Residual-based test for fractional cointegration

- We propose a residual-based test for fractional cointegration.
- The integration orders can be real-valued and the resulting cointegrating error can be stationary or nonstationary.
- The proposed test is simple to implement, has standard asymptotics and does not require a prescribed bandwidth.
- The proposed test has better power than the GPH test for unit-root series and has satisfactory sizes when other tests fail.

By allowing deviations from equilibrium to follow a fractionally integrated process, the notion of fractional cointegration analysis encompasses a wide range of mean-reverting behaviors. For fractional cointegrations, asymptotic theories have been extensively studied, and numerous empirical studies have been conducted in finance and economics. But as far as testing for fractional cointegration is concerned, most of the testing procedures have restrictions on the integration orders of observed time series or integrating error and some tests involve determination of bandwidth. In this paper, a general fractional cointegration model with the observed series and the cointegrating error being fractional processes is considered, and a residual-based testing procedure for fractional cointegration is proposed. Under some regularity conditions, the test statistic has an asymptotic standard normal distribution under the null hypothesis of no fractional cointegration and diverges under the alternatives. This test procedure is easy to implement and works well in finite samples, as reported in a Monte Carlo experiment.