critical values for T=100 from Table 1 of Yang & Bewley (1996) have been used for these tests.