Abstract
Many economic events involve initial observations that substantially deviate from long-run steady state. Such initial conditions are known to affect the power of univariate unit root tests diversely, whereas their impact on multivariate tests is largely unknown. This paper investigates the impact of the initial condition on the power of tests for cointegration rank, such as Johansen’s widely used likelihood ratio test, tests with prior adjustment for deterministic terms, and a test based on the eigenvalues of the companion matrix. We find that the power of the likelihood ratio test is increasing in the magnitude of the initial condition, whereas the power of the other tests is generally decreasing. We exploit these findings in an application to price convergence.
Similar content being viewed by others
References
Ahlgren N, Nyblom J (2008) Tests against stationary and explosive alternatives in vector autoregressive models. J Time Ser Anal 29: 421–443
Bernard AB, Durlauf SN (1996) Interpreting tests of the convergence hypothesis. J Econom 71: 161–173
Brada JC, Kutan AM, Zhou S (2005) Real and monetary convergence between the European Union’s core and recent member countries: a rolling cointegration approach. J Bank Finance 29: 249–270
Busetti F, Fabiani S, Harvey A (2006) Convergence of prices and rates of inflation. Oxf Bull Econ Stat 68(Suppl): 863–877
Busetti F, Forni L, Harvey A, Venditti F (2007) Inflation convergence and divergence within the European Monetary Union. Int J Cent Bank 3: 95–121
Catani P (2008) Unit root tests and the initial condition (in Swedish). Mimeo, Hanken School of Economics, Helsinki
Elliott G, Müller UK (2006) Minimizing the impact of the initial condition on testing for unit roots. J Econom 135: 285–310
Elliott G, Rothenberg T, Stock J (1996) Efficient tests for an autoregressive unit root. Econometrica 64: 813–836
Hansen PR (2005) Granger’s representation theorem: a closed-form expression for I(1) processes. Econom J 8: 23–38
Harvey DI, Leybourne SJ (2005) On testing for unit roots and the initial observation. Econom J 8: 97–111
Harvey DI, Leybourne SJ (2006) Power of a unit-root test and the initial condition. J Time Ser Anal 27: 739–752
Harvey DI, Leybourne SJ, Taylor AMR (2009) Unit root testing in practice: dealing with uncertainty over the trend and initial condition. Econom Theory 25: 587–636
Horvath MTK, Watson MW (1995) Testing for cointegration when some of the cointegrating vectors are prespecified. Econom Theory 11: 984–1014
Hubrich K, Lütkepohl H, Saikkonen P (2001) A review of systems cointegration tests. Econom Rev 20: 247–318
Johansen S (1991) The power function of the likelihood ratio test for cointegration. In: Gruber J (eds) Econometric decision models: new methods of modelling and applications. Springer, New York, pp 323–335
Johansen S (1996) Likelihood-based inference in cointegrated vector autoregressive models. Oxford University Press, Oxford
Johansen S, Juselius K (1990) Maximum likelihood estimation and inference on cointegration: with applications to the demand for money. Oxf Bull Econ Stat 52: 169–210
Koukouritakis M (2009) Testing the purchasing power parity: evidence from the new EU countries. Appl Econ Lett 16: 39–44
Lütkepohl H, Saikkonen P (2000) Testing for the cointegrating rank of a VAR process with a time trend. J Econom 95: 177–198
Monacelli T (2005) Monetary policy in a low pass-through environment. J Money Credit Bank 37: 1047–1066
Müller UK, Elliott G (2003) Tests for unit roots and the initial condition. Econometrica 71: 1269–1286
Palomba G, Sarno E, Zazzaro A (2009) Testing similarities of short-run inflation dynamics among EU-25 countries after the Euro. Empir Econ 37: 231–270
Pedroni P (2001) Purchasing power parity tests in cointegrated panels. Rev Econ Stat 83: 727–731
Perron P, Campbell JY (1993) A note on Johansen’s cointegration procedure when trends are present. Empir Econ 18: 777–789
Phillips PCB, Magdalinos T (2009) Unit root and cointegrating limit theory when initialization is in the infinite past. Econom Theory 25: 1682–1715
Rahbek AC (1994) The power of some multivariate cointegration tests. Mimeo, Discussion Paper, Institute of Mathematical Statistics, University of Copenhagen, Copenhagen
Saikkonen P, Lütkepohl H (1999) Local power of likelihood ratio tests for the cointegrating rank of a VAR process. Econom Theory 15: 50–78
Saikkonen P, Lütkepohl H (2000) Trend adjustment prior to testing for the cointegrating rank of a vector autoregressive process. J Time Ser Anal 21: 435–456
Saikkonen P, Luukkonen R (1997) Testing cointegration in infinite order vector autoregressive processes. J Econom 81: 93–126
Solakoglu EG (2006) Testing purchasing power parity hypothesis for transition economies. Appl Financ Econ 16: 561–568
Tanaka K (1996) Time series analysis. Nonstationary and noninvertible distribution theory. Wiley, New York
Taylor MP, Peel DA, Sarno L (2001) Nonlinear mean-reversion in real exchange rates: toward a solution to the purchasing power parity puzzles. Int Econ Rev 42: 1015–1042
Toda HY (1994) Finite sample properties of likelihood ratio tests for cointegrating ranks when linear trends are present. Rev Econ Stat 76: 66–79
Toda HY (1995) Finite sample performance of likelihood ratio tests for cointegrating ranks in vector autoregressions. Econom Theory 11: 1015–1032
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Ahlgren, N., Juselius, M. Tests for cointegration rank and the initial condition. Empir Econ 42, 667–691 (2012). https://doi.org/10.1007/s00181-010-0442-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00181-010-0442-z