We respond to Sornette and Johansen’s criticisms of our findings regarding log-periodic precursors to financial crashes. Included in this paper are discussions of the Sornette-Johansen theoretical paradigm, traditional methods of identifying log-periodic precursors, the behavior of the first differences of a log-periodic price series, and the distribution of drawdowns for a securities price.

This paper suggests a bootstrap testing procedure for determining the rank of cointegrated systems. The properties of the new testing procedure are investigated using Monte Carlo techniques. The performance of the test compares favourably to that of the widely used procedures for determining cointegration rank proposed by Johansen (1988). JEL classi cation: C12; C15; C32.

Suvi Larjavaara*, Joachim Schüz, Anthony Swerdlow, Maria Feychting, Christoffer Johansen, Susanna Lagorio, Tore Tynes, Lars Klaeboe, Sven Reidar Tonjer, Maria Blettner, Gabriele Berg-Beckhoff, Brigitte Schlehofer, Minouk Schoemaker, Juliet Britton, Riitta Mäntylä, Stefan Lönn, Anders Ahlbom, Olof Flodmark, Anders Lilja, Stefano Martini, Emanuela Rastelli, Antonello Vidiri, Veikko Kähärä, Jani R...

We show that the order of integration of a vector autoregressive process is equal to the difference between the multiplicity of the unit root in the characteristic equation and the multiplicity of the unit root in the adjoint matrix polynomial. The equivalence with the standard I(1) and I(2) conditions (Johansen, 1996) is proved and polynomial cointegration discussed in the general setup.

This paper studies the effects of increasing the frequency of observation and the data span on the Johansen cointegration tests. The ability of the tests to detect cointegration depends more on the total sample length than the number of observations.  2000 Elsevier Science S.A. All rights reserved.