Econometrica, Vol. 77, No. 5 (September, 2009), 1703–1704
CORRIGENDUM TO “BOOTSTRAP ALGORITHMS FOR TESTING AND DETERMINING THE COINTEGRATION RANK IN VAR MODELS” BY ANDERS RYGH SWENSEN1 THE CLAIM on page 1712 of my paper (Swensen (2006)) that the eigenvalues of the matrices Φj j = 0 1 r − 1 must be of modulus less than 1 because the eigenvalues of Φ = Φr have this property is false, as the following example shows. Let the vector autoregression (VAR) be given by −04 05 00 Xt = (1 1)Xt−1 + Xt−1 + εt −04 00 14 Then the matrix Φ=
02 05 14 −04 05 00 −04 00 14
has eigenvalues with modulus 093 087, and 087, while the matrix 05 00 Φ0 = Γ1 = 00 14 of course has eigenvalues 05 and 14. This affects the conclusions of Proposition 2, Corollary 1, Lemma 2, and Lemma 4. To make the conclusions of these enunciations correct, the following assumption must be included in addition to Assumption 1. ASSUMPTION 2: The eigenvalues of the matrices Φj j = 0 1 r − 1 defined in (A.9) must have modulus less than 1. It may be worth pointing out that the model considered for the numerical simulations reported in Section 4 is covered by Assumption 2. Then r = 1 and the eigenvalues of Φ0 equal zero. Another example is choosing Γ1 in the example above as any matrix having eigenvalues with modulus less than zero. Assumption 2 does, therefore, not imply that Proposition 2 is vacuous. Hence two natural responses arise. First, to determine how restrictive the additional condition is and, second, to formulate a bootstrap algorithm that is consistent only under Assumption 1 of the paper. For some thoughts on the second option, see Remarks 3.2 and 4.7 in Cavaliere, Rabek, and Taylor (2008). 1
Giuseppe Cavaliere and Anders Rahbek kindly made the author aware of the mistake.
© 2009 The Econometric Society
DOI: 10.3982/ECTA8201
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ANDERS RYGH SWENSEN REFERENCES
CAVALIERE, G., A. RABEK, AND A. M. R. TAYLOR (2008): “Testing for Co-Integration in Vector Autoregressions With Non-Stationary Volatility,” CREATES Report RP2008-50, Aarhus University. [1703] SWENSEN, A. R. (2006): “Bootstrap Algorithms for Testing and Determining the Cointegration Rank in VAR Models,” Econometrica, 74, 1699–1714. [1703]
Dept. of Mathematics, University of Oslo, P.O. Box 1053 Blindern, Oslo NO0316, Norway;
[email protected]. Manuscript received October, 2008; final revision received January, 2009.
