I(2) Cointegration Analysis in the Presence of Deterministic Shifts

This paper investigates limit theory for the likelihood analysis of an I(2) cointegrated vector autoregressive (VAR) model in the presence of deterministic shifts. A log likelihood ratio (logLR) test statistic for integration indices is considered, and it is demonstrated that the asymptotic distribution of the statistic is given in the form of a generalised Dicky-Fuller type distribution. A logLR test statistic for restrictions on long-run and slope coe¢ cients is also investigated, and it is proved that the statistic is asymptotically 2 distributed. Finally, an empirical analysis of macroeconomic data in Japan is performed using the I(2) VAR model subject to a deterministic shift. KEY WORDS: Cointegration, I(2), Maximum Likelihood, Likelihood Ratio Test, Deterministic Shifts. 1 Introduction The main objectives of this paper are twofold. The …rst objective is to investigate limit theory for the likelihood analysis of an I(2) cointegrated vector autoregressive (VAR) model in the presence of deterministic shifts. The second is to conduct an empirical analysis of macroeconomic data in Japan using the I(2) cointegrated VAR model. The introductory section gives an overview of the literature on I(2) cointegrated VAR models, and then describes the most signi…cant aspects of the paper. As a statistical representation of integrated time series data, an I(1) cointegrated VAR model is introduced by Johansen (1988, 1996), and it has gained great popularity in theoretical as well as applied econometrics. See Juselius (2006) for extensive empirical research using cointegrated VAR models, and also see Doornik and Hendry (2006) and Kurita (2007) for modelling dynamic econometric systems based on I(1) cointegration analysis. Correspondence to: Faculty of Economics, Fukuoka University, Bunkei Center Building, 8-19-1 Nanakuma, Johnanku, Fukuoka, 814-0180, Japan. E-mail: [email protected]