Can you do the wrong thing and still be right? Hypothesis testing in I(2) and near-I(2) cointegrated VARs

ABSTRACT

In this paper, we investigate the small-sample performance of LR tests on long-run coefficients in the I(2) model; we focus on a comparison between I(2) and near-I(2) data, i.e. I(1) data with a second root very close to unity, and report the results of some Monte Carlo experiments. With near-I(2) data, the finite-sample properties of the tests are (i) similar to those found with genuine I(2) data, (ii) systematically superior to those of the analogous tests constructed in the I(1) model, even if the latter is, in principle, correctly specified and the former is not. Therefore, there seems to be strong support to the idea that, in practice, modelling near-I(2) data using the I(2) model may be a good idea, despite the inherent misspecification.

KEYWORDS:

• Cointegration
• I(2)
• near-I(2)
• hypothesis testing

JEL CLASSIFICATION:

• C12
• C13

Acknowledgement

We would like to thank Gunnar Bårdsen, Luca Fanelli, Massimo Franchi, Søren Johansen, Katarina Juselius and two anonymous referees for their suggestions and criticisms. The usual disclaimers, of course, apply.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

¹ See, e.g. Juselius (2006), p. 333. From the empirical point of view this may be important, e.g. in asset markets, where error-increasing medium-run relations may be associated with error-correcting long-run ones.

² A similar question has been addressed in the I(1) case by (Johansen 2006a) in his critical appraisal of the standard methods used to estimate empirical DSGE models.

³ Applications of the I(2) model are rather scant: besides Bacchiocchi and Fanelli (2005), some examples are Kongsted (2003), Nielsen and Bowdler (2006), and more recently the empirical section in Kurita, Nielsen, and Rahbek (2011).

⁴ Note also that univariate unit root tests may have a low ability to detect a double unit root when the shocks to the drift term of the differenced process (which generate the second unit root) are actually small compared to those to the differenced process (see Juselius 2013). Hence, these tests are likely to be somehow biased against the I(2) hypothesis.

⁵ To exclude orders of integration higher than two, an additional rank condition has to hold; see Johansen (1992).

⁶ Clearly, greater generality can easily be achieved by adding deterministic terms and/or lags of Δ²X_t to the right-hand side of Equation (1), but this is totally unnecessary in the present context.

⁷ The phrase ‘unrestricted roots’ is meant as a short form of ‘roots of the autoregressive polynomial I(1 – z)^{2 –} Π₁z – Γz (1 – z) when estimated unrestrictedly’.

⁸ We thank an anonymous referee for drawing our attention to this point

⁹ For instance, since any change of the long-run parameters would affect in an unknown way the adjustment coefficients, in this case we will not be able to evaluate the power of the test.

¹⁰ In Elliot (1998), the size distortions in the tests on the cointegration vectors are dependent on the cross correlation in the error term as this leads to endogeneity bias. However, inference on the cointegration vectors in the VAR is invariant to multiplication by full rank matrices.

¹¹ For instance, to the best of our knowledge, all simulation studies of tests on the cointegrating coefficients in the I(1) VAR take the cointegrating rank as known, and this is also typically the case for the type of deterministic kernel.

¹² We thank Gunnar Bårdsen for his suggestion to try this.

¹³ The second set of restrictions, on the polynomial cointegrating vector, is required in the I(2) case to ensures that the (reduced) rank of the Π matrix in the I(2) system is preserved in the real-transformed system.

¹⁴ Note: we do not estimate the model exactly as Equation (11); instead of the β₁ vector (which is orthogonal to β) what we estimate is some other γ vector (not necessarily orthogonal to β). For the purposes of testing the hypothesis of our interest, though, this is irrelevant.

¹⁵ Power was not evaluated for this DGP: changing the long-run parameters would of course require a corresponding modification in the adjustment coefficients; the issue is quite complex and was left for future research.

Additional information

Funding

Research supported by ‘La Sapienza’ [grant n. 2011C26A1145RM] and MIUR [PRIN grant 2010J3LZEN].