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Testing for No Cointegration in Vector Autoregressions with Estimated Degree of Fractional Integration

M. Demetrescu,Vladimir V. Kusin,Nazarii Salish

Published 2021 in Economic Modelling

ABSTRACT

The persistence of fractionally integrated VAR processes is characterized by the common integration parameter d and, under cointegration, by the persistence of the deviations from equilibrium. Finding long-run relations between fractionally integrated series is thus more difficult compared to classical (integer) cointegration since the two persistence parameters are typically unknown in practice. Moreover, even when d is known, the parameter characterizing the persistence of the departures from potential equilibrium strength of the potential return to equilibrium is not identified under no cointegration. To deal with the lack of identification under the null when testing, we resort to a linearization of the fractional difference filter, which leads to regression-based tests of no error-correction (if choosing a single-equation approach) and of no cointegration (if choosing a system approach). Short-run dynamics is accounted for via lag augmentation. When d is known, all test statistics possess standard (normal or χ2) asymptotic limiting distributions under the respective null hypothesis and under local alternatives. For the case of not knowing d, we discuss conditions under which plugging in some consistent estimator does not affect the asymptotics. Estimation error of d induces size distortions even when no shortrun dynamics is present; the key insight is that letting the number of lags increase as the sample size goes to infinity will account for the estimation error of d. Monte Carlo experiments illustrate the usefulness of the new tests and compare the single-equation no fractional cointegration tests with system fractional cointegration tests for both known and estimated fractional integration parameter in finite samples.

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